1,1,104,102,0.2988551,"\int \tan ^5(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]","-\frac{i a \tan ^{-1}(\tan (c+d x))}{d}+\frac{i a \tan ^5(c+d x)}{5 d}-\frac{i a \tan ^3(c+d x)}{3 d}+\frac{i a \tan (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{i a \tan ^5(c+d x)}{5 d}+\frac{a \tan ^4(c+d x)}{4 d}-\frac{i a \tan ^3(c+d x)}{3 d}-\frac{a \tan ^2(c+d x)}{2 d}+\frac{i a \tan (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}-i a x",1,"((-I)*a*ArcTan[Tan[c + d*x]])/d + (I*a*Tan[c + d*x])/d - ((I/3)*a*Tan[c + d*x]^3)/d + ((I/5)*a*Tan[c + d*x]^5)/d - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)","A",1
2,1,81,83,0.1971096,"\int \tan ^4(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^4*(a + I*a*Tan[c + d*x]),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{i a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}","\frac{i a \tan ^4(c+d x)}{4 d}+\frac{a \tan ^3(c+d x)}{3 d}-\frac{i a \tan ^2(c+d x)}{2 d}-\frac{a \tan (c+d x)}{d}-\frac{i a \log (\cos (c+d x))}{d}+a x",1,"(a*ArcTan[Tan[c + d*x]])/d - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d) - ((I/4)*a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/d","A",1
3,1,74,67,0.154509,"\int \tan ^3(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x]),x]","\frac{i a \tan ^{-1}(\tan (c+d x))}{d}+\frac{i a \tan ^3(c+d x)}{3 d}-\frac{i a \tan (c+d x)}{d}+\frac{a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{i a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^2(c+d x)}{2 d}-\frac{i a \tan (c+d x)}{d}+\frac{a \log (\cos (c+d x))}{d}+i a x",1,"(I*a*ArcTan[Tan[c + d*x]])/d - (I*a*Tan[c + d*x])/d + ((I/3)*a*Tan[c + d*x]^3)/d + (a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
4,1,53,49,0.1009812,"\int \tan ^2(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x]),x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{i a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{i a \tan ^2(c+d x)}{2 d}+\frac{a \tan (c+d x)}{d}+\frac{i a \log (\cos (c+d x))}{d}-a x",1,"-((a*ArcTan[Tan[c + d*x]])/d) + (a*Tan[c + d*x])/d + ((I/2)*a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/d","A",1
5,1,43,34,0.017294,"\int \tan (c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x]),x]","-\frac{i a \tan ^{-1}(\tan (c+d x))}{d}+\frac{i a \tan (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{i a \tan (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}-i a x",1,"((-I)*a*ArcTan[Tan[c + d*x]])/d - (a*Log[Cos[c + d*x]])/d + (I*a*Tan[c + d*x])/d","A",1
6,1,19,19,0.0044664,"\int (a+i a \tan (c+d x)) \, dx","Integrate[a + I*a*Tan[c + d*x],x]","a x-\frac{i a \log (\cos (c+d x))}{d}","a x-\frac{i a \log (\cos (c+d x))}{d}",1,"a*x - (I*a*Log[Cos[c + d*x]])/d","A",1
7,1,27,19,0.0178964,"\int \cot (c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x]),x]","\frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a x","\frac{a \log (\sin (c+d x))}{d}+i a x",1,"I*a*x + (a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
8,1,54,32,0.1116961,"\int \cot ^2(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[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{i a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}","-\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}-a x",1,"-((a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d) + (I*a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
9,1,68,50,0.1647533,"\int \cot ^3(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[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{i a \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}","-\frac{a \cot ^2(c+d x)}{2 d}-\frac{i a \cot (c+d x)}{d}-\frac{a \log (\sin (c+d x))}{d}-i a x",1,"((-I)*a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (a*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d)","C",1
10,1,72,64,0.1989862,"\int \cot ^4(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[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{i a \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{i a \cot ^2(c+d x)}{2 d}+\frac{a \cot (c+d x)}{d}-\frac{i a \log (\sin (c+d x))}{d}+a x",1,"-1/3*(a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d - ((I/2)*a*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/d","C",1
11,1,84,83,0.3609007,"\int \cot ^5(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[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{i 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 \cot ^4(c+d x)}{4 d}-\frac{i a \cot ^3(c+d x)}{3 d}+\frac{a \cot ^2(c+d x)}{2 d}+\frac{i a \cot (c+d x)}{d}+\frac{a \log (\sin (c+d x))}{d}+i a x",1,"((-1/3*I)*a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/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)","C",1
12,1,84,100,0.3501664,"\int \cot ^6(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + I*a*Tan[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{i 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 \cot ^5(c+d x)}{5 d}-\frac{i a \cot ^4(c+d x)}{4 d}+\frac{a \cot ^3(c+d x)}{3 d}+\frac{i a \cot ^2(c+d x)}{2 d}-\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}-a x",1,"-1/5*(a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/d + ((I/4)*a*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/d","C",1
13,1,108,112,0.300583,"\int \tan ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^4*(a + I*a*Tan[c + d*x])^2,x]","\frac{2 a^2 \tan ^{-1}(\tan (c+d x))}{d}-\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{2 a^2 \tan ^3(c+d x)}{3 d}-\frac{2 a^2 \tan (c+d x)}{d}-\frac{i a^2 \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{2 d}","-\frac{a^2 \tan ^5(c+d x)}{5 d}+\frac{i a^2 \tan ^4(c+d x)}{2 d}+\frac{2 a^2 \tan ^3(c+d x)}{3 d}-\frac{i a^2 \tan ^2(c+d x)}{d}-\frac{2 a^2 \tan (c+d x)}{d}-\frac{2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x",1,"(2*a^2*ArcTan[Tan[c + d*x]])/d - (2*a^2*Tan[c + d*x])/d + (2*a^2*Tan[c + d*x]^3)/(3*d) - (a^2*Tan[c + d*x]^5)/(5*d) - ((I/2)*a^2*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/d","A",1
14,1,73,93,0.1999732,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \left(24 i \tan ^{-1}(\tan (c+d x))-3 \tan ^4(c+d x)+8 i \tan ^3(c+d x)+12 \tan ^2(c+d x)-24 i \tan (c+d x)+24 \log (\cos (c+d x))\right)}{12 d}","-\frac{a^2 \tan ^4(c+d x)}{4 d}+\frac{2 i a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan ^2(c+d x)}{d}-\frac{2 i a^2 \tan (c+d x)}{d}+\frac{2 a^2 \log (\cos (c+d x))}{d}+2 i a^2 x",1,"(a^2*((24*I)*ArcTan[Tan[c + d*x]] + 24*Log[Cos[c + d*x]] - (24*I)*Tan[c + d*x] + 12*Tan[c + d*x]^2 + (8*I)*Tan[c + d*x]^3 - 3*Tan[c + d*x]^4))/(12*d)","A",1
15,1,76,64,0.2432515,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^2,x]","-\frac{2 a^2 \tan ^{-1}(\tan (c+d x))}{d}-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{i a^2 \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{d}","\frac{a^2 \tan (c+d x)}{d}+\frac{2 i a^2 \log (\cos (c+d x))}{d}-2 a^2 x-\frac{i (a+i a \tan (c+d x))^3}{3 a d}",1,"(-2*a^2*ArcTan[Tan[c + d*x]])/d + (2*a^2*Tan[c + d*x])/d - (a^2*Tan[c + d*x]^3)/(3*d) + (I*a^2*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/d","A",1
16,1,51,62,0.151824,"\int \tan (c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \left(-4 i \tan ^{-1}(\tan (c+d x))-\tan ^2(c+d x)+4 i \tan (c+d x)-4 \log (\cos (c+d x))\right)}{2 d}","\frac{i a^2 \tan (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-2 i a^2 x+\frac{(a+i a \tan (c+d x))^2}{2 d}",1,"(a^2*((-4*I)*ArcTan[Tan[c + d*x]] - 4*Log[Cos[c + d*x]] + (4*I)*Tan[c + d*x] - Tan[c + d*x]^2))/(2*d)","A",1
17,1,100,38,0.847024,"\int (a+i a \tan (c+d x))^2 \, dx","Integrate[(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \sec (c) \sec (c+d x) \left(-4 d x \cos (2 c+d x)+\cos (d x) \left(-4 d x+i \log \left(\cos ^2(c+d x)\right)\right)+i \cos (2 c+d x) \log \left(\cos ^2(c+d x)\right)+4 \cos (c) \cos (c+d x) \tan ^{-1}(\tan (3 c+d x))+2 \sin (d x)\right)}{2 d}","-\frac{a^2 \tan (c+d x)}{d}-\frac{2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x",1,"-1/2*(a^2*Sec[c]*Sec[c + d*x]*(4*ArcTan[Tan[3*c + d*x]]*Cos[c]*Cos[c + d*x] - 4*d*x*Cos[2*c + d*x] + Cos[d*x]*(-4*d*x + I*Log[Cos[c + d*x]^2]) + I*Cos[2*c + d*x]*Log[Cos[c + d*x]^2] + 2*Sin[d*x]))/d","B",1
18,1,30,37,0.0416887,"\int \cot (c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 (\log (\tan (c+d x))+2 \log (\cos (c+d x))+2 i d x)}{d}","\frac{a^2 \log (\sin (c+d x))}{d}+\frac{a^2 \log (\cos (c+d x))}{d}+2 i a^2 x",1,"(a^2*((2*I)*d*x + 2*Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
19,1,100,38,0.7941975,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2,x]","\frac{a^2 \csc (c) \csc (c+d x) \left(4 d x \cos (2 c+d x)+4 \sin (c) \sin (c+d x) \tan ^{-1}(\tan (3 c+d x))-i \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+\cos (d x) \left(-4 d x+i \log \left(\sin ^2(c+d x)\right)\right)+2 \sin (d x)\right)}{2 d}","-\frac{a^2 \cot (c+d x)}{d}+\frac{2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x",1,"(a^2*Csc[c]*Csc[c + d*x]*(4*d*x*Cos[2*c + d*x] + Cos[d*x]*(-4*d*x + I*Log[Sin[c + d*x]^2]) - I*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + 2*Sin[d*x] + 4*ArcTan[Tan[3*c + d*x]]*Sin[c]*Sin[c + d*x]))/(2*d)","B",1
20,1,64,58,0.2033184,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \left(4 i \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)+\cot ^2(c+d x)+4 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right)}{2 d}","-\frac{a^2 \cot ^2(c+d x)}{2 d}-\frac{2 i a^2 \cot (c+d x)}{d}-\frac{2 a^2 \log (\sin (c+d x))}{d}-2 i a^2 x",1,"-1/2*(a^2*(Cot[c + d*x]^2 + (4*I)*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 4*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
21,1,105,74,0.4654989,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \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^2 \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}-\frac{i a^2 \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{d}","-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{i a^2 \cot ^2(c+d x)}{d}+\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 i a^2 \log (\sin (c+d x))}{d}+2 a^2 x",1,"-1/3*(a^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d + (a^2*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (I*a^2*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/d","C",1
22,1,79,93,0.3826253,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \left(3 \left(\cot ^4(c+d x)-4 \cot ^2(c+d x)-8 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right)+8 i \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)\right)}{12 d}","-\frac{a^2 \cot ^4(c+d x)}{4 d}-\frac{2 i a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^2(c+d x)}{d}+\frac{2 i a^2 \cot (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d}+2 i a^2 x",1,"-1/12*(a^2*((8*I)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 3*(-4*Cot[c + d*x]^2 + Cot[c + d*x]^4 - 8*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))))/d","C",1
23,1,124,112,0.8064323,"\int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \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^2 \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}+\frac{i a^2 \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)}{2 d}","-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{i a^2 \cot ^4(c+d x)}{2 d}+\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{i a^2 \cot ^2(c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}+\frac{2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x",1,"-1/5*(a^2*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/d + (a^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + ((I/2)*a^2*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/d","C",1
24,1,296,126,1.845303,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^5(c+d x) \left(360 i \sin (2 c+d x)-280 i \sin (2 c+3 d x)+135 i \sin (4 c+3 d x)-83 i \sin (4 c+5 d x)+150 i d x \cos (2 c+3 d x)+105 \cos (2 c+3 d x)+150 i d x \cos (4 c+3 d x)+105 \cos (4 c+3 d x)+30 i d x \cos (4 c+5 d x)+30 i d x \cos (6 c+5 d x)+75 \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+75 \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)+15 \cos (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)+15 \cos (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)+75 \cos (d x) \left(2 \log \left(\cos ^2(c+d x)\right)+4 i d x+3\right)+75 \cos (2 c+d x) \left(2 \log \left(\cos ^2(c+d x)\right)+4 i d x+3\right)-470 i \sin (d x)\right)}{240 d}","-\frac{\tan ^4(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{5 d}-\frac{11 a^3 \tan ^4(c+d x)}{20 d}+\frac{4 i a^3 \tan ^3(c+d x)}{3 d}+\frac{2 a^3 \tan ^2(c+d x)}{d}-\frac{4 i a^3 \tan (c+d x)}{d}+\frac{4 a^3 \log (\cos (c+d x))}{d}+4 i a^3 x",1,"(a^3*Sec[c]*Sec[c + d*x]^5*(105*Cos[2*c + 3*d*x] + (150*I)*d*x*Cos[2*c + 3*d*x] + 105*Cos[4*c + 3*d*x] + (150*I)*d*x*Cos[4*c + 3*d*x] + (30*I)*d*x*Cos[4*c + 5*d*x] + (30*I)*d*x*Cos[6*c + 5*d*x] + 75*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] + 75*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] + 15*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] + 15*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] + 75*Cos[d*x]*(3 + (4*I)*d*x + 2*Log[Cos[c + d*x]^2]) + 75*Cos[2*c + d*x]*(3 + (4*I)*d*x + 2*Log[Cos[c + d*x]^2]) - (470*I)*Sin[d*x] + (360*I)*Sin[2*c + d*x] - (280*I)*Sin[2*c + 3*d*x] + (135*I)*Sin[4*c + 3*d*x] - (83*I)*Sin[4*c + 5*d*x]))/(240*d)","B",1
25,1,228,90,1.6756051,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^3,x]","-\frac{a^3 \sec (c) \sec ^4(c+d x) \left(-13 \sin (c+2 d x)+7 \sin (3 c+2 d x)-5 \sin (3 c+4 d x)+8 d x \cos (3 c+2 d x)-5 i \cos (3 c+2 d x)+2 d x \cos (3 c+4 d x)+2 d x \cos (5 c+4 d x)-4 i \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+2 \cos (c) \left(-3 i \log \left(\cos ^2(c+d x)\right)+6 d x-4 i\right)+\cos (c+2 d x) \left(-4 i \log \left(\cos ^2(c+d x)\right)+8 d x-5 i\right)-i \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-i \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)+15 \sin (c)\right)}{8 d}","\frac{2 a^3 \tan (c+d x)}{d}+\frac{4 i a^3 \log (\cos (c+d x))}{d}-4 a^3 x-\frac{i (a+i a \tan (c+d x))^4}{4 a d}-\frac{i a (a+i a \tan (c+d x))^2}{2 d}",1,"-1/8*(a^3*Sec[c]*Sec[c + d*x]^4*((-5*I)*Cos[3*c + 2*d*x] + 8*d*x*Cos[3*c + 2*d*x] + 2*d*x*Cos[3*c + 4*d*x] + 2*d*x*Cos[5*c + 4*d*x] + 2*Cos[c]*(-4*I + 6*d*x - (3*I)*Log[Cos[c + d*x]^2]) + Cos[c + 2*d*x]*(-5*I + 8*d*x - (4*I)*Log[Cos[c + d*x]^2]) - (4*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - I*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - I*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 15*Sin[c] - 13*Sin[c + 2*d*x] + 7*Sin[3*c + 2*d*x] - 5*Sin[3*c + 4*d*x]))/d","B",1
26,1,178,85,1.2138165,"\int \tan (c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^3,x]","-\frac{i a^3 \sec (c) \sec ^3(c+d x) \left(15 \sin (2 c+d x)-13 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)-3 i \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+9 \cos (d x) \left(-i \log \left(\cos ^2(c+d x)\right)+2 d x-i\right)+9 \cos (2 c+d x) \left(-i \log \left(\cos ^2(c+d x)\right)+2 d x-i\right)-3 i \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-24 \sin (d x)\right)}{12 d}","\frac{2 i a^3 \tan (c+d x)}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d}-4 i a^3 x+\frac{a (a+i a \tan (c+d x))^2}{2 d}+\frac{(a+i a \tan (c+d x))^3}{3 d}",1,"((-1/12*I)*a^3*Sec[c]*Sec[c + d*x]^3*(6*d*x*Cos[2*c + 3*d*x] + 6*d*x*Cos[4*c + 3*d*x] + 9*Cos[d*x]*(-I + 2*d*x - I*Log[Cos[c + d*x]^2]) + 9*Cos[2*c + d*x]*(-I + 2*d*x - I*Log[Cos[c + d*x]^2]) - (3*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 24*Sin[d*x] + 15*Sin[2*c + d*x] - 13*Sin[2*c + 3*d*x]))/d","B",1
27,1,119,63,1.0195327,"\int (a+i a \tan (c+d x))^3 \, dx","Integrate[(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec ^2(c+d x) \left(-3 \sin (c+2 d x)+2 d x \cos (3 c+2 d x)-i \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+\cos (c+2 d x) \left(2 d x-i \log \left(\cos ^2(c+d x)\right)\right)+\cos (c) \left(-2 i \log \left(\cos ^2(c+d x)\right)+4 d x-i\right)+3 \sin (c)\right)}{2 d}","-\frac{2 a^3 \tan (c+d x)}{d}-\frac{4 i a^3 \log (\cos (c+d x))}{d}+4 a^3 x+\frac{i a (a+i a \tan (c+d x))^2}{2 d}",1,"(a^3*Sec[c]*Sec[c + d*x]^2*(2*d*x*Cos[3*c + 2*d*x] + Cos[c + 2*d*x]*(2*d*x - I*Log[Cos[c + d*x]^2]) + Cos[c]*(-I + 4*d*x - (2*I)*Log[Cos[c + d*x]^2]) - I*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] + 3*Sin[c] - 3*Sin[c + 2*d*x]))/(2*d)","A",1
28,1,95,60,1.3244933,"\int \cot (c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \sec (c) \sec (c+d x) \left(\cos (d x) \left(\log \left(\sin ^2(c+d x)\right)+3 \log \left(\cos ^2(c+d x)\right)+8 i d x\right)+\cos (2 c+d x) \left(\log \left(\sin ^2(c+d x)\right)+3 \log \left(\cos ^2(c+d x)\right)+8 i d x\right)-4 i \sin (d x)\right)}{4 d}","-\frac{a^3+i a^3 \tan (c+d x)}{d}+\frac{a^3 \log (\sin (c+d x))}{d}+\frac{3 a^3 \log (\cos (c+d x))}{d}+4 i a^3 x",1,"(a^3*Sec[c]*Sec[c + d*x]*(Cos[d*x]*((8*I)*d*x + 3*Log[Cos[c + d*x]^2] + Log[Sin[c + d*x]^2]) + Cos[2*c + d*x]*((8*I)*d*x + 3*Log[Cos[c + d*x]^2] + Log[Sin[c + d*x]^2]) - (4*I)*Sin[d*x]))/(4*d)","A",1
29,1,144,69,1.5997229,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc (c+d x) \left(14 d x \cos (2 c+d x)+12 \sin (c) \sin (c+d x) \tan ^{-1}(\tan (4 c+d x))-i \cos (2 c+d x) \log \left(\cos ^2(c+d x)\right)+\cos (d x) \left(3 i \log \left(\sin ^2(c+d x)\right)+i \log \left(\cos ^2(c+d x)\right)-14 d x\right)-3 i \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+4 \sin (d x)\right)}{8 d}","\frac{3 i a^3 \log (\sin (c+d x))}{d}+\frac{i a^3 \log (\cos (c+d x))}{d}-\frac{\cot (c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{d}-4 a^3 x",1,"(a^3*Csc[c/2]*Csc[c + d*x]*Sec[c/2]*(14*d*x*Cos[2*c + d*x] - I*Cos[2*c + d*x]*Log[Cos[c + d*x]^2] + Cos[d*x]*(-14*d*x + I*Log[Cos[c + d*x]^2] + (3*I)*Log[Sin[c + d*x]^2]) - (3*I)*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + 4*Sin[d*x] + 12*ArcTan[Tan[4*c + d*x]]*Sin[c]*Sin[c + d*x]))/(8*d)","B",1
30,1,126,71,1.0794778,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^2(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left(-3 i \cos (c+2 d x)+\sin (c) \left(-2 \log \left(\sin ^2(c+d x)\right)+2 \cos (2 (c+d x)) \left(\log \left(\sin ^2(c+d x)\right)+2 i d x\right)-4 i d x-1\right)+3 i \cos (c)\right)}{4 d (\cos (d x)+i \sin (d x))^3}","-\frac{2 i a^3 \cot (c+d x)}{d}-\frac{4 a^3 \log (\sin (c+d x))}{d}-4 i a^3 x-\frac{a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}",1,"(a^3*Csc[c/2]*Csc[c + d*x]^2*Sec[c/2]*((3*I)*Cos[c] - (3*I)*Cos[c + 2*d*x] + (-1 - (4*I)*d*x - 2*Log[Sin[c + d*x]^2] + 2*Cos[2*(c + d*x)]*((2*I)*d*x + Log[Sin[c + d*x]^2]))*Sin[c])*(Cos[3*d*x] + I*Sin[3*d*x]))/(4*d*(Cos[d*x] + I*Sin[d*x])^3)","A",1
31,1,251,101,1.3283409,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^3(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left(-15 \sin (2 c+d x)+13 \sin (2 c+3 d x)-36 d x \cos (2 c+d x)+9 i \cos (2 c+d x)-12 d x \cos (2 c+3 d x)+12 d x \cos (4 c+3 d x)-48 \sin (c) \sin ^3(c+d x) \tan ^{-1}(\tan (4 c+d x))+9 \cos (d x) \left(-i \log \left(\sin ^2(c+d x)\right)+4 d x-i\right)+9 i \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+3 i \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-3 i \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)-24 \sin (d x)\right)}{24 d (\cos (d x)+i \sin (d x))^3}","\frac{2 a^3 \cot (c+d x)}{d}-\frac{4 i a^3 \log (\sin (c+d x))}{d}+4 a^3 x-\frac{\cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{i a \cot ^2(c+d x) (a+i a \tan (c+d x))^2}{2 d}",1,"(a^3*Csc[c/2]*Csc[c + d*x]^3*Sec[c/2]*(Cos[3*d*x] + I*Sin[3*d*x])*((9*I)*Cos[2*c + d*x] - 36*d*x*Cos[2*c + d*x] - 12*d*x*Cos[2*c + 3*d*x] + 12*d*x*Cos[4*c + 3*d*x] + 9*Cos[d*x]*(-I + 4*d*x - I*Log[Sin[c + d*x]^2]) + (9*I)*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + (3*I)*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] - 24*Sin[d*x] - 48*ArcTan[Tan[4*c + d*x]]*Sin[c]*Sin[c + d*x]^3 - 15*Sin[2*c + d*x] + 13*Sin[2*c + 3*d*x]))/(24*d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
32,1,254,108,1.4186125,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^4(c+d x) \left(12 i d x \sin (c)+8 i d x \sin (c+2 d x)+5 \sin (c+2 d x)-8 i d x \sin (3 c+2 d x)-5 \sin (3 c+2 d x)-2 i d x \sin (3 c+4 d x)+2 i d x \sin (5 c+4 d x)+13 i \cos (c+2 d x)+7 i \cos (3 c+2 d x)-5 i \cos (3 c+4 d x)+6 \sin (c) \log \left(\sin ^2(c+d x)\right)+4 \sin (c+2 d x) \log \left(\sin ^2(c+d x)\right)-4 \sin (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)-\sin (3 c+4 d x) \log \left(\sin ^2(c+d x)\right)+\sin (5 c+4 d x) \log \left(\sin ^2(c+d x)\right)+8 \sin (c)-15 i \cos (c)\right)}{16 d}","-\frac{3 i a^3 \cot ^3(c+d x)}{4 d}+\frac{2 a^3 \cot ^2(c+d x)}{d}+\frac{4 i a^3 \cot (c+d x)}{d}+\frac{4 a^3 \log (\sin (c+d x))}{d}-\frac{\cot ^4(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{4 d}+4 i a^3 x",1,"(a^3*Csc[c/2]*Csc[c + d*x]^4*Sec[c/2]*((-15*I)*Cos[c] + (13*I)*Cos[c + 2*d*x] + (7*I)*Cos[3*c + 2*d*x] - (5*I)*Cos[3*c + 4*d*x] + 8*Sin[c] + (12*I)*d*x*Sin[c] + 6*Log[Sin[c + d*x]^2]*Sin[c] + 5*Sin[c + 2*d*x] + (8*I)*d*x*Sin[c + 2*d*x] + 4*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] - 5*Sin[3*c + 2*d*x] - (8*I)*d*x*Sin[3*c + 2*d*x] - 4*Log[Sin[c + d*x]^2]*Sin[3*c + 2*d*x] - (2*I)*d*x*Sin[3*c + 4*d*x] - Log[Sin[c + d*x]^2]*Sin[3*c + 4*d*x] + (2*I)*d*x*Sin[5*c + 4*d*x] + Log[Sin[c + d*x]^2]*Sin[5*c + 4*d*x]))/(16*d)","B",1
33,1,359,126,2.1184727,"\int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]","\frac{a^3 \csc (c) \csc ^5(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left(360 \sin (2 c+d x)-280 \sin (2 c+3 d x)-135 \sin (4 c+3 d x)+83 \sin (4 c+5 d x)+600 d x \cos (2 c+d x)-225 i \cos (2 c+d x)+300 d x \cos (2 c+3 d x)-105 i \cos (2 c+3 d x)-300 d x \cos (4 c+3 d x)+105 i \cos (4 c+3 d x)-60 d x \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)+960 \sin (c) \sin ^5(c+d x) \tan ^{-1}(\tan (4 c+d x))-75 \cos (d x) \left(-2 i \log \left(\sin ^2(c+d x)\right)+8 d x-3 i\right)-150 i \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)-75 i \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)+75 i \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)+15 i \cos (4 c+5 d x) \log \left(\sin ^2(c+d x)\right)-15 i \cos (6 c+5 d x) \log \left(\sin ^2(c+d x)\right)+470 \sin (d x)\right)}{240 d (\cos (d x)+i \sin (d x))^3}","-\frac{11 i a^3 \cot ^4(c+d x)}{20 d}+\frac{4 a^3 \cot ^3(c+d x)}{3 d}+\frac{2 i a^3 \cot ^2(c+d x)}{d}-\frac{4 a^3 \cot (c+d x)}{d}+\frac{4 i a^3 \log (\sin (c+d x))}{d}-\frac{\cot ^5(c+d x) \left(a^3+i a^3 \tan (c+d x)\right)}{5 d}-4 a^3 x",1,"(a^3*Csc[c]*Csc[c + d*x]^5*(Cos[3*d*x] + I*Sin[3*d*x])*((-225*I)*Cos[2*c + d*x] + 600*d*x*Cos[2*c + d*x] - (105*I)*Cos[2*c + 3*d*x] + 300*d*x*Cos[2*c + 3*d*x] + (105*I)*Cos[4*c + 3*d*x] - 300*d*x*Cos[4*c + 3*d*x] - 60*d*x*Cos[4*c + 5*d*x] + 60*d*x*Cos[6*c + 5*d*x] - 75*Cos[d*x]*(-3*I + 8*d*x - (2*I)*Log[Sin[c + d*x]^2]) - (150*I)*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] - (75*I)*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + (75*I)*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] + (15*I)*Cos[4*c + 5*d*x]*Log[Sin[c + d*x]^2] - (15*I)*Cos[6*c + 5*d*x]*Log[Sin[c + d*x]^2] + 470*Sin[d*x] + 960*ArcTan[Tan[4*c + d*x]]*Sin[c]*Sin[c + d*x]^5 + 360*Sin[2*c + d*x] - 280*Sin[2*c + 3*d*x] - 135*Sin[4*c + 3*d*x] + 83*Sin[4*c + 5*d*x]))/(240*d*(Cos[d*x] + I*Sin[d*x])^3)","B",1
34,1,349,160,2.3960507,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \sec (c) \sec ^6(c+d x) \left(-780 i \sin (c+2 d x)+510 i \sin (3 c+2 d x)-366 i \sin (3 c+4 d x)+150 i \sin (5 c+4 d x)-86 i \sin (5 c+6 d x)+450 i d x \cos (3 c+2 d x)+345 \cos (3 c+2 d x)+180 i d x \cos (3 c+4 d x)+120 \cos (3 c+4 d x)+180 i d x \cos (5 c+4 d x)+120 \cos (5 c+4 d x)+30 i d x \cos (5 c+6 d x)+30 i d x \cos (7 c+6 d x)+225 \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+90 \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)+90 \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)+15 \cos (5 c+6 d x) \log \left(\cos ^2(c+d x)\right)+15 \cos (7 c+6 d x) \log \left(\cos ^2(c+d x)\right)+15 \cos (c+2 d x) \left(15 \log \left(\cos ^2(c+d x)\right)+30 i d x+23\right)+10 \cos (c) \left(30 \log \left(\cos ^2(c+d x)\right)+60 i d x+49\right)+860 i \sin (c)\right)}{240 d}","-\frac{67 a^4 \tan ^4(c+d x)}{60 d}-\frac{7 \tan ^4(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{15 d}+\frac{8 i a^4 \tan ^3(c+d x)}{3 d}+\frac{4 a^4 \tan ^2(c+d x)}{d}-\frac{8 i a^4 \tan (c+d x)}{d}+\frac{8 a^4 \log (\cos (c+d x))}{d}+8 i a^4 x-\frac{\tan ^4(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{6 d}",1,"(a^4*Sec[c]*Sec[c + d*x]^6*(345*Cos[3*c + 2*d*x] + (450*I)*d*x*Cos[3*c + 2*d*x] + 120*Cos[3*c + 4*d*x] + (180*I)*d*x*Cos[3*c + 4*d*x] + 120*Cos[5*c + 4*d*x] + (180*I)*d*x*Cos[5*c + 4*d*x] + (30*I)*d*x*Cos[5*c + 6*d*x] + (30*I)*d*x*Cos[7*c + 6*d*x] + 225*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] + 90*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] + 90*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 15*Cos[5*c + 6*d*x]*Log[Cos[c + d*x]^2] + 15*Cos[7*c + 6*d*x]*Log[Cos[c + d*x]^2] + 15*Cos[c + 2*d*x]*(23 + (30*I)*d*x + 15*Log[Cos[c + d*x]^2]) + 10*Cos[c]*(49 + (60*I)*d*x + 30*Log[Cos[c + d*x]^2]) + (860*I)*Sin[c] - (780*I)*Sin[c + 2*d*x] + (510*I)*Sin[3*c + 2*d*x] - (366*I)*Sin[3*c + 4*d*x] + (150*I)*Sin[5*c + 4*d*x] - (86*I)*Sin[5*c + 6*d*x]))/(240*d)","B",1
35,1,294,116,2.7998451,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]","-\frac{a^4 \sec (c) \sec ^5(c+d x) \left(345 \sin (2 c+d x)-275 \sin (2 c+3 d x)+120 \sin (4 c+3 d x)-79 \sin (4 c+5 d x)+150 d x \cos (2 c+3 d x)-90 i \cos (2 c+3 d x)+150 d x \cos (4 c+3 d x)-90 i \cos (4 c+3 d x)+30 d x \cos (4 c+5 d x)+30 d x \cos (6 c+5 d x)-75 i \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+30 \cos (d x) \left(-5 i \log \left(\cos ^2(c+d x)\right)+10 d x-7 i\right)+30 \cos (2 c+d x) \left(-5 i \log \left(\cos ^2(c+d x)\right)+10 d x-7 i\right)-75 i \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-15 i \cos (4 c+5 d x) \log \left(\cos ^2(c+d x)\right)-15 i \cos (6 c+5 d x) \log \left(\cos ^2(c+d x)\right)-445 \sin (d x)\right)}{120 d}","\frac{4 a^4 \tan (c+d x)}{d}+\frac{8 i a^4 \log (\cos (c+d x))}{d}-8 a^4 x-\frac{i \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\frac{i a (a+i a \tan (c+d x))^3}{3 d}",1,"-1/120*(a^4*Sec[c]*Sec[c + d*x]^5*((-90*I)*Cos[2*c + 3*d*x] + 150*d*x*Cos[2*c + 3*d*x] - (90*I)*Cos[4*c + 3*d*x] + 150*d*x*Cos[4*c + 3*d*x] + 30*d*x*Cos[4*c + 5*d*x] + 30*d*x*Cos[6*c + 5*d*x] + 30*Cos[d*x]*(-7*I + 10*d*x - (5*I)*Log[Cos[c + d*x]^2]) + 30*Cos[2*c + d*x]*(-7*I + 10*d*x - (5*I)*Log[Cos[c + d*x]^2]) - (75*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - (75*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (15*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - (15*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - 445*Sin[d*x] + 345*Sin[2*c + d*x] - 275*Sin[2*c + 3*d*x] + 120*Sin[4*c + 3*d*x] - 79*Sin[4*c + 5*d*x]))/d","B",1
36,1,231,108,1.2630701,"\int \tan (c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^4,x]","-\frac{i a^4 \sec (c) \sec ^4(c+d x) \left(-38 \sin (c+2 d x)+18 \sin (3 c+2 d x)-14 \sin (3 c+4 d x)+24 d x \cos (3 c+2 d x)-12 i \cos (3 c+2 d x)+6 d x \cos (3 c+4 d x)+6 d x \cos (5 c+4 d x)-12 i \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+12 \cos (c+2 d x) \left(-i \log \left(\cos ^2(c+d x)\right)+2 d x-i\right)+3 \cos (c) \left(-6 i \log \left(\cos ^2(c+d x)\right)+12 d x-7 i\right)-3 i \cos (3 c+4 d x) \log \left(\cos ^2(c+d x)\right)-3 i \cos (5 c+4 d x) \log \left(\cos ^2(c+d x)\right)+42 \sin (c)\right)}{12 d}","\frac{4 i a^4 \tan (c+d x)}{d}-\frac{8 a^4 \log (\cos (c+d x))}{d}-8 i a^4 x+\frac{\left(a^2+i a^2 \tan (c+d x)\right)^2}{d}+\frac{a (a+i a \tan (c+d x))^3}{3 d}+\frac{(a+i a \tan (c+d x))^4}{4 d}",1,"((-1/12*I)*a^4*Sec[c]*Sec[c + d*x]^4*((-12*I)*Cos[3*c + 2*d*x] + 24*d*x*Cos[3*c + 2*d*x] + 6*d*x*Cos[3*c + 4*d*x] + 6*d*x*Cos[5*c + 4*d*x] + 12*Cos[c + 2*d*x]*(-I + 2*d*x - I*Log[Cos[c + d*x]^2]) + 3*Cos[c]*(-7*I + 12*d*x - (6*I)*Log[Cos[c + d*x]^2]) - (12*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - (3*I)*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - (3*I)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 42*Sin[c] - 38*Sin[c + 2*d*x] + 18*Sin[3*c + 2*d*x] - 14*Sin[3*c + 4*d*x]))/d","B",1
37,1,176,89,1.1713379,"\int (a+i a \tan (c+d x))^4 \, dx","Integrate[(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \sec (c) \sec ^3(c+d x) \left(12 \sin (2 c+d x)-11 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)-3 i \cos (2 c+3 d x) \log \left(\cos ^2(c+d x)\right)+3 \cos (d x) \left(-3 i \log \left(\cos ^2(c+d x)\right)+6 d x-2 i\right)+3 \cos (2 c+d x) \left(-3 i \log \left(\cos ^2(c+d x)\right)+6 d x-2 i\right)-3 i \cos (4 c+3 d x) \log \left(\cos ^2(c+d x)\right)-21 \sin (d x)\right)}{6 d}","-\frac{4 a^4 \tan (c+d x)}{d}-\frac{8 i a^4 \log (\cos (c+d x))}{d}+8 a^4 x+\frac{i \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}+\frac{i a (a+i a \tan (c+d x))^3}{3 d}",1,"(a^4*Sec[c]*Sec[c + d*x]^3*(6*d*x*Cos[2*c + 3*d*x] + 6*d*x*Cos[4*c + 3*d*x] + 3*Cos[d*x]*(-2*I + 6*d*x - (3*I)*Log[Cos[c + d*x]^2]) + 3*Cos[2*c + d*x]*(-2*I + 6*d*x - (3*I)*Log[Cos[c + d*x]^2]) - (3*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 21*Sin[d*x] + 12*Sin[2*c + d*x] - 11*Sin[2*c + 3*d*x]))/(6*d)","A",1
38,1,159,86,1.9600203,"\int \cot (c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \sec (c) \sec ^2(c+d x) \left(-16 i \sin (c+2 d x)+16 i d x \cos (3 c+2 d x)+7 \cos (3 c+2 d x) \log \left(\cos ^2(c+d x)\right)+\cos (c+2 d x) \left(\log \left(\sin ^2(c+d x)\right)+7 \log \left(\cos ^2(c+d x)\right)+16 i d x\right)+2 \cos (c) \left(\log \left(\sin ^2(c+d x)\right)+7 \log \left(\cos ^2(c+d x)\right)+16 i d x+2\right)+\cos (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)+16 i \sin (c)\right)}{8 d}","-\frac{3 \left(a^4+i a^4 \tan (c+d x)\right)}{d}+\frac{a^4 \log (\sin (c+d x))}{d}+\frac{7 a^4 \log (\cos (c+d x))}{d}+8 i a^4 x-\frac{\left(a^2+i a^2 \tan (c+d x)\right)^2}{2 d}",1,"(a^4*Sec[c]*Sec[c + d*x]^2*((16*I)*d*x*Cos[3*c + 2*d*x] + 7*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] + Cos[3*c + 2*d*x]*Log[Sin[c + d*x]^2] + Cos[c + 2*d*x]*((16*I)*d*x + 7*Log[Cos[c + d*x]^2] + Log[Sin[c + d*x]^2]) + 2*Cos[c]*(2 + (16*I)*d*x + 7*Log[Cos[c + d*x]^2] + Log[Sin[c + d*x]^2]) + (16*I)*Sin[c] - (16*I)*Sin[c + 2*d*x]))/(8*d)","A",1
39,1,151,71,2.2727615,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \csc (c) \sec (c) \csc (c+d x) \sec (c+d x) \left(6 d x \cos (4 c+2 d x)+4 \sin (2 c) \sin (2 (c+d x)) \tan ^{-1}(\tan (5 c+d x))-i \cos (4 c+2 d x) \log \left(\cos ^2(c+d x)\right)+\cos (2 d x) \left(i \log \left(\sin ^2(c+d x)\right)+i \log \left(\cos ^2(c+d x)\right)-6 d x\right)-i \cos (4 c+2 d x) \log \left(\sin ^2(c+d x)\right)+2 \sin (2 d x)\right)}{4 d}","\frac{4 i a^4 \log (\sin (c+d x))}{d}+\frac{4 i a^4 \log (\cos (c+d x))}{d}-8 a^4 x-\frac{\cot (c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}",1,"(a^4*Csc[c]*Csc[c + d*x]*Sec[c]*Sec[c + d*x]*(6*d*x*Cos[4*c + 2*d*x] - I*Cos[4*c + 2*d*x]*Log[Cos[c + d*x]^2] + Cos[2*d*x]*(-6*d*x + I*Log[Cos[c + d*x]^2] + I*Log[Sin[c + d*x]^2]) - I*Cos[4*c + 2*d*x]*Log[Sin[c + d*x]^2] + 2*Sin[2*d*x] + 4*ArcTan[Tan[5*c + d*x]]*Sin[2*c]*Sin[2*(c + d*x)]))/(4*d)","B",1
40,1,133,103,2.0182961,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \csc ^2(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left(-7 \log \left(\sin ^2(c+d x)\right)-\log \left(\cos ^2(c+d x)\right)-8 i \csc (c) \cos (c+2 d x)+\cos (2 (c+d x)) \left(7 \log \left(\sin ^2(c+d x)\right)+\log \left(\cos ^2(c+d x)\right)+16 i d x\right)+8 i \cot (c)-16 i d x-2\right)}{4 d (\cos (d x)+i \sin (d x))^4}","-\frac{7 a^4 \log (\sin (c+d x))}{d}-\frac{a^4 \log (\cos (c+d x))}{d}-\frac{3 i \cot (c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{d}-8 i a^4 x-\frac{\cot ^2(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{2 d}",1,"(a^4*Csc[c + d*x]^2*(-2 - (16*I)*d*x + (8*I)*Cot[c] - (8*I)*Cos[c + 2*d*x]*Csc[c] - Log[Cos[c + d*x]^2] - 7*Log[Sin[c + d*x]^2] + Cos[2*(c + d*x)]*((16*I)*d*x + Log[Cos[c + d*x]^2] + 7*Log[Sin[c + d*x]^2]))*(Cos[4*d*x] + I*Sin[4*d*x]))/(4*d*(Cos[d*x] + I*Sin[d*x])^4)","A",1
41,1,240,103,1.0917198,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \csc (c) \csc ^3(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left(-12 \sin (2 c+d x)+11 \sin (2 c+3 d x)-36 d x \cos (2 c+d x)+6 i \cos (2 c+d x)-12 d x \cos (2 c+3 d x)+12 d x \cos (4 c+3 d x)-48 \sin (c) \sin ^3(c+d x) \tan ^{-1}(\tan (5 c+d x))+\cos (d x) \left(-9 i \log \left(\sin ^2(c+d x)\right)+36 d x-6 i\right)+9 i \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+3 i \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-3 i \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)-21 \sin (d x)\right)}{6 d (\cos (d x)+i \sin (d x))^4}","\frac{4 a^4 \cot (c+d x)}{d}-\frac{8 i a^4 \log (\sin (c+d x))}{d}+8 a^4 x-\frac{i \cot ^2(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}-\frac{a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}",1,"(a^4*Csc[c]*Csc[c + d*x]^3*(Cos[4*d*x] + I*Sin[4*d*x])*((6*I)*Cos[2*c + d*x] - 36*d*x*Cos[2*c + d*x] - 12*d*x*Cos[2*c + 3*d*x] + 12*d*x*Cos[4*c + 3*d*x] + Cos[d*x]*(-6*I + 36*d*x - (9*I)*Log[Sin[c + d*x]^2]) + (9*I)*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + (3*I)*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] - 21*Sin[d*x] - 48*ArcTan[Tan[5*c + d*x]]*Sin[c]*Sin[c + d*x]^3 - 12*Sin[2*c + d*x] + 11*Sin[2*c + 3*d*x]))/(6*d*(Cos[d*x] + I*Sin[d*x])^4)","B",1
42,1,245,134,1.0881914,"\int \cot ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \csc (c) \csc ^4(c+d x) \left(36 i d x \sin (c)+24 i d x \sin (c+2 d x)+12 \sin (c+2 d x)-24 i d x \sin (3 c+2 d x)-12 \sin (3 c+2 d x)-6 i d x \sin (3 c+4 d x)+6 i d x \sin (5 c+4 d x)+38 i \cos (c+2 d x)+18 i \cos (3 c+2 d x)-14 i \cos (3 c+4 d x)+18 \sin (c) \log \left(\sin ^2(c+d x)\right)+12 \sin (c+2 d x) \log \left(\sin ^2(c+d x)\right)-12 \sin (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)-3 \sin (3 c+4 d x) \log \left(\sin ^2(c+d x)\right)+3 \sin (5 c+4 d x) \log \left(\sin ^2(c+d x)\right)+21 \sin (c)-42 i \cos (c)\right)}{12 d}","\frac{4 i a^4 \cot (c+d x)}{d}+\frac{8 a^4 \log (\sin (c+d x))}{d}+8 i a^4 x+\frac{\cot ^2(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}",1,"(a^4*Csc[c]*Csc[c + d*x]^4*((-42*I)*Cos[c] + (38*I)*Cos[c + 2*d*x] + (18*I)*Cos[3*c + 2*d*x] - (14*I)*Cos[3*c + 4*d*x] + 21*Sin[c] + (36*I)*d*x*Sin[c] + 18*Log[Sin[c + d*x]^2]*Sin[c] + 12*Sin[c + 2*d*x] + (24*I)*d*x*Sin[c + 2*d*x] + 12*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] - 12*Sin[3*c + 2*d*x] - (24*I)*d*x*Sin[3*c + 2*d*x] - 12*Log[Sin[c + d*x]^2]*Sin[3*c + 2*d*x] - (6*I)*d*x*Sin[3*c + 4*d*x] - 3*Log[Sin[c + d*x]^2]*Sin[3*c + 4*d*x] + (6*I)*d*x*Sin[5*c + 4*d*x] + 3*Log[Sin[c + d*x]^2]*Sin[5*c + 4*d*x]))/(12*d)","A",1
43,1,359,142,3.3958138,"\int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \csc (c) \csc ^5(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left(345 \sin (2 c+d x)-275 \sin (2 c+3 d x)-120 \sin (4 c+3 d x)+79 \sin (4 c+5 d x)+600 d x \cos (2 c+d x)-210 i \cos (2 c+d x)+300 d x \cos (2 c+3 d x)-90 i \cos (2 c+3 d x)-300 d x \cos (4 c+3 d x)+90 i \cos (4 c+3 d x)-60 d x \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)+960 \sin (c) \sin ^5(c+d x) \tan ^{-1}(\tan (5 c+d x))-30 \cos (d x) \left(-5 i \log \left(\sin ^2(c+d x)\right)+20 d x-7 i\right)-150 i \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)-75 i \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)+75 i \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)+15 i \cos (4 c+5 d x) \log \left(\sin ^2(c+d x)\right)-15 i \cos (6 c+5 d x) \log \left(\sin ^2(c+d x)\right)+445 \sin (d x)\right)}{120 d (\cos (d x)+i \sin (d x))^4}","\frac{23 a^4 \cot ^3(c+d x)}{15 d}+\frac{4 i a^4 \cot ^2(c+d x)}{d}-\frac{8 a^4 \cot (c+d x)}{d}+\frac{8 i a^4 \log (\sin (c+d x))}{d}-\frac{3 i \cot ^4(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{5 d}-8 a^4 x-\frac{\cot ^5(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{5 d}",1,"(a^4*Csc[c]*Csc[c + d*x]^5*(Cos[4*d*x] + I*Sin[4*d*x])*((-210*I)*Cos[2*c + d*x] + 600*d*x*Cos[2*c + d*x] - (90*I)*Cos[2*c + 3*d*x] + 300*d*x*Cos[2*c + 3*d*x] + (90*I)*Cos[4*c + 3*d*x] - 300*d*x*Cos[4*c + 3*d*x] - 60*d*x*Cos[4*c + 5*d*x] + 60*d*x*Cos[6*c + 5*d*x] - 30*Cos[d*x]*(-7*I + 20*d*x - (5*I)*Log[Sin[c + d*x]^2]) - (150*I)*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] - (75*I)*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + (75*I)*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] + (15*I)*Cos[4*c + 5*d*x]*Log[Sin[c + d*x]^2] - (15*I)*Cos[6*c + 5*d*x]*Log[Sin[c + d*x]^2] + 445*Sin[d*x] + 960*ArcTan[Tan[5*c + d*x]]*Sin[c]*Sin[c + d*x]^5 + 345*Sin[2*c + d*x] - 275*Sin[2*c + 3*d*x] - 120*Sin[4*c + 3*d*x] + 79*Sin[4*c + 5*d*x]))/(120*d*(Cos[d*x] + I*Sin[d*x])^4)","B",1
44,1,363,162,2.0624212,"\int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4,x]","\frac{a^4 \csc (c) \csc ^6(c+d x) \left(-600 i d x \sin (c)-450 i d x \sin (c+2 d x)-345 \sin (c+2 d x)+450 i d x \sin (3 c+2 d x)+345 \sin (3 c+2 d x)+180 i d x \sin (3 c+4 d x)+120 \sin (3 c+4 d x)-180 i d x \sin (5 c+4 d x)-120 \sin (5 c+4 d x)-30 i d x \sin (5 c+6 d x)+30 i d x \sin (7 c+6 d x)-780 i \cos (c+2 d x)-510 i \cos (3 c+2 d x)+366 i \cos (3 c+4 d x)+150 i \cos (5 c+4 d x)-86 i \cos (5 c+6 d x)-300 \sin (c) \log \left(\sin ^2(c+d x)\right)-225 \sin (c+2 d x) \log \left(\sin ^2(c+d x)\right)+225 \sin (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)+90 \sin (3 c+4 d x) \log \left(\sin ^2(c+d x)\right)-90 \sin (5 c+4 d x) \log \left(\sin ^2(c+d x)\right)-15 \sin (5 c+6 d x) \log \left(\sin ^2(c+d x)\right)+15 \sin (7 c+6 d x) \log \left(\sin ^2(c+d x)\right)-490 \sin (c)+860 i \cos (c)\right)}{240 d}","\frac{67 a^4 \cot ^4(c+d x)}{60 d}+\frac{8 i a^4 \cot ^3(c+d x)}{3 d}-\frac{4 a^4 \cot ^2(c+d x)}{d}-\frac{8 i a^4 \cot (c+d x)}{d}-\frac{8 a^4 \log (\sin (c+d x))}{d}-\frac{7 i \cot ^5(c+d x) \left(a^4+i a^4 \tan (c+d x)\right)}{15 d}-8 i a^4 x-\frac{\cot ^6(c+d x) \left(a^2+i a^2 \tan (c+d x)\right)^2}{6 d}",1,"(a^4*Csc[c]*Csc[c + d*x]^6*((860*I)*Cos[c] - (780*I)*Cos[c + 2*d*x] - (510*I)*Cos[3*c + 2*d*x] + (366*I)*Cos[3*c + 4*d*x] + (150*I)*Cos[5*c + 4*d*x] - (86*I)*Cos[5*c + 6*d*x] - 490*Sin[c] - (600*I)*d*x*Sin[c] - 300*Log[Sin[c + d*x]^2]*Sin[c] - 345*Sin[c + 2*d*x] - (450*I)*d*x*Sin[c + 2*d*x] - 225*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] + 345*Sin[3*c + 2*d*x] + (450*I)*d*x*Sin[3*c + 2*d*x] + 225*Log[Sin[c + d*x]^2]*Sin[3*c + 2*d*x] + 120*Sin[3*c + 4*d*x] + (180*I)*d*x*Sin[3*c + 4*d*x] + 90*Log[Sin[c + d*x]^2]*Sin[3*c + 4*d*x] - 120*Sin[5*c + 4*d*x] - (180*I)*d*x*Sin[5*c + 4*d*x] - 90*Log[Sin[c + d*x]^2]*Sin[5*c + 4*d*x] - (30*I)*d*x*Sin[5*c + 6*d*x] - 15*Log[Sin[c + d*x]^2]*Sin[5*c + 6*d*x] + (30*I)*d*x*Sin[7*c + 6*d*x] + 15*Log[Sin[c + d*x]^2]*Sin[7*c + 6*d*x]))/(240*d)","B",1
45,1,840,130,6.5217514,"\int \frac{\tan ^6(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^6/(a + I*a*Tan[c + d*x]),x]","\frac{\left(\frac{\sin (c)}{4}-\frac{1}{4} i \cos (c)\right) (\cos (d x)+i \sin (d x)) \sec ^5(c+d x)}{d (i \tan (c+d x) a+a)}-\frac{i (\cos (d x)+i \sin (d x)) (-\cos (c-d x)+\cos (c+d x)-i \sin (c-d x)+i \sin (c+d x)) \sec ^4(c+d x)}{6 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (i \tan (c+d x) a+a)}+\frac{\left(\frac{1}{6} i \cos (c)-\frac{\sin (c)}{6}\right) (9 \cos (c)-2 i \sin (c)) (\cos (d x)+i \sin (d x)) \sec ^3(c+d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (i \tan (c+d x) a+a)}+\frac{7 i (\cos (d x)+i \sin (d x)) (-\cos (c-d x)+\cos (c+d x)-i \sin (c-d x)+i \sin (c+d x)) \sec ^2(c+d x)}{6 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) (i \tan (c+d x) a+a)}+\frac{5 x \cos (c) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{2 (i \tan (c+d x) a+a)}+\frac{3 \tan ^{-1}(\tan (d x)) \cos (c) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{d (i \tan (c+d x) a+a)}+\frac{3 i \cos (c) \log \left(\cos ^2(c+d x)\right) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{2 d (i \tan (c+d x) a+a)}+\frac{\cos (2 d x) \left(-\frac{1}{4} i \cos (c)-\frac{\sin (c)}{4}\right) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{d (i \tan (c+d x) a+a)}+\frac{5 i x \sin (c) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{2 (i \tan (c+d x) a+a)}+\frac{3 i \tan ^{-1}(\tan (d x)) \sin (c) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{d (i \tan (c+d x) a+a)}-\frac{3 \log \left(\cos ^2(c+d x)\right) \sin (c) (\cos (d x)+i \sin (d x)) \sec (c+d x)}{2 d (i \tan (c+d x) a+a)}+\frac{\left(\frac{1}{4} i \sin (c)-\frac{\cos (c)}{4}\right) (\cos (d x)+i \sin (d x)) \sin (2 d x) \sec (c+d x)}{d (i \tan (c+d x) a+a)}+\frac{x (\cos (d x)+i \sin (d x)) (-3 \sec (c)-i (3 \cos (c)+3 i \sin (c)) \tan (c)) \sec (c+d x)}{i \tan (c+d x) a+a}","-\frac{\tan ^5(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{3 i \tan ^4(c+d x)}{4 a d}+\frac{5 \tan ^3(c+d x)}{6 a d}+\frac{3 i \tan ^2(c+d x)}{2 a d}-\frac{5 \tan (c+d x)}{2 a d}+\frac{3 i \log (\cos (c+d x))}{a d}+\frac{5 x}{2 a}",1,"(5*x*Cos[c]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x]))/(2*(a + I*a*Tan[c + d*x])) + (3*ArcTan[Tan[d*x]]*Cos[c]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x]))/(d*(a + I*a*Tan[c + d*x])) + (((3*I)/2)*Cos[c]*Log[Cos[c + d*x]^2]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x]))/(d*(a + I*a*Tan[c + d*x])) + (Cos[2*d*x]*Sec[c + d*x]*((-1/4*I)*Cos[c] - Sin[c]/4)*(Cos[d*x] + I*Sin[d*x]))/(d*(a + I*a*Tan[c + d*x])) + (Sec[c + d*x]^3*((I/6)*Cos[c] - Sin[c]/6)*(9*Cos[c] - (2*I)*Sin[c])*(Cos[d*x] + I*Sin[d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(a + I*a*Tan[c + d*x])) + (Sec[c + d*x]^5*((-1/4*I)*Cos[c] + Sin[c]/4)*(Cos[d*x] + I*Sin[d*x]))/(d*(a + I*a*Tan[c + d*x])) + (((5*I)/2)*x*Sec[c + d*x]*Sin[c]*(Cos[d*x] + I*Sin[d*x]))/(a + I*a*Tan[c + d*x]) + ((3*I)*ArcTan[Tan[d*x]]*Sec[c + d*x]*Sin[c]*(Cos[d*x] + I*Sin[d*x]))/(d*(a + I*a*Tan[c + d*x])) - (3*Log[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c]*(Cos[d*x] + I*Sin[d*x]))/(2*d*(a + I*a*Tan[c + d*x])) + (Sec[c + d*x]*(-1/4*Cos[c] + (I/4)*Sin[c])*(Cos[d*x] + I*Sin[d*x])*Sin[2*d*x])/(d*(a + I*a*Tan[c + d*x])) + (((7*I)/6)*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])*(-Cos[c - d*x] + Cos[c + d*x] - I*Sin[c - d*x] + I*Sin[c + d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(a + I*a*Tan[c + d*x])) - ((I/6)*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])*(-Cos[c - d*x] + Cos[c + d*x] - I*Sin[c - d*x] + I*Sin[c + d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(a + I*a*Tan[c + d*x])) + (x*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])*(-3*Sec[c] - I*(3*Cos[c] + (3*I)*Sin[c])*Tan[c]))/(a + I*a*Tan[c + d*x])","B",1
46,1,235,109,3.581972,"\int \frac{\tan ^5(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + I*a*Tan[c + d*x]),x]","\frac{\sec (c+d x) (\cos (d x)+i \sin (d x)) \left(6 d x \sin (c)+3 \sin (c) \sin (2 d x)+3 i \sin (c) \cos (2 d x)-4 i \sin (d x) \sec ^3(c+d x)+2 i \sin (c) \sec ^2(c+d x)+28 i \sin (d x) \sec (c+d x)+12 i \sin (c) \log \left(\cos ^2(c+d x)\right)+24 (\sin (c)-i \cos (c)) \tan ^{-1}(\tan (d x))+4 \tan (c) \sin (d x) \sec ^3(c+d x)+4 \sin (c) \tan (c) \sec ^2(c+d x)-28 \tan (c) \sin (d x) \sec (c+d x)+3 \cos (c) \left(2 \sec ^2(c+d x)+4 \log \left(\cos ^2(c+d x)\right)-2 i d x+i \sin (2 d x)-\cos (2 d x)\right)\right)}{12 d (a+i a \tan (c+d x))}","-\frac{\tan ^4(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{5 i \tan ^3(c+d x)}{6 a d}+\frac{\tan ^2(c+d x)}{a d}+\frac{5 i \tan (c+d x)}{2 a d}+\frac{2 \log (\cos (c+d x))}{a d}-\frac{5 i x}{2 a}",1,"(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])*(6*d*x*Sin[c] + (3*I)*Cos[2*d*x]*Sin[c] + (12*I)*Log[Cos[c + d*x]^2]*Sin[c] + (2*I)*Sec[c + d*x]^2*Sin[c] + 24*ArcTan[Tan[d*x]]*((-I)*Cos[c] + Sin[c]) + (28*I)*Sec[c + d*x]*Sin[d*x] - (4*I)*Sec[c + d*x]^3*Sin[d*x] + 3*Cos[c]*((-2*I)*d*x - Cos[2*d*x] + 4*Log[Cos[c + d*x]^2] + 2*Sec[c + d*x]^2 + I*Sin[2*d*x]) + 3*Sin[c]*Sin[2*d*x] + 4*Sec[c + d*x]^2*Sin[c]*Tan[c] - 28*Sec[c + d*x]*Sin[d*x]*Tan[c] + 4*Sec[c + d*x]^3*Sin[d*x]*Tan[c]))/(12*d*(a + I*a*Tan[c + d*x]))","B",1
47,1,196,90,1.8369877,"\int \frac{\tan ^4(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x]),x]","\frac{\cos (c) \sec (c+d x) (\cos (d x)+i \sin (d x)) \left((-8-8 i \tan (c)) \tan ^{-1}(\tan (d x))-8 d x \tan ^2(c)+2 i d x \tan (c)+8 d x \sec ^2(c)-2 i \sec ^2(c+d x)-i \tan (c) \sin (2 d x)-4 i \log \left(\cos ^2(c+d x)\right)+(\tan (c)+i) \cos (2 d x)+2 \tan (c) \sec ^2(c+d x)+4 \sec (c) \sin (d x) \sec (c+d x)+4 \tan (c) \log \left(\cos ^2(c+d x)\right)+4 i \tan (c) \sec (c) \sin (d x) \sec (c+d x)-6 d x+\sin (2 d x)\right)}{4 d (a+i a \tan (c+d x))}","-\frac{\tan ^3(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{i \tan ^2(c+d x)}{a d}+\frac{3 \tan (c+d x)}{2 a d}-\frac{2 i \log (\cos (c+d x))}{a d}-\frac{3 x}{2 a}",1,"(Cos[c]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])*(-6*d*x - (4*I)*Log[Cos[c + d*x]^2] + 8*d*x*Sec[c]^2 - (2*I)*Sec[c + d*x]^2 + 4*Sec[c]*Sec[c + d*x]*Sin[d*x] + Sin[2*d*x] + ArcTan[Tan[d*x]]*(-8 - (8*I)*Tan[c]) + (2*I)*d*x*Tan[c] + 4*Log[Cos[c + d*x]^2]*Tan[c] + 2*Sec[c + d*x]^2*Tan[c] + (4*I)*Sec[c]*Sec[c + d*x]*Sin[d*x]*Tan[c] - I*Sin[2*d*x]*Tan[c] - 8*d*x*Tan[c]^2 + Cos[2*d*x]*(I + Tan[c])))/(4*d*(a + I*a*Tan[c + d*x]))","B",1
48,1,174,74,1.1386013,"\int \frac{\tan ^3(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x]),x]","-\frac{i \cos (c) \sec (c+d x) (\cos (d x)+i \sin (d x)) \left((-4-4 i \tan (c)) \tan ^{-1}(\tan (d x))-4 d x \tan ^2(c)-2 i d x \tan (c)+4 d x \sec ^2(c)-i \tan (c) \sin (2 d x)-2 i \log \left(\cos ^2(c+d x)\right)+(\tan (c)+i) \cos (2 d x)+4 \sec (c) \sin (d x) \sec (c+d x)+2 \tan (c) \log \left(\cos ^2(c+d x)\right)+4 i \tan (c) \sec (c) \sin (d x) \sec (c+d x)-6 d x+\sin (2 d x)\right)}{4 d (a+i a \tan (c+d x))}","-\frac{\tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{3 i \tan (c+d x)}{2 a d}-\frac{\log (\cos (c+d x))}{a d}+\frac{3 i x}{2 a}",1,"((-1/4*I)*Cos[c]*Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])*(-6*d*x - (2*I)*Log[Cos[c + d*x]^2] + 4*d*x*Sec[c]^2 + 4*Sec[c]*Sec[c + d*x]*Sin[d*x] + Sin[2*d*x] + ArcTan[Tan[d*x]]*(-4 - (4*I)*Tan[c]) - (2*I)*d*x*Tan[c] + 2*Log[Cos[c + d*x]^2]*Tan[c] + (4*I)*Sec[c]*Sec[c + d*x]*Sin[d*x]*Tan[c] - I*Sin[2*d*x]*Tan[c] - 4*d*x*Tan[c]^2 + Cos[2*d*x]*(I + Tan[c])))/(d*(a + I*a*Tan[c + d*x]))","B",1
49,1,86,50,0.3187711,"\int \frac{\tan ^2(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x]),x]","\frac{4 \tan ^{-1}(\tan (d x)) (\tan (c+d x)-i)+2 \log \left(\cos ^2(c+d x)\right)+\tan (c+d x) \left(2 i \log \left(\cos ^2(c+d x)\right)-2 d x+i\right)+2 i d x-1}{4 a d (\tan (c+d x)-i)}","-\frac{i}{2 d (a+i a \tan (c+d x))}+\frac{i \log (\cos (c+d x))}{a d}+\frac{x}{2 a}",1,"(-1 + (2*I)*d*x + 2*Log[Cos[c + d*x]^2] + (I - 2*d*x + (2*I)*Log[Cos[c + d*x]^2])*Tan[c + d*x] + 4*ArcTan[Tan[d*x]]*(-I + Tan[c + d*x]))/(4*a*d*(-I + Tan[c + d*x]))","A",1
50,1,45,33,0.0977529,"\int \frac{\tan (c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x]),x]","\frac{(1-2 i d x) \tan (c+d x)-2 d x+i}{4 a d (\tan (c+d x)-i)}","-\frac{1}{2 d (a+i a \tan (c+d x))}-\frac{i x}{2 a}",1,"(I - 2*d*x + (1 - (2*I)*d*x)*Tan[c + d*x])/(4*a*d*(-I + Tan[c + d*x]))","A",1
51,1,45,33,0.1063642,"\int \frac{1}{a+i a \tan (c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-1),x]","\frac{(2 d x-i) \tan (c+d x)-2 i d x+1}{4 a d (\tan (c+d x)-i)}","\frac{x}{2 a}+\frac{i}{2 d (a+i a \tan (c+d x))}",1,"(1 - (2*I)*d*x + (-I + 2*d*x)*Tan[c + d*x])/(4*a*d*(-I + Tan[c + d*x]))","A",1
52,1,87,47,0.3257836,"\int \frac{\cot (c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x]),x]","\frac{\tan ^{-1}(\tan (d x)) (-4-4 i \tan (c+d x))-2 i \log \left(\sin ^2(c+d x)\right)+\tan (c+d x) \left(2 \log \left(\sin ^2(c+d x)\right)+2 i d x-1\right)+2 d x-i}{4 a d (\tan (c+d x)-i)}","\frac{1}{2 d (a+i a \tan (c+d x))}+\frac{\log (\sin (c+d x))}{a d}-\frac{i x}{2 a}",1,"(-I + 2*d*x - (2*I)*Log[Sin[c + d*x]^2] + ArcTan[Tan[d*x]]*(-4 - (4*I)*Tan[c + d*x]) + (-1 + (2*I)*d*x + 2*Log[Sin[c + d*x]^2])*Tan[c + d*x])/(4*a*d*(-I + Tan[c + d*x]))","A",1
53,1,286,70,0.6610585,"\int \frac{\cot ^2(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x]),x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc (c+d x) \sec (c+d x) \left(-4 d x \sin (c)-2 d x \sin (c+2 d x)-7 i \sin (c+2 d x)+2 d x \sin (3 c+2 d x)-i \sin (3 c+2 d x)+2 i d x \cos (c+2 d x)-9 \cos (c+2 d x)-2 i d x \cos (3 c+2 d x)+\cos (3 c+2 d x)-4 i \sin (c) \log \left(\sin ^2(c+d x)\right)-2 i \sin (c+2 d x) \log \left(\sin ^2(c+d x)\right)+2 i \sin (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)-2 \cos (c+2 d x) \log \left(\sin ^2(c+d x)\right)+2 \cos (3 c+2 d x) \log \left(\sin ^2(c+d x)\right)+16 i \sin (c) \tan ^{-1}(\tan (d x)) \sin (c+d x) (\cos (c+d x)+i \sin (c+d x))+10 i \sin (c)+8 \cos (c)\right)}{32 a d (\tan (c+d x)-i)}","-\frac{3 \cot (c+d x)}{2 a d}-\frac{i \log (\sin (c+d x))}{a d}+\frac{\cot (c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{3 x}{2 a}",1,"(Csc[c/2]*Csc[c + d*x]*Sec[c/2]*Sec[c + d*x]*(8*Cos[c] - 9*Cos[c + 2*d*x] + (2*I)*d*x*Cos[c + 2*d*x] + Cos[3*c + 2*d*x] - (2*I)*d*x*Cos[3*c + 2*d*x] - 2*Cos[c + 2*d*x]*Log[Sin[c + d*x]^2] + 2*Cos[3*c + 2*d*x]*Log[Sin[c + d*x]^2] + (10*I)*Sin[c] - 4*d*x*Sin[c] - (4*I)*Log[Sin[c + d*x]^2]*Sin[c] + (16*I)*ArcTan[Tan[d*x]]*Sin[c]*(Cos[c + d*x] + I*Sin[c + d*x])*Sin[c + d*x] - (7*I)*Sin[c + 2*d*x] - 2*d*x*Sin[c + 2*d*x] - (2*I)*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] - I*Sin[3*c + 2*d*x] + 2*d*x*Sin[3*c + 2*d*x] + (2*I)*Log[Sin[c + d*x]^2]*Sin[3*c + 2*d*x]))/(32*a*d*(-I + Tan[c + d*x]))","B",1
54,1,414,90,0.8751345,"\int \frac{\cot ^3(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + I*a*Tan[c + d*x]),x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^2(c+d x) \sec (c+d x) \left(-2 d x \sin (2 c+d x)+i \sin (2 c+d x)-2 d x \sin (2 c+3 d x)+9 i \sin (2 c+3 d x)+2 d x \sin (4 c+3 d x)-i \sin (4 c+3 d x)+6 i d x \cos (2 c+d x)-3 \cos (2 c+d x)+2 i d x \cos (2 c+3 d x)+7 \cos (2 c+3 d x)-2 i d x \cos (4 c+3 d x)+\cos (4 c+3 d x)-4 i \sin (d x) \log \left(\sin ^2(c+d x)\right)+4 i \sin (2 c+d x) \log \left(\sin ^2(c+d x)\right)+4 i \sin (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-4 i \sin (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)+\cos (d x) \left(-12 \log \left(\sin ^2(c+d x)\right)-6 i d x-5\right)+12 \cos (2 c+d x) \log \left(\sin ^2(c+d x)\right)+4 \cos (2 c+3 d x) \log \left(\sin ^2(c+d x)\right)-4 \cos (4 c+3 d x) \log \left(\sin ^2(c+d x)\right)+64 \sin (c) \tan ^{-1}(\tan (d x)) \sin ^2(c+d x) (\cos (c+d x)+i \sin (c+d x))+2 d x \sin (d x)-25 i \sin (d x)\right)}{64 a d (\tan (c+d x)-i)}","-\frac{\cot ^2(c+d x)}{a d}+\frac{3 i \cot (c+d x)}{2 a d}-\frac{2 \log (\sin (c+d x))}{a d}+\frac{\cot ^2(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{3 i x}{2 a}",1,"(Csc[c/2]*Csc[c + d*x]^2*Sec[c/2]*Sec[c + d*x]*(-3*Cos[2*c + d*x] + (6*I)*d*x*Cos[2*c + d*x] + 7*Cos[2*c + 3*d*x] + (2*I)*d*x*Cos[2*c + 3*d*x] + Cos[4*c + 3*d*x] - (2*I)*d*x*Cos[4*c + 3*d*x] + Cos[d*x]*(-5 - (6*I)*d*x - 12*Log[Sin[c + d*x]^2]) + 12*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] + 4*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] - 4*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]^2] - (25*I)*Sin[d*x] + 2*d*x*Sin[d*x] - (4*I)*Log[Sin[c + d*x]^2]*Sin[d*x] + 64*ArcTan[Tan[d*x]]*Sin[c]*(Cos[c + d*x] + I*Sin[c + d*x])*Sin[c + d*x]^2 + I*Sin[2*c + d*x] - 2*d*x*Sin[2*c + d*x] + (4*I)*Log[Sin[c + d*x]^2]*Sin[2*c + d*x] + (9*I)*Sin[2*c + 3*d*x] - 2*d*x*Sin[2*c + 3*d*x] + (4*I)*Log[Sin[c + d*x]^2]*Sin[2*c + 3*d*x] - I*Sin[4*c + 3*d*x] + 2*d*x*Sin[4*c + 3*d*x] - (4*I)*Log[Sin[c + d*x]^2]*Sin[4*c + 3*d*x]))/(64*a*d*(-I + Tan[c + d*x]))","B",1
55,1,365,108,3.1473959,"\int \frac{\cot ^4(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + I*a*Tan[c + d*x]),x]","\frac{\csc (c) (\cos (d x)+i \sin (d x)) \left(14 \csc (c+d x)-24 d x \sec (c+d x)+14 i \sec (c+d x)+24 d x \cos ^2(c) \sec (c+d x)-28 \cos (c-d x) \csc (2 (c+d x))-28 i \sin (c-d x) \csc (2 (c+d x))+30 d x \sin ^2(c) \sec (c+d x)-3 \sin ^2(c) \sin (2 d x) \sec (c+d x)-3 i d x \sin (2 c) \sec (c+d x)+12 i \sin ^2(c) \sec (c+d x) \log \left(\sin ^2(c+d x)\right)+6 \sin (2 c) \sec (c+d x) \log \left(\sin ^2(c+d x)\right)-3 i \sin ^2(c) \cos (2 d x) \sec (c+d x)+3 \sin (c) \cos (c) \cos (2 d x) \sec (c+d x)-3 i \sin (c) \cos (c) \sin (2 d x) \sec (c+d x)+2 \csc ^3(c+d x) (\cos (c-d x) \sec (c+d x)+i \sin (c-d x) \sec (c+d x)-1)+2 (\cos (c)+i \sin (c)) (2 \sin (c)+i \cos (c)) \csc ^2(c+d x) \sec (c+d x)+24 \sin (c) (\sin (c)-i \cos (c)) \tan ^{-1}(\tan (d x)) \sec (c+d x)\right)}{12 a d (\tan (c+d x)-i)}","-\frac{5 \cot ^3(c+d x)}{6 a d}+\frac{i \cot ^2(c+d x)}{a d}+\frac{5 \cot (c+d x)}{2 a d}+\frac{2 i \log (\sin (c+d x))}{a d}+\frac{\cot ^3(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{5 x}{2 a}",1,"(Csc[c]*(Cos[d*x] + I*Sin[d*x])*(14*Csc[c + d*x] - 28*Cos[c - d*x]*Csc[2*(c + d*x)] + (14*I)*Sec[c + d*x] - 24*d*x*Sec[c + d*x] + 24*d*x*Cos[c]^2*Sec[c + d*x] + 3*Cos[c]*Cos[2*d*x]*Sec[c + d*x]*Sin[c] + 30*d*x*Sec[c + d*x]*Sin[c]^2 - (3*I)*Cos[2*d*x]*Sec[c + d*x]*Sin[c]^2 + (12*I)*Log[Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c]^2 + 24*ArcTan[Tan[d*x]]*Sec[c + d*x]*Sin[c]*((-I)*Cos[c] + Sin[c]) + 2*Csc[c + d*x]^2*Sec[c + d*x]*(Cos[c] + I*Sin[c])*(I*Cos[c] + 2*Sin[c]) - (3*I)*d*x*Sec[c + d*x]*Sin[2*c] + 6*Log[Sin[c + d*x]^2]*Sec[c + d*x]*Sin[2*c] - (3*I)*Cos[c]*Sec[c + d*x]*Sin[c]*Sin[2*d*x] - 3*Sec[c + d*x]*Sin[c]^2*Sin[2*d*x] - (28*I)*Csc[2*(c + d*x)]*Sin[c - d*x] + 2*Csc[c + d*x]^3*(-1 + Cos[c - d*x]*Sec[c + d*x] + I*Sec[c + d*x]*Sin[c - d*x])))/(12*a*d*(-I + Tan[c + d*x]))","B",1
56,1,882,142,6.4719166,"\int \frac{\tan ^6(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^6/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \sec (c) (\cos (d x)+i \sin (d x))^2 (-\cos (2 c-d x)+\cos (2 c+d x)-i \sin (2 c-d x)+i \sin (2 c+d x)) \sec ^5(c+d x)}{6 d (i \tan (c+d x) a+a)^2}+\frac{\sec (c) (3 \cos (c)-i \sin (c)) \left(\frac{1}{3} \sin (2 c)-\frac{1}{3} i \cos (2 c)\right) (\cos (d x)+i \sin (d x))^2 \sec ^4(c+d x)}{d (i \tan (c+d x) a+a)^2}-\frac{13 i \sec (c) (\cos (d x)+i \sin (d x))^2 (-\cos (2 c-d x)+\cos (2 c+d x)-i \sin (2 c-d x)+i \sin (2 c+d x)) \sec ^3(c+d x)}{6 d (i \tan (c+d x) a+a)^2}-\frac{25 x \cos (2 c) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{4 (i \tan (c+d x) a+a)^2}-\frac{6 \tan ^{-1}(\tan (d x)) \cos (2 c) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{d (i \tan (c+d x) a+a)^2}+\frac{5 i \cos (2 d x) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{4 d (i \tan (c+d x) a+a)^2}-\frac{3 i \cos (2 c) \log \left(\cos ^2(c+d x)\right) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{d (i \tan (c+d x) a+a)^2}+\frac{\cos (4 d x) \left(-\frac{1}{16} i \cos (2 c)-\frac{1}{16} \sin (2 c)\right) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{d (i \tan (c+d x) a+a)^2}-\frac{25 i x \sin (2 c) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{4 (i \tan (c+d x) a+a)^2}-\frac{6 i \tan ^{-1}(\tan (d x)) \sin (2 c) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{d (i \tan (c+d x) a+a)^2}+\frac{3 \log \left(\cos ^2(c+d x)\right) \sin (2 c) (\cos (d x)+i \sin (d x))^2 \sec ^2(c+d x)}{d (i \tan (c+d x) a+a)^2}+\frac{5 (\cos (d x)+i \sin (d x))^2 \sin (2 d x) \sec ^2(c+d x)}{4 d (i \tan (c+d x) a+a)^2}+\frac{\left(\frac{1}{16} i \sin (2 c)-\frac{1}{16} \cos (2 c)\right) (\cos (d x)+i \sin (d x))^2 \sin (4 d x) \sec ^2(c+d x)}{d (i \tan (c+d x) a+a)^2}+\frac{x (\cos (d x)+i \sin (d x))^2 (i (6 \cos (2 c)+6 i \sin (2 c)) \tan (c)+6 i \tan (c)+6) \sec ^2(c+d x)}{(i \tan (c+d x) a+a)^2}","\frac{3 i \tan ^4(c+d x)}{2 a^2 d (1+i \tan (c+d x))}-\frac{25 \tan ^3(c+d x)}{12 a^2 d}-\frac{3 i \tan ^2(c+d x)}{a^2 d}+\frac{25 \tan (c+d x)}{4 a^2 d}-\frac{6 i \log (\cos (c+d x))}{a^2 d}-\frac{25 x}{4 a^2}-\frac{\tan ^5(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(-25*x*Cos[2*c]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2)/(4*(a + I*a*Tan[c + d*x])^2) - (6*ArcTan[Tan[d*x]]*Cos[2*c]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) + (((5*I)/4)*Cos[2*d*x]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) - ((3*I)*Cos[2*c]*Log[Cos[c + d*x]^2]*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) + (Cos[4*d*x]*Sec[c + d*x]^2*((-1/16*I)*Cos[2*c] - Sin[2*c]/16)*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) + (Sec[c]*Sec[c + d*x]^4*(3*Cos[c] - I*Sin[c])*((-1/3*I)*Cos[2*c] + Sin[2*c]/3)*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) - (((25*I)/4)*x*Sec[c + d*x]^2*Sin[2*c]*(Cos[d*x] + I*Sin[d*x])^2)/(a + I*a*Tan[c + d*x])^2 - ((6*I)*ArcTan[Tan[d*x]]*Sec[c + d*x]^2*Sin[2*c]*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) + (3*Log[Cos[c + d*x]^2]*Sec[c + d*x]^2*Sin[2*c]*(Cos[d*x] + I*Sin[d*x])^2)/(d*(a + I*a*Tan[c + d*x])^2) + (5*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*Sin[2*d*x])/(4*d*(a + I*a*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(-1/16*Cos[2*c] + (I/16)*Sin[2*c])*(Cos[d*x] + I*Sin[d*x])^2*Sin[4*d*x])/(d*(a + I*a*Tan[c + d*x])^2) - (((13*I)/6)*Sec[c]*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^2*(-Cos[2*c - d*x] + Cos[2*c + d*x] - I*Sin[2*c - d*x] + I*Sin[2*c + d*x]))/(d*(a + I*a*Tan[c + d*x])^2) + ((I/6)*Sec[c]*Sec[c + d*x]^5*(Cos[d*x] + I*Sin[d*x])^2*(-Cos[2*c - d*x] + Cos[2*c + d*x] - I*Sin[2*c - d*x] + I*Sin[2*c + d*x]))/(d*(a + I*a*Tan[c + d*x])^2) + (x*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*(6 + (6*I)*Tan[c] + I*(6*Cos[2*c] + (6*I)*Sin[2*c])*Tan[c]))/(a + I*a*Tan[c + d*x])^2","B",1
57,1,300,124,1.6866521,"\int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^2 \left(-128 i d x \sin ^2(c)+60 d x \sin (2 c)-\sin (2 c) \sin (4 d x)-64 d x \tan (c)-i \sin (2 c) \cos (4 d x)-16 \sec (c) \cos (2 c-d x) \sec (c+d x)+16 \sec (c) \cos (2 c+d x) \sec (c+d x)+8 i \sin (2 c) \sec ^2(c+d x)-16 i \sec (c) \sin (2 c-d x) \sec (c+d x)+16 i \sec (c) \sin (2 c+d x) \sec (c+d x)+32 i \sin (2 c) \log \left(\cos ^2(c+d x)\right)+64 (\sin (2 c)-i \cos (2 c)) \tan ^{-1}(\tan (d x))+\cos (2 c) \left(-64 d x \tan (c)+8 \sec ^2(c+d x)+32 \log \left(\cos ^2(c+d x)\right)-60 i d x-i \sin (4 d x)+\cos (4 d x)\right)+64 i d x+16 i \sin (2 d x)-16 \cos (2 d x)\right)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{5 i \tan ^3(c+d x)}{4 a^2 d (1+i \tan (c+d x))}-\frac{2 \tan ^2(c+d x)}{a^2 d}-\frac{15 i \tan (c+d x)}{4 a^2 d}-\frac{4 \log (\cos (c+d x))}{a^2 d}+\frac{15 i x}{4 a^2}-\frac{\tan ^4(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*((64*I)*d*x - 16*Cos[2*d*x] - 16*Cos[2*c - d*x]*Sec[c]*Sec[c + d*x] + 16*Cos[2*c + d*x]*Sec[c]*Sec[c + d*x] - (128*I)*d*x*Sin[c]^2 + 60*d*x*Sin[2*c] - I*Cos[4*d*x]*Sin[2*c] + (32*I)*Log[Cos[c + d*x]^2]*Sin[2*c] + (8*I)*Sec[c + d*x]^2*Sin[2*c] + 64*ArcTan[Tan[d*x]]*((-I)*Cos[2*c] + Sin[2*c]) + (16*I)*Sin[2*d*x] - Sin[2*c]*Sin[4*d*x] - (16*I)*Sec[c]*Sec[c + d*x]*Sin[2*c - d*x] + (16*I)*Sec[c]*Sec[c + d*x]*Sin[2*c + d*x] - 64*d*x*Tan[c] + Cos[2*c]*((-60*I)*d*x + Cos[4*d*x] + 32*Log[Cos[c + d*x]^2] + 8*Sec[c + d*x]^2 - I*Sin[4*d*x] - 64*d*x*Tan[c])))/(16*a^2*d*(-I + Tan[c + d*x])^2)","B",1
58,1,273,104,1.5256168,"\int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^2 \left(64 d x \sin ^2(c)+36 i d x \sin (2 c)-i \sin (2 c) \sin (4 d x)-32 i d x \tan (c)+\sin (2 c) \cos (4 d x)-8 i \sec (c) \cos (2 c-d x) \sec (c+d x)+8 i \sec (c) \cos (2 c+d x) \sec (c+d x)+8 \sec (c) \sin (2 c-d x) \sec (c+d x)-8 \sec (c) \sin (2 c+d x) \sec (c+d x)-16 \sin (2 c) \log \left(\cos ^2(c+d x)\right)+32 (\cos (2 c)+i \sin (2 c)) \tan ^{-1}(\tan (d x))+\cos (2 c) \left(-32 i d x \tan (c)+16 i \log \left(\cos ^2(c+d x)\right)+36 d x+\sin (4 d x)+i \cos (4 d x)\right)-32 d x-12 \sin (2 d x)-12 i \cos (2 d x)\right)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{i \tan ^2(c+d x)}{a^2 d (1+i \tan (c+d x))}-\frac{9 \tan (c+d x)}{4 a^2 d}+\frac{2 i \log (\cos (c+d x))}{a^2 d}+\frac{9 x}{4 a^2}-\frac{\tan ^3(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"-1/16*(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*(-32*d*x - (12*I)*Cos[2*d*x] - (8*I)*Cos[2*c - d*x]*Sec[c]*Sec[c + d*x] + (8*I)*Cos[2*c + d*x]*Sec[c]*Sec[c + d*x] + 64*d*x*Sin[c]^2 + 32*ArcTan[Tan[d*x]]*(Cos[2*c] + I*Sin[2*c]) + (36*I)*d*x*Sin[2*c] + Cos[4*d*x]*Sin[2*c] - 16*Log[Cos[c + d*x]^2]*Sin[2*c] - 12*Sin[2*d*x] - I*Sin[2*c]*Sin[4*d*x] + 8*Sec[c]*Sec[c + d*x]*Sin[2*c - d*x] - 8*Sec[c]*Sec[c + d*x]*Sin[2*c + d*x] - (32*I)*d*x*Tan[c] + Cos[2*c]*(36*d*x + I*Cos[4*d*x] + (16*I)*Log[Cos[c + d*x]^2] + Sin[4*d*x] - (32*I)*d*x*Tan[c])))/(a^2*d*(-I + Tan[c + d*x])^2)","B",1
59,1,135,79,0.383784,"\int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) \left(4 d x \sin (2 (c+d x))+i \sin (2 (c+d x))+\cos (2 (c+d x)) \left(-8 \log \left(\cos ^2(c+d x)\right)-4 i d x-1\right)-8 i \sin (2 (c+d x)) \log \left(\cos ^2(c+d x)\right)+16 i \tan ^{-1}(\tan (d x)) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+8\right)}{16 a^2 d (\tan (c+d x)-i)^2}","-\frac{3}{4 a^2 d (1+i \tan (c+d x))}+\frac{\log (\cos (c+d x))}{a^2 d}-\frac{3 i x}{4 a^2}-\frac{\tan ^2(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(8 + Cos[2*(c + d*x)]*(-1 - (4*I)*d*x - 8*Log[Cos[c + d*x]^2]) + (16*I)*ArcTan[Tan[d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + I*Sin[2*(c + d*x)] + 4*d*x*Sin[2*(c + d*x)] - (8*I)*Log[Cos[c + d*x]^2]*Sin[2*(c + d*x)]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","A",1
60,1,68,59,0.1999976,"\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) ((1+4 i d x) \sin (2 (c+d x))+(4 d x+i) \cos (2 (c+d x))-4 i)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{3 i}{4 a^2 d (1+i \tan (c+d x))}-\frac{x}{4 a^2}-\frac{i}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(-4*I + (I + 4*d*x)*Cos[2*(c + d*x)] + (1 + (4*I)*d*x)*Sin[2*(c + d*x)]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","A",1
61,1,66,59,0.0936715,"\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) ((1+4 i d x) \cos (2 (c+d x))-(4 d x+i) \sin (2 (c+d x)))}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{1}{4 d \left(a^2+i a^2 \tan (c+d x)\right)}-\frac{i x}{4 a^2}-\frac{1}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*((1 + (4*I)*d*x)*Cos[2*(c + d*x)] - (I + 4*d*x)*Sin[2*(c + d*x)]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","A",1
62,1,68,61,0.1679021,"\int \frac{1}{(a+i a \tan (c+d x))^2} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-2),x]","-\frac{\sec ^2(c+d x) ((1+4 i d x) \sin (2 (c+d x))+(4 d x+i) \cos (2 (c+d x))+4 i)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{i}{4 d \left(a^2+i a^2 \tan (c+d x)\right)}+\frac{x}{4 a^2}+\frac{i}{4 d (a+i a \tan (c+d x))^2}",1,"-1/16*(Sec[c + d*x]^2*(4*I + (I + 4*d*x)*Cos[2*(c + d*x)] + (1 + (4*I)*d*x)*Sin[2*(c + d*x)]))/(a^2*d*(-I + Tan[c + d*x])^2)","A",1
63,1,135,71,0.3448038,"\int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) \left(4 d x \sin (2 (c+d x))+i \sin (2 (c+d x))-8 i \sin (2 (c+d x)) \log \left(\sin ^2(c+d x)\right)+\cos (2 (c+d x)) \left(-8 \log \left(\sin ^2(c+d x)\right)-4 i d x-1\right)+16 i \tan ^{-1}(\tan (d x)) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-8\right)}{16 a^2 d (\tan (c+d x)-i)^2}","\frac{3}{4 a^2 d (1+i \tan (c+d x))}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{3 i x}{4 a^2}+\frac{1}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(-8 + Cos[2*(c + d*x)]*(-1 - (4*I)*d*x - 8*Log[Sin[c + d*x]^2]) + (16*I)*ArcTan[Tan[d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + I*Sin[2*(c + d*x)] + 4*d*x*Sin[2*(c + d*x)] - (8*I)*Log[Sin[c + d*x]^2]*Sin[2*(c + d*x)]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","A",1
64,1,276,97,1.8150653,"\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^2 \left(-36 i d x \sin (2 c)+i \sin (2 c) \sin (4 d x)+64 d x \cos ^2(c)+32 i d x \cot (c)+16 \sin (2 c) \log \left(\sin ^2(c+d x)\right)-\sin (2 c) \cos (4 d x)+8 i \csc (c) \cos (2 c-d x) \csc (c+d x)-8 i \csc (c) \cos (2 c+d x) \csc (c+d x)-8 \csc (c) \sin (2 c-d x) \csc (c+d x)+8 \csc (c) \sin (2 c+d x) \csc (c+d x)-32 (\cos (2 c)+i \sin (2 c)) \tan ^{-1}(\tan (d x))-i \cos (2 c) \left(32 d x \cot (c)-i \left(16 i \log \left(\sin ^2(c+d x)\right)+36 d x+\sin (4 d x)\right)+\cos (4 d x)\right)-32 d x-12 \sin (2 d x)-12 i \cos (2 d x)\right)}{16 a^2 d (\tan (c+d x)-i)^2}","-\frac{9 \cot (c+d x)}{4 a^2 d}-\frac{2 i \log (\sin (c+d x))}{a^2 d}+\frac{\cot (c+d x)}{a^2 d (1+i \tan (c+d x))}-\frac{9 x}{4 a^2}+\frac{\cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"-1/16*(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*(-32*d*x + 64*d*x*Cos[c]^2 - (12*I)*Cos[2*d*x] + (32*I)*d*x*Cot[c] + (8*I)*Cos[2*c - d*x]*Csc[c]*Csc[c + d*x] - (8*I)*Cos[2*c + d*x]*Csc[c]*Csc[c + d*x] - 32*ArcTan[Tan[d*x]]*(Cos[2*c] + I*Sin[2*c]) - (36*I)*d*x*Sin[2*c] - Cos[4*d*x]*Sin[2*c] + 16*Log[Sin[c + d*x]^2]*Sin[2*c] - 12*Sin[2*d*x] + I*Sin[2*c]*Sin[4*d*x] - I*Cos[2*c]*(Cos[4*d*x] + 32*d*x*Cot[c] - I*(36*d*x + (16*I)*Log[Sin[c + d*x]^2] + Sin[4*d*x])) - 8*Csc[c]*Csc[c + d*x]*Sin[2*c - d*x] + 8*Csc[c]*Csc[c + d*x]*Sin[2*c + d*x]))/(a^2*d*(-I + Tan[c + d*x])^2)","B",1
65,1,319,122,1.4945533,"\int \frac{\cot ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^3/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^2 \left(60 d x \sin (2 c)-\sin (2 c) \sin (4 d x)+128 i d x \cos ^2(c)-60 i d x \cos (2 c)+\cos (2 c) \cos (4 d x)-64 d x \cot (c)+32 i \sin (2 c) \log \left(\sin ^2(c+d x)\right)-i \sin (2 c) \cos (4 d x)-i \cos (2 c) \sin (4 d x)+64 d x \cos (2 c) \cot (c)+8 \cos (2 c) \csc ^2(c+d x)-16 \csc (c) \cos (2 c-d x) \csc (c+d x)+16 \csc (c) \cos (2 c+d x) \csc (c+d x)+8 i \sin (2 c) \csc ^2(c+d x)-16 i \csc (c) \sin (2 c-d x) \csc (c+d x)+16 i \csc (c) \sin (2 c+d x) \csc (c+d x)+32 \cos (2 c) \log \left(\sin ^2(c+d x)\right)+64 (\sin (2 c)-i \cos (2 c)) \tan ^{-1}(\tan (d x))-64 i d x-16 i \sin (2 d x)+16 \cos (2 d x)\right)}{16 a^2 d (\tan (c+d x)-i)^2}","-\frac{2 \cot ^2(c+d x)}{a^2 d}+\frac{15 i \cot (c+d x)}{4 a^2 d}-\frac{4 \log (\sin (c+d x))}{a^2 d}+\frac{5 \cot ^2(c+d x)}{4 a^2 d (1+i \tan (c+d x))}+\frac{15 i x}{4 a^2}+\frac{\cot ^2(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*((-64*I)*d*x + (128*I)*d*x*Cos[c]^2 - (60*I)*d*x*Cos[2*c] + 16*Cos[2*d*x] + Cos[2*c]*Cos[4*d*x] - 64*d*x*Cot[c] + 64*d*x*Cos[2*c]*Cot[c] - 16*Cos[2*c - d*x]*Csc[c]*Csc[c + d*x] + 16*Cos[2*c + d*x]*Csc[c]*Csc[c + d*x] + 8*Cos[2*c]*Csc[c + d*x]^2 + 32*Cos[2*c]*Log[Sin[c + d*x]^2] + 60*d*x*Sin[2*c] - I*Cos[4*d*x]*Sin[2*c] + (8*I)*Csc[c + d*x]^2*Sin[2*c] + (32*I)*Log[Sin[c + d*x]^2]*Sin[2*c] + 64*ArcTan[Tan[d*x]]*((-I)*Cos[2*c] + Sin[2*c]) - (16*I)*Sin[2*d*x] - I*Cos[2*c]*Sin[4*d*x] - Sin[2*c]*Sin[4*d*x] - (16*I)*Csc[c]*Csc[c + d*x]*Sin[2*c - d*x] + (16*I)*Csc[c]*Csc[c + d*x]*Sin[2*c + d*x]))/(16*a^2*d*(-I + Tan[c + d*x])^2)","B",1
66,1,264,161,3.9351306,"\int \frac{\tan ^6(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^6/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 \left(660 i d x \sin (3 c)-234 i \sin (c) \sin (2 d x)-27 i \sin (c) \sin (4 d x)+2 i \sin (3 c) \sin (6 d x)+234 \sin (c) \cos (2 d x)+27 \sin (c) \cos (4 d x)-2 \sin (3 c) \cos (6 d x)+9 \cos (c) (29 \sin (d x)-23 i \cos (d x)) (\cos (3 d x)-i \sin (3 d x))-48 \sin (3 c) \sec ^2(c+d x)-288 i \sin (3 c) \sec (c) \sin (d x) \sec (c+d x)-672 \sin (3 c) \log (\cos (c+d x))+\cos (3 c) \left(48 i \sec ^2(c+d x)+672 i \log (\cos (c+d x))-288 \sec (c) \sin (d x) \sec (c+d x)+660 d x-2 \sin (6 d x)-2 i \cos (6 d x)\right)\right)}{96 d (a+i a \tan (c+d x))^3}","\frac{55 \tan ^3(c+d x)}{24 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{7 i \tan ^2(c+d x)}{2 a^3 d}-\frac{55 \tan (c+d x)}{8 a^3 d}+\frac{7 i \log (\cos (c+d x))}{a^3 d}+\frac{55 x}{8 a^3}-\frac{\tan ^5(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{13 i \tan ^4(c+d x)}{24 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*(234*Cos[2*d*x]*Sin[c] + 27*Cos[4*d*x]*Sin[c] + (660*I)*d*x*Sin[3*c] - 2*Cos[6*d*x]*Sin[3*c] - 672*Log[Cos[c + d*x]]*Sin[3*c] - 48*Sec[c + d*x]^2*Sin[3*c] - (288*I)*Sec[c]*Sec[c + d*x]*Sin[3*c]*Sin[d*x] - (234*I)*Sin[c]*Sin[2*d*x] + 9*Cos[c]*((-23*I)*Cos[d*x] + 29*Sin[d*x])*(Cos[3*d*x] - I*Sin[3*d*x]) - (27*I)*Sin[c]*Sin[4*d*x] + Cos[3*c]*(660*d*x - (2*I)*Cos[6*d*x] + (672*I)*Log[Cos[c + d*x]] + (48*I)*Sec[c + d*x]^2 - 288*Sec[c]*Sec[c + d*x]*Sin[d*x] - 2*Sin[6*d*x]) + (2*I)*Sin[3*c]*Sin[6*d*x]))/(96*d*(a + I*a*Tan[c + d*x])^3)","A",1
67,1,239,143,2.4599634,"\int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 (-138 \sin (c) \sin (2 d x)-21 \sin (c) \sin (4 d x)+300 d x \sin (3 c)+2 \sin (3 c) \sin (6 d x)-138 i \sin (c) \cos (2 d x)-21 i \sin (c) \cos (4 d x)+2 i \sin (3 c) \cos (6 d x)+\cos (c) (39 \cos (d x)+53 i \sin (d x)) (-3 \cos (3 d x)+3 i \sin (3 d x))-96 \sin (3 c) \sec (c) \sin (d x) \sec (c+d x)+288 i \sin (3 c) \log (\cos (c+d x))+\cos (3 c) (288 \log (\cos (c+d x))+96 i \sec (c) \sin (d x) \sec (c+d x)-300 i d x+2 i \sin (6 d x)-2 \cos (6 d x)))}{96 d (a+i a \tan (c+d x))^3}","\frac{3 \tan ^2(c+d x)}{2 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{25 i \tan (c+d x)}{8 a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{25 i x}{8 a^3}-\frac{\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*((-138*I)*Cos[2*d*x]*Sin[c] - (21*I)*Cos[4*d*x]*Sin[c] + 300*d*x*Sin[3*c] + (2*I)*Cos[6*d*x]*Sin[3*c] + (288*I)*Log[Cos[c + d*x]]*Sin[3*c] - 96*Sec[c]*Sec[c + d*x]*Sin[3*c]*Sin[d*x] - 138*Sin[c]*Sin[2*d*x] + Cos[c]*(39*Cos[d*x] + (53*I)*Sin[d*x])*(-3*Cos[3*d*x] + (3*I)*Sin[3*d*x]) - 21*Sin[c]*Sin[4*d*x] + Cos[3*c]*((-300*I)*d*x - 2*Cos[6*d*x] + 288*Log[Cos[c + d*x]] + (96*I)*Sec[c]*Sec[c + d*x]*Sin[d*x] + (2*I)*Sin[6*d*x]) + 2*Sin[3*c]*Sin[6*d*x]))/(96*d*(a + I*a*Tan[c + d*x])^3)","A",1
68,1,118,119,0.4436183,"\int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-81 i \sin (c+d x)+84 d x \sin (3 (c+d x))+2 i \sin (3 (c+d x))-51 \cos (c+d x)+\cos (3 (c+d x)) (96 \log (\cos (c+d x))-84 i d x-2)+96 i \sin (3 (c+d x)) \log (\cos (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{7 i}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{i \log (\cos (c+d x))}{a^3 d}-\frac{7 x}{8 a^3}-\frac{\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^3*(-51*Cos[c + d*x] + Cos[3*(c + d*x)]*(-2 - (84*I)*d*x + 96*Log[Cos[c + d*x]]) - (81*I)*Sin[c + d*x] + (2*I)*Sin[3*(c + d*x)] + 84*d*x*Sin[3*(c + d*x)] + (96*I)*Log[Cos[c + d*x]]*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
69,1,91,92,0.2476819,"\int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^3,x]","-\frac{\sec ^3(c+d x) (27 \sin (c+d x)+12 i d x \sin (3 (c+d x))-2 \sin (3 (c+d x))-9 i \cos (c+d x)+2 (6 d x-i) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{3}{8 a^3 d (1+i \tan (c+d x))}+\frac{i x}{8 a^3}+\frac{i \tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{1}{8 a d (a+i a \tan (c+d x))^2}",1,"-1/96*(Sec[c + d*x]^3*((-9*I)*Cos[c + d*x] + 2*(-I + 6*d*x)*Cos[3*(c + d*x)] + 27*Sin[c + d*x] - 2*Sin[3*(c + d*x)] + (12*I)*d*x*Sin[3*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
70,1,91,88,0.4724935,"\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-3 i \sin (c+d x)+12 d x \sin (3 (c+d x))-2 i \sin (3 (c+d x))-9 \cos (c+d x)+2 (1-6 i d x) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","-\frac{i}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{x}{8 a^3}+\frac{3 i}{8 a d (a+i a \tan (c+d x))^2}-\frac{i}{6 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*(-9*Cos[c + d*x] + 2*(1 - (6*I)*d*x)*Cos[3*(c + d*x)] - (3*I)*Sin[c + d*x] - (2*I)*Sin[3*(c + d*x)] + 12*d*x*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
71,1,91,84,0.3991053,"\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-9 \sin (c+d x)+12 i d x \sin (3 (c+d x))-2 \sin (3 (c+d x))+3 i \cos (c+d x)+2 (6 d x-i) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{1}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{i x}{8 a^3}+\frac{1}{8 a d (a+i a \tan (c+d x))^2}-\frac{1}{6 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*((3*I)*Cos[c + d*x] + 2*(-I + 6*d*x)*Cos[3*(c + d*x)] - 9*Sin[c + d*x] - 2*Sin[3*(c + d*x)] + (12*I)*d*x*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
72,1,93,88,0.2184624,"\int \frac{1}{(a+i a \tan (c+d x))^3} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-3),x]","\frac{i \sec ^3(c+d x) (-9 \sin (c+d x)+12 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))+27 i \cos (c+d x)+2 (6 d x+i) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{i}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{x}{8 a^3}+\frac{i}{8 a d (a+i a \tan (c+d x))^2}+\frac{i}{6 d (a+i a \tan (c+d x))^3}",1,"((I/96)*Sec[c + d*x]^3*((27*I)*Cos[c + d*x] + 2*(I + 6*d*x)*Cos[3*(c + d*x)] - 9*Sin[c + d*x] + 2*Sin[3*(c + d*x)] + (12*I)*d*x*Sin[3*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
73,1,118,98,0.4179504,"\int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (-51 \sin (c+d x)+84 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))+81 i \cos (c+d x)-96 \sin (3 (c+d x)) \log (\sin (c+d x))+\cos (3 (c+d x)) (96 i \log (\sin (c+d x))+84 d x+2 i))}{96 a^3 d (\tan (c+d x)-i)^3}","\frac{7}{8 d \left(a^3+i a^3 \tan (c+d x)\right)}+\frac{\log (\sin (c+d x))}{a^3 d}-\frac{7 i x}{8 a^3}+\frac{3}{8 a d (a+i a \tan (c+d x))^2}+\frac{1}{6 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*((81*I)*Cos[c + d*x] + Cos[3*(c + d*x)]*(2*I + 84*d*x + (96*I)*Log[Sin[c + d*x]]) - 51*Sin[c + d*x] + 2*Sin[3*(c + d*x)] + (84*I)*d*x*Sin[3*(c + d*x)] - 96*Log[Sin[c + d*x]]*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","A",1
74,1,379,133,4.4413405,"\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 \left(288 d x \sin (c)+300 d x \sin (3 c)+138 \sin (c) \sin (2 d x)-21 \sin (c) \sin (4 d x)-2 \sin (3 c) \sin (6 d x)-300 i d x \cos (3 c)+2 \cos (3 c) \cos (6 d x)+144 i \sin (3 c) \log \left(\sin ^2(c+d x)\right)-2 i \cos (3 c) \sin (6 d x)+138 i \sin (c) \cos (2 d x)-21 i \sin (c) \cos (4 d x)-2 i \sin (3 c) \cos (6 d x)+288 d x \cos (3 c) \cot (c)+288 i d x \sin (3 c) \cot (c)-48 i \csc (c) \sin (3 c-d x) \csc (c+d x)+48 i \csc (c) \sin (3 c+d x) \csc (c+d x)+144 \cos (3 c) \log \left(\sin ^2(c+d x)\right)+288 (\sin (3 c)-i \cos (3 c)) \tan ^{-1}(\tan (d x))-3 \cos (c) (96 d x \cot (c)+192 i d x+46 i \sin (2 d x)+7 i \sin (4 d x)-46 \cos (2 d x)-7 \cos (4 d x))-24 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (3 c-d x) \csc (c+d x)+24 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (3 c+d x) \csc (c+d x)\right)}{96 a^3 d (\tan (c+d x)-i)^3}","-\frac{25 \cot (c+d x)}{8 a^3 d}-\frac{3 i \log (\sin (c+d x))}{a^3 d}+\frac{3 \cot (c+d x)}{2 d \left(a^3+i a^3 \tan (c+d x)\right)}-\frac{25 x}{8 a^3}+\frac{11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac{\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*((-300*I)*d*x*Cos[3*c] + 2*Cos[3*c]*Cos[6*d*x] + 288*d*x*Cos[3*c]*Cot[c] + 144*Cos[3*c]*Log[Sin[c + d*x]^2] - 24*Cos[3*c - d*x]*Csc[c/2]*Csc[c + d*x]*Sec[c/2] + 24*Cos[3*c + d*x]*Csc[c/2]*Csc[c + d*x]*Sec[c/2] + 288*d*x*Sin[c] + (138*I)*Cos[2*d*x]*Sin[c] - (21*I)*Cos[4*d*x]*Sin[c] + 300*d*x*Sin[3*c] - (2*I)*Cos[6*d*x]*Sin[3*c] + (288*I)*d*x*Cot[c]*Sin[3*c] + (144*I)*Log[Sin[c + d*x]^2]*Sin[3*c] + 288*ArcTan[Tan[d*x]]*((-I)*Cos[3*c] + Sin[3*c]) + 138*Sin[c]*Sin[2*d*x] - 3*Cos[c]*((192*I)*d*x - 46*Cos[2*d*x] - 7*Cos[4*d*x] + 96*d*x*Cot[c] + (46*I)*Sin[2*d*x] + (7*I)*Sin[4*d*x]) - 21*Sin[c]*Sin[4*d*x] - (2*I)*Cos[3*c]*Sin[6*d*x] - 2*Sin[3*c]*Sin[6*d*x] - (48*I)*Csc[c]*Csc[c + d*x]*Sin[3*c - d*x] + (48*I)*Csc[c]*Csc[c + d*x]*Sin[3*c + d*x]))/(96*a^3*d*(-I + Tan[c + d*x])^3)","B",1
75,1,429,171,0.8224684,"\int \frac{\tan ^6(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^6/(a + I*a*Tan[c + d*x])^4,x]","-\frac{\sec (c) \sec ^5(c+d x) (832 \sin (2 c+d x)+1560 i d x \sin (2 c+3 d x)+835 \sin (2 c+3 d x)+1560 i d x \sin (4 c+3 d x)+1603 \sin (4 c+3 d x)+1560 i d x \sin (4 c+5 d x)-765 \sin (4 c+5 d x)+1560 i d x \sin (6 c+5 d x)+3 \sin (6 c+5 d x)-536 i \cos (2 c+d x)+1560 d x \cos (2 c+3 d x)-893 i \cos (2 c+3 d x)+1560 d x \cos (4 c+3 d x)-1661 i \cos (4 c+3 d x)+1560 d x \cos (4 c+5 d x)+771 i \cos (4 c+5 d x)+1560 d x \cos (6 c+5 d x)+3 i \cos (6 c+5 d x)+1536 i \cos (2 c+3 d x) \log (\cos (c+d x))+1536 i \cos (4 c+3 d x) \log (\cos (c+d x))+1536 i \cos (4 c+5 d x) \log (\cos (c+d x))+1536 i \cos (6 c+5 d x) \log (\cos (c+d x))-1536 \sin (2 c+3 d x) \log (\cos (c+d x))-1536 \sin (4 c+3 d x) \log (\cos (c+d x))-1536 \sin (4 c+5 d x) \log (\cos (c+d x))-1536 \sin (6 c+5 d x) \log (\cos (c+d x))+832 \sin (d x)-536 i \cos (d x))}{1536 a^4 d (\tan (c+d x)-i)^4}","\frac{31 \tan ^3(c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{2 i \tan ^2(c+d x)}{a^4 d (1+i \tan (c+d x))}+\frac{65 \tan (c+d x)}{16 a^4 d}-\frac{4 i \log (\cos (c+d x))}{a^4 d}-\frac{65 x}{16 a^4}-\frac{\tan ^5(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{7 i \tan ^4(c+d x)}{24 a d (a+i a \tan (c+d x))^3}",1,"-1/1536*(Sec[c]*Sec[c + d*x]^5*((-536*I)*Cos[d*x] - (536*I)*Cos[2*c + d*x] - (893*I)*Cos[2*c + 3*d*x] + 1560*d*x*Cos[2*c + 3*d*x] - (1661*I)*Cos[4*c + 3*d*x] + 1560*d*x*Cos[4*c + 3*d*x] + (771*I)*Cos[4*c + 5*d*x] + 1560*d*x*Cos[4*c + 5*d*x] + (3*I)*Cos[6*c + 5*d*x] + 1560*d*x*Cos[6*c + 5*d*x] + (1536*I)*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]] + (1536*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]] + (1536*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]] + (1536*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]] + 832*Sin[d*x] + 832*Sin[2*c + d*x] + 835*Sin[2*c + 3*d*x] + (1560*I)*d*x*Sin[2*c + 3*d*x] - 1536*Log[Cos[c + d*x]]*Sin[2*c + 3*d*x] + 1603*Sin[4*c + 3*d*x] + (1560*I)*d*x*Sin[4*c + 3*d*x] - 1536*Log[Cos[c + d*x]]*Sin[4*c + 3*d*x] - 765*Sin[4*c + 5*d*x] + (1560*I)*d*x*Sin[4*c + 5*d*x] - 1536*Log[Cos[c + d*x]]*Sin[4*c + 5*d*x] + 3*Sin[6*c + 5*d*x] + (1560*I)*d*x*Sin[6*c + 5*d*x] - 1536*Log[Cos[c + d*x]]*Sin[6*c + 5*d*x]))/(a^4*d*(-I + Tan[c + d*x])^4)","B",1
76,1,126,147,0.4409908,"\int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (112 \cos (2 (c+d x))+i (96 \sin (2 (c+d x))+120 i d x \sin (4 (c+d x))+\sin (4 (c+d x))+\cos (4 (c+d x)) (128 i \log (\cos (c+d x))+120 d x+i)-128 \sin (4 (c+d x)) \log (\cos (c+d x))+32 i))}{128 a^4 d (\tan (c+d x)-i)^4}","\frac{7 \tan ^2(c+d x)}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{15}{16 a^4 d (1+i \tan (c+d x))}-\frac{\log (\cos (c+d x))}{a^4 d}+\frac{15 i x}{16 a^4}-\frac{\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{i \tan ^3(c+d x)}{4 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(112*Cos[2*(c + d*x)] + I*(32*I + Cos[4*(c + d*x)]*(I + 120*d*x + (128*I)*Log[Cos[c + d*x]]) + 96*Sin[2*(c + d*x)] + Sin[4*(c + d*x)] + (120*I)*d*x*Sin[4*(c + d*x)] - 128*Log[Cos[c + d*x]]*Sin[4*(c + d*x)])))/(128*a^4*d*(-I + Tan[c + d*x])^4)","A",1
77,1,98,128,0.2276951,"\int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (32 \sin (2 (c+d x))+24 i d x \sin (4 (c+d x))+3 \sin (4 (c+d x))-64 i \cos (2 (c+d x))+3 (8 d x+i) \cos (4 (c+d x))+36 i)}{384 a^4 d (\tan (c+d x)-i)^4}","-\frac{3 i}{16 a^4 d (1+i \tan (c+d x))}+\frac{x}{16 a^4}+\frac{i}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{i \tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{\tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(36*I - (64*I)*Cos[2*(c + d*x)] + 3*(I + 8*d*x)*Cos[4*(c + d*x)] + 32*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)] + (24*I)*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
78,1,95,126,0.5445853,"\int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (32 i \sin (2 (c+d x))-24 d x \sin (4 (c+d x))-3 i \sin (4 (c+d x))+16 \cos (2 (c+d x))+3 (1+8 i d x) \cos (4 (c+d x)))}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{3}{16 a^4 d (1+i \tan (c+d x))}+\frac{i x}{16 a^4}-\frac{1}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac{i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(16*Cos[2*(c + d*x)] + 3*(1 + (8*I)*d*x)*Cos[4*(c + d*x)] + (32*I)*Sin[2*(c + d*x)] - (3*I)*Sin[4*(c + d*x)] - 24*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
79,1,69,116,0.3990035,"\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^4,x]","-\frac{(\cos (4 (c+d x))-i \sin (4 (c+d x))) ((1+8 i d x) \sin (4 (c+d x))+(8 d x+i) \cos (4 (c+d x))-4 i)}{128 a^4 d}","-\frac{i}{16 d \left(a^4+i a^4 \tan (c+d x)\right)}-\frac{x}{16 a^4}-\frac{i}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{i}{4 a d (a+i a \tan (c+d x))^3}-\frac{i}{8 d (a+i a \tan (c+d x))^4}",1,"-1/128*((Cos[4*(c + d*x)] - I*Sin[4*(c + d*x)])*(-4*I + (I + 8*d*x)*Cos[4*(c + d*x)] + (1 + (8*I)*d*x)*Sin[4*(c + d*x)]))/(a^4*d)","A",1
80,1,94,110,0.3664712,"\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (32 i \sin (2 (c+d x))+24 d x \sin (4 (c+d x))+3 i \sin (4 (c+d x))+16 \cos (2 (c+d x))+(-3-24 i d x) \cos (4 (c+d x)))}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{1}{16 d \left(a^4+i a^4 \tan (c+d x)\right)}-\frac{i x}{16 a^4}+\frac{1}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{1}{12 a d (a+i a \tan (c+d x))^3}-\frac{1}{8 d (a+i a \tan (c+d x))^4}",1,"(Sec[c + d*x]^4*(16*Cos[2*(c + d*x)] + (-3 - (24*I)*d*x)*Cos[4*(c + d*x)] + (32*I)*Sin[2*(c + d*x)] + (3*I)*Sin[4*(c + d*x)] + 24*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
81,1,98,116,0.2348259,"\int \frac{1}{(a+i a \tan (c+d x))^4} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-4),x]","\frac{\sec ^4(c+d x) (-32 \sin (2 (c+d x))+24 i d x \sin (4 (c+d x))+3 \sin (4 (c+d x))+64 i \cos (2 (c+d x))+3 (8 d x+i) \cos (4 (c+d x))+36 i)}{384 a^4 d (\tan (c+d x)-i)^4}","\frac{i}{16 d \left(a^4+i a^4 \tan (c+d x)\right)}+\frac{x}{16 a^4}+\frac{i}{16 d \left(a^2+i a^2 \tan (c+d x)\right)^2}+\frac{i}{12 a d (a+i a \tan (c+d x))^3}+\frac{i}{8 d (a+i a \tan (c+d x))^4}",1,"(Sec[c + d*x]^4*(36*I + (64*I)*Cos[2*(c + d*x)] + 3*(I + 8*d*x)*Cos[4*(c + d*x)] - 32*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)] + (24*I)*d*x*Sin[4*(c + d*x)]))/(384*a^4*d*(-I + Tan[c + d*x])^4)","A",1
82,1,123,120,0.3995233,"\int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (96 i \sin (2 (c+d x))+120 d x \sin (4 (c+d x))-i \sin (4 (c+d x))+112 \cos (2 (c+d x))+128 i \sin (4 (c+d x)) \log (\sin (c+d x))+\cos (4 (c+d x)) (128 \log (\sin (c+d x))-120 i d x+1)+32)}{128 a^4 d (\tan (c+d x)-i)^4}","\frac{15}{16 a^4 d (1+i \tan (c+d x))}+\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{15 i x}{16 a^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{1}{8 d (a+i a \tan (c+d x))^4}",1,"(Sec[c + d*x]^4*(32 + 112*Cos[2*(c + d*x)] + Cos[4*(c + d*x)]*(1 - (120*I)*d*x + 128*Log[Sin[c + d*x]]) + (96*I)*Sin[2*(c + d*x)] - I*Sin[4*(c + d*x)] + 120*d*x*Sin[4*(c + d*x)] + (128*I)*Log[Sin[c + d*x]]*Sin[4*(c + d*x)]))/(128*a^4*d*(-I + Tan[c + d*x])^4)","A",1
83,1,444,159,2.6440483,"\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^4,x]","\frac{i \csc (c) \sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4 \left(1536 d x \cos ^3(c)+4608 i d x \sin (c) \cos ^2(c)-64 \cos (c) \left(24 i d x \sin (4 c)+24 d x \cos (4 c)+\sin ^2(c) (72 d x-\sin (6 d x)-i \cos (6 d x))\right)+1536 i \sin (c) (\cos (4 c)+i \sin (4 c)) \tan ^{-1}(\tan (d x))+i \left(-1536 d x \sin ^3(c)+1560 i d x \sin (4 c) \sin (c)+864 i \sin (2 c) \sin (c) \sin (2 d x)+180 \sin (c) \sin (4 d x)-3 i \sin (4 c) \sin (c) \sin (8 d x)-768 \sin (4 c) \sin (c) \log \left(\sin ^2(c+d x)\right)+1560 d x \sin (c) \cos (4 c)+864 i \sin (c) \cos (2 c) \cos (2 d x)+180 i \sin (c) \cos (4 d x)+32 i \sin (c) \cos (2 c) \cos (6 d x)+3 i \sin (c) \cos (4 c) \cos (8 d x)-864 \sin (2 c) \sin (c) \cos (2 d x)+3 \sin (4 c) \sin (c) \cos (8 d x)+864 \sin (c) \cos (2 c) \sin (2 d x)+32 \sin (c) \cos (2 c) \sin (6 d x)+3 \sin (c) \cos (4 c) \sin (8 d x)-192 i \cos (4 c-d x) \csc (c+d x)+192 i \cos (4 c+d x) \csc (c+d x)+192 \sin (4 c-d x) \csc (c+d x)-192 \sin (4 c+d x) \csc (c+d x)+768 i \sin (c) \cos (4 c) \log \left(\sin ^2(c+d x)\right)\right)\right)}{384 a^4 d (\tan (c+d x)-i)^4}","-\frac{65 \cot (c+d x)}{16 a^4 d}-\frac{4 i \log (\sin (c+d x))}{a^4 d}+\frac{2 \cot (c+d x)}{a^4 d (1+i \tan (c+d x))}+\frac{31 \cot (c+d x)}{48 a^4 d (1+i \tan (c+d x))^2}-\frac{65 x}{16 a^4}+\frac{7 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^3}+\frac{\cot (c+d x)}{8 d (a+i a \tan (c+d x))^4}",1,"((I/384)*Csc[c]*Sec[c + d*x]^4*(Cos[d*x] + I*Sin[d*x])^4*(1536*d*x*Cos[c]^3 + (4608*I)*d*x*Cos[c]^2*Sin[c] + (1536*I)*ArcTan[Tan[d*x]]*Sin[c]*(Cos[4*c] + I*Sin[4*c]) - 64*Cos[c]*(24*d*x*Cos[4*c] + (24*I)*d*x*Sin[4*c] + Sin[c]^2*(72*d*x - I*Cos[6*d*x] - Sin[6*d*x])) + I*((-192*I)*Cos[4*c - d*x]*Csc[c + d*x] + (192*I)*Cos[4*c + d*x]*Csc[c + d*x] + 1560*d*x*Cos[4*c]*Sin[c] + (864*I)*Cos[2*c]*Cos[2*d*x]*Sin[c] + (180*I)*Cos[4*d*x]*Sin[c] + (32*I)*Cos[2*c]*Cos[6*d*x]*Sin[c] + (3*I)*Cos[4*c]*Cos[8*d*x]*Sin[c] + (768*I)*Cos[4*c]*Log[Sin[c + d*x]^2]*Sin[c] - 1536*d*x*Sin[c]^3 - 864*Cos[2*d*x]*Sin[c]*Sin[2*c] + (1560*I)*d*x*Sin[c]*Sin[4*c] + 3*Cos[8*d*x]*Sin[c]*Sin[4*c] - 768*Log[Sin[c + d*x]^2]*Sin[c]*Sin[4*c] + 864*Cos[2*c]*Sin[c]*Sin[2*d*x] + (864*I)*Sin[c]*Sin[2*c]*Sin[2*d*x] + 180*Sin[c]*Sin[4*d*x] + 32*Cos[2*c]*Sin[c]*Sin[6*d*x] + 3*Cos[4*c]*Sin[c]*Sin[8*d*x] - (3*I)*Sin[c]*Sin[4*c]*Sin[8*d*x] + 192*Csc[c + d*x]*Sin[4*c - d*x] - 192*Csc[c + d*x]*Sin[4*c + d*x])))/(a^4*d*(-I + Tan[c + d*x])^4)","B",1
84,1,105,168,2.1827634,"\int \tan ^4(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{a+i a \tan (c+d x)} \left(\frac{2}{105} \left(3 (5 \tan (c+d x)-i) \sec ^2(c+d x)-46 (\tan (c+d x)-i)\right)-i e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d}","\frac{2 \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{2 i \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}+\frac{62 i (a+i a \tan (c+d x))^{3/2}}{105 a d}+\frac{8 i \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"(Sqrt[a + I*a*Tan[c + d*x]]*(((-I)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/E^(I*(c + d*x)) + (2*(-46*(-I + Tan[c + d*x]) + 3*Sec[c + d*x]^2*(-I + 5*Tan[c + d*x])))/105))/d","A",1
85,1,95,127,1.6288823,"\int \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \left(-i \sin (2 (c+d x))-16 \cos (2 (c+d x))+\frac{30 \cos ^3(c+d x) \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-10\right)}{15 d}","\frac{2 \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{2 (a+i a \tan (c+d x))^{3/2}}{15 a d}-\frac{8 \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"(Sec[c + d*x]^2*(-10 + (30*ArcSinh[E^(I*(c + d*x))]*Cos[c + d*x]^3)/Sqrt[1 + E^((2*I)*(c + d*x))] - 16*Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(15*d)","A",1
86,1,97,76,0.7051963,"\int \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \left(4 e^{3 i (c+d x)}-3 \left(1+e^{2 i (c+d x)}\right)^{3/2} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{3 d \left(1+e^{2 i (c+d x)}\right)}","\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d}",1,"((-1/3*I)*(4*E^((3*I)*(c + d*x)) - 3*(1 + E^((2*I)*(c + d*x)))^(3/2)*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x))))","A",1
87,1,77,67,0.420585,"\int \tan (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^{-i (c+d x)} \left(2 e^{i (c+d x)}-\sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{d}","\frac{2 \sqrt{a+i a \tan (c+d x)}}{d}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((2*E^(I*(c + d*x)) - Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
88,1,64,46,0.1814204,"\int \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)}}{d}","-\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((-I)*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
89,0,0,78,0.6441198,"\int \cot (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]],x]","\int \cot (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}",1,"Integrate[Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
90,1,197,111,4.1016451,"\int \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{a+i a \tan (c+d x)} \left(-4 \cot (c+d x)+i e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(\sqrt{2} \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)+\log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-\log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+e^{i (c+d x)}+1\right)\right)+4 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)\right)}{4 d}","-\frac{i \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{\cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-4*Cot[c + d*x] + (I*Sqrt[1 + E^((2*I)*(c + d*x))]*(4*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))] + Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - Log[1 + E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]])))/E^(I*(c + d*x)))*Sqrt[a + I*a*Tan[c + d*x]])/(4*d)","A",1
91,1,144,145,1.9967417,"\int \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{a+i a \tan (c+d x)} \left(-4 \csc ^2(c+d x)+2 i \csc (c) \sin (d x) \csc (c+d x)+e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(7 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)-8 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)-2 i \cot (c)+4\right)}{8 d}","\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{\cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}-\frac{i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}",1,"((4 + (Sqrt[1 + E^((2*I)*(c + d*x))]*(-8*ArcSinh[E^(I*(c + d*x))] + 7*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^(I*(c + d*x)) - (2*I)*Cot[c] - 4*Csc[c + d*x]^2 + (2*I)*Csc[c]*Csc[c + d*x]*Sin[d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(8*d)","A",1
92,1,166,199,2.6602274,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{(a+i a \tan (c+d x))^{3/2} \left(\frac{2 \sqrt{2} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2}}+\frac{1}{210} (-1+i \tan (c+d x)) \sec ^{\frac{5}{2}}(c+d x) (-7 i \sin (c+d x)+53 i \sin (3 (c+d x))+378 \cos (c+d x)+158 \cos (3 (c+d x)))\right)}{d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^2 \tan ^4(c+d x)}{7 d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i a^2 \tan ^3(c+d x)}{7 d \sqrt{a+i a \tan (c+d x)}}+\frac{16 a \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 d}-\frac{76 (a+i a \tan (c+d x))^{3/2}}{105 d}-\frac{64 a \sqrt{a+i a \tan (c+d x)}}{35 d}",1,"(((2*Sqrt[2]*ArcSinh[E^(I*(c + d*x))])/((E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)) + (Sec[c + d*x]^(5/2)*(378*Cos[c + d*x] + 158*Cos[3*(c + d*x)] - (7*I)*Sin[c + d*x] + (53*I)*Sin[3*(c + d*x)])*(-1 + I*Tan[c + d*x]))/210)*(a + I*a*Tan[c + d*x])^(3/2))/(d*Sec[c + d*x]^(3/2))","A",1
93,1,162,101,1.3933898,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a e^{-\frac{1}{2} i (2 c+3 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\sin \left(\frac{d x}{2}\right)-i \cos \left(\frac{d x}{2}\right)\right) \left(\sqrt{1+e^{2 i (c+d x)}} \sec ^3(c+d x) (2 i \sin (2 (c+d x))+7 \cos (2 (c+d x))+5)-20 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{5 \sqrt{2} d}","\frac{2 i \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}-\frac{2 i a \sqrt{a+i a \tan (c+d x)}}{d}",1,"(a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*((-I)*Cos[(d*x)/2] + Sin[(d*x)/2])*(-20*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^3*(5 + 7*Cos[2*(c + d*x)] + (2*I)*Sin[2*(c + d*x)])))/(5*Sqrt[2]*d*E^((I/2)*(2*c + 3*d*x)))","A",1
94,1,148,92,0.9289615,"\int \tan (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{2} a e^{-\frac{1}{2} i (2 c+3 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\cos \left(\frac{d x}{2}\right)+i \sin \left(\frac{d x}{2}\right)\right) \left(\sqrt{1+e^{2 i (c+d x)}} (4+i \tan (c+d x)) \sec (c+d x)-6 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{3 d}","-\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{2 a \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(Sqrt[2]*a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(Cos[(d*x)/2] + I*Sin[(d*x)/2])*(-6*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]*(4 + I*Tan[c + d*x])))/(3*d*E^((I/2)*(2*c + 3*d*x)))","A",1
95,1,79,72,0.480655,"\int (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{2 i a e^{-i (c+d x)} \left(e^{i (c+d x)}-\sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right) \sqrt{a+i a \tan (c+d x)}}{d}","\frac{2 i a \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 i \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((2*I)*a*(E^(I*(c + d*x)) - Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
96,1,201,79,1.1624694,"\int \cot (c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-3 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(\sqrt{2} \left(\log \left(1-e^{i (c+d x)}\right)-\log \left(1+e^{i (c+d x)}\right)+\log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-\log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}+e^{i (c+d x)}+1\right)\right)+4 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{\sqrt{2} d}","\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}",1,"(((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(4*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))] + Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - Log[1 + E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]])))/(Sqrt[2]*d*E^((3*I)*(c + d*x)))","B",1
97,1,178,141,1.3276587,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a e^{-\frac{1}{2} i (2 c+3 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\sin \left(\frac{d x}{2}\right)-i \cos \left(\frac{d x}{2}\right)\right) \left(-i \sqrt{1+e^{2 i (c+d x)}} \csc (c+d x)-4 \sinh ^{-1}\left(e^{i (c+d x)}\right)+3 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} d}","-\frac{3 i a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{2 i \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{i a^2}{d \sqrt{a+i a \tan (c+d x)}}-\frac{a^2 \cot (c+d x)}{d \sqrt{a+i a \tan (c+d x)}}",1,"(a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-4*ArcSinh[E^(I*(c + d*x))] + 3*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] - I*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c + d*x])*((-I)*Cos[(d*x)/2] + Sin[(d*x)/2]))/(Sqrt[2]*d*E^((I/2)*(2*c + 3*d*x)))","A",1
98,1,190,184,1.9367397,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{a e^{-\frac{1}{2} i (2 c+3 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\cos \left(\frac{d x}{2}\right)+i \sin \left(\frac{d x}{2}\right)\right) \left(16 \sinh ^{-1}\left(e^{i (c+d x)}\right)-11 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+\sqrt{1+e^{2 i (c+d x)}} (2 \cot (c+d x)+5 i) \csc (c+d x)\right)}{4 \sqrt{2} d}","\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 \cot ^2(c+d x)}{2 d \sqrt{a+i a \tan (c+d x)}}-\frac{i a^2 \cot (c+d x)}{2 d \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}",1,"-1/4*(a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(16*ArcSinh[E^(I*(c + d*x))] - 11*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] + Sqrt[1 + E^((2*I)*(c + d*x))]*(5*I + 2*Cot[c + d*x])*Csc[c + d*x])*(Cos[(d*x)/2] + I*Sin[(d*x)/2]))/(Sqrt[2]*d*E^((I/2)*(2*c + 3*d*x)))","A",1
99,1,176,204,2.1795496,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x)) \left(\sqrt{1+e^{2 i (c+d x)}} \sec ^5(c+d x) (282 i \sin (2 (c+d x))+331 i \sin (4 (c+d x))+3012 \cos (2 (c+d x))+961 \cos (4 (c+d x))+2331)-10080 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{1260 \sqrt{2} d}","\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^2 \tan ^4(c+d x) \sqrt{a+i a \tan (c+d x)}}{9 d}+\frac{38 i a^2 \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{63 d}+\frac{92 a^2 \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{368 a^2 \sqrt{a+i a \tan (c+d x)}}{105 d}-\frac{472 a (a+i a \tan (c+d x))^{3/2}}{315 d}",1,"-1/1260*(a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(Cos[d*x] + I*Sin[d*x])*(-10080*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^5*(2331 + 3012*Cos[2*(c + d*x)] + 961*Cos[4*(c + d*x)] + (282*I)*Sin[2*(c + d*x)] + (331*I)*Sin[4*(c + d*x)])))/(Sqrt[2]*d*E^(I*(c + 2*d*x)))","A",1
100,1,170,130,2.6623172,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\sin (d x)-i \cos (d x)) \left(\sqrt{1+e^{2 i (c+d x)}} \sec ^3(c+d x) (122 \cos (2 (c+d x))+7 i \tan (c+d x)+19 i \sin (3 (c+d x)) \sec (c+d x)+86)-336 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{42 \sqrt{2} d}","\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{4 i a^2 \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a d}-\frac{2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*((-I)*Cos[d*x] + Sin[d*x])*(-336*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^3*(86 + 122*Cos[2*(c + d*x)] + (19*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (7*I)*Tan[c + d*x])))/(42*Sqrt[2]*d*E^(I*(c + 2*d*x)))","A",1
101,1,154,119,1.2798946,"\int \tan (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x)) \left(\sqrt{1+e^{2 i (c+d x)}} \sec ^3(c+d x) (11 i \sin (2 (c+d x))+41 \cos (2 (c+d x))+35)-120 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{15 \sqrt{2} d}","-\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{4 a^2 \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 a (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 (a+i a \tan (c+d x))^{5/2}}{5 d}",1,"(a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(Cos[d*x] + I*Sin[d*x])*(-120*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^3*(35 + 41*Cos[2*(c + d*x)] + (11*I)*Sin[2*(c + d*x)])))/(15*Sqrt[2]*d*E^(I*(c + 2*d*x)))","A",1
102,1,140,101,0.9494444,"\int (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{\sqrt{2} a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x)) \left(12 i \sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{1+e^{2 i (c+d x)}} (\tan (c+d x)-7 i) \sec (c+d x)\right)}{3 d}","-\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{4 i a^2 \sqrt{a+i a \tan (c+d x)}}{d}+\frac{2 i a (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"-1/3*(Sqrt[2]*a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(Cos[d*x] + I*Sin[d*x])*((12*I)*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]*(-7*I + Tan[c + d*x])))/(d*E^(I*(c + 2*d*x)))","A",1
103,1,148,104,1.1728306,"\int \cot (c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{\sqrt{2} a^2 e^{-i (c+d x)} \sqrt{a+i a \tan (c+d x)} \left(\sqrt{2} e^{i (c+d x)}-2 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{d}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{d}",1,"-((Sqrt[2]*a^2*(Sqrt[2]*E^(I*(c + d*x)) - 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))))","A",1
104,1,170,114,1.4210204,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\sin (d x)-i \cos (d x)) \left(-i \sqrt{1+e^{2 i (c+d x)}} \csc (c+d x)-8 \sinh ^{-1}\left(e^{i (c+d x)}\right)+5 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} d}","-\frac{5 i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{d}",1,"(a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-8*ArcSinh[E^(I*(c + d*x))] + 5*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] - I*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c + d*x])*((-I)*Cos[d*x] + Sin[d*x]))/(Sqrt[2]*d*E^(I*(c + 2*d*x)))","A",1
105,1,182,151,1.994143,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x)) \left(32 \sinh ^{-1}\left(e^{i (c+d x)}\right)-23 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+\sqrt{1+e^{2 i (c+d x)}} (2 \cot (c+d x)+9 i) \csc (c+d x)\right)}{4 \sqrt{2} d}","\frac{23 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}-\frac{9 i a^2 \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}",1,"-1/4*(a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(32*ArcSinh[E^(I*(c + d*x))] - 23*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] + Sqrt[1 + E^((2*I)*(c + d*x))]*(9*I + 2*Cot[c + d*x])*Csc[c + d*x])*(Cos[d*x] + I*Sin[d*x]))/(Sqrt[2]*d*E^(I*(c + 2*d*x)))","A",1
106,1,200,190,2.4179392,"\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{a^2 e^{-i (c+2 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} (\cos (d x)+i \sin (d x)) \left(-384 i \sinh ^{-1}\left(e^{i (c+d x)}\right)+270 i \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+\sqrt{1+e^{2 i (c+d x)}} \csc ^3(c+d x) (-26 i \sin (2 (c+d x))-65 \cos (2 (c+d x))+49)\right)}{48 \sqrt{2} d}","\frac{45 i a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{8 d}-\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}-\frac{13 i a^2 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d}+\frac{19 a^2 \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{8 d}",1,"(a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(Cos[d*x] + I*Sin[d*x])*((-384*I)*ArcSinh[E^(I*(c + d*x))] + (270*I)*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] + Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c + d*x]^3*(49 - 65*Cos[2*(c + d*x)] - (26*I)*Sin[2*(c + d*x)])))/(48*Sqrt[2]*d*E^(I*(c + 2*d*x)))","A",1
107,1,166,130,1.7327439,"\int (a+i a \tan (c+d x))^{7/2} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(7/2),x]","\frac{i \sqrt{2} a^3 e^{-\frac{1}{2} i (2 c+5 d x)} \sqrt{1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\cos \left(\frac{3 d x}{2}\right)+i \sin \left(\frac{3 d x}{2}\right)\right) \left(\sqrt{1+e^{2 i (c+d x)}} \sec ^3(c+d x) (8 i \sin (2 (c+d x))+38 \cos (2 (c+d x))+35)-120 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{15 d}","-\frac{8 i \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{8 i a^3 \sqrt{a+i a \tan (c+d x)}}{d}+\frac{4 i a^2 (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac{2 i a (a+i a \tan (c+d x))^{5/2}}{5 d}",1,"((I/15)*Sqrt[2]*a^3*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(Cos[(3*d*x)/2] + I*Sin[(3*d*x)/2])*(-120*ArcSinh[E^(I*(c + d*x))] + Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^3*(35 + 38*Cos[2*(c + d*x)] + (8*I)*Sin[2*(c + d*x)])))/(d*E^((I/2)*(2*c + 5*d*x)))","A",1
108,1,123,201,1.6497534,"\int \frac{\tan ^5(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^5/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{-\frac{840 e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-\left(\sec ^4(c+d x) (224 i \sin (2 (c+d x))+124 i \sin (4 (c+d x))+1484 \cos (2 (c+d x))+229 \cos (4 (c+d x))+1015)\right)}{840 d \sqrt{a+i a \tan (c+d x)}}","\frac{223 (a+i a \tan (c+d x))^{3/2}}{105 a^2 d}-\frac{\tan ^4(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{9 i \tan ^3(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 a d}+\frac{47 \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{35 a d}-\frac{188 \sqrt{a+i a \tan (c+d x)}}{35 a d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"((-840*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - Sec[c + d*x]^4*(1015 + 1484*Cos[2*(c + d*x)] + 229*Cos[4*(c + d*x)] + (224*I)*Sin[2*(c + d*x)] + (124*I)*Sin[4*(c + d*x)]))/(840*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
109,1,122,172,1.7893392,"\int \frac{\tan ^4(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^4/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i \sec ^3(c+d x) (20 i \sin (c+d x)+44 i \sin (3 (c+d x))+185 \cos (c+d x)+59 \cos (3 (c+d x)))-\frac{60 i e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}}{60 d \sqrt{a+i a \tan (c+d x)}}","-\frac{23 i (a+i a \tan (c+d x))^{3/2}}{15 a^2 d}-\frac{\tan ^3(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{7 i \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 a d}+\frac{28 i \sqrt{a+i a \tan (c+d x)}}{5 a d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(((-60*I)*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + I*Sec[c + d*x]^3*(185*Cos[c + d*x] + 59*Cos[3*(c + d*x)] + (20*I)*Sin[c + d*x] + (44*I)*Sin[3*(c + d*x)]))/(60*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
110,1,129,126,1.1130028,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{18 e^{2 i (c+d x)}+7 e^{4 i (c+d x)}+3 e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sinh ^{-1}\left(e^{i (c+d x)}\right)+3}{3 \sqrt{2} d \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{5 (a+i a \tan (c+d x))^{3/2}}{3 a^2 d}-\frac{\tan ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{4 \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(3 + 18*E^((2*I)*(c + d*x)) + 7*E^((4*I)*(c + d*x)) + 3*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(3/2)*ArcSinh[E^(I*(c + d*x))])/(3*Sqrt[2]*d*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2)","A",1
111,1,113,98,0.7942748,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i \left(\sqrt{1+e^{2 i (c+d x)}} \left(1+5 e^{2 i (c+d x)}\right)-e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d \left(1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{a+i a \tan (c+d x)}}","-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a d}-\frac{i}{d \sqrt{a+i a \tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"((-I)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(1 + 5*E^((2*I)*(c + d*x))) - E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*ArcSinh[E^(I*(c + d*x))]))/(d*(1 + E^((2*I)*(c + d*x)))^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])","A",1
112,1,83,67,0.244159,"\int \frac{\tan (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{-\sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","-\frac{1}{d \sqrt{a+i a \tan (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(-Sqrt[1 + E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
113,1,84,71,0.2089503,"\int \frac{1}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i \left(\sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","\frac{i}{d \sqrt{a+i a \tan (c+d x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(I*(Sqrt[1 + E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
114,1,133,99,1.1139922,"\int \frac{\cot (c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{1+e^{2 i (c+d x)}}+e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-2 \sqrt{2} e^{i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","\frac{1}{d \sqrt{a+i a \tan (c+d x)}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}",1,"(Sqrt[1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 2*Sqrt[2]*E^(I*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
115,1,153,141,2.1869744,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i \sec (c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{2} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+i \csc (c+d x) (2 i \sin (2 (c+d x))+\cos (2 (c+d x))+1)\right)}{2 d \sqrt{a+i a \tan (c+d x)}}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{2 \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{\cot (c+d x)}{d \sqrt{a+i a \tan (c+d x)}}",1,"((I/2)*Sec[c + d*x]*(Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] + I*Csc[c + d*x]*(1 + Cos[2*(c + d*x)] + (2*I)*Sin[2*(c + d*x)])))/(d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
116,1,170,180,2.5749347,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{-8 e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)+22 \sqrt{2} e^{i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+\sqrt{1+e^{2 i (c+d x)}} \csc ^2(c+d x) (i \sin (2 (c+d x))+5 \cos (2 (c+d x))-9)}{8 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{3 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}+\frac{\cot ^2(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a d}",1,"(-8*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] + 22*Sqrt[2]*E^(I*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]] + Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c + d*x]^2*(-9 + 5*Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))/(8*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
117,1,149,205,1.833062,"\int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-2 i (c+d x)} \left(115 e^{2 i (c+d x)}+855 e^{4 i (c+d x)}+1105 e^{6 i (c+d x)}+466 e^{8 i (c+d x)}-15 e^{3 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \sinh ^{-1}\left(e^{i (c+d x)}\right)-5\right)}{30 a d \left(1+e^{2 i (c+d x)}\right)^3 \sqrt{a+i a \tan (c+d x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{151 (a+i a \tan (c+d x))^{3/2}}{30 a^3 d}-\frac{39 \tan ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{10 a^2 d}+\frac{78 \sqrt{a+i a \tan (c+d x)}}{5 a^2 d}-\frac{\tan ^4(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{19 i \tan ^3(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"(-5 + 115*E^((2*I)*(c + d*x)) + 855*E^((4*I)*(c + d*x)) + 1105*E^((6*I)*(c + d*x)) + 466*E^((8*I)*(c + d*x)) - 15*E^((3*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(5/2)*ArcSinh[E^(I*(c + d*x))])/(30*a*d*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
118,1,165,174,1.2669796,"\int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i e^{-4 i (c+d x)} \sec ^2(c+d x) \left(\left(18 e^{2 i (c+d x)}+87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}-1\right) \sqrt{1+e^{2 i (c+d x)}}+3 e^{3 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^2 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{24 a d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a+i a \tan (c+d x)}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}-\frac{10 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}-\frac{\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{5 i \tan ^2(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/24*I)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-1 + 18*E^((2*I)*(c + d*x)) + 87*E^((4*I)*(c + d*x)) + 52*E^((6*I)*(c + d*x))) + 3*E^((3*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^2*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(a*d*E^((4*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a + I*a*Tan[c + d*x]])","A",1
119,1,123,133,1.2002999,"\int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-2 i (c+d x)} \left(-13 e^{2 i (c+d x)}-38 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+1\right)}{6 a d \left(1+e^{2 i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{3 a^2 d}-\frac{\tan ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}-\frac{11}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"(1 - 13*E^((2*I)*(c + d*x)) - 38*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/(6*a*d*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
120,1,124,104,1.0165677,"\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-1}{3 a d \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 i}{2 a d \sqrt{a+i a \tan (c+d x)}}-\frac{i}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(-1 + 7*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/(3*a*d*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
121,1,124,98,0.5859138,"\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i \left(e^{2 i (c+d x)}+2 e^{4 i (c+d x)}-3 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-1\right)}{3 a d \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{1}{2 a d \sqrt{a+i a \tan (c+d x)}}-\frac{1}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/3*I)*(-1 + E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x)) - 3*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))]))/(a*d*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
122,1,124,104,0.579771,"\int \frac{1}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-3/2),x]","\frac{5 e^{2 i (c+d x)}+4 e^{4 i (c+d x)}-3 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+1}{3 a d \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{i}{2 a d \sqrt{a+i a \tan (c+d x)}}+\frac{i}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(1 + 5*E^((2*I)*(c + d*x)) + 4*E^((4*I)*(c + d*x)) - 3*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/(3*a*d*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
123,1,196,132,1.9868168,"\int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i \left(11 e^{2 i (c+d x)}+10 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-12 \sqrt{2} e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+1\right)}{3 a d \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3}{2 a d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/3*I)*(1 + 11*E^((2*I)*(c + d*x)) + 10*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] - 12*Sqrt[2]*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]))/(a*d*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
124,1,214,181,1.8964275,"\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{e^{-4 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \csc (2 (c+d x)) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-15 e^{2 i (c+d x)}+28 e^{4 i (c+d x)}-1\right)-3 e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \sinh ^{-1}\left(e^{i (c+d x)}\right)-18 \sqrt{2} e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{12 a d \sqrt{a+i a \tan (c+d x)}}","\frac{3 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{7 \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{13 \cot (c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\cot (c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"-1/12*(Sqrt[1 + E^((2*I)*(c + d*x))]*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-1 - 15*E^((2*I)*(c + d*x)) + 28*E^((4*I)*(c + d*x))) - 3*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*ArcSinh[E^(I*(c + d*x))] - 18*Sqrt[2]*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Csc[2*(c + d*x)])/(a*d*E^((4*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
125,1,214,220,4.0609685,"\int \frac{\cot ^3(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{2} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2} \left(23 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)-2 \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)-\frac{1}{3} \csc ^2(c+d x) \sqrt{\sec (c+d x)} (27 i \sin (2 (c+d x))-18 i \sin (4 (c+d x))+6 \cos (2 (c+d x))-19 \cos (4 (c+d x))+25)\right)}{8 d (a+i a \tan (c+d x))^{3/2}}","\frac{23 \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{11 \cot ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 a^2 d}+\frac{21 i \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a^2 d}+\frac{17 \cot ^2(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\cot ^2(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(Sec[c + d*x]^(3/2)*(Sqrt[2]*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(-2*ArcSinh[E^(I*(c + d*x))] + 23*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]) - (Csc[c + d*x]^2*Sqrt[Sec[c + d*x]]*(25 + 6*Cos[2*(c + d*x)] - 19*Cos[4*(c + d*x)] + (27*I)*Sin[2*(c + d*x)] - (18*I)*Sin[4*(c + d*x)]))/3))/(8*d*(a + I*a*Tan[c + d*x])^(3/2))","A",1
126,1,149,205,1.9427885,"\int \frac{\tan ^5(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{e^{-4 i (c+d x)} \left(-33 e^{2 i (c+d x)}+348 e^{4 i (c+d x)}+1527 e^{6 i (c+d x)}+983 e^{8 i (c+d x)}+15 e^{5 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sinh ^{-1}\left(e^{i (c+d x)}\right)+3\right)}{60 a^2 d \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{a+i a \tan (c+d x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{361 (a+i a \tan (c+d x))^{3/2}}{60 a^4 d}-\frac{89 \sqrt{a+i a \tan (c+d x)}}{5 a^3 d}+\frac{89 \tan ^2(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\tan ^4(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 i \tan ^3(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}",1,"-1/60*(3 - 33*E^((2*I)*(c + d*x)) + 348*E^((4*I)*(c + d*x)) + 1527*E^((6*I)*(c + d*x)) + 983*E^((8*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(3/2)*ArcSinh[E^(I*(c + d*x))])/(a^2*d*E^((4*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
127,1,149,176,1.5704146,"\int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 i (c+d x)} \left(i \left(1+e^{2 i (c+d x)}\right) \left(-26 e^{2 i (c+d x)}+194 e^{4 i (c+d x)}+463 e^{6 i (c+d x)}+3\right) \sec ^2(c+d x)-\frac{60 i e^{7 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{83 i \sqrt{a+i a \tan (c+d x)}}{30 a^3 d}+\frac{151 i}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{17 i \tan ^2(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"(((-60*I)*E^((7*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + I*(1 + E^((2*I)*(c + d*x)))*(3 - 26*E^((2*I)*(c + d*x)) + 194*E^((4*I)*(c + d*x)) + 463*E^((6*I)*(c + d*x)))*Sec[c + d*x]^2)/(240*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
128,1,135,133,1.2511162,"\int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-19 e^{2 i (c+d x)}+83 e^{4 i (c+d x)}+3\right)+15 e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{31}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\tan ^2(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}-\frac{13}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(3 - 19*E^((2*I)*(c + d*x)) + 83*E^((4*I)*(c + d*x))) + 15*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(240*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
129,1,135,133,1.1801318,"\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-6 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right)-5 e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{80 a^2 d \sqrt{a+i a \tan (c+d x)}}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i}{2 a d (a+i a \tan (c+d x))^{3/2}}-\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/80*I)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(1 - 3*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x))) - 5*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
130,1,135,125,0.7421629,"\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(-e^{2 i (c+d x)}+17 e^{4 i (c+d x)}-3\right)-15 e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{1}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{6 a d (a+i a \tan (c+d x))^{3/2}}-\frac{1}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(-3 - E^((2*I)*(c + d*x)) + 17*E^((4*I)*(c + d*x))) - 15*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(240*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
131,1,137,133,0.7562006,"\int \frac{1}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-5/2),x]","\frac{i e^{-6 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(11 e^{2 i (c+d x)}+23 e^{4 i (c+d x)}+3\right)-15 e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{240 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{i}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((I/240)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(3 + 11*E^((2*I)*(c + d*x)) + 23*E^((4*I)*(c + d*x))) - 15*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))])*Sec[c + d*x]^2)/(a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
132,1,188,159,1.9356253,"\int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{e^{-6 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \sec ^2(c+d x) \left(\sqrt{1+e^{2 i (c+d x)}} \left(7 e^{2 i (c+d x)}+41 e^{4 i (c+d x)}+1\right)+5 e^{5 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-40 \sqrt{2} e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{80 a^2 d \sqrt{a+i a \tan (c+d x)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}+\frac{7}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{2 a d (a+i a \tan (c+d x))^{3/2}}+\frac{1}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((1 + E^((2*I)*(c + d*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(c + d*x))]*(1 + 7*E^((2*I)*(c + d*x)) + 41*E^((4*I)*(c + d*x))) + 5*E^((5*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 40*Sqrt[2]*E^((5*I)*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^2)/(80*a^2*d*E^((6*I)*(c + d*x))*Sqrt[a + I*a*Tan[c + d*x]])","A",1
133,1,263,214,3.4626868,"\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i \left(31 e^{2 i (c+d x)}+280 e^{4 i (c+d x)}-151 e^{6 i (c+d x)}-403 e^{8 i (c+d x)}+15 e^{5 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+300 \sqrt{2} e^{5 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{1+e^{2 i (c+d x)}}}\right)+3\right)}{15 a^2 d \left(-1+e^{2 i (c+d x)}\right) \left(1+e^{2 i (c+d x)}\right)^3 (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{5 i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{21 \cot (c+d x) \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}+\frac{41 \cot (c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{19 \cot (c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\cot (c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/15*I)*(3 + 31*E^((2*I)*(c + d*x)) + 280*E^((4*I)*(c + d*x)) - 151*E^((6*I)*(c + d*x)) - 403*E^((8*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + 300*Sqrt[2]*E^((5*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[1 + E^((2*I)*(c + d*x))]]))/(a^2*d*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^3*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
134,1,150,162,2.0555231,"\int \frac{1}{(a+i a \tan (c+d x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-7/2),x]","-\frac{81 e^{2 i (c+d x)}+188 e^{4 i (c+d x)}+298 e^{6 i (c+d x)}+176 e^{8 i (c+d x)}-105 e^{7 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+15}{105 a^3 d \left(1+e^{2 i (c+d x)}\right)^4 (\tan (c+d x)-i)^3 \sqrt{a+i a \tan (c+d x)}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} a^{7/2} d}+\frac{i}{8 a^3 d \sqrt{a+i a \tan (c+d x)}}+\frac{i}{12 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{i}{10 a d (a+i a \tan (c+d x))^{5/2}}+\frac{i}{7 d (a+i a \tan (c+d x))^{7/2}}",1,"-1/105*(15 + 81*E^((2*I)*(c + d*x)) + 188*E^((4*I)*(c + d*x)) + 298*E^((6*I)*(c + d*x)) + 176*E^((8*I)*(c + d*x)) - 105*E^((7*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/(a^3*d*(1 + E^((2*I)*(c + d*x)))^4*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]])","A",1
135,1,125,107,2.4357127,"\int (d \tan (e+f x))^{5/2} (a+i a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x]),x]","\frac{a d^2 \sqrt{d \tan (e+f x)} \left(30 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+\sqrt{i \tan (e+f x)} \sec ^2(e+f x) (5 \sin (2 (e+f x))-18 i \cos (2 (e+f x))-12 i)\right)}{15 f \sqrt{i \tan (e+f x)}}","-\frac{2 (-1)^{3/4} a d^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{2 i a d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a d (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 i a (d \tan (e+f x))^{5/2}}{5 f}",1,"(a*d^2*((30*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] + Sec[e + f*x]^2*(-12*I - (18*I)*Cos[2*(e + f*x)] + 5*Sin[2*(e + f*x)])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(15*f*Sqrt[I*Tan[e + f*x]])","A",1
136,1,148,82,1.0402774,"\int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x]),x]","-\frac{2 a d \left(2 e^{2 i (e+f x)}-4 e^{4 i (e+f x)}+3 \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} \left(1+e^{2 i (e+f x)}\right)^2 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+2\right) \sqrt{d \tan (e+f x)}}{3 f \left(-1+e^{4 i (e+f x)}\right)}","\frac{2 \sqrt[4]{-1} a d^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f}",1,"(-2*a*d*(2 + 2*E^((2*I)*(e + f*x)) - 4*E^((4*I)*(e + f*x)) + 3*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*(1 + E^((2*I)*(e + f*x)))^2*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]])*Sqrt[d*Tan[e + f*x]])/(3*(-1 + E^((4*I)*(e + f*x)))*f)","A",1
137,1,85,61,0.8056101,"\int \sqrt{d \tan (e+f x)} (a+i a \tan (e+f x)) \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x]),x]","\frac{2 i a \sqrt{d \tan (e+f x)} \left(\sqrt{i \tan (e+f x)}-\tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{f \sqrt{i \tan (e+f x)}}","\frac{2 (-1)^{3/4} a \sqrt{d} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}+\frac{2 i a \sqrt{d \tan (e+f x)}}{f}",1,"((2*I)*a*(-ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] + Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[I*Tan[e + f*x]])","A",1
138,1,87,40,0.7627369,"\int \frac{a+i a \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]],x]","-\frac{2 i a \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)}{f \sqrt{d \tan (e+f x)}}","-\frac{2 \sqrt[4]{-1} a \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{d} f}",1,"((-2*I)*a*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]])/(f*Sqrt[d*Tan[e + f*x]])","C",1
139,1,138,62,1.157407,"\int \frac{a+i a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2),x]","-\frac{2 a e^{-i (e+f x)} \sin (e+f x) (\tan (e+f x)-i) \left(i \sqrt{i \tan (e+f x)}+\tan (e+f x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} (d \tan (e+f x))^{3/2}}","-\frac{2 (-1)^{3/4} a \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{3/2} f}-\frac{2 a}{d f \sqrt{d \tan (e+f x)}}",1,"(-2*a*Sin[e + f*x]*(-I + Tan[e + f*x])*(I*Sqrt[I*Tan[e + f*x]] + ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Tan[e + f*x]))/(E^(I*(e + f*x))*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f*(d*Tan[e + f*x])^(3/2))","C",1
140,1,140,87,1.3291642,"\int \frac{a+i a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2),x]","\frac{2 a e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right) \cos (e+f x) (\tan (e+f x)-i) \left(3 i \tan (e+f x)-3 (i \tan (e+f x))^{3/2} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+1\right)}{3 d^2 f \left(-1+e^{2 i (e+f x)}\right) \sqrt{d \tan (e+f x)}}","\frac{2 \sqrt[4]{-1} a \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{5/2} f}-\frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}",1,"(2*a*(1 + E^((2*I)*(e + f*x)))*Cos[e + f*x]*(1 - 3*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*(I*Tan[e + f*x])^(3/2) + (3*I)*Tan[e + f*x])*(-I + Tan[e + f*x]))/(3*d^2*E^(I*(e + f*x))*(-1 + E^((2*I)*(e + f*x)))*f*Sqrt[d*Tan[e + f*x]])","A",1
141,1,105,110,4.2233543,"\int \frac{a+i a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2),x]","-\frac{2 a \left(\cot (e+f x) (18 \cot (2 (e+f x))+5 i)-6 \csc ^2(e+f x)+15 \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{15 d^3 f \sqrt{d \tan (e+f x)}}","\frac{2 (-1)^{3/4} a \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{7/2} f}+\frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{2 i a}{3 d^2 f (d \tan (e+f x))^{3/2}}-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}",1,"(-2*a*(Cot[e + f*x]*(5*I + 18*Cot[2*(e + f*x)]) - 6*Csc[e + f*x]^2 + 15*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Sqrt[I*Tan[e + f*x]]))/(15*d^3*f*Sqrt[d*Tan[e + f*x]])","A",1
142,1,90,107,0.5089587,"\int (d \tan (e+f x))^{5/2} (a-i a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a - I*a*Tan[e + f*x]),x]","\frac{2 a (d \tan (e+f x))^{5/2} \left(15 (-1)^{3/4} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)+\sqrt{\tan (e+f x)} \left(-3 i \tan ^2(e+f x)+5 \tan (e+f x)+15 i\right)\right)}{15 f \tan ^{\frac{5}{2}}(e+f x)}","\frac{2 (-1)^{3/4} a d^{5/2} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}+\frac{2 i a d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a d (d \tan (e+f x))^{3/2}}{3 f}-\frac{2 i a (d \tan (e+f x))^{5/2}}{5 f}",1,"(2*a*(d*Tan[e + f*x])^(5/2)*(15*(-1)^(3/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + Sqrt[Tan[e + f*x]]*(15*I + 5*Tan[e + f*x] - (3*I)*Tan[e + f*x]^2)))/(15*f*Tan[e + f*x]^(5/2))","A",1
143,1,78,82,0.2222061,"\int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a - I*a*Tan[e + f*x]),x]","\frac{2 a (d \tan (e+f x))^{3/2} \left(3 \sqrt[4]{-1} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)+(3-i \tan (e+f x)) \sqrt{\tan (e+f x)}\right)}{3 f \tan ^{\frac{3}{2}}(e+f x)}","\frac{2 \sqrt[4]{-1} a d^{3/2} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}-\frac{2 i a (d \tan (e+f x))^{3/2}}{3 f}",1,"(2*a*(3*(-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + (3 - I*Tan[e + f*x])*Sqrt[Tan[e + f*x]])*(d*Tan[e + f*x])^(3/2))/(3*f*Tan[e + f*x]^(3/2))","A",1
144,1,64,61,0.0615854,"\int \sqrt{d \tan (e+f x)} (a-i a \tan (e+f x)) \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a - I*a*Tan[e + f*x]),x]","-\frac{2 i a \sqrt{d \tan (e+f x)} \left(\sqrt{\tan (e+f x)}+\sqrt[4]{-1} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)\right)}{f \sqrt{\tan (e+f x)}}","-\frac{2 (-1)^{3/4} a \sqrt{d} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{2 i a \sqrt{d \tan (e+f x)}}{f}",1,"((-2*I)*a*((-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + Sqrt[Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[Tan[e + f*x]])","A",1
145,1,50,40,0.0438693,"\int \frac{a-i a \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a - I*a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]],x]","-\frac{2 \sqrt[4]{-1} a \sqrt{\tan (e+f x)} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)}{f \sqrt{d \tan (e+f x)}}","-\frac{2 \sqrt[4]{-1} a \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{d} f}",1,"(-2*(-1)^(1/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]]*Sqrt[Tan[e + f*x]])/(f*Sqrt[d*Tan[e + f*x]])","A",1
146,1,39,62,0.1068793,"\int \frac{a-i a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a - I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2),x]","-\frac{2 a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-i \tan (e+f x)\right)}{d f \sqrt{d \tan (e+f x)}}","\frac{2 (-1)^{3/4} a \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{3/2} f}-\frac{2 a}{d f \sqrt{d \tan (e+f x)}}",1,"(-2*a*Hypergeometric2F1[-1/2, 1, 1/2, (-I)*Tan[e + f*x]])/(d*f*Sqrt[d*Tan[e + f*x]])","C",1
147,1,41,87,0.1120498,"\int \frac{a-i a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a - I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2),x]","-\frac{2 a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-i \tan (e+f x)\right)}{3 d f (d \tan (e+f x))^{3/2}}","\frac{2 \sqrt[4]{-1} a \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{5/2} f}+\frac{2 i a}{d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}",1,"(-2*a*Hypergeometric2F1[-3/2, 1, -1/2, (-I)*Tan[e + f*x]])/(3*d*f*(d*Tan[e + f*x])^(3/2))","C",1
148,1,41,110,0.1634942,"\int \frac{a-i a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx","Integrate[(a - I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2),x]","-\frac{2 a \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-i \tan (e+f x)\right)}{5 d f (d \tan (e+f x))^{5/2}}","-\frac{2 (-1)^{3/4} a \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{7/2} f}+\frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}+\frac{2 i a}{3 d^2 f (d \tan (e+f x))^{3/2}}-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}",1,"(-2*a*Hypergeometric2F1[-5/2, 1, -3/2, (-I)*Tan[e + f*x]])/(5*d*f*(d*Tan[e + f*x])^(5/2))","C",1
149,1,145,140,2.9808981,"\int (d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2 \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])^2,x]","\frac{a^2 d^2 \sqrt{d \tan (e+f x)} \left(840 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)-i \sqrt{i \tan (e+f x)} \sec ^3(e+f x) (25 i \sin (e+f x)+85 i \sin (3 (e+f x))+588 \cos (e+f x)+252 \cos (3 (e+f x)))\right)}{210 f \sqrt{i \tan (e+f x)}}","-\frac{4 (-1)^{3/4} a^2 d^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{4 i a^2 d^2 \sqrt{d \tan (e+f x)}}{f}-\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{4 i a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac{4 a^2 d (d \tan (e+f x))^{3/2}}{3 f}",1,"(a^2*d^2*((840*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] - I*Sec[e + f*x]^3*(588*Cos[e + f*x] + 252*Cos[3*(e + f*x)] + (25*I)*Sin[e + f*x] + (85*I)*Sin[3*(e + f*x)])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(210*f*Sqrt[I*Tan[e + f*x]])","A",1
150,1,127,113,3.4441311,"\int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^2,x]","\frac{a^2 d^2 \left(120 i \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+\sec ^2(e+f x) (-20 i \cos (2 (e+f x))+21 \tan (e+f x)+33 \sin (3 (e+f x)) \sec (e+f x)+20 i)\right)}{30 f \sqrt{d \tan (e+f x)}}","\frac{4 \sqrt[4]{-1} a^2 d^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac{4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac{4 a^2 d \sqrt{d \tan (e+f x)}}{f}",1,"(a^2*d^2*((120*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Sqrt[I*Tan[e + f*x]] + Sec[e + f*x]^2*(20*I - (20*I)*Cos[2*(e + f*x)] + 33*Sec[e + f*x]*Sin[3*(e + f*x)] + 21*Tan[e + f*x])))/(30*f*Sqrt[d*Tan[e + f*x]])","A",1
151,1,102,90,2.056916,"\int \sqrt{d \tan (e+f x)} (a+i a \tan (e+f x))^2 \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2,x]","\frac{2 i a^2 \sqrt{d \tan (e+f x)} \left((6+i \tan (e+f x)) \sqrt{i \tan (e+f x)}-6 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{3 f \sqrt{i \tan (e+f x)}}","\frac{4 (-1)^{3/4} a^2 \sqrt{d} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{4 i a^2 \sqrt{d \tan (e+f x)}}{f}",1,"(((2*I)/3)*a^2*(-6*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] + (6 + I*Tan[e + f*x])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[I*Tan[e + f*x]])","A",1
152,1,157,66,1.946146,"\int \frac{(a+i a \tan (e+f x))^2}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/Sqrt[d*Tan[e + f*x]],x]","-\frac{2 a^2 e^{-2 i (e+f x)} \sqrt{\tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \left(\tan (e+f x)+2 i \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{f \sqrt{-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}} \sqrt{d \tan (e+f x)}}","-\frac{4 \sqrt[4]{-1} a^2 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{d} f}-\frac{2 a^2 \sqrt{d \tan (e+f x)}}{d f}",1,"(-2*a^2*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*Sqrt[Tan[e + f*x]]*((2*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Sqrt[I*Tan[e + f*x]] + Tan[e + f*x]))/(E^((2*I)*(e + f*x))*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]*f*Sqrt[d*Tan[e + f*x]])","C",1
153,1,147,66,1.4985157,"\int \frac{(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2),x]","-\frac{2 a^2 e^{-2 i (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \left(\sqrt{i \tan (e+f x)}-2 i \tan (e+f x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{d f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{d \tan (e+f x)}}","-\frac{4 (-1)^{3/4} a^2 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{3/2} f}-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}",1,"(-2*a^2*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(Sqrt[I*Tan[e + f*x]] - (2*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Tan[e + f*x]))/(d*E^((2*I)*(e + f*x))*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f*Sqrt[d*Tan[e + f*x]])","C",1
154,1,87,93,2.0484786,"\int \frac{(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2),x]","-\frac{2 a^2 \left(\cot (e+f x)-6 i \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+6 i\right)}{3 d^2 f \sqrt{d \tan (e+f x)}}","\frac{4 \sqrt[4]{-1} a^2 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{5/2} f}-\frac{4 i a^2}{d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 a^2}{3 d f (d \tan (e+f x))^{3/2}}",1,"(-2*a^2*(6*I + Cot[e + f*x] - (6*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Sqrt[I*Tan[e + f*x]]))/(3*d^2*f*Sqrt[d*Tan[e + f*x]])","A",1
155,1,381,118,7.8355761,"\int \frac{(a+i a \tan (e+f x))^2}{(d \tan (e+f x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(7/2),x]","\frac{\sin ^2(e+f x) \tan ^2(e+f x) (a+i a \tan (e+f x))^2 \left(\csc (e) \left(\frac{2}{5} \cos (2 e)-\frac{2}{5} i \sin (2 e)\right) \sin (f x) \csc ^3(e+f x)+\csc (e) (3 \cos (e)+10 i \sin (e)) \left(-\frac{2}{15} \cos (2 e)+\frac{2}{15} i \sin (2 e)\right) \csc ^2(e+f x)+\csc (e) \left(-\frac{22}{5} \cos (2 e)+\frac{22}{5} i \sin (2 e)\right) \sin (f x) \csc (e+f x)+\csc (e) (33 \cos (e)+10 i \sin (e)) \left(\frac{2}{15} \cos (2 e)-\frac{2}{15} i \sin (2 e)\right)\right)}{f (\cos (f x)+i \sin (f x))^2 (d \tan (e+f x))^{7/2}}-\frac{4 i e^{-2 i e} \sqrt{-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}} \cos ^2(e+f x) \tan ^{\frac{7}{2}}(e+f x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right) (a+i a \tan (e+f x))^2}{f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} (\cos (f x)+i \sin (f x))^2 (d \tan (e+f x))^{7/2}}","\frac{4 (-1)^{3/4} a^2 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{7/2} f}+\frac{4 a^2}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{4 i a^2}{3 d^2 f (d \tan (e+f x))^{3/2}}-\frac{2 a^2}{5 d f (d \tan (e+f x))^{5/2}}",1,"((Csc[e]*(33*Cos[e] + (10*I)*Sin[e])*((2*Cos[2*e])/15 - ((2*I)/15)*Sin[2*e]) + Csc[e]*Csc[e + f*x]^2*(3*Cos[e] + (10*I)*Sin[e])*((-2*Cos[2*e])/15 + ((2*I)/15)*Sin[2*e]) + Csc[e]*Csc[e + f*x]^3*((2*Cos[2*e])/5 - ((2*I)/5)*Sin[2*e])*Sin[f*x] + Csc[e]*Csc[e + f*x]*((-22*Cos[2*e])/5 + ((22*I)/5)*Sin[2*e])*Sin[f*x])*Sin[e + f*x]^2*Tan[e + f*x]^2*(a + I*a*Tan[e + f*x])^2)/(f*(Cos[f*x] + I*Sin[f*x])^2*(d*Tan[e + f*x])^(7/2)) - ((4*I)*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Cos[e + f*x]^2*Tan[e + f*x]^(7/2)*(a + I*a*Tan[e + f*x])^2)/(E^((2*I)*e)*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f*(Cos[f*x] + I*Sin[f*x])^2*(d*Tan[e + f*x])^(7/2))","B",1
156,1,150,179,3.6209614,"\int (d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])^3,x]","\frac{a^3 d^2 \sqrt{d \tan (e+f x)} \left(10080 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)-i \sqrt{i \tan (e+f x)} \sec ^4(e+f x) (570 i \sin (2 (e+f x))+555 i \sin (4 (e+f x))+4900 \cos (2 (e+f x))+1547 \cos (4 (e+f x))+3633)\right)}{1260 f \sqrt{i \tan (e+f x)}}","-\frac{8 (-1)^{3/4} a^3 d^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{8 i a^3 d^2 \sqrt{d \tan (e+f x)}}{f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) (d \tan (e+f x))^{7/2}}{9 d f}-\frac{40 a^3 (d \tan (e+f x))^{7/2}}{63 d f}+\frac{8 i a^3 (d \tan (e+f x))^{5/2}}{5 f}+\frac{8 a^3 d (d \tan (e+f x))^{3/2}}{3 f}",1,"(a^3*d^2*((10080*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] - I*Sec[e + f*x]^4*(3633 + 4900*Cos[2*(e + f*x)] + 1547*Cos[4*(e + f*x)] + (570*I)*Sin[2*(e + f*x)] + (555*I)*Sin[4*(e + f*x)])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(1260*f*Sqrt[I*Tan[e + f*x]])","A",1
157,1,138,152,3.1789195,"\int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^3,x]","\frac{a^3 d \sqrt{d \tan (e+f x)} \left(\sqrt{i \tan (e+f x)} \sec ^3(e+f x) (95 i \sin (e+f x)+155 i \sin (3 (e+f x))+1197 \cos (e+f x)+483 \cos (3 (e+f x)))-1680 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{210 f \sqrt{i \tan (e+f x)}}","\frac{8 \sqrt[4]{-1} a^3 d^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac{8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac{8 a^3 d \sqrt{d \tan (e+f x)}}{f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) (d \tan (e+f x))^{5/2}}{7 d f}",1,"(a^3*d*(-1680*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] + Sec[e + f*x]^3*(1197*Cos[e + f*x] + 483*Cos[3*(e + f*x)] + (95*I)*Sin[e + f*x] + (155*I)*Sin[3*(e + f*x)])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(210*f*Sqrt[I*Tan[e + f*x]])","A",1
158,1,122,129,4.0062936,"\int \sqrt{d \tan (e+f x)} (a+i a \tan (e+f x))^3 \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^3,x]","\frac{i a^3 \sqrt{d \tan (e+f x)} \left(\sqrt{i \tan (e+f x)} \sec ^2(e+f x) (5 i \sin (2 (e+f x))+21 \cos (2 (e+f x))+19)-40 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{5 f \sqrt{i \tan (e+f x)}}","\frac{8 (-1)^{3/4} a^3 \sqrt{d} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{8 a^3 (d \tan (e+f x))^{3/2}}{5 d f}+\frac{8 i a^3 \sqrt{d \tan (e+f x)}}{f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) (d \tan (e+f x))^{3/2}}{5 d f}",1,"((I/5)*a^3*(-40*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] + Sec[e + f*x]^2*(19 + 21*Cos[2*(e + f*x)] + (5*I)*Sin[2*(e + f*x)])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[I*Tan[e + f*x]])","A",1
159,1,154,107,3.1764001,"\int \frac{(a+i a \tan (e+f x))^3}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]],x]","-\frac{2 a^3 e^{-3 i (e+f x)} \sqrt{d \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) \left((9+i \tan (e+f x)) \sqrt{i \tan (e+f x)}-12 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{3 d f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}}","-\frac{8 \sqrt[4]{-1} a^3 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{d} f}-\frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) \sqrt{d \tan (e+f x)}}{3 d f}",1,"(-2*a^3*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(-12*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]] + (9 + I*Tan[e + f*x])*Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(3*d*E^((3*I)*(e + f*x))*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f)","A",1
160,1,156,80,2.3502849,"\int \frac{(a+i a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(3/2),x]","\frac{2 a^3 e^{-3 i (e+f x)} (\sin (3 (e+f x))-i \cos (3 (e+f x))) \left(\sqrt{i \tan (e+f x)} (\tan (e+f x)-i)-4 \tan (e+f x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{d f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{d \tan (e+f x)}}","-\frac{8 (-1)^{3/4} a^3 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{3/2} f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right)}{d f \sqrt{d \tan (e+f x)}}",1,"(2*a^3*((-I)*Cos[3*(e + f*x)] + Sin[3*(e + f*x)])*(-4*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Tan[e + f*x] + Sqrt[I*Tan[e + f*x]]*(-I + Tan[e + f*x])))/(d*E^((3*I)*(e + f*x))*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f*Sqrt[d*Tan[e + f*x]])","A",1
161,1,87,109,3.3765317,"\int \frac{(a+i a \tan (e+f x))^3}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(5/2),x]","-\frac{2 a^3 \left(\cot (e+f x)-12 i \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+9 i\right)}{3 d^2 f \sqrt{d \tan (e+f x)}}","\frac{8 \sqrt[4]{-1} a^3 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{5/2} f}-\frac{16 i a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right)}{3 d f (d \tan (e+f x))^{3/2}}",1,"(-2*a^3*(9*I + Cot[e + f*x] - (12*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Sqrt[I*Tan[e + f*x]]))/(3*d^2*f*Sqrt[d*Tan[e + f*x]])","A",1
162,1,109,132,5.8502127,"\int \frac{(a+i a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(7/2),x]","-\frac{a^3 \left(40 \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+\csc ^2(e+f x) (5 i \sin (2 (e+f x))+21 \cos (2 (e+f x))-19)\right)}{5 d^3 f \sqrt{d \tan (e+f x)}}","\frac{8 (-1)^{3/4} a^3 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{7/2} f}+\frac{8 a^3}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{8 i a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right)}{5 d f (d \tan (e+f x))^{5/2}}",1,"-1/5*(a^3*(Csc[e + f*x]^2*(-19 + 21*Cos[2*(e + f*x)] + (5*I)*Sin[2*(e + f*x)]) + 40*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Sqrt[I*Tan[e + f*x]]))/(d^3*f*Sqrt[d*Tan[e + f*x]])","A",1
163,1,416,159,9.3886924,"\int \frac{(a+i a \tan (e+f x))^3}{(d \tan (e+f x))^{9/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(9/2),x]","\frac{8 e^{-3 i e} \sqrt{-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}} \cos ^3(e+f x) \tan ^{\frac{9}{2}}(e+f x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right) (a+i a \tan (e+f x))^3}{f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} (\cos (f x)+i \sin (f x))^3 (d \tan (e+f x))^{9/2}}+\frac{\sin ^3(e+f x) \tan ^2(e+f x) (a+i a \tan (e+f x))^3 \left(\left(-\frac{2}{7} \cos (3 e)+\frac{2}{7} i \sin (3 e)\right) \csc ^4(e+f x)+i \csc (e) \left(\frac{6}{5} \cos (3 e)-\frac{6}{5} i \sin (3 e)\right) \sin (f x) \csc ^3(e+f x)+\csc (e) (63 \cos (e)+170 i \sin (e)) \left(-\frac{2}{105} \sin (3 e)-\frac{2}{105} i \cos (3 e)\right) \csc ^2(e+f x)-i \csc (e) \left(\frac{46}{5} \cos (3 e)-\frac{46}{5} i \sin (3 e)\right) \sin (f x) \csc (e+f x)+i \csc (e) (483 \cos (e)+155 i \sin (e)) \left(\frac{2}{105} \cos (3 e)-\frac{2}{105} i \sin (3 e)\right)\right)}{f (\cos (f x)+i \sin (f x))^3 (d \tan (e+f x))^{9/2}}","-\frac{8 \sqrt[4]{-1} a^3 \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{9/2} f}+\frac{8 i a^3}{d^4 f \sqrt{d \tan (e+f x)}}+\frac{8 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}-\frac{32 i a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right)}{7 d f (d \tan (e+f x))^{7/2}}",1,"((Csc[e]*Csc[e + f*x]^2*(63*Cos[e] + (170*I)*Sin[e])*(((-2*I)/105)*Cos[3*e] - (2*Sin[3*e])/105) + I*Csc[e]*(483*Cos[e] + (155*I)*Sin[e])*((2*Cos[3*e])/105 - ((2*I)/105)*Sin[3*e]) + Csc[e + f*x]^4*((-2*Cos[3*e])/7 + ((2*I)/7)*Sin[3*e]) + I*Csc[e]*Csc[e + f*x]^3*((6*Cos[3*e])/5 - ((6*I)/5)*Sin[3*e])*Sin[f*x] - I*Csc[e]*Csc[e + f*x]*((46*Cos[3*e])/5 - ((46*I)/5)*Sin[3*e])*Sin[f*x])*Sin[e + f*x]^3*Tan[e + f*x]^2*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3*(d*Tan[e + f*x])^(9/2)) + (8*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*Cos[e + f*x]^3*Tan[e + f*x]^(9/2)*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f*(Cos[f*x] + I*Sin[f*x])^3*(d*Tan[e + f*x])^(9/2))","B",1
164,1,275,312,1.7972503,"\int \frac{(d \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^(7/2)/(a + I*a*Tan[e + f*x]),x]","-\frac{d^4 \sec ^3(e+f x) \left(54 i \sin (e+f x)+22 i \sin (3 (e+f x))-16 \cos (e+f x)+16 \cos (3 (e+f x))+(42+30 i) \sqrt{\sin (2 (e+f x))} \cos (e+f x) \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (e+f x)+i \sin (e+f x))+(15+21 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(21-15 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(21-15 i) \sqrt{\sin (2 (e+f x))} \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{48 a f (\tan (e+f x)-i) \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{5}{4}-\frac{7 i}{4}\right) d^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a f}-\frac{\left(\frac{5}{4}-\frac{7 i}{4}\right) d^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a f}+\frac{\left(\frac{5}{8}+\frac{7 i}{8}\right) d^{7/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a f}-\frac{\left(\frac{5}{8}+\frac{7 i}{8}\right) d^{7/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a f}+\frac{5 d^3 \sqrt{d \tan (e+f x)}}{2 a f}-\frac{7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac{d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}",1,"-1/48*(d^4*Sec[e + f*x]^3*(-16*Cos[e + f*x] + 16*Cos[3*(e + f*x)] + (54*I)*Sin[e + f*x] + (21 - 15*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (21 - 15*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (42 + 30*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x])*Sqrt[Sin[2*(e + f*x)]] + (15 + 21*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) + (22*I)*Sin[3*(e + f*x)]))/(a*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x]))","A",0
165,1,164,287,1.8882925,"\int \frac{(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) d^2 \csc (e+f x) \sqrt{d \tan (e+f x)} \left(-(2+2 i) \sin (e+f x) (4 \tan (e+f x)-5 i)+(-4-i) \sqrt{\sin (2 (e+f x))} (\tan (e+f x)-i) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+(1+4 i) \sqrt{\sin (2 (e+f x))} (\tan (e+f x)-i) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{a f (\tan (e+f x)-i)}","-\frac{\left(\frac{3}{4}+\frac{5 i}{4}\right) d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a f}+\frac{\left(\frac{3}{4}+\frac{5 i}{4}\right) d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a f}+\frac{\left(\frac{3}{8}-\frac{5 i}{8}\right) d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a f}-\frac{\left(\frac{3}{8}-\frac{5 i}{8}\right) d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a f}-\frac{5 i d^2 \sqrt{d \tan (e+f x)}}{2 a f}-\frac{d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}",1,"((1/8 + I/8)*d^2*Csc[e + f*x]*Sqrt[d*Tan[e + f*x]]*((-4 - I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*(-I + Tan[e + f*x]) + (1 + 4*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]]*(-I + Tan[e + f*x]) - (2 + 2*I)*Sin[e + f*x]*(-5*I + 4*Tan[e + f*x])))/(a*f*(-I + Tan[e + f*x]))","A",1
166,1,150,260,0.9342594,"\int \frac{(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x]),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) d \sqrt{\sin (2 (e+f x))} \csc (e+f x) \sqrt{d \tan (e+f x)} \left((1+i) \sqrt{\sin (2 (e+f x))} \sec (e+f x)+(1+2 i) (\tan (e+f x)-i) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+(2+i) (\tan (e+f x)-i) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{a f (\tan (e+f x)-i)}","-\frac{\left(\frac{1}{4}-\frac{3 i}{4}\right) d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a f}+\frac{\left(\frac{1}{4}-\frac{3 i}{4}\right) d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a f}-\frac{\left(\frac{1}{8}+\frac{3 i}{8}\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a f}+\frac{\left(\frac{1}{8}+\frac{3 i}{8}\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a f}-\frac{d \sqrt{d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}",1,"((1/8 + I/8)*d*Csc[e + f*x]*Sqrt[Sin[2*(e + f*x)]]*Sqrt[d*Tan[e + f*x]]*((1 + I)*Sec[e + f*x]*Sqrt[Sin[2*(e + f*x)]] + (1 + 2*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*(-I + Tan[e + f*x]) + (2 + I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*(-I + Tan[e + f*x])))/(a*f*(-I + Tan[e + f*x]))","A",1
167,1,129,81,1.0707232,"\int \frac{\sqrt{d \tan (e+f x)}}{a+i a \tan (e+f x)} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x]),x]","\frac{\sqrt{d \tan (e+f x)} \left(\sqrt{i \tan (e+f x)}+(-1-i \tan (e+f x)) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)\right)}{2 a f \sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}} (\tan (e+f x)-i)}","\frac{(-1)^{3/4} \sqrt{d} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a f}+\frac{i \sqrt{d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}",1,"((ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*(-1 - I*Tan[e + f*x]) + Sqrt[I*Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(2*a*Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]*f*(-I + Tan[e + f*x]))","A",1
168,1,147,262,0.8176284,"\int \frac{1}{\sqrt{d \tan (e+f x)} (a+i a \tan (e+f x))} \, dx","Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])),x]","\frac{\sqrt{\sin (2 (e+f x))} \sec (e+f x) \left(-2 i \sqrt{\sin (2 (e+f x))} \sec (e+f x)+(1+3 i) (1+i \tan (e+f x)) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+(3+i) (\tan (e+f x)-i) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{8 a f (\tan (e+f x)-i) \sqrt{d \tan (e+f x)}}","-\frac{\left(\frac{3}{4}-\frac{i}{4}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a \sqrt{d} f}+\frac{\left(\frac{3}{4}-\frac{i}{4}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a \sqrt{d} f}+\frac{\sqrt{d \tan (e+f x)}}{2 d f (a+i a \tan (e+f x))}-\frac{\left(\frac{3}{8}+\frac{i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a \sqrt{d} f}+\frac{\left(\frac{3}{8}+\frac{i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a \sqrt{d} f}",1,"(Sec[e + f*x]*Sqrt[Sin[2*(e + f*x)]]*((-2*I)*Sec[e + f*x]*Sqrt[Sin[2*(e + f*x)]] + (1 + 3*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*(1 + I*Tan[e + f*x]) + (3 + I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*(-I + Tan[e + f*x])))/(8*a*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x]))","A",1
169,1,155,287,1.3168259,"\int \frac{1}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))} \, dx","Integrate[1/((d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])),x]","\frac{-20 \tan (e+f x)+(5+3 i) \sqrt{\sin (2 (e+f x))} (\tan (e+f x)-i) \sec (e+f x) \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+(5-3 i) \sqrt{\sin (2 (e+f x))} (\tan (e+f x)-i) \sec (e+f x) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+16 i}{8 a d f (\tan (e+f x)-i) \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{5}{4}+\frac{3 i}{4}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a d^{3/2} f}-\frac{\left(\frac{5}{4}+\frac{3 i}{4}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a d^{3/2} f}-\frac{\left(\frac{5}{8}-\frac{3 i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a d^{3/2} f}+\frac{\left(\frac{5}{8}-\frac{3 i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a d^{3/2} f}-\frac{5}{2 a d f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (a+i a \tan (e+f x)) \sqrt{d \tan (e+f x)}}",1,"(16*I - 20*Tan[e + f*x] + (5 + 3*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sec[e + f*x]*Sqrt[Sin[2*(e + f*x)]]*(-I + Tan[e + f*x]) + (5 - 3*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sec[e + f*x]*Sqrt[Sin[2*(e + f*x)]]*(-I + Tan[e + f*x]))/(8*a*d*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x]))","A",1
170,1,273,314,2.3598965,"\int \frac{1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))} \, dx","Integrate[1/((d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])),x]","\frac{\sec ^3(e+f x) \left(16 \sin (e+f x)+16 \sin (3 (e+f x))+54 i \cos (e+f x)-22 i \cos (3 (e+f x))-(15+21 i) \left(\sin (2 (e+f x))+2 i \sin ^2(e+f x)\right) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+(-15+21 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(21+15 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-(21+15 i) \sqrt{\sin (2 (e+f x))} \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{48 a d f (\tan (e+f x)-i) (d \tan (e+f x))^{3/2}}","\frac{\left(\frac{7}{4}-\frac{5 i}{4}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a d^{5/2} f}-\frac{\left(\frac{7}{4}-\frac{5 i}{4}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a d^{5/2} f}+\frac{\left(\frac{7}{8}+\frac{5 i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a d^{5/2} f}-\frac{\left(\frac{7}{8}+\frac{5 i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a d^{5/2} f}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (a+i a \tan (e+f x)) (d \tan (e+f x))^{3/2}}-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^3*((54*I)*Cos[e + f*x] - (22*I)*Cos[3*(e + f*x)] + 16*Sin[e + f*x] - (21 + 15*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (21 + 15*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - (15 - 21*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) - (15 + 21*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*((2*I)*Sin[e + f*x]^2 + Sin[2*(e + f*x)]) + 16*Sin[3*(e + f*x)]))/(48*a*d*f*(d*Tan[e + f*x])^(3/2)*(-I + Tan[e + f*x]))","A",1
171,1,346,353,2.2450978,"\int \frac{(d \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^(9/2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{d^5 \sec ^4(e+f x) \left(142 i \sin (2 (e+f x))+199 i \sin (4 (e+f x))+64 \cos (2 (e+f x))+205 \cos (4 (e+f x))+(294+270 i) \sqrt{\sin (2 (e+f x))} \cos (e+f x) \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))+(135+147 i) \sqrt{\sin (2 (e+f x))} \sin (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(147-135 i) \sqrt{\sin (2 (e+f x))} \cos (e+f x) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(147-135 i) \sqrt{\sin (2 (e+f x))} \cos (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(135+147 i) \sin (e+f x) \sqrt{\sin (2 (e+f x))} \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-269\right)}{192 a^2 f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","-\frac{\left(\frac{49}{16}+\frac{45 i}{16}\right) d^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{49}{16}+\frac{45 i}{16}\right) d^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{49}{32}-\frac{45 i}{32}\right) d^{9/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{49}{32}-\frac{45 i}{32}\right) d^{9/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}-\frac{45 i d^4 \sqrt{d \tan (e+f x)}}{8 a^2 f}-\frac{49 d^3 (d \tan (e+f x))^{3/2}}{24 a^2 f}+\frac{9 i d^2 (d \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}-\frac{d (d \tan (e+f x))^{7/2}}{4 f (a+i a \tan (e+f x))^2}",1,"(d^5*Sec[e + f*x]^4*(-269 + 64*Cos[2*(e + f*x)] + 205*Cos[4*(e + f*x)] + (147 - 135*I)*Cos[e + f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (147 - 135*I)*Cos[3*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (135 + 147*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[e + f*x]*Sqrt[Sin[2*(e + f*x)]] + (294 + 270*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Cos[e + f*x]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*Sqrt[Sin[2*(e + f*x)]] + (142*I)*Sin[2*(e + f*x)] + (135 + 147*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]]*Sin[3*(e + f*x)] + (199*I)*Sin[4*(e + f*x)]))/(192*a^2*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",1
172,1,233,326,1.3983993,"\int \frac{(d \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^(7/2)/(a + I*a*Tan[e + f*x])^2,x]","-\frac{i d^4 \sec ^3(e+f x) \left(-23 i \sin (e+f x)+41 i \sin (3 (e+f x))-43 \cos (e+f x)+43 \cos (3 (e+f x))+(21+25 i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))+(25+21 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(21-25 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{32 a^2 f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","-\frac{\left(\frac{25}{16}-\frac{21 i}{16}\right) d^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{25}{16}-\frac{21 i}{16}\right) d^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{25}{32}+\frac{21 i}{32}\right) d^{7/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{25}{32}+\frac{21 i}{32}\right) d^{7/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}-\frac{25 d^3 \sqrt{d \tan (e+f x)}}{8 a^2 f}+\frac{7 i d^2 (d \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}-\frac{d (d \tan (e+f x))^{5/2}}{4 f (a+i a \tan (e+f x))^2}",1,"((-1/32*I)*d^4*Sec[e + f*x]^3*(-43*Cos[e + f*x] + 43*Cos[3*(e + f*x)] - (23*I)*Sin[e + f*x] + (21 - 25*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (21 + 25*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*Sqrt[Sin[2*(e + f*x)]] + (25 + 21*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) + (41*I)*Sin[3*(e + f*x)]))/(a^2*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",1
173,1,231,301,1.0973921,"\int \frac{(d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^2,x]","-\frac{d^3 \sec ^3(e+f x) \left(5 i \sin (e+f x)+5 i \sin (3 (e+f x))-7 \cos (e+f x)+7 \cos (3 (e+f x))+(9+5 i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))+(5+9 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(9-5 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{32 a^2 f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{9}{16}+\frac{5 i}{16}\right) d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{9}{16}+\frac{5 i}{16}\right) d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{9}{32}-\frac{5 i}{32}\right) d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{9}{32}-\frac{5 i}{32}\right) d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}+\frac{5 i d^2 \sqrt{d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac{d (d \tan (e+f x))^{3/2}}{4 f (a+i a \tan (e+f x))^2}",1,"-1/32*(d^3*Sec[e + f*x]^3*(-7*Cos[e + f*x] + 7*Cos[3*(e + f*x)] + (5*I)*Sin[e + f*x] + (9 - 5*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (9 + 5*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*Sqrt[Sin[2*(e + f*x)]] + (5 + 9*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) + (5*I)*Sin[3*(e + f*x)]))/(a^2*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",1
174,1,231,297,1.0710124,"\int \frac{(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{d^2 \sec ^3(e+f x) \left(-\sin (e+f x)-\sin (3 (e+f x))-3 i \cos (e+f x)+3 i \cos (3 (e+f x))+(3-i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))+(3+i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(1-3 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{32 a^2 f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{1}{16}+\frac{3 i}{16}\right) d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{1}{16}+\frac{3 i}{16}\right) d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{1}{32}-\frac{3 i}{32}\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{1}{32}-\frac{3 i}{32}\right) d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}+\frac{3 d \sqrt{d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac{d \sqrt{d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}",1,"(d^2*Sec[e + f*x]^3*((-3*I)*Cos[e + f*x] + (3*I)*Cos[3*(e + f*x)] - Sin[e + f*x] + (1 - 3*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (3 + I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) + (3 - I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]) - Sin[3*(e + f*x)]))/(32*a^2*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",1
175,1,227,299,0.9305799,"\int \frac{\sqrt{d \tan (e+f x)}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^2,x]","-\frac{d \sec ^3(e+f x) \left(3 i \sin (e+f x)+3 i \sin (3 (e+f x))-\cos (e+f x)+\cos (3 (e+f x))-(3+i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))+(3-i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-(1+3 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{32 a^2 f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","-\frac{\left(\frac{1}{16}-\frac{3 i}{16}\right) \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 f}+\frac{\left(\frac{1}{16}-\frac{3 i}{16}\right) \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 f}+\frac{i \sqrt{d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}+\frac{\left(\frac{1}{32}+\frac{3 i}{32}\right) \sqrt{d} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}-\frac{\left(\frac{1}{32}+\frac{3 i}{32}\right) \sqrt{d} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 f}+\frac{i \sqrt{d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}",1,"-1/32*(d*Sec[e + f*x]^3*(-Cos[e + f*x] + Cos[3*(e + f*x)] + (3*I)*Sin[e + f*x] - (1 + 3*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (3 - I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) - (3 + I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]) + (3*I)*Sin[3*(e + f*x)]))/(a^2*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",1
176,1,228,301,1.069227,"\int \frac{1}{\sqrt{d \tan (e+f x)} (a+i a \tan (e+f x))^2} \, dx","Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2),x]","\frac{\sec ^3(e+f x) \left(-7 \sin (e+f x)-7 \sin (3 (e+f x))-5 i \cos (e+f x)+5 i \cos (3 (e+f x))+(5+9 i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))+(5-9 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-(9+5 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{32 a^2 f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","-\frac{\left(\frac{9}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 \sqrt{d} f}+\frac{\left(\frac{9}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 \sqrt{d} f}+\frac{5 \sqrt{d \tan (e+f x)}}{8 a^2 d f (1+i \tan (e+f x))}-\frac{\left(\frac{9}{32}+\frac{5 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 \sqrt{d} f}+\frac{\left(\frac{9}{32}+\frac{5 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 \sqrt{d} f}+\frac{\sqrt{d \tan (e+f x)}}{4 d f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^3*((-5*I)*Cos[e + f*x] + (5*I)*Cos[3*(e + f*x)] - 7*Sin[e + f*x] - (9 + 5*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (5 - 9*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) + (5 + 9*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]) - 7*Sin[3*(e + f*x)]))/(32*a^2*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",0
177,1,231,326,1.1869319,"\int \frac{1}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2} \, dx","Integrate[1/((d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^2),x]","\frac{\sec ^3(e+f x) \left(43 i \sin (e+f x)+43 i \sin (3 (e+f x))+23 \cos (e+f x)+41 \cos (3 (e+f x))+(21-25 i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))+(-21-25 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-(25-21 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)\right)}{32 a^2 d f (\tan (e+f x)-i)^2 \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{25}{16}+\frac{21 i}{16}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 d^{3/2} f}-\frac{\left(\frac{25}{16}+\frac{21 i}{16}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 d^{3/2} f}-\frac{\left(\frac{25}{32}-\frac{21 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 d^{3/2} f}+\frac{\left(\frac{25}{32}-\frac{21 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 d^{3/2} f}-\frac{25}{8 a^2 d f \sqrt{d \tan (e+f x)}}+\frac{7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt{d \tan (e+f x)}}+\frac{1}{4 d f (a+i a \tan (e+f x))^2 \sqrt{d \tan (e+f x)}}",1,"(Sec[e + f*x]^3*(23*Cos[e + f*x] + 41*Cos[3*(e + f*x)] + (43*I)*Sin[e + f*x] - (25 - 21*I)*Cos[2*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - (21 + 25*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[2*(e + f*x)]^(3/2) + (21 - 25*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]) + (43*I)*Sin[3*(e + f*x)]))/(32*a^2*d*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^2)","A",1
178,1,346,353,2.3676211,"\int \frac{1}{(d \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^2} \, dx","Integrate[1/((d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])^2),x]","-\frac{\sec ^4(e+f x) \left(-142 i \sin (2 (e+f x))+199 i \sin (4 (e+f x))-64 \cos (2 (e+f x))+205 \cos (4 (e+f x))+(270+294 i) \sin (e+f x) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))-(147+135 i) \sqrt{\sin (2 (e+f x))} \sin (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(135-147 i) \sqrt{\sin (2 (e+f x))} \cos (e+f x) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-(135-147 i) \sqrt{\sin (2 (e+f x))} \cos (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(147+135 i) \sin (e+f x) \sqrt{\sin (2 (e+f x))} \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-269\right)}{192 a^2 d f (\tan (e+f x)-i)^2 (d \tan (e+f x))^{3/2}}","\frac{\left(\frac{49}{16}-\frac{45 i}{16}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^2 d^{5/2} f}-\frac{\left(\frac{49}{16}-\frac{45 i}{16}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^2 d^{5/2} f}+\frac{\left(\frac{49}{32}+\frac{45 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 d^{5/2} f}-\frac{\left(\frac{49}{32}+\frac{45 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^2 d^{5/2} f}+\frac{45 i}{8 a^2 d^2 f \sqrt{d \tan (e+f x)}}-\frac{49}{24 a^2 d f (d \tan (e+f x))^{3/2}}+\frac{9}{8 a^2 d f (1+i \tan (e+f x)) (d \tan (e+f x))^{3/2}}+\frac{1}{4 d f (a+i a \tan (e+f x))^2 (d \tan (e+f x))^{3/2}}",1,"-1/192*(Sec[e + f*x]^4*(-269 - 64*Cos[2*(e + f*x)] + 205*Cos[4*(e + f*x)] + (135 - 147*I)*Cos[e + f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - (135 - 147*I)*Cos[3*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + (147 + 135*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sin[e + f*x]*Sqrt[Sin[2*(e + f*x)]] - (142*I)*Sin[2*(e + f*x)] + (270 + 294*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sin[e + f*x]*Sqrt[Sin[2*(e + f*x)]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]) - (147 + 135*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]]*Sin[3*(e + f*x)] + (199*I)*Sin[4*(e + f*x)]))/(a^2*d*f*(d*Tan[e + f*x])^(3/2)*(-I + Tan[e + f*x])^2)","A",1
179,1,236,370,3.5097062,"\int \frac{(d \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(9/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{i d^4 e^{-6 i (e+f x)} \sqrt{d \tan (e+f x)} \left(9 e^{2 i (e+f x)}-49 e^{4 i (e+f x)}-105 e^{6 i (e+f x)}+146 e^{8 i (e+f x)}-87 e^{6 i (e+f x)} \sqrt{-1+e^{4 i (e+f x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (e+f x)}}\right)-6 e^{6 i (e+f x)} \sqrt{-1+e^{2 i (e+f x)}} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)-1\right)}{48 a^3 f \left(-1+e^{2 i (e+f x)}\right)}","\frac{\left(\frac{7}{4}+\frac{15 i}{8}\right) d^{9/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^3 f}-\frac{\left(\frac{7}{4}+\frac{15 i}{8}\right) d^{9/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^3 f}-\frac{\left(\frac{7}{8}-\frac{15 i}{16}\right) d^{9/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 f}+\frac{\left(\frac{7}{8}-\frac{15 i}{16}\right) d^{9/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 f}+\frac{15 i d^4 \sqrt{d \tan (e+f x)}}{4 a^3 f}+\frac{7 d^3 (d \tan (e+f x))^{3/2}}{6 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{5 i d^2 (d \tan (e+f x))^{5/2}}{12 a f (a+i a \tan (e+f x))^2}-\frac{d (d \tan (e+f x))^{7/2}}{6 f (a+i a \tan (e+f x))^3}",1,"((I/48)*d^4*(-1 + 9*E^((2*I)*(e + f*x)) - 49*E^((4*I)*(e + f*x)) - 105*E^((6*I)*(e + f*x)) + 146*E^((8*I)*(e + f*x)) - 87*E^((6*I)*(e + f*x))*Sqrt[-1 + E^((4*I)*(e + f*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(e + f*x))]] - 6*E^((6*I)*(e + f*x))*Sqrt[-1 + E^((2*I)*(e + f*x))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]])*Sqrt[d*Tan[e + f*x]])/(a^3*E^((6*I)*(e + f*x))*(-1 + E^((2*I)*(e + f*x)))*f)","A",0
180,1,234,343,1.3862441,"\int \frac{(d \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(7/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{d^4 \sec ^4(e+f x) \left(-12 i \sin (2 (e+f x))+21 i \sin (4 (e+f x))+19 \cos (4 (e+f x))+(21+15 i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))+i \sin (3 (e+f x)))+(15+21 i) \sqrt{\sin (2 (e+f x))} \sin (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(21-15 i) \sqrt{\sin (2 (e+f x))} \cos (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)-19\right)}{96 a^3 f (\tan (e+f x)-i)^3 \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{5}{16}-\frac{7 i}{16}\right) d^{7/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^3 f}-\frac{\left(\frac{5}{16}-\frac{7 i}{16}\right) d^{7/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^3 f}+\frac{\left(\frac{5}{32}+\frac{7 i}{32}\right) d^{7/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 f}-\frac{\left(\frac{5}{32}+\frac{7 i}{32}\right) d^{7/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 f}+\frac{5 d^3 \sqrt{d \tan (e+f x)}}{8 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{i d^2 (d \tan (e+f x))^{3/2}}{3 a f (a+i a \tan (e+f x))^2}-\frac{d (d \tan (e+f x))^{5/2}}{6 f (a+i a \tan (e+f x))^3}",1,"(d^4*Sec[e + f*x]^4*(-19 + 19*Cos[4*(e + f*x)] + (21 - 15*I)*Cos[3*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - (12*I)*Sin[2*(e + f*x)] + (21 + 15*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)]) + (15 + 21*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]]*Sin[3*(e + f*x)] + (21*I)*Sin[4*(e + f*x)]))/(96*a^3*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^3)","A",1
181,1,232,329,1.0852759,"\int \frac{(d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{d^3 \sec ^4(e+f x) \left(-6 \sin (2 (e+f x))+3 \sin (4 (e+f x))-i \cos (4 (e+f x))+6 i \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))+i \sin (3 (e+f x)))-6 \sqrt{\sin (2 (e+f x))} \sin (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+6 i \sqrt{\sin (2 (e+f x))} \cos (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+i\right)}{96 a^3 f (\tan (e+f x)-i)^3 \sqrt{d \tan (e+f x)}}","\frac{d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 \sqrt{2} a^3 f}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{8 \sqrt{2} a^3 f}-\frac{d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{16 \sqrt{2} a^3 f}+\frac{d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{16 \sqrt{2} a^3 f}-\frac{i d^2 \sqrt{d \tan (e+f x)}}{4 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{i d^2 \sqrt{d \tan (e+f x)}}{4 a f (a+i a \tan (e+f x))^2}-\frac{d (d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}",1,"(d^3*Sec[e + f*x]^4*(I - I*Cos[4*(e + f*x)] + (6*I)*Cos[3*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - 6*Sin[2*(e + f*x)] + (6*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)]) - 6*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]]*Sin[3*(e + f*x)] + 3*Sin[4*(e + f*x)]))/(96*a^3*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^3)","A",1
182,1,158,157,2.2484143,"\int \frac{(d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{d^2 (\sin (3 (e+f x))+i \cos (3 (e+f x))) \left(3 i \sin (e+f x)-3 i \sin (3 (e+f x))+5 \cos (e+f x)-5 \cos (3 (e+f x))+6 \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right) (\cos (3 (e+f x))+i \sin (3 (e+f x)))\right)}{48 a^3 f \sqrt{d \tan (e+f x)}}","\frac{\sqrt[4]{-1} d^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 f}+\frac{d \sqrt{d \tan (e+f x)}}{8 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{d \sqrt{d \tan (e+f x)}}{6 a f (a+i a \tan (e+f x))^2}-\frac{d \sqrt{d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}",1,"(d^2*(I*Cos[3*(e + f*x)] + Sin[3*(e + f*x)])*(5*Cos[e + f*x] - 5*Cos[3*(e + f*x)] + (3*I)*Sin[e + f*x] - (3*I)*Sin[3*(e + f*x)] + 6*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*Sqrt[I*Tan[e + f*x]]))/(48*a^3*f*Sqrt[d*Tan[e + f*x]])","A",1
183,1,225,292,2.4868974,"\int \frac{\sqrt{d \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3,x]","-\frac{d (\cos (3 (e+f x))-i \sin (3 (e+f x))) \left(-3 i \sin (e+f x)-3 i \sin (3 (e+f x))+\cos (e+f x)-\cos (3 (e+f x))+6 \tan ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right) \sqrt{i \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x)))+6 \sqrt{i \tan (e+f x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right) (\cos (3 (e+f x))+i \sin (3 (e+f x)))\right)}{48 a^3 f \sqrt{d \tan (e+f x)}}","\frac{i \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 \sqrt{2} a^3 f}-\frac{i \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{8 \sqrt{2} a^3 f}+\frac{i \sqrt{d} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{16 \sqrt{2} a^3 f}-\frac{i \sqrt{d} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{16 \sqrt{2} a^3 f}+\frac{i \sqrt{d \tan (e+f x)}}{12 a f (a+i a \tan (e+f x))^2}+\frac{i \sqrt{d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}",1,"-1/48*(d*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)])*(Cos[e + f*x] - Cos[3*(e + f*x)] - (3*I)*Sin[e + f*x] - (3*I)*Sin[3*(e + f*x)] + 6*ArcTan[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*Sqrt[I*Tan[e + f*x]] + 6*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*Sqrt[I*Tan[e + f*x]]))/(a^3*f*Sqrt[d*Tan[e + f*x]])","A",1
184,1,234,343,1.2324778,"\int \frac{1}{\sqrt{d \tan (e+f x)} (a+i a \tan (e+f x))^3} \, dx","Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^3),x]","\frac{\sec ^4(e+f x) \left(19 \cos (4 (e+f x))-(15+21 i) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))+i \sin (3 (e+f x)))+i \left(12 \sin (2 (e+f x))+21 \sin (4 (e+f x))+(-15+21 i) \sqrt{\sin (2 (e+f x))} \sin (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+(21+15 i) \sqrt{\sin (2 (e+f x))} \cos (3 (e+f x)) \log \left(\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right)+19 i\right)\right)}{96 a^3 f (\tan (e+f x)-i)^3 \sqrt{d \tan (e+f x)}}","-\frac{\left(\frac{7}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^3 \sqrt{d} f}+\frac{\left(\frac{7}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^3 \sqrt{d} f}+\frac{5 \sqrt{d \tan (e+f x)}}{8 d f \left(a^3+i a^3 \tan (e+f x)\right)}-\frac{\left(\frac{7}{32}+\frac{5 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 \sqrt{d} f}+\frac{\left(\frac{7}{32}+\frac{5 i}{32}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 \sqrt{d} f}+\frac{\sqrt{d \tan (e+f x)}}{3 a d f (a+i a \tan (e+f x))^2}+\frac{\sqrt{d \tan (e+f x)}}{6 d f (a+i a \tan (e+f x))^3}",1,"(Sec[e + f*x]^4*(19*Cos[4*(e + f*x)] - (15 + 21*I)*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Sqrt[Sin[2*(e + f*x)]]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)]) + I*(19*I + (21 + 15*I)*Cos[3*(e + f*x)]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] + 12*Sin[2*(e + f*x)] - (15 - 21*I)*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2*(e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]]*Sin[3*(e + f*x)] + 21*Sin[4*(e + f*x)])))/(96*a^3*f*Sqrt[d*Tan[e + f*x]]*(-I + Tan[e + f*x])^3)","A",1
185,1,234,368,2.0942669,"\int \frac{1}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^3} \, dx","Integrate[1/((d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^3),x]","\frac{e^{-6 i (e+f x)} \left(9 e^{2 i (e+f x)}+49 e^{4 i (e+f x)}-105 e^{6 i (e+f x)}-146 e^{8 i (e+f x)}-87 e^{6 i (e+f x)} \sqrt{-1+e^{4 i (e+f x)}} \tan ^{-1}\left(\sqrt{-1+e^{4 i (e+f x)}}\right)+6 e^{6 i (e+f x)} \sqrt{-1+e^{2 i (e+f x)}} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right)+1\right)}{48 a^3 d f \left(1+e^{2 i (e+f x)}\right) \sqrt{d \tan (e+f x)}}","\frac{\left(\frac{15}{8}+\frac{7 i}{4}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{2} a^3 d^{3/2} f}-\frac{\left(\frac{15}{8}+\frac{7 i}{4}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{2} a^3 d^{3/2} f}-\frac{\left(\frac{15}{16}-\frac{7 i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 d^{3/2} f}+\frac{\left(\frac{15}{16}-\frac{7 i}{8}\right) \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} a^3 d^{3/2} f}-\frac{15}{4 a^3 d f \sqrt{d \tan (e+f x)}}+\frac{7}{6 d f \left(a^3+i a^3 \tan (e+f x)\right) \sqrt{d \tan (e+f x)}}+\frac{5}{12 a d f (a+i a \tan (e+f x))^2 \sqrt{d \tan (e+f x)}}+\frac{1}{6 d f (a+i a \tan (e+f x))^3 \sqrt{d \tan (e+f x)}}",1,"(1 + 9*E^((2*I)*(e + f*x)) + 49*E^((4*I)*(e + f*x)) - 105*E^((6*I)*(e + f*x)) - 146*E^((8*I)*(e + f*x)) - 87*E^((6*I)*(e + f*x))*Sqrt[-1 + E^((4*I)*(e + f*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(e + f*x))]] + 6*E^((6*I)*(e + f*x))*Sqrt[-1 + E^((2*I)*(e + f*x))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]])/(48*a^3*d*E^((6*I)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))*f*Sqrt[d*Tan[e + f*x]])","A",0
186,1,267,176,3.8395916,"\int \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)} \left(\frac{\sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(1-3 e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^2}+4 \sqrt{2} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-7 \sqrt{\tan (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{4 \sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}}}","\frac{7 (-1)^{3/4} \sqrt{a} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}-\frac{i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{(1+i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((I/4)*(1 + E^((2*I)*(c + d*x)))*(4*Sqrt[2]*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + (Sqrt[2]*E^(I*(c + d*x))*(1 - 3*E^((2*I)*(c + d*x)))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[Tan[c + d*x]])/(1 + E^((2*I)*(c + d*x)))^2 - 7*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Sqrt[Tan[c + d*x]])*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
187,1,249,135,3.723488,"\int \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{a+i a \tan (c+d x)} \left(8 \sqrt{\tan (c+d x)}+\frac{i e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(8 \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+\sqrt{2} \left(\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}\right)}{8 d}","-\frac{\sqrt[4]{-1} \sqrt{a} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{\sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}-\frac{(1-i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"(((I*Sqrt[-1 + E^((2*I)*(c + d*x))]*(8*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(-Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + 8*Sqrt[Tan[c + d*x]])*Sqrt[a + I*a*Tan[c + d*x]])/(8*d)","A",1
188,1,229,104,1.1665685,"\int \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\sqrt{2} \left(\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)-4 \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)\right) \cos (c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d \sqrt{-1+e^{2 i (c+d x)}}}","-\frac{2 (-1)^{3/4} \sqrt{a} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(1+i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((I/2)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]*(-4*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] - Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]]))*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[-1 + E^((2*I)*(c + d*x))])","B",1
189,0,0,49,1.0421796,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[Tan[c + d*x]],x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx","\frac{(1-i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[Tan[c + d*x]], x]","F",-1
190,0,0,82,2.1826016,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(3/2),x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","\frac{(1+i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(3/2), x]","F",-1
191,0,0,120,2.5152929,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(5/2),x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","-\frac{2 \sqrt{a+i a \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}-\frac{(1-i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(5/2), x]","F",-1
192,0,0,154,3.4062051,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(7/2),x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+i a \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{26 \sqrt{a+i a \tan (c+d x)}}{15 d \sqrt{\tan (c+d x)}}-\frac{(1+i) \sqrt{a} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(7/2), x]","F",-1
193,1,213,254,2.8650298,"\int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a \sqrt{a+i a \tan (c+d x)} \left(\frac{6 e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(32 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-23 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+2 \sqrt{\tan (c+d x)} \sec ^2(c+d x) (14 \sin (2 (c+d x))-35 i \cos (2 (c+d x))-19 i)\right)}{96 d}","\frac{23 (-1)^{3/4} a^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{8 d}+\frac{(2+2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \tan ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a+i a \tan (c+d x)}}+\frac{i a^2 \tan ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a+i a \tan (c+d x)}}+\frac{7 a \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d}-\frac{9 i a \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{8 d}",1,"(a*((6*Sqrt[-1 + E^((2*I)*(c + d*x))]*(32*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 23*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + 2*Sec[c + d*x]^2*(-19*I - (35*I)*Cos[2*(c + d*x)] + 14*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])*Sqrt[a + I*a*Tan[c + d*x]])/(96*d)","A",1
194,1,234,217,3.1473057,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{a (5+2 i \tan (c+d x)) \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{i a e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(16 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-11 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{4 \sqrt{2} d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}","-\frac{11 \sqrt[4]{-1} a^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}-\frac{(2-2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i a^2 \tan ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a+i a \tan (c+d x)}}+\frac{5 a \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}",1,"((I/4)*a*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(16*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 11*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + (a*(5 + (2*I)*Tan[c + d*x])*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d)","A",1
195,1,203,176,1.746342,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{i a e^{-i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-2 \sqrt{2} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3 \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{\sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}}}","-\frac{3 (-1)^{3/4} a^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(2+2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{i a^2 \sqrt{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}",1,"(I*a*(Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))] - 2*Sqrt[2]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 3*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
196,1,255,104,1.9373648,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Sqrt[Tan[c + d*x]],x]","-\frac{i a e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(8 \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+\sqrt{2} \left(\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)}{2 \sqrt{2} d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}","\frac{2 \sqrt[4]{-1} a^{3/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(2-2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((-1/2*I)*a*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(8*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(-Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))])","B",1
197,1,160,83,1.6514283,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(3/2),x]","-\frac{2 i \sqrt{2} a^2 e^{i (c+d x)} \sqrt{\tan (c+d x)} \left(e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-\left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{d \left(-1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{(2+2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"((-2*I)*Sqrt[2]*a^2*E^(I*(c + d*x))*(E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))] - (-1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(d*(-1 + E^((2*I)*(c + d*x)))^(3/2)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
198,1,160,119,1.7468705,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(5/2),x]","-\frac{2 i \sqrt{2} a e^{-i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(e^{i (c+d x)} \left(-3+5 e^{2 i (c+d x)}\right)-3 \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{3 d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)}}","-\frac{(2-2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 (a+i a \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 i a \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"(((-2*I)/3)*Sqrt[2]*a*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x))*(-3 + 5*E^((2*I)*(c + d*x))) - 3*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[Tan[c + d*x]])","A",1
199,1,166,198,2.5962052,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(7/2),x]","\frac{2 i a e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)} \left(e^{i (c+d x)} \left(-10 e^{2 i (c+d x)}+9 e^{4 i (c+d x)}+5\right)-5 \left(-1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{5 d \left(-1+e^{2 i (c+d x)}\right)^3}","-\frac{(2+2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 i a^2}{5 d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{2 a^2}{5 d \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{5 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{12 a \sqrt{a+i a \tan (c+d x)}}{5 d \sqrt{\tan (c+d x)}}",1,"(((2*I)/5)*a*(1 + E^((2*I)*(c + d*x)))*(E^(I*(c + d*x))*(5 - 10*E^((2*I)*(c + d*x)) + 9*E^((4*I)*(c + d*x))) - 5*(-1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^3)","A",1
200,1,224,235,3.1540955,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(9/2),x]","-\frac{2 i \sqrt{2} a e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\frac{a \csc ^3(c+d x) \sqrt{a+i a \tan (c+d x)} (-378 i \sin (c+d x)+158 i \sin (3 (c+d x))+7 \cos (c+d x)+53 \cos (3 (c+d x)))}{210 d \sqrt{\tan (c+d x)}}","\frac{(2-2 i) a^{3/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 i a^2}{7 d \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{2 a^2}{7 d \tan ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{76 a \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{16 i a \sqrt{a+i a \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{268 i a \sqrt{a+i a \tan (c+d x)}}{105 d \sqrt{\tan (c+d x)}}",1,"((-2*I)*Sqrt[2]*a*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - (a*Csc[c + d*x]^3*(7*Cos[c + d*x] + 53*Cos[3*(c + d*x)] - (378*I)*Sin[c + d*x] + (158*I)*Sin[3*(c + d*x)])*Sqrt[a + I*a*Tan[c + d*x]])/(210*d*Sqrt[Tan[c + d*x]])","A",1
201,1,232,258,4.0800453,"\int \tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{a^2 \sqrt{a+i a \tan (c+d x)} \left(\frac{6 e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(512 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-363 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-i \sqrt{\tan (c+d x)} \sec ^3(c+d x) (70 i \sin (c+d x)+262 i \sin (3 (c+d x))+1205 \cos (c+d x)+583 \cos (3 (c+d x)))\right)}{768 d}","\frac{363 (-1)^{3/4} a^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{64 d}+\frac{(4+4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \tan ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{4 d}+\frac{17 i a^2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{24 d}+\frac{107 a^2 \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{96 d}-\frac{149 i a^2 \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{64 d}",1,"(a^2*((6*Sqrt[-1 + E^((2*I)*(c + d*x))]*(512*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 363*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - I*Sec[c + d*x]^3*(1205*Cos[c + d*x] + 583*Cos[3*(c + d*x)] + (70*I)*Sin[c + d*x] + (262*I)*Sin[3*(c + d*x)])*Sqrt[Tan[c + d*x]])*Sqrt[a + I*a*Tan[c + d*x]])/(768*d)","A",1
202,1,258,219,3.0902734,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i a^2 e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(64 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-45 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{8 \sqrt{2} d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}+\frac{a^2 \sqrt{\tan (c+d x)} \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} (26 i \sin (2 (c+d x))+65 \cos (2 (c+d x))+49)}{48 d}","-\frac{45 \sqrt[4]{-1} a^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{8 d}-\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}+\frac{13 i a^2 \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d}+\frac{19 a^2 \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{8 d}",1,"((I/8)*a^2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(64*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 45*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) + (a^2*Sec[c + d*x]^2*(49 + 65*Cos[2*(c + d*x)] + (26*I)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(48*d)","A",1
203,1,192,182,3.1706401,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{a^2 \sqrt{a+i a \tan (c+d x)} \left(\frac{e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(23 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-32 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-2 \sqrt{\tan (c+d x)} (2 \tan (c+d x)-9 i)\right)}{8 d}","-\frac{23 (-1)^{3/4} a^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}-\frac{(4+4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 d}+\frac{9 i a^2 \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 d}",1,"(a^2*Sqrt[a + I*a*Tan[c + d*x]]*((Sqrt[-1 + E^((2*I)*(c + d*x))]*(-32*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 23*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - 2*Sqrt[Tan[c + d*x]]*(-9*I + 2*Tan[c + d*x])))/(8*d)","A",1
204,1,267,139,2.5332231,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Sqrt[Tan[c + d*x]],x]","\frac{2 i \sqrt{2} a^2 e^{i (c+d x)} \cos ^2(c+d x) \left(\sqrt{2} e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)-4 \sqrt{2} \left(1+e^{2 i (c+d x)}\right) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+5 \left(1+e^{2 i (c+d x)}\right) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right)^3}","\frac{5 \sqrt[4]{-1} a^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"((2*I)*Sqrt[2]*a^2*E^(I*(c + d*x))*(Sqrt[2]*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))) - 4*Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 5*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^3)","A",1
205,1,215,139,2.2095838,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(3/2),x]","-\frac{i \sqrt{2} a^3 e^{i (c+d x)} \sqrt{\tan (c+d x)} \left(2 e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-4 \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\sqrt{2} \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{d \left(-1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{2 (-1)^{3/4} a^{5/2} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(4+4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}",1,"((-I)*Sqrt[2]*a^3*E^(I*(c + d*x))*(2*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))] - 4*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(d*(-1 + E^((2*I)*(c + d*x)))^(3/2)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
206,1,162,122,1.8782149,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(5/2),x]","-\frac{4 i \sqrt{2} a^2 e^{-i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(e^{i (c+d x)} \left(-3+4 e^{2 i (c+d x)}\right)-3 \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{3 d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)}}","-\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{4 i a^2 \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}-\frac{2 a (a+i a \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(((-4*I)/3)*Sqrt[2]*a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x))*(-3 + 4*E^((2*I)*(c + d*x))) - 3*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[Tan[c + d*x]])","A",1
207,1,187,156,3.5636058,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(7/2),x]","-\frac{4 i a^2 e^{-i (c+d x)} \cot (c+d x) \left(e^{i (c+d x)} \left(-35 e^{2 i (c+d x)}+26 e^{4 i (c+d x)}+15\right)-15 \left(-1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{15 d \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(-1+e^{4 i (c+d x)}\right)}","-\frac{(4+4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{4 a^2 \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}-\frac{2 i a (a+i a \tan (c+d x))^{3/2}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 (a+i a \tan (c+d x))^{5/2}}{5 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(((-4*I)/15)*a^2*(E^(I*(c + d*x))*(15 - 35*E^((2*I)*(c + d*x)) + 26*E^((4*I)*(c + d*x))) - 15*(-1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((4*I)*(c + d*x))))","A",1
208,1,188,202,2.8796023,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(9/2),x]","\frac{4 i \sqrt{2} a^2 e^{-i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(e^{i (c+d x)} \left(70 e^{2 i (c+d x)}-77 e^{4 i (c+d x)}+40 e^{6 i (c+d x)}-21\right)-21 \left(-1+e^{2 i (c+d x)}\right)^{7/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{21 d \left(-1+e^{2 i (c+d x)}\right)^3 \sqrt{\tan (c+d x)}}","\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}",1,"(((4*I)/21)*Sqrt[2]*a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x))*(-21 + 70*E^((2*I)*(c + d*x)) - 77*E^((4*I)*(c + d*x)) + 40*E^((6*I)*(c + d*x))) - 21*(-1 + E^((2*I)*(c + d*x)))^(7/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^3*Sqrt[Tan[c + d*x]])","A",1
209,1,207,239,4.6404332,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(11/2),x]","\frac{a^2 \sqrt{a+i a \tan (c+d x)} \left(\frac{10080 e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}}}-\sqrt{\tan (c+d x)} \csc ^5(c+d x) (-282 i \sin (c+d x)+49 i \sin (3 (c+d x))+331 i \sin (5 (c+d x))+1650 \cos (c+d x)-2051 \cos (3 (c+d x))+961 \cos (5 (c+d x)))\right)}{2520 d}","\frac{(4+4 i) a^{5/2} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{472 i a^2 \sqrt{a+i a \tan (c+d x)}}{315 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{92 a^2 \sqrt{a+i a \tan (c+d x)}}{105 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{38 i a^2 \sqrt{a+i a \tan (c+d x)}}{63 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{1576 a^2 \sqrt{a+i a \tan (c+d x)}}{315 d \sqrt{\tan (c+d x)}}",1,"(a^2*((10080*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])/(E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]) - Csc[c + d*x]^5*(1650*Cos[c + d*x] - 2051*Cos[3*(c + d*x)] + 961*Cos[5*(c + d*x)] - (282*I)*Sin[c + d*x] + (49*I)*Sin[3*(c + d*x)] + (331*I)*Sin[5*(c + d*x)])*Sqrt[Tan[c + d*x]])*Sqrt[a + I*a*Tan[c + d*x]])/(2520*d)","A",1
210,1,224,218,3.2669911,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(7/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sec ^2(c+d x) (i \sin (2 (c+d x))+5 \cos (2 (c+d x))+9)-\frac{2 e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \left(4 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-11 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^2}}{8 d \sqrt{a+i a \tan (c+d x)}}","\frac{11 \sqrt[4]{-1} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 \sqrt{a} d}-\frac{\tan ^{\frac{5}{2}}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{3 i \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a d}+\frac{7 \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a d}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((-2*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*(4*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 11*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/((((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^2) + Sec[c + d*x]^2*(9 + 5*Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/(8*d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
211,1,222,177,2.1718556,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(5/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} (\tan (c+d x)-2 i)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{i e^{i (c+d x)} \sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(\tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{(-1)^{3/4} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}-\frac{\tan ^{\frac{3}{2}}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*E^(I*(c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]) + (Sqrt[Tan[c + d*x]]*(-2*I + Tan[c + d*x]))/(d*Sqrt[a + I*a*Tan[c + d*x]])","A",1
212,1,210,140,1.8814766,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(3/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(e^{2 i (c+d x)}+e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)}}","-\frac{2 \sqrt[4]{-1} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((2*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a*d*E^((2*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
213,1,132,88,1.3421288,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}}-e^{i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{i \sqrt{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*(Sqrt[-1 + E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
214,1,140,85,1.5716243,"\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]","-\frac{i e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(e^{2 i (c+d x)}+e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{\sqrt{2} a d \sqrt{\tan (c+d x)}}","\frac{\sqrt{\tan (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((-I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((2*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a*d*E^((2*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
215,1,160,120,1.5768639,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{i \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(1-5 e^{2 i (c+d x)}\right)+e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{2} d \left(-1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{3 \sqrt{a+i a \tan (c+d x)}}{a d \sqrt{\tan (c+d x)}}+\frac{1}{d \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*((1 - 5*E^((2*I)*(c + d*x)))*Sqrt[-1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d*(-1 + E^((2*I)*(c + d*x)))^(3/2)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
216,1,159,161,1.8503802,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{i \left(-18 e^{2 i (c+d x)}+7 e^{4 i (c+d x)}+3 e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3\right)}{3 \sqrt{2} d \left(-1+e^{4 i (c+d x)}\right) \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\tan (c+d x)}}","-\frac{5 \sqrt{a+i a \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{1}{d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \sqrt{a+i a \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((I/3)*(3 - 18*E^((2*I)*(c + d*x)) + 7*E^((4*I)*(c + d*x)) + 3*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((4*I)*(c + d*x)))*Sqrt[Tan[c + d*x]])","A",1
217,1,191,198,1.358665,"\int \frac{1}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{i \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(165 e^{2 i (c+d x)}-205 e^{4 i (c+d x)}+103 e^{6 i (c+d x)}-15\right)-15 e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^3 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{15 \sqrt{2} d \left(-1+e^{2 i (c+d x)}\right)^{7/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{23 i \sqrt{a+i a \tan (c+d x)}}{15 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{1}{d \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{61 \sqrt{a+i a \tan (c+d x)}}{15 a d \sqrt{\tan (c+d x)}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((I/15)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(-15 + 165*E^((2*I)*(c + d*x)) - 205*E^((4*I)*(c + d*x)) + 103*E^((6*I)*(c + d*x))) - 15*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^3*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d*(-1 + E^((2*I)*(c + d*x)))^(7/2)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
218,1,240,218,4.064452,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{i \sqrt{\tan (c+d x)} \sec ^2(c+d x) (29 i \sin (2 (c+d x))+27 \cos (2 (c+d x))+15)-\frac{12 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(\tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+6 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{\sqrt{-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}} \left(1+e^{2 i (c+d x)}\right)^2}}{12 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}","-\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{7 \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}-\frac{\tan ^{\frac{5}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{13 i \tan ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-12*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 6*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2) + I*Sec[c + d*x]^2*(15 + 27*Cos[2*(c + d*x)] + (29*I)*Sin[2*(c + d*x)])*Sqrt[Tan[c + d*x]])/(12*a*d*(-I + Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])","A",1
219,1,217,181,2.0517121,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(1-10 e^{2 i (c+d x)}\right)-3 e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+12 \sqrt{2} e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{6 \sqrt{2} a d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{2 (-1)^{3/4} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{\tan ^{\frac{3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{3 i \sqrt{\tan (c+d x)}}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/6*I)*((1 - 10*E^((2*I)*(c + d*x)))*Sqrt[-1 + E^((2*I)*(c + d*x))] - 3*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 12*Sqrt[2]*E^((3*I)*(c + d*x))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
220,1,158,127,1.8129992,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(-5 e^{2 i (c+d x)}+4 e^{4 i (c+d x)}-3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{6 \sqrt{2} a^2 d \sqrt{\tan (c+d x)}}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{i \tan ^{\frac{3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)}}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"((-1/6*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 - 5*E^((2*I)*(c + d*x)) + 4*E^((4*I)*(c + d*x)) - 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
221,1,164,127,1.533971,"\int \frac{\sqrt{\tan (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{i e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(1+2 e^{2 i (c+d x)}\right)-3 e^{3 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{6 \sqrt{2} a d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{\tan ^{\frac{3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac{i \sqrt{\tan (c+d x)}}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"((I/6)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + 2*E^((2*I)*(c + d*x))) - 3*E^((3*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
222,1,158,125,1.7988871,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)),x]","-\frac{i e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(-7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{6 \sqrt{2} a^2 d \sqrt{\tan (c+d x)}}","\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{7 \sqrt{\tan (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\sqrt{\tan (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/6*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 - 7*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
223,1,190,162,1.9142297,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{i e^{-2 i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(13 e^{2 i (c+d x)}-38 e^{4 i (c+d x)}+1\right)+3 e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{6 \sqrt{2} a d \left(-1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{25 \sqrt{a+i a \tan (c+d x)}}{6 a^2 d \sqrt{\tan (c+d x)}}+\frac{11}{6 a d \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{1}{3 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}",1,"((I/6)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + 13*E^((2*I)*(c + d*x)) - 38*E^((4*I)*(c + d*x))) + 3*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
224,1,186,201,1.2765406,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{i e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(18 e^{2 i (c+d x)}-87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}+3 e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{6 \sqrt{2} a^2 d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)}}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{13 i \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\tan (c+d x)}}+\frac{5}{2 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{1}{3 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}",1,"((I/6)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + 18*E^((2*I)*(c + d*x)) - 87*E^((4*I)*(c + d*x)) + 52*E^((6*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[Tan[c + d*x]])","A",1
225,1,270,257,3.8182655,"\int \frac{\tan ^{\frac{9}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(9/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\tan (c+d x)} \left(-\sqrt{-1+e^{2 i (c+d x)}} \left(-28 e^{2 i (c+d x)}+252 e^{4 i (c+d x)}+403 e^{6 i (c+d x)}+3\right)+15 e^{5 i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+300 \sqrt{2} e^{5 i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 \sqrt{2} a^3 d \sqrt{-1+e^{2 i (c+d x)}}}","\frac{5 (-1)^{3/4} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{21 i \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{4 a^3 d}+\frac{41 \tan ^{\frac{3}{2}}(c+d x)}{12 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\tan ^{\frac{7}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{19 i \tan ^{\frac{5}{2}}(c+d x)}{30 a d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/60*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-(Sqrt[-1 + E^((2*I)*(c + d*x))]*(3 - 28*E^((2*I)*(c + d*x)) + 252*E^((4*I)*(c + d*x)) + 403*E^((6*I)*(c + d*x)))) + 15*E^((5*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 300*Sqrt[2]*E^((5*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))])","A",1
226,1,241,218,2.7327683,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(-8 e^{2 i (c+d x)}+48 e^{4 i (c+d x)}-41 e^{6 i (c+d x)}-5 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+40 \sqrt{2} e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{20 \sqrt{2} a^3 d \sqrt{\tan (c+d x)}}","\frac{2 \sqrt[4]{-1} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{7 \sqrt{\tan (c+d x)}}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{\tan ^{\frac{5}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i \tan ^{\frac{3}{2}}(c+d x)}{2 a d (a+i a \tan (c+d x))^{3/2}}",1,"((I/20)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 - 8*E^((2*I)*(c + d*x)) + 48*E^((4*I)*(c + d*x)) - 41*E^((6*I)*(c + d*x)) - 5*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 40*Sqrt[2]*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
227,1,177,166,2.0754568,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i e^{-4 i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(11 e^{2 i (c+d x)}-23 e^{4 i (c+d x)}-3\right)+15 e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 \sqrt{2} a^2 d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{i \sqrt{\tan (c+d x)}}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i \tan ^{\frac{5}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{\tan ^{\frac{3}{2}}(c+d x)}{6 a d (a+i a \tan (c+d x))^{3/2}}",1,"((I/60)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(-3 + 11*E^((2*I)*(c + d*x)) - 23*E^((4*I)*(c + d*x))) + 15*E^((5*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
228,1,171,164,2.0240526,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(-4 e^{2 i (c+d x)}-16 e^{4 i (c+d x)}+17 e^{6 i (c+d x)}-15 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3\right)}{60 \sqrt{2} a^3 d \sqrt{\tan (c+d x)}}","-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{\sqrt{\tan (c+d x)}}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{\tan ^{\frac{5}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i \tan ^{\frac{3}{2}}(c+d x)}{6 a d (a+i a \tan (c+d x))^{3/2}}",1,"((-1/60*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(3 - 4*E^((2*I)*(c + d*x)) - 16*E^((4*I)*(c + d*x)) + 17*E^((6*I)*(c + d*x)) - 15*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
229,1,175,168,1.7839234,"\int \frac{\sqrt{\tan (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i e^{-4 i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(3 e^{2 i (c+d x)}+e^{4 i (c+d x)}+1\right)-5 e^{5 i (c+d x)} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{20 \sqrt{2} a^2 d \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{i \sqrt{\tan (c+d x)}}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{i \sqrt{\tan (c+d x)}}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i \sqrt{\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((I/20)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + 3*E^((2*I)*(c + d*x)) + E^((4*I)*(c + d*x))) - 5*E^((5*I)*(c + d*x))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
230,1,171,162,2.2143622,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)),x]","-\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(-16 e^{2 i (c+d x)}-64 e^{4 i (c+d x)}+83 e^{6 i (c+d x)}+15 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-3\right)}{60 \sqrt{2} a^3 d \sqrt{\tan (c+d x)}}","\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{67 \sqrt{\tan (c+d x)}}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{13 \sqrt{\tan (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((-1/60*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-3 - 16*E^((2*I)*(c + d*x)) - 64*E^((4*I)*(c + d*x)) + 83*E^((6*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x))*Sqrt[Tan[c + d*x]])","A",1
231,1,203,199,1.3652291,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{i e^{-4 i (c+d x)} \sqrt{\tan (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \left(26 e^{2 i (c+d x)}+194 e^{4 i (c+d x)}-463 e^{6 i (c+d x)}+3\right)+15 e^{5 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 \sqrt{2} a^2 d \left(-1+e^{2 i (c+d x)}\right)^{3/2} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{317 \sqrt{a+i a \tan (c+d x)}}{60 a^3 d \sqrt{\tan (c+d x)}}+\frac{151}{60 a^2 d \sqrt{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{17}{30 a d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{1}{5 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^{5/2}}",1,"((I/60)*(Sqrt[-1 + E^((2*I)*(c + d*x))]*(3 + 26*E^((2*I)*(c + d*x)) + 194*E^((4*I)*(c + d*x)) - 463*E^((6*I)*(c + d*x))) + 15*E^((5*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])","A",1
232,1,199,238,1.5269793,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(33 e^{2 i (c+d x)}+348 e^{4 i (c+d x)}-1527 e^{6 i (c+d x)}+983 e^{8 i (c+d x)}+15 e^{5 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3\right)}{60 \sqrt{2} a^3 d \left(-1+e^{2 i (c+d x)}\right) \sqrt{\tan (c+d x)}}","-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{361 \sqrt{a+i a \tan (c+d x)}}{60 a^3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{707 i \sqrt{a+i a \tan (c+d x)}}{60 a^3 d \sqrt{\tan (c+d x)}}+\frac{89}{20 a^2 d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{7}{10 a d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{1}{5 d \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}",1,"((I/60)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(3 + 33*E^((2*I)*(c + d*x)) + 348*E^((4*I)*(c + d*x)) - 1527*E^((6*I)*(c + d*x)) + 983*E^((8*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Sqrt[Tan[c + d*x]])","A",1
233,1,181,343,1.6790554,"\int \frac{\tan ^{\frac{10}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(10/3)/(a + I*a*Tan[c + d*x]),x]","-\frac{i \sqrt[3]{\tan (c+d x)} \sec ^2(c+d x) \left(3\ 2^{2/3} \left(1+e^{2 i (c+d x)}\right)^{4/3} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)-34 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (i \sin (2 (c+d x))+\cos (2 (c+d x))+1)+18 i \sin (2 (c+d x))+22 \cos (2 (c+d x))+34\right)}{16 a d (\tan (c+d x)-i)}","\frac{7 \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}-\frac{7 \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}-\frac{7 \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}-\frac{\tan ^{\frac{7}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{5 i \tan ^{\frac{4}{3}}(c+d x)}{4 a d}-\frac{5 i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 \sqrt{3} a d}+\frac{7 \sqrt[3]{\tan (c+d x)}}{2 a d}-\frac{5 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{6 a d}+\frac{7 \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{7 \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}+\frac{5 i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{12 a d}",1,"((-1/16*I)*Sec[c + d*x]^2*(34 + 22*Cos[2*(c + d*x)] + 3*2^(2/3)*(1 + E^((2*I)*(c + d*x)))^(4/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2] - 34*Hypergeometric2F1[1/3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (18*I)*Sin[2*(c + d*x)])*Tan[c + d*x]^(1/3))/(a*d*(-I + Tan[c + d*x]))","C",1
234,1,163,319,1.1335257,"\int \frac{\tan ^{\frac{8}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(8/3)/(a + I*a*Tan[c + d*x]),x]","\frac{i e^{-2 i (c+d x)} \left(3 \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+26 e^{2 i (c+d x)} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-28 e^{2 i (c+d x)}-4\right) \tan ^{\frac{2}{3}}(c+d x)}{16 a d}","-\frac{5 \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}+\frac{5 \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}+\frac{5 \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{a d}-\frac{2 i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{\sqrt{3} a d}+\frac{2 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{3 a d}+\frac{5 \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{5 \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{3 a d}",1,"((I/16)*(-4 - 28*E^((2*I)*(c + d*x)) + 3*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 26*E^((2*I)*(c + d*x))*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))])*Tan[c + d*x]^(2/3))/(a*d*E^((2*I)*(c + d*x)))","C",1
235,1,162,299,1.0543309,"\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x]),x]","-\frac{e^{-2 i (c+d x)} \left(3\ 2^{2/3} e^{2 i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+2 \left(-5 e^{2 i (c+d x)} \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+e^{2 i (c+d x)}+1\right)\right) \sqrt[3]{\tan (c+d x)}}{8 a d}","-\frac{\tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}+\frac{\tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}+\frac{\tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}+\frac{i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{\sqrt{3} a d}-\frac{\sqrt[3]{\tan (c+d x)}}{2 d (a+i a \tan (c+d x))}+\frac{i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{3 a d}-\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}+\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{6 a d}",1,"-1/8*((3*2^(2/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2] + 2*(1 + E^((2*I)*(c + d*x)) - 5*E^((2*I)*(c + d*x))*Hypergeometric2F1[1/3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]))*Tan[c + d*x]^(1/3))/(a*d*E^((2*I)*(c + d*x)))","C",1
236,1,163,303,0.9833039,"\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x]),x]","\frac{i e^{-2 i (c+d x)} \left(-3 \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)-2 e^{2 i (c+d x)} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+4 e^{2 i (c+d x)}+4\right) \tan ^{\frac{2}{3}}(c+d x)}{16 a d}","-\frac{\tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}+\frac{\tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}+\frac{\tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}+\frac{i \tan ^{\frac{2}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 \sqrt{3} a d}-\frac{i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{6 a d}+\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}+\frac{i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{12 a d}",1,"((I/16)*(4 + 4*E^((2*I)*(c + d*x)) - 3*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] - 2*E^((2*I)*(c + d*x))*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))])*Tan[c + d*x]^(2/3))/(a*d*E^((2*I)*(c + d*x)))","C",1
237,1,161,303,1.0363589,"\int \frac{1}{\sqrt[3]{\tan (c+d x)} (a+i a \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])),x]","\frac{e^{-2 i (c+d x)} \left(3 \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+10 e^{2 i (c+d x)} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+4 e^{2 i (c+d x)}+4\right) \tan ^{\frac{2}{3}}(c+d x)}{16 a d}","\frac{i \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}-\frac{i \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}-\frac{i \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}+\frac{\tan ^{\frac{2}{3}}(c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{\tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{\sqrt{3} a d}+\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{3 a d}-\frac{i \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}+\frac{i \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{\log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{6 a d}",1,"((4 + 4*E^((2*I)*(c + d*x)) + 3*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 10*E^((2*I)*(c + d*x))*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))])*Tan[c + d*x]^(2/3))/(16*a*d*E^((2*I)*(c + d*x)))","C",1
238,1,201,321,1.5468647,"\int \frac{1}{\tan ^{\frac{5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(5/3)*(a + I*a*Tan[c + d*x])),x]","\frac{i \sqrt[3]{\tan (c+d x)} \csc (c+d x) \sec (c+d x) \left(2 \left(13 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (i \sin (2 (c+d x))+\cos (2 (c+d x))-1)+8 i \sin (2 (c+d x))+6 \cos (2 (c+d x))+6\right)-3\ 2^{2/3} \left(-1+e^{2 i (c+d x)}\right) \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)\right)}{16 a d (\tan (c+d x)-i)}","\frac{5 i \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}-\frac{5 i \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}-\frac{5 i \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}-\frac{2}{a d \tan ^{\frac{2}{3}}(c+d x)}+\frac{1}{2 d \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))}+\frac{2 \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{\sqrt{3} a d}+\frac{2 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{3 a d}+\frac{5 i \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{5 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{\log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{3 a d}",1,"((I/16)*Csc[c + d*x]*Sec[c + d*x]*(-3*2^(2/3)*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2] + 2*(6 + 6*Cos[2*(c + d*x)] + 13*Hypergeometric2F1[1/3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (8*I)*Sin[2*(c + d*x)]))*Tan[c + d*x]^(1/3))/(a*d*(-I + Tan[c + d*x]))","C",1
239,1,236,347,1.8158483,"\int \frac{1}{\tan ^{\frac{7}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(7/3)*(a + I*a*Tan[c + d*x])),x]","-\frac{i \tan ^{\frac{2}{3}}(c+d x) \csc ^2(c+d x) \sec (c+d x) \left(3 \sqrt[3]{2} e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \left(-1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+4 \left(-34 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) \sin ^2(c+d x) (\cos (c+d x)+i \sin (c+d x))+9 i \sin (c+d x)+9 i \sin (3 (c+d x))-23 \cos (c+d x)+11 \cos (3 (c+d x))\right)\right)}{64 a d (\tan (c+d x)-i)}","-\frac{7 i \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{12 a d}+\frac{7 i \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{12 a d}+\frac{7 i \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{6 a d}+\frac{1}{2 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac{5}{4 a d \tan ^{\frac{4}{3}}(c+d x)}+\frac{5 \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 \sqrt{3} a d}+\frac{7 i}{2 a d \sqrt[3]{\tan (c+d x)}}-\frac{5 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{6 a d}+\frac{7 i \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}-\frac{7 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{8 \sqrt{3} a d}+\frac{5 \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{12 a d}",1,"((-1/64*I)*Csc[c + d*x]^2*Sec[c + d*x]*((3*2^(1/3)*(-1 + E^((2*I)*(c + d*x)))^2*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2])/E^(I*(c + d*x)) + 4*(-23*Cos[c + d*x] + 11*Cos[3*(c + d*x)] + (9*I)*Sin[c + d*x] - 34*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[c + d*x] + I*Sin[c + d*x])*Sin[c + d*x]^2 + (9*I)*Sin[3*(c + d*x)]))*Tan[c + d*x]^(2/3))/(a*d*(-I + Tan[c + d*x]))","C",1
240,1,210,379,3.2811125,"\int \frac{\tan ^{\frac{14}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(14/3)/(a + I*a*Tan[c + d*x])^2,x]","\frac{\tan ^{\frac{2}{3}}(c+d x) \sec ^2(c+d x) \left(90 i \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+4 \left(1165 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+776 i \cos (2 (c+d x))-547 \tan (c+d x)-403 \sin (3 (c+d x)) \sec (c+d x)+344 i\right)\right)}{960 a^2 d (\tan (c+d x)-i)^2}","-\frac{121 \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}+\frac{121 \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}+\frac{121 \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}+\frac{7 i \tan ^{\frac{8}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}-\frac{121 \tan ^{\frac{5}{3}}(c+d x)}{60 a^2 d}-\frac{14 i \tan ^{\frac{2}{3}}(c+d x)}{3 a^2 d}-\frac{14 i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}+\frac{14 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}+\frac{121 \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{121 \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{7 i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}-\frac{\tan ^{\frac{11}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*((90*I)*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 4*(344*I + (776*I)*Cos[2*(c + d*x)] + 1165*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]) - 403*Sec[c + d*x]*Sin[3*(c + d*x)] - 547*Tan[c + d*x]))*Tan[c + d*x]^(2/3))/(960*a^2*d*(-I + Tan[c + d*x])^2)","C",1
241,1,192,357,1.7356497,"\int \frac{\tan ^{\frac{10}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(10/3)/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\sqrt[3]{\tan (c+d x)} \sec ^2(c+d x) \left(9\ 2^{2/3} e^{2 i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+2 \left(89 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-88 i \sin (2 (c+d x))-85 \cos (2 (c+d x))-13\right)\right)}{48 a^2 d (\tan (c+d x)-i)^2}","-\frac{49 \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}+\frac{49 \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}+\frac{49 \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}+\frac{5 i \tan ^{\frac{4}{3}}(c+d x)}{6 a^2 d (1+i \tan (c+d x))}+\frac{5 i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}-\frac{49 \sqrt[3]{\tan (c+d x)}}{12 a^2 d}+\frac{5 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}-\frac{49 \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}+\frac{49 \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{5 i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{18 a^2 d}-\frac{\tan ^{\frac{7}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"-1/48*(Sec[c + d*x]^2*(9*2^(2/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2] + 2*(-13 - 85*Cos[2*(c + d*x)] + 89*Hypergeometric2F1[1/3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - (88*I)*Sin[2*(c + d*x)]))*Tan[c + d*x]^(1/3))/(a^2*d*(-I + Tan[c + d*x])^2)","C",1
242,1,194,337,1.5748548,"\int \frac{\tan ^{\frac{8}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(8/3)/(a + I*a*Tan[c + d*x])^2,x]","-\frac{i \tan ^{\frac{2}{3}}(c+d x) \sec ^2(c+d x) \left(9 \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)-82 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+4 (11 i \sin (2 (c+d x))+8 \cos (2 (c+d x))+8)\right)}{96 a^2 d (\tan (c+d x)-i)^2}","\frac{25 \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}-\frac{25 \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}-\frac{25 \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}+\frac{2 i \tan ^{\frac{2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}+\frac{2 i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}-\frac{2 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}-\frac{25 \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}+\frac{25 \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}+\frac{i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}-\frac{\tan ^{\frac{5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"((-1/96*I)*Sec[c + d*x]^2*(9*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] - 82*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + 4*(8 + 8*Cos[2*(c + d*x)] + (11*I)*Sin[2*(c + d*x)]))*Tan[c + d*x]^(2/3))/(a^2*d*(-I + Tan[c + d*x])^2)","C",1
243,1,190,335,1.5528906,"\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^2,x]","\frac{\sqrt[3]{\tan (c+d x)} \sec ^2(c+d x) \left(9\ 2^{2/3} e^{2 i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)-2 \left(7 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+4 i \sin (2 (c+d x))+\cos (2 (c+d x))+1\right)\right)}{48 a^2 d (\tan (c+d x)-i)^2}","\frac{\tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}-\frac{\tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}-\frac{\tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}+\frac{i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}+\frac{\sqrt[3]{\tan (c+d x)}}{3 a^2 d (1+i \tan (c+d x))}+\frac{i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}+\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{18 a^2 d}-\frac{\sqrt[3]{\tan (c+d x)}}{4 d (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]^2*(9*2^(2/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2] - 2*(1 + Cos[2*(c + d*x)] + 7*Hypergeometric2F1[1/3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (4*I)*Sin[2*(c + d*x)]))*Tan[c + d*x]^(1/3))/(48*a^2*d*(-I + Tan[c + d*x])^2)","C",1
244,1,194,337,1.4536183,"\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x])^2,x]","\frac{i \tan ^{\frac{2}{3}}(c+d x) \sec ^2(c+d x) \left(9 \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+14 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-4 (i \sin (2 (c+d x))+4 \cos (2 (c+d x))+4)\right)}{96 a^2 d (\tan (c+d x)-i)^2}","-\frac{\tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}+\frac{\tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}+\frac{\tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}+\frac{i \tan ^{\frac{2}{3}}(c+d x)}{3 a^2 d (1+i \tan (c+d x))}+\frac{i \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}-\frac{i \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}+\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{\log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}+\frac{i \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{18 a^2 d}+\frac{\tan ^{\frac{5}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"((I/96)*Sec[c + d*x]^2*(9*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 14*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - 4*(4 + 4*Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))*Tan[c + d*x]^(2/3))/(a^2*d*(-I + Tan[c + d*x])^2)","C",1
245,1,189,339,1.5309074,"\int \frac{1}{\sqrt[3]{\tan (c+d x)} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^2),x]","-\frac{\tan ^{\frac{2}{3}}(c+d x) \sec ^2(c+d x) \left(9 \sqrt[3]{2} e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+46 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+28 i \sin (2 (c+d x))+40 \cos (2 (c+d x))+40\right)}{96 a^2 d (\tan (c+d x)-i)^2}","\frac{7 i \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}-\frac{7 i \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}-\frac{7 i \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}+\frac{7 \tan ^{\frac{2}{3}}(c+d x)}{12 a^2 d (1+i \tan (c+d x))}-\frac{2 \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}+\frac{2 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}-\frac{7 i \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}+\frac{7 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{\log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}+\frac{\tan ^{\frac{2}{3}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}",1,"-1/96*(Sec[c + d*x]^2*(40 + 40*Cos[2*(c + d*x)] + 9*2^(1/3)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 46*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (28*I)*Sin[2*(c + d*x)])*Tan[c + d*x]^(2/3))/(a^2*d*(-I + Tan[c + d*x])^2)","C",1
246,1,205,359,3.72792,"\int \frac{1}{\tan ^{\frac{5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Tan[c + d*x]^(5/3)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\sqrt[3]{\tan (c+d x)} \sec (c+d x) \left(-\frac{36 i 2^{2/3} e^{3 i (c+d x)} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}+476 i \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x)) \sec (c+d x)+2 i \sin (c+d x))+4 \csc (c+d x) (53 i \sin (2 (c+d x))+50 \cos (2 (c+d x))-14)\right)}{96 a^2 d (\tan (c+d x)-i)^2}","\frac{55 i \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}-\frac{55 i \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}-\frac{55 i \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}-\frac{8}{3 a^2 d \tan ^{\frac{2}{3}}(c+d x)}+\frac{11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac{2}{3}}(c+d x)}+\frac{8 \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{3 \sqrt{3} a^2 d}+\frac{8 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}+\frac{55 i \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{55 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{4 \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{9 a^2 d}+\frac{1}{4 d \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^2}",1,"(Sec[c + d*x]*(((-36*I)*2^(2/3)*E^((3*I)*(c + d*x))*Hypergeometric2F1[1/3, 1/3, 4/3, (1 - E^((2*I)*(c + d*x)))/2])/(1 + E^((2*I)*(c + d*x)))^(2/3) + (476*I)*Hypergeometric2F1[1/3, 1, 4/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))]*(Cos[2*(c + d*x)]*Sec[c + d*x] + (2*I)*Sin[c + d*x]) + 4*Csc[c + d*x]*(-14 + 50*Cos[2*(c + d*x)] + (53*I)*Sin[2*(c + d*x)]))*Tan[c + d*x]^(1/3))/(96*a^2*d*(-I + Tan[c + d*x])^2)","C",1
247,1,281,381,1.9408078,"\int \frac{1}{\tan ^{\frac{7}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Tan[c + d*x]^(7/3)*(a + I*a*Tan[c + d*x])^2),x]","\frac{e^{-2 i (c+2 d x)} \left(9 \sqrt[3]{2} e^{4 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \left(-1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right)+382 e^{4 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+76 e^{2 i (c+d x)}-736 e^{4 i (c+d x)}-220 e^{6 i (c+d x)}+586 e^{8 i (c+d x)}+6\right) \tan ^{\frac{2}{3}}(c+d x) \sec ^2(c+d x) (\cos (d x)+i \sin (d x))^2}{96 a^2 d \left(-1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i)^2}","-\frac{91 i \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{72 a^2 d}+\frac{91 i \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{72 a^2 d}+\frac{91 i \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{36 a^2 d}-\frac{25}{12 a^2 d \tan ^{\frac{4}{3}}(c+d x)}+\frac{13}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac{4}{3}}(c+d x)}+\frac{25 \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{6 \sqrt{3} a^2 d}+\frac{91 i}{12 a^2 d \sqrt[3]{\tan (c+d x)}}-\frac{25 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{18 a^2 d}+\frac{91 i \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}-\frac{91 i \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{48 \sqrt{3} a^2 d}+\frac{25 \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{36 a^2 d}+\frac{1}{4 d \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))^2}",1,"((6 + 76*E^((2*I)*(c + d*x)) - 736*E^((4*I)*(c + d*x)) - 220*E^((6*I)*(c + d*x)) + 586*E^((8*I)*(c + d*x)) + 9*2^(1/3)*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^2*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (1 - E^((2*I)*(c + d*x)))/2] + 382*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^2*Hypergeometric2F1[2/3, 1, 5/3, -((-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))])*Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^2*Tan[c + d*x]^(2/3))/(96*a^2*d*E^((2*I)*(c + 2*d*x))*(-1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])^2)","C",1
248,0,0,82,5.7000561,"\int \tan ^{\frac{4}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(4/3)*Sqrt[a + I*a*Tan[c + d*x]],x]","\int \tan ^{\frac{4}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","\frac{3 a \sqrt{1+i \tan (c+d x)} \tan ^{\frac{7}{3}}(c+d x) F_1\left(\frac{7}{3};\frac{1}{2},1;\frac{10}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{7 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(4/3)*Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
249,0,0,82,16.6759715,"\int \tan ^{\frac{2}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(2/3)*Sqrt[a + I*a*Tan[c + d*x]],x]","\int \tan ^{\frac{2}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","\frac{3 a \sqrt{1+i \tan (c+d x)} \tan ^{\frac{5}{3}}(c+d x) F_1\left(\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{5 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(2/3)*Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
250,0,0,82,1.0410043,"\int \sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]],x]","\int \sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \, dx","\frac{3 a \sqrt{1+i \tan (c+d x)} \tan ^{\frac{4}{3}}(c+d x) F_1\left(\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{4 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
251,0,0,82,1.023706,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt[3]{\tan (c+d x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(1/3),x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt[3]{\tan (c+d x)}} \, dx","\frac{3 a \sqrt{1+i \tan (c+d x)} \tan ^{\frac{2}{3}}(c+d x) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{2 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(1/3), x]","F",-1
252,0,0,80,0.990326,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{2}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(2/3),x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{2}{3}}(c+d x)} \, dx","\frac{3 a \sqrt{1+i \tan (c+d x)} \sqrt[3]{\tan (c+d x)} F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(2/3), x]","F",-1
253,0,0,80,12.3930979,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{4}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(4/3),x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\tan ^{\frac{4}{3}}(c+d x)} \, dx","-\frac{3 a \sqrt{1+i \tan (c+d x)} F_1\left(-\frac{1}{3};\frac{1}{2},1;\frac{2}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(4/3), x]","F",-1
254,0,0,82,8.0997134,"\int \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x])^(3/2),x]","\int \tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","\frac{3 a \tan ^{\frac{7}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} F_1\left(\frac{7}{3};-\frac{1}{2},1;\frac{10}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{7 d \sqrt{1+i \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
255,0,0,82,14.2339177,"\int \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2),x]","\int \tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","\frac{3 a \tan ^{\frac{5}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} F_1\left(\frac{5}{3};-\frac{1}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{5 d \sqrt{1+i \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
256,0,0,82,7.5381328,"\int \sqrt[3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^(3/2),x]","\int \sqrt[3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2} \, dx","\frac{3 a \tan ^{\frac{4}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} F_1\left(\frac{4}{3};-\frac{1}{2},1;\frac{7}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{4 d \sqrt{1+i \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
257,0,0,82,16.8817651,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\sqrt[3]{\tan (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(1/3),x]","\int \frac{(a+i a \tan (c+d x))^{3/2}}{\sqrt[3]{\tan (c+d x)}} \, dx","\frac{3 a \tan ^{\frac{2}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)} F_1\left(\frac{2}{3};-\frac{1}{2},1;\frac{5}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{2 d \sqrt{1+i \tan (c+d x)}}",1,"Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(1/3), x]","F",-1
258,0,0,80,3.53866,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{2}{3}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(2/3),x]","\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{2}{3}}(c+d x)} \, dx","\frac{3 a \sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)} F_1\left(\frac{1}{3};-\frac{1}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{1+i \tan (c+d x)}}",1,"Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(2/3), x]","F",-1
259,0,0,80,14.8234245,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{4}{3}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(4/3),x]","\int \frac{(a+i a \tan (c+d x))^{3/2}}{\tan ^{\frac{4}{3}}(c+d x)} \, dx","-\frac{3 a \sqrt{a+i a \tan (c+d x)} F_1\left(-\frac{1}{3};-\frac{1}{2},1;\frac{2}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{1+i \tan (c+d x)} \sqrt[3]{\tan (c+d x)}}",1,"Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(4/3), x]","F",-1
260,0,0,81,6.1990905,"\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(4/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{7}{3}}(c+d x) F_1\left(\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{7 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(4/3)/Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
261,0,0,81,2.615194,"\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(2/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{5}{3}}(c+d x) F_1\left(\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{5 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(2/3)/Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
262,0,0,81,5.6137867,"\int \frac{\sqrt[3]{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(1/3)/Sqrt[a + I*a*Tan[c + d*x]],x]","\int \frac{\sqrt[3]{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{4}{3}}(c+d x) F_1\left(\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{4 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(1/3)/Sqrt[a + I*a*Tan[c + d*x]], x]","F",-1
263,0,0,81,3.5048491,"\int \frac{1}{\sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\int \frac{1}{\sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{2}{3}}(c+d x) F_1\left(\frac{2}{3};\frac{3}{2},1;\frac{5}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{2 d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[1/(Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]), x]","F",-1
264,0,0,79,6.4160969,"\int \frac{1}{\tan ^{\frac{2}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(2/3)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\int \frac{1}{\tan ^{\frac{2}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \sqrt[3]{\tan (c+d x)} F_1\left(\frac{1}{3};\frac{3}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[1/(Tan[c + d*x]^(2/3)*Sqrt[a + I*a*Tan[c + d*x]]), x]","F",-1
265,0,0,79,11.8127359,"\int \frac{1}{\tan ^{\frac{4}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(4/3)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\int \frac{1}{\tan ^{\frac{4}{3}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","-\frac{3 \sqrt{1+i \tan (c+d x)} F_1\left(-\frac{1}{3};\frac{3}{2},1;\frac{2}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[1/(Tan[c + d*x]^(4/3)*Sqrt[a + I*a*Tan[c + d*x]]), x]","F",-1
266,0,0,84,6.7785443,"\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^(3/2),x]","\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{7}{3}}(c+d x) F_1\left(\frac{7}{3};\frac{5}{2},1;\frac{10}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{7 a d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
267,0,0,84,12.4753985,"\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x])^(3/2),x]","\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{5}{3}}(c+d x) F_1\left(\frac{5}{3};\frac{5}{2},1;\frac{8}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{5 a d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
268,0,0,84,6.4705248,"\int \frac{\sqrt[3]{\tan (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(1/3)/(a + I*a*Tan[c + d*x])^(3/2),x]","\int \frac{\sqrt[3]{\tan (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{4}{3}}(c+d x) F_1\left(\frac{4}{3};\frac{5}{2},1;\frac{7}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{4 a d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^(1/3)/(a + I*a*Tan[c + d*x])^(3/2), x]","F",-1
269,0,0,84,11.7769416,"\int \frac{1}{\sqrt[3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\int \frac{1}{\sqrt[3]{\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \tan ^{\frac{2}{3}}(c+d x) F_1\left(\frac{2}{3};\frac{5}{2},1;\frac{5}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{2 a d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^(3/2)), x]","F",-1
270,0,0,82,7.3101992,"\int \frac{1}{\tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\int \frac{1}{\tan ^{\frac{2}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","\frac{3 \sqrt{1+i \tan (c+d x)} \sqrt[3]{\tan (c+d x)} F_1\left(\frac{1}{3};\frac{5}{2},1;\frac{4}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{a d \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[1/(Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2)), x]","F",-1
271,0,0,82,12.5197095,"\int \frac{1}{\tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\int \frac{1}{\tan ^{\frac{4}{3}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","-\frac{3 \sqrt{1+i \tan (c+d x)} F_1\left(-\frac{1}{3};\frac{5}{2},1;\frac{2}{3};-i \tan (c+d x),i \tan (c+d x)\right)}{a d \sqrt[3]{\tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"Integrate[1/(Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x])^(3/2)), x]","F",-1
272,-1,0,234,180.0038781,"\int \tan ^3(c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(1/3),x]","\text{\$Aborted}","\frac{\sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}+\frac{3 \tan ^2(c+d x) \sqrt[3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac{18 \sqrt[3]{a+i a \tan (c+d x)}}{7 d}-\frac{3 \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}-\frac{\sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{i \sqrt[3]{a} x}{2\ 2^{2/3}}",1,"$Aborted","F",-1
273,-1,0,185,180.0032944,"\int \tan ^2(c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3),x]","\text{\$Aborted}","\frac{i \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}-\frac{3 i (a+i a \tan (c+d x))^{4/3}}{4 a d}-\frac{3 i \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}-\frac{i \sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{\sqrt[3]{a} x}{2\ 2^{2/3}}",1,"$Aborted","F",-1
274,-1,0,174,180.0059966,"\int \tan (c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3),x]","\text{\$Aborted}","-\frac{\sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}+\frac{3 \sqrt[3]{a+i a \tan (c+d x)}}{d}+\frac{3 \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}+\frac{\sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac{i \sqrt[3]{a} x}{2\ 2^{2/3}}",1,"$Aborted","F",-1
275,1,223,156,0.9484651,"\int \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(1/3),x]","-\frac{i e^{-\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \sqrt[3]{a+i a \tan (c+d x)} \left(-2 \log \left(1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)+\log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)\right)}{4 d}","-\frac{i \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}+\frac{3 i \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}+\frac{i \sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\sqrt[3]{a} x}{2\ 2^{2/3}}",1,"((-1/4*I)*(1 + E^((2*I)*(c + d*x)))^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 2*Log[1 - E^(((2*I)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] + Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)])*(a + I*a*Tan[c + d*x])^(1/3))/(d*E^(((2*I)/3)*(c + d*x)))","A",1
276,0,0,260,0.6554259,"\int \cot (c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3),x]","\int \cot (c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","-\frac{\sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}+\frac{\sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}-\frac{\sqrt[3]{a} \log (\tan (c+d x))}{2 d}+\frac{3 \sqrt[3]{a} \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 d}-\frac{3 \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}-\frac{\sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{i \sqrt[3]{a} x}{2\ 2^{2/3}}",1,"Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3), x]","F",-1
277,-1,0,299,180.0070372,"\int \cot ^2(c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3),x]","\text{\$Aborted}","-\frac{i \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} d}+\frac{i \sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}-\frac{i \sqrt[3]{a} \log (\tan (c+d x))}{6 d}+\frac{i \sqrt[3]{a} \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 d}-\frac{3 i \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}-\frac{i \sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\cot (c+d x) \sqrt[3]{a+i a \tan (c+d x)}}{d}+\frac{\sqrt[3]{a} x}{2\ 2^{2/3}}",1,"$Aborted","F",-1
278,-1,0,327,180.0032477,"\int \cot ^3(c+d x) \sqrt[3]{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(1/3),x]","\text{\$Aborted}","\frac{8 \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} d}-\frac{\sqrt{3} \sqrt[3]{a} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2^{2/3} d}+\frac{4 \sqrt[3]{a} \log (\tan (c+d x))}{9 d}-\frac{4 \sqrt[3]{a} \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{3 d}+\frac{3 \sqrt[3]{a} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2\ 2^{2/3} d}+\frac{\sqrt[3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac{\cot ^2(c+d x) \sqrt[3]{a+i a \tan (c+d x)}}{2 d}-\frac{i \cot (c+d x) \sqrt[3]{a+i a \tan (c+d x)}}{6 d}+\frac{i \sqrt[3]{a} x}{2\ 2^{2/3}}",1,"$Aborted","F",-1
279,1,81,156,0.3894846,"\int (a+i a \tan (c+d x))^{2/3} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(2/3),x]","-\frac{3 i \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)}{2 \sqrt[3]{2} d}","\frac{i \sqrt{3} a^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{2} d}+\frac{3 i a^{2/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{2} d}+\frac{i a^{2/3} \log (\cos (c+d x))}{2 \sqrt[3]{2} d}-\frac{a^{2/3} x}{2 \sqrt[3]{2}}",1,"(((-3*I)/2)*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))])/(2^(1/3)*d)","C",1
280,-1,0,251,180.0042315,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(4/3),x]","\text{\$Aborted}","\frac{\sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}-\frac{3 a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{i a^{4/3} x}{2^{2/3}}+\frac{3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac{6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac{9 (a+i a \tan (c+d x))^{4/3}}{20 d}-\frac{3 a \sqrt[3]{a+i a \tan (c+d x)}}{d}",1,"$Aborted","F",-1
281,-1,0,203,180.0051335,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3),x]","\text{\$Aborted}","\frac{i \sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}-\frac{3 i a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}-\frac{3 i (a+i a \tan (c+d x))^{7/3}}{7 a d}-\frac{3 i a \sqrt[3]{a+i a \tan (c+d x)}}{d}",1,"$Aborted","F",-1
282,-1,0,192,180.0054734,"\int \tan (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3),x]","\text{\$Aborted}","-\frac{\sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}+\frac{3 a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}+\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{i a^{4/3} x}{2^{2/3}}+\frac{3 a \sqrt[3]{a+i a \tan (c+d x)}}{d}+\frac{3 (a+i a \tan (c+d x))^{4/3}}{4 d}",1,"$Aborted","F",-1
283,1,294,175,1.241975,"\int (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{i a e^{\frac{1}{3} i (c+d x)} \cos (c+d x) \sqrt[3]{a+i a \tan (c+d x)} \left(6 e^{\frac{2}{3} i (c+d x)}+2 \sqrt[3]{1+e^{2 i (c+d x)}} \log \left(1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)-\sqrt[3]{1+e^{2 i (c+d x)}} \log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)-2 \sqrt{3} \sqrt[3]{1+e^{2 i (c+d x)}} \tan ^{-1}\left(\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)\right)}{d \left(1+e^{2 i (c+d x)}\right)}","-\frac{i \sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}+\frac{3 i a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}+\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{a^{4/3} x}{2^{2/3}}+\frac{3 i a \sqrt[3]{a+i a \tan (c+d x)}}{d}",1,"(I*a*E^((I/3)*(c + d*x))*Cos[c + d*x]*(6*E^(((2*I)/3)*(c + d*x)) - 2*Sqrt[3]*(1 + E^((2*I)*(c + d*x)))^(1/3)*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] + 2*(1 + E^((2*I)*(c + d*x)))^(1/3)*Log[1 - E^(((2*I)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] - (1 + E^((2*I)*(c + d*x)))^(1/3)*Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)])*(a + I*a*Tan[c + d*x])^(1/3))/(d*(1 + E^((2*I)*(c + d*x))))","A",1
284,1,411,254,4.4212584,"\int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{a e^{-\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \sqrt[3]{a+i a \tan (c+d x)} \left(-4 \log \left(1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)+2\ 2^{2/3} \log \left(1-\frac{\sqrt[3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)-2^{2/3} \log \left(\frac{\sqrt[3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}+\frac{2^{2/3} e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}+1\right)+2 \log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)+4 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)-2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)\right)}{4 d}","-\frac{\sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}+\frac{\sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}-\frac{a^{4/3} \log (\tan (c+d x))}{2 d}+\frac{3 a^{4/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 d}-\frac{3 a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}-\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac{i a^{4/3} x}{2^{2/3}}",1,"(a*(1 + E^((2*I)*(c + d*x)))^(1/3)*(4*Sqrt[3]*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 4*Log[1 - E^(((2*I)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] + 2*2^(2/3)*Log[1 - (2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3)] - 2^(2/3)*Log[1 + (2^(2/3)*E^(((4*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(2/3) + (2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3)] + 2*Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)])*(a + I*a*Tan[c + d*x])^(1/3))/(4*d*E^(((2*I)/3)*(c + d*x)))","A",1
285,1,587,315,7.7214623,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{i \sqrt[3]{e^{i d x}} e^{-\frac{1}{3} i (5 c+2 d x)} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt[3]{1+e^{2 i (c+d x)}} (a+i a \tan (c+d x))^{4/3} \left(-6 \log \left(1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)+4\ 2^{2/3} \log \left(1-\frac{\sqrt[3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)+3 \log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)-2\ 2^{2/3} \log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+\sqrt[3]{2} e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+2^{2/3} e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)+6 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)-4\ 2^{2/3} \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{2} e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)\right)}{3\ 2^{2/3} d \sec ^{\frac{4}{3}}(c+d x) (\cos (d x)+i \sin (d x))^{4/3}}+\frac{\cos (c+d x) (a+i a \tan (c+d x))^{4/3} (\csc (c) (\cos (c)-i \sin (c)) \sin (d x) \csc (c+d x)+\cot (c) (-\cos (c)+i \sin (c)))}{d (\cos (d x)+i \sin (d x))}","-\frac{4 i a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} d}+\frac{i \sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}-\frac{2 i a^{4/3} \log (\tan (c+d x))}{3 d}+\frac{2 i a^{4/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{d}-\frac{3 i a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}-\frac{i a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{a^{4/3} x}{2^{2/3}}+\frac{i a \sqrt[3]{a+i a \tan (c+d x)}}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^{4/3}}{d}",1,"((I/3)*(E^(I*d*x))^(1/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*(6*Sqrt[3]*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 4*2^(2/3)*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] - 6*Log[1 - E^(((2*I)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] + 4*2^(2/3)*Log[1 - (2^(1/3)*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3)] + 3*Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)] - 2*2^(2/3)*Log[(2^(2/3)*E^(((4*I)/3)*(c + d*x)) + 2^(1/3)*E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)])*(a + I*a*Tan[c + d*x])^(4/3))/(2^(2/3)*d*E^((I/3)*(5*c + 2*d*x))*Sec[c + d*x]^(4/3)*(Cos[d*x] + I*Sin[d*x])^(4/3)) + (Cos[c + d*x]*(Cot[c]*(-Cos[c] + I*Sin[c]) + Csc[c]*Csc[c + d*x]*(Cos[c] - I*Sin[c])*Sin[d*x])*(a + I*a*Tan[c + d*x])^(4/3))/(d*(Cos[d*x] + I*Sin[d*x]))","A",1
286,-1,0,321,180.0044261,"\int \cot ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx","Integrate[Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(4/3),x]","\text{\$Aborted}","\frac{11 a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} d}-\frac{\sqrt[3]{2} \sqrt{3} a^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}+\frac{11 a^{4/3} \log (\tan (c+d x))}{18 d}-\frac{11 a^{4/3} \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{6 d}+\frac{3 a^{4/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2^{2/3} d}+\frac{a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}+\frac{i a^{4/3} x}{2^{2/3}}-\frac{\cot ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{2 d}-\frac{2 i a \cot (c+d x) \sqrt[3]{a+i a \tan (c+d x)}}{3 d}",1,"$Aborted","F",-1
287,1,82,177,0.6893973,"\int (a+i a \tan (c+d x))^{5/3} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/3),x]","-\frac{3 i a \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(-1+\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt[3]{2} d}","\frac{i 2^{2/3} \sqrt{3} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d}+\frac{3 i a^{5/3} \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{\sqrt[3]{2} d}+\frac{i a^{5/3} \log (\cos (c+d x))}{\sqrt[3]{2} d}-\frac{a^{5/3} x}{\sqrt[3]{2}}+\frac{3 i a (a+i a \tan (c+d x))^{2/3}}{2 d}",1,"((-3*I)*a*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(-1 + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]))/(2^(1/3)*d)","C",1
288,-1,0,83,180.0010965,"\int \frac{\tan ^m(c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(1/3),x]","\text{\$Aborted}","\frac{\sqrt[3]{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{4}{3},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{d (m+1) \sqrt[3]{a+i a \tan (c+d x)}}",1,"$Aborted","F",-1
289,0,0,81,5.1713632,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + I*a*Tan[c + d*x])^(1/3),x]","\int \frac{\sqrt{\tan (c+d x)}}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","\frac{2 \sqrt[3]{1+i \tan (c+d x)} \tan ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{3 d \sqrt[3]{a+i a \tan (c+d x)}}",1,"Integrate[Sqrt[Tan[c + d*x]]/(a + I*a*Tan[c + d*x])^(1/3), x]","F",-1
290,1,125,282,1.364097,"\int \frac{\tan ^4(c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(1/3),x]","\frac{15 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\tan (c+d x)-i)+3 i \sec ^3(c+d x) (2 i \sin (c+d x)+7 i \sin (3 (c+d x))+37 \cos (c+d x)+12 \cos (3 (c+d x)))}{40 d \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{39 i (a+i a \tan (c+d x))^{5/3}}{20 a^2 d}+\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{3 \tan ^3(c+d x)}{8 d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{15 i \tan ^2(c+d x)}{8 d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{45 i (a+i a \tan (c+d x))^{2/3}}{8 a d}+\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"((3*I)*Sec[c + d*x]^3*(37*Cos[c + d*x] + 12*Cos[3*(c + d*x)] + (2*I)*Sin[c + d*x] + (7*I)*Sin[3*(c + d*x)]) + 15*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-I + Tan[c + d*x]))/(40*d*(a + I*a*Tan[c + d*x])^(1/3))","C",1
291,1,115,237,0.8854287,"\int \frac{\tan ^3(c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(1/3),x]","\frac{3 \sec ^2(c+d x) \left(5 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (i \sin (2 (c+d x))+\cos (2 (c+d x))+1)+4 i \sin (2 (c+d x))+24 \cos (2 (c+d x))+40\right)}{80 d \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{3 \tan ^2(c+d x)}{5 d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{3 (a+i a \tan (c+d x))^{2/3}}{10 a d}+\frac{21}{10 d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{\log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{i x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(3*Sec[c + d*x]^2*(40 + 24*Cos[2*(c + d*x)] + 5*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (4*I)*Sin[2*(c + d*x)]))/(80*d*(a + I*a*Tan[c + d*x])^(1/3))","C",1
292,1,82,213,0.7131297,"\int \frac{\tan ^2(c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(1/3),x]","-\frac{3 \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\tan (c+d x)-i)-4 \tan (c+d x)+8 i\right)}{8 d \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{3 i (a+i a \tan (c+d x))^{2/3}}{2 a d}-\frac{3 i}{2 d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{i \log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(-3*(8*I - 4*Tan[c + d*x] + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-I + Tan[c + d*x])))/(8*d*(a + I*a*Tan[c + d*x])^(1/3))","C",1
293,1,140,178,0.630149,"\int \frac{\tan (c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^(1/3),x]","-\frac{3 \left(e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)}{4 d \sqrt[3]{a+i a \tan (c+d x)} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{3}{2 d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{\log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{i x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(-3*(2*(1 + E^((2*I)*d*x))*Cos[c] + E^((2*I)*d*x)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[c] + I*Sin[c]) + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c]))/(4*d*((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
294,1,141,184,0.439594,"\int \frac{1}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-1/3),x]","\frac{3 \left(e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-2 i \sin (c) \left(-1+e^{2 i d x}\right)-2 \cos (c) \left(1+e^{2 i d x}\right)\right)}{4 d \sqrt[3]{a+i a \tan (c+d x)} \left(i \cos (c) \left(1+e^{2 i d x}\right)-\sin (c) \left(-1+e^{2 i d x}\right)\right)}","\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{3 i}{2 d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{i \log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(3*(-2*(1 + E^((2*I)*d*x))*Cos[c] + E^((2*I)*d*x)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[c] + I*Sin[c]) - (2*I)*(-1 + E^((2*I)*d*x))*Sin[c]))/(4*d*(I*(1 + E^((2*I)*d*x))*Cos[c] - (-1 + E^((2*I)*d*x))*Sin[c])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
295,1,195,286,1.2533119,"\int \frac{\cot (c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(1/3),x]","\frac{3 \left(e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)-4 e^{2 i d x} (\cos (c)+i \sin (c)) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)}{4 d \sqrt[3]{a+i a \tan (c+d x)} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{a} d}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}+\frac{3}{2 d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{\log (\tan (c+d x))}{2 \sqrt[3]{a} d}+\frac{3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{a} d}-\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{\log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{i x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(3*(2*(1 + E^((2*I)*d*x))*Cos[c] + E^((2*I)*d*x)*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[c] + I*Sin[c]) - 4*E^((2*I)*d*x)*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(Cos[c] + I*Sin[c]) + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c]))/(4*d*((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
296,1,179,327,1.449554,"\int \frac{\cot ^2(c+d x)}{\sqrt[3]{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(1/3),x]","\frac{\csc (c+d x) \sec (c+d x) \left(3 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (i \sin (2 (c+d x))+\cos (2 (c+d x))-1)+4 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (i \sin (2 (c+d x))+\cos (2 (c+d x))-1)-20 i \sin (2 (c+d x))-8 \cos (2 (c+d x))-8\right)}{16 d \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{i \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d}-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{5 i}{2 d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{i \log (\tan (c+d x))}{6 \sqrt[3]{a} d}-\frac{i \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 \sqrt[3]{a} d}-\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{i \log (\cos (c+d x))}{4 \sqrt[3]{2} \sqrt[3]{a} d}-\frac{\cot (c+d x)}{d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{x}{4 \sqrt[3]{2} \sqrt[3]{a}}",1,"(Csc[c + d*x]*Sec[c + d*x]*(-8 - 8*Cos[2*(c + d*x)] + 3*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + 4*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - (20*I)*Sin[2*(c + d*x)]))/(16*d*(a + I*a*Tan[c + d*x])^(1/3))","C",1
297,1,290,184,1.0759778,"\int \frac{1}{(a+i a \tan (c+d x))^{2/3}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-2/3),x]","\frac{i \left(3 \left(1+e^{2 i (c+d x)}\right)^{2/3}+2 e^{\frac{4}{3} i (c+d x)} \log \left(1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)-e^{\frac{4}{3} i (c+d x)} \log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)-2 \sqrt{3} e^{\frac{4}{3} i (c+d x)} \tan ^{-1}\left(\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)\right)}{4\ 2^{2/3} d \left(1+e^{2 i (c+d x)}\right)^{2/3} \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3}}","-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{2\ 2^{2/3} a^{2/3} d}+\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{4\ 2^{2/3} a^{2/3} d}+\frac{i \log (\cos (c+d x))}{4\ 2^{2/3} a^{2/3} d}-\frac{x}{4\ 2^{2/3} a^{2/3}}+\frac{3 i}{4 d (a+i a \tan (c+d x))^{2/3}}",1,"((I/4)*(3*(1 + E^((2*I)*(c + d*x)))^(2/3) - 2*Sqrt[3]*E^(((4*I)/3)*(c + d*x))*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] + 2*E^(((4*I)/3)*(c + d*x))*Log[1 - E^(((2*I)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] - E^(((4*I)/3)*(c + d*x))*Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)]))/(2^(2/3)*d*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3))","A",1
298,0,0,86,34.8897561,"\int \frac{\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(4/3),x]","\int \frac{\tan ^m(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","\frac{\sqrt[3]{1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left(m+1;\frac{7}{3},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right)}{a d (m+1) \sqrt[3]{a+i a \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(4/3), x]","F",-1
299,0,0,84,15.2931227,"\int \frac{\sqrt{\tan (c+d x)}}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + I*a*Tan[c + d*x])^(4/3),x]","\int \frac{\sqrt{\tan (c+d x)}}{(a+i a \tan (c+d x))^{4/3}} \, dx","\frac{2 \sqrt[3]{1+i \tan (c+d x)} \tan ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{2};\frac{7}{3},1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{3 a d \sqrt[3]{a+i a \tan (c+d x)}}",1,"Integrate[Sqrt[Tan[c + d*x]]/(a + I*a*Tan[c + d*x])^(4/3), x]","F",-1
300,1,145,282,1.5959962,"\int \frac{\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(4/3),x]","-\frac{3 \sec ^2(c+d x) \left(5 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+113 \cos (2 (c+d x))+75 i \tan (c+d x)+59 i \sin (3 (c+d x)) \sec (c+d x)+81\right)}{80 a d (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}+\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}+\frac{i \log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}-\frac{x}{8 \sqrt[3]{2} a^{4/3}}-\frac{87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}+\frac{3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac{39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}-\frac{51 i}{10 a d \sqrt[3]{a+i a \tan (c+d x)}}",1,"(-3*Sec[c + d*x]^2*(81 + 113*Cos[2*(c + d*x)] + 5*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (59*I)*Sec[c + d*x]*Sin[3*(c + d*x)] + (75*I)*Tan[c + d*x]))/(80*a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
301,1,130,237,0.983679,"\int \frac{\tan ^3(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{3 \sec ^2(c+d x) \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\sin (2 (c+d x))-i \cos (2 (c+d x)))-18 \sin (2 (c+d x))+17 i \cos (2 (c+d x))+9 i\right)}{16 a d (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}-\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}-\frac{\log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}-\frac{i x}{8 \sqrt[3]{2} a^{4/3}}+\frac{3 \tan ^2(c+d x)}{2 d (a+i a \tan (c+d x))^{4/3}}-\frac{27}{4 a d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{15}{8 d (a+i a \tan (c+d x))^{4/3}}",1,"(3*Sec[c + d*x]^2*(9*I + (17*I)*Cos[2*(c + d*x)] - 18*Sin[2*(c + d*x)] + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])))/(16*a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
302,1,128,213,0.7956041,"\int \frac{\tan ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{3 \sec ^2(c+d x) \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+6 i \sin (2 (c+d x))+5 \cos (2 (c+d x))+5\right)}{16 a d (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}-\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}-\frac{i \log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}+\frac{x}{8 \sqrt[3]{2} a^{4/3}}+\frac{9 i}{4 a d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{3 i}{8 d (a+i a \tan (c+d x))^{4/3}}",1,"(3*Sec[c + d*x]^2*(5 + 5*Cos[2*(c + d*x)] + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (6*I)*Sin[2*(c + d*x)]))/(16*a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
303,1,130,205,0.6618777,"\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Tan[c + d*x]/(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{3 i \sec ^2(c+d x) \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i \sin (2 (c+d x))-\cos (2 (c+d x))-1\right)}{16 a d (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}+\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}+\frac{\log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}+\frac{i x}{8 \sqrt[3]{2} a^{4/3}}+\frac{3}{4 a d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}",1,"(((3*I)/16)*Sec[c + d*x]^2*(-1 - Cos[2*(c + d*x)] + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - (2*I)*Sin[2*(c + d*x)]))/(a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
304,1,128,213,0.45846,"\int \frac{1}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-4/3),x]","-\frac{3 \sec ^2(c+d x) \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i \sin (2 (c+d x))-3 \cos (2 (c+d x))-3\right)}{16 a d (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}+\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}+\frac{i \log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}-\frac{x}{8 \sqrt[3]{2} a^{4/3}}+\frac{3 i}{4 a d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{3 i}{8 d (a+i a \tan (c+d x))^{4/3}}",1,"(-3*Sec[c + d*x]^2*(-3 - 3*Cos[2*(c + d*x)] + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - (2*I)*Sin[2*(c + d*x)]))/(16*a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
305,1,189,313,1.6254682,"\int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(4/3),x]","-\frac{3 i \sec ^2(c+d x) \left(\, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-8 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+6 i \sin (2 (c+d x))+7 \cos (2 (c+d x))+7\right)}{16 a d (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{4/3} d}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}-\frac{\log (\tan (c+d x))}{2 a^{4/3} d}+\frac{3 \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{2 a^{4/3} d}-\frac{3 \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}-\frac{\log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}-\frac{i x}{8 \sqrt[3]{2} a^{4/3}}+\frac{9}{4 a d \sqrt[3]{a+i a \tan (c+d x)}}+\frac{3}{8 d (a+i a \tan (c+d x))^{4/3}}",1,"(((-3*I)/16)*Sec[c + d*x]^2*(7 + 7*Cos[2*(c + d*x)] + Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) - 8*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]) + (6*I)*Sin[2*(c + d*x)]))/(a*d*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
306,1,233,354,2.0695359,"\int \frac{\cot ^2(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx","Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(4/3),x]","\frac{6 e^{4 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+64 e^{4 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{2 e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)+63 e^{2 i (c+d x)}-35 e^{4 i (c+d x)}-95 e^{6 i (c+d x)}+3}{8 a d \left(-1+e^{2 i (c+d x)}\right) \left(1+e^{2 i (c+d x)}\right)^2 (\tan (c+d x)-i) \sqrt[3]{a+i a \tan (c+d x)}}","-\frac{4 i \tan ^{-1}\left(\frac{\sqrt[3]{a}+2 \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{4/3} d}-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4 \sqrt[3]{2} a^{4/3} d}+\frac{2 i \log (\tan (c+d x))}{3 a^{4/3} d}-\frac{2 i \log \left(\sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{a^{4/3} d}-\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8 \sqrt[3]{2} a^{4/3} d}-\frac{i \log (\cos (c+d x))}{8 \sqrt[3]{2} a^{4/3} d}+\frac{x}{8 \sqrt[3]{2} a^{4/3}}-\frac{19 i}{4 a d \sqrt[3]{a+i a \tan (c+d x)}}-\frac{11 i}{8 d (a+i a \tan (c+d x))^{4/3}}-\frac{\cot (c+d x)}{d (a+i a \tan (c+d x))^{4/3}}",1,"(3 + 63*E^((2*I)*(c + d*x)) - 35*E^((4*I)*(c + d*x)) - 95*E^((6*I)*(c + d*x)) + 6*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[2/3, 1, 5/3, E^((2*I)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))] + 64*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[2/3, 1, 5/3, (2*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))])/(8*a*d*(-1 + E^((2*I)*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^2*(-I + Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(1/3))","C",1
307,1,331,213,1.5137609,"\int \frac{1}{(a+i a \tan (c+d x))^{5/3}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(-5/3),x]","\frac{i e^{-2 i (c+d x)} \sec ^2(c+d x) \left(27 e^{2 i (c+d x)}+21 e^{4 i (c+d x)}+10 e^{\frac{10}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \log \left(1-\frac{e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}\right)-5 e^{\frac{10}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \log \left(\frac{\left(1+e^{2 i (c+d x)}\right)^{2/3}+e^{\frac{2}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}}+e^{\frac{4}{3} i (c+d x)}}{\left(1+e^{2 i (c+d x)}\right)^{2/3}}\right)-10 \sqrt{3} e^{\frac{10}{3} i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \tan ^{-1}\left(\frac{1+\frac{2 e^{\frac{2}{3} i (c+d x)}}{\sqrt[3]{1+e^{2 i (c+d x)}}}}{\sqrt{3}}\right)+6\right)}{80 d (a+i a \tan (c+d x))^{5/3}}","-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}+2^{2/3} \sqrt[3]{a+i a \tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{4\ 2^{2/3} a^{5/3} d}+\frac{3 i \log \left(\sqrt[3]{2} \sqrt[3]{a}-\sqrt[3]{a+i a \tan (c+d x)}\right)}{8\ 2^{2/3} a^{5/3} d}+\frac{i \log (\cos (c+d x))}{8\ 2^{2/3} a^{5/3} d}-\frac{x}{8\ 2^{2/3} a^{5/3}}+\frac{3 i}{8 a d (a+i a \tan (c+d x))^{2/3}}+\frac{3 i}{10 d (a+i a \tan (c+d x))^{5/3}}",1,"((I/80)*(6 + 27*E^((2*I)*(c + d*x)) + 21*E^((4*I)*(c + d*x)) - 10*Sqrt[3]*E^(((10*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*ArcTan[(1 + (2*E^(((2*I)/3)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))^(1/3))/Sqrt[3]] + 10*E^(((10*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Log[1 - E^(((2*I)/3)*(c + d*x))/(1 + E^((2*I)*(c + d*x)))^(1/3)] - 5*E^(((10*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Log[(E^(((4*I)/3)*(c + d*x)) + E^(((2*I)/3)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3) + (1 + E^((2*I)*(c + d*x)))^(2/3))/(1 + E^((2*I)*(c + d*x)))^(2/3)])*Sec[c + d*x]^2)/(d*E^((2*I)*(c + d*x))*(a + I*a*Tan[c + d*x])^(5/3))","A",1
308,1,159,43,1.0276563,"\int (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx","Integrate[(e*Tan[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]","\frac{a e^{-i c} 2^{-m-1} \left(-\frac{i \left(-1+e^{2 i (c+d x)}\right)}{1+e^{2 i (c+d x)}}\right)^{m+1} \left(1+e^{2 i (c+d x)}\right)^{m+1} \cos (c+d x) (1+i \tan (c+d x)) \, _2F_1\left(m+1,m+1;m+2;\frac{1}{2} \left(1-e^{2 i (c+d x)}\right)\right) \tan ^{-m}(c+d x) (e \tan (c+d x))^m}{d (m+1) (\cos (d x)+i \sin (d x))}","\frac{a (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d e (m+1)}",1,"(2^(-1 - m)*a*(((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x))))^(1 + m)*(1 + E^((2*I)*(c + d*x)))^(1 + m)*Cos[c + d*x]*Hypergeometric2F1[1 + m, 1 + m, 2 + m, (1 - E^((2*I)*(c + d*x)))/2]*(1 + I*Tan[c + d*x])*(e*Tan[c + d*x])^m)/(d*E^(I*c)*(1 + m)*(Cos[d*x] + I*Sin[d*x])*Tan[c + d*x]^m)","B",1
309,1,44,43,0.0674135,"\int (e \tan (c+d x))^m (a-i a \tan (c+d x)) \, dx","Integrate[(e*Tan[c + d*x])^m*(a - I*a*Tan[c + d*x]),x]","\frac{a \tan (c+d x) (e \tan (c+d x))^m \, _2F_1(1,m+1;m+2;-i \tan (c+d x))}{d (m+1)}","\frac{a (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;-i \tan (c+d x))}{d e (m+1)}",1,"(a*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*Tan[c + d*x]]*Tan[c + d*x]*(e*Tan[c + d*x])^m)/(d*(1 + m))","A",1
310,1,1065,189,9.9449454,"\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^4 \, dx","Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4,x]","\frac{i 2^{3-n} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^n \cos ^4(e+f x) \left(2^n \, _2F_1\left(1,n;n+1;-\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)-\left(1+e^{2 i (e+f x)}\right)^n \, _2F_1\left(n,n;n+1;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^4 \tan ^{-n}(e+f x)}{\left(e^{2 i e}+e^{4 i e}\right) f n (\cos (f x)+i \sin (f x))^4}-\frac{8 i e^{-4 i e} \left(-1+e^{2 i (e+f x)}\right)^n \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^n \left(\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-n} \cos ^4(e+f x) \left(-\frac{\left(1+e^{2 i e}\right) \left(-1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,n+1;n+2;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{-n-1}}{n+1}-\frac{\, _2F_1\left(1,n;n+1;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{-n}}{n}+\frac{2^{-n} \, _2F_1\left(n,n;n+1;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)}{n}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^4 \tan ^{-n}(e+f x)}{\left(1+e^{2 i e}\right) f (\cos (f x)+i \sin (f x))^4}+\frac{\cos ^4(e+f x) \left(\frac{(\cos (2 e)+2 i \sin (2 e)-1) (2 i \cos (4 e)+2 \sin (4 e)) \sec ^2(e)}{n+1}+\frac{\sec (e+f x) (2 i \cos (4 e)+2 \sin (4 e)) (-\cos (e-f x)+\cos (e+f x)-2 i \sin (e-f x)+2 i \sin (e+f x)) \sec ^2(e)}{n+1}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^4}{f (\cos (f x)+i \sin (f x))^4}+\frac{\cos ^4(e+f x) \left(\frac{\sec (e) (\cos (4 e)-i \sin (4 e)) \sin (f x) \sec ^3(e+f x)}{n+3}+\frac{(\cos (4 e)-i \sin (4 e)) \tan (e) \sec ^2(e+f x)}{n+3}+\frac{\sec (e) (2 \cos (4 e)-2 i \sin (4 e)) \sin (f x) \sec (e+f x)}{(n+1) (n+3)}+\frac{(2 \cos (4 e)-2 i \sin (4 e)) \tan (e)}{(n+1) (n+3)}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^4}{f (\cos (f x)+i \sin (f x))^4}+\frac{\cos ^4(e+f x) \left(\frac{(-2 n+\cos (2 e)-3) (-2 i \cos (4 e)-2 \sin (4 e)) \sec ^2(e)}{(n+1) (n+2)}+\frac{(\cos (e+f x)-\cos (e-f x)) \sec (e+f x) (-2 i \cos (4 e)-2 \sin (4 e)) \sec ^2(e)}{n+1}+\frac{\sec ^2(e+f x) (-4 i \cos (4 e)-4 \sin (4 e))}{n+2}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^4}{f (\cos (f x)+i \sin (f x))^4}","\frac{8 a^4 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)}-\frac{2 a^4 \left(2 n^2+11 n+16\right) (d \tan (e+f x))^{n+1}}{d f (n+1) (n+2) (n+3)}-\frac{2 (n+4) \left(a^4+i a^4 \tan (e+f x)\right) (d \tan (e+f x))^{n+1}}{d f (n+2) (n+3)}-\frac{\left(a^2+i a^2 \tan (e+f x)\right)^2 (d \tan (e+f x))^{n+1}}{d f (n+3)}",1,"(Cos[e + f*x]^4*((Sec[e + f*x]^2*((-4*I)*Cos[4*e] - 4*Sin[4*e]))/(2 + n) + ((-3 - 2*n + Cos[2*e])*Sec[e]^2*((-2*I)*Cos[4*e] - 2*Sin[4*e]))/((1 + n)*(2 + n)) + ((-Cos[e - f*x] + Cos[e + f*x])*Sec[e]^2*Sec[e + f*x]*((-2*I)*Cos[4*e] - 2*Sin[4*e]))/(1 + n))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4)/(f*(Cos[f*x] + I*Sin[f*x])^4) + (Cos[e + f*x]^4*((Sec[e]^2*(-1 + Cos[2*e] + (2*I)*Sin[2*e])*((2*I)*Cos[4*e] + 2*Sin[4*e]))/(1 + n) + (Sec[e]^2*Sec[e + f*x]*((2*I)*Cos[4*e] + 2*Sin[4*e])*(-Cos[e - f*x] + Cos[e + f*x] - (2*I)*Sin[e - f*x] + (2*I)*Sin[e + f*x]))/(1 + n))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4)/(f*(Cos[f*x] + I*Sin[f*x])^4) + (Cos[e + f*x]^4*((Sec[e]*Sec[e + f*x]^3*(Cos[4*e] - I*Sin[4*e])*Sin[f*x])/(3 + n) + (Sec[e]*Sec[e + f*x]*(2*Cos[4*e] - (2*I)*Sin[4*e])*Sin[f*x])/((1 + n)*(3 + n)) + (Sec[e + f*x]^2*(Cos[4*e] - I*Sin[4*e])*Tan[e])/(3 + n) + ((2*Cos[4*e] - (2*I)*Sin[4*e])*Tan[e])/((1 + n)*(3 + n)))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4)/(f*(Cos[f*x] + I*Sin[f*x])^4) + (I*2^(3 - n)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^4*(2^n*Hypergeometric2F1[1, n, 1 + n, -((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))] - (1 + E^((2*I)*(e + f*x)))^n*Hypergeometric2F1[n, n, 1 + n, (1 - E^((2*I)*(e + f*x)))/2])*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4)/((E^((2*I)*e) + E^((4*I)*e))*f*n*(Cos[f*x] + I*Sin[f*x])^4*Tan[e + f*x]^n) - ((8*I)*(-1 + E^((2*I)*(e + f*x)))^n*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^4*(-(Hypergeometric2F1[1, n, 1 + n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]/((1 + E^((2*I)*(e + f*x)))^n*n)) - ((1 + E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(-1 - n)*Hypergeometric2F1[1, 1 + n, 2 + n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/(1 + n) + Hypergeometric2F1[n, n, 1 + n, (1 - E^((2*I)*(e + f*x)))/2]/(2^n*n))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4)/(E^((4*I)*e)*(1 + E^((2*I)*e))*((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^n*f*(Cos[f*x] + I*Sin[f*x])^4*Tan[e + f*x]^n)","B",0
311,1,900,127,9.0848815,"\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^3,x]","\frac{i 2^{2-n} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^n \cos ^3(e+f x) \left(2^n \, _2F_1\left(1,n;n+1;-\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)-\left(1+e^{2 i (e+f x)}\right)^n \, _2F_1\left(n,n;n+1;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^3 \tan ^{-n}(e+f x)}{\left(e^{i e}+e^{3 i e}\right) f n (\cos (f x)+i \sin (f x))^3}-\frac{4 i e^{-3 i e} \left(-1+e^{2 i (e+f x)}\right)^n \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^n \left(\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-n} \cos ^3(e+f x) \left(-\frac{\left(1+e^{2 i e}\right) \left(-1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,n+1;n+2;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{-n-1}}{n+1}-\frac{\, _2F_1\left(1,n;n+1;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{-n}}{n}+\frac{2^{-n} \, _2F_1\left(n,n;n+1;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)}{n}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^3 \tan ^{-n}(e+f x)}{\left(1+e^{2 i e}\right) f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\frac{(\cos (2 e)+3 i \sin (2 e)-1) \left(\frac{1}{2} i \cos (3 e)+\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{n+1}+\frac{\sec (e+f x) \left(\frac{1}{2} i \cos (3 e)+\frac{1}{2} \sin (3 e)\right) (-\cos (e-f x)+\cos (e+f x)-3 i \sin (e-f x)+3 i \sin (e+f x)) \sec ^2(e)}{n+1}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\frac{(-2 n+\cos (2 e)-3) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{(n+1) (n+2)}+\frac{(\cos (e+f x)-\cos (e-f x)) \sec (e+f x) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{n+1}+\frac{\sec ^2(e+f x) (-i \cos (3 e)-\sin (3 e))}{n+2}\right) (d \tan (e+f x))^n (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)}-\frac{a^3 (2 n+5) (d \tan (e+f x))^{n+1}}{d f (n+1) (n+2)}-\frac{\left(a^3+i a^3 \tan (e+f x)\right) (d \tan (e+f x))^{n+1}}{d f (n+2)}",1,"(Cos[e + f*x]^3*((Sec[e + f*x]^2*((-I)*Cos[3*e] - Sin[3*e]))/(2 + n) + ((-3 - 2*n + Cos[2*e])*Sec[e]^2*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/((1 + n)*(2 + n)) + ((-Cos[e - f*x] + Cos[e + f*x])*Sec[e]^2*Sec[e + f*x]*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/(1 + n))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*((Sec[e]^2*(-1 + Cos[2*e] + (3*I)*Sin[2*e])*((I/2)*Cos[3*e] + Sin[3*e]/2))/(1 + n) + (Sec[e]^2*Sec[e + f*x]*((I/2)*Cos[3*e] + Sin[3*e]/2)*(-Cos[e - f*x] + Cos[e + f*x] - (3*I)*Sin[e - f*x] + (3*I)*Sin[e + f*x]))/(1 + n))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (I*2^(2 - n)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^3*(2^n*Hypergeometric2F1[1, n, 1 + n, -((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))] - (1 + E^((2*I)*(e + f*x)))^n*Hypergeometric2F1[n, n, 1 + n, (1 - E^((2*I)*(e + f*x)))/2])*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^3)/((E^(I*e) + E^((3*I)*e))*f*n*(Cos[f*x] + I*Sin[f*x])^3*Tan[e + f*x]^n) - ((4*I)*(-1 + E^((2*I)*(e + f*x)))^n*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^3*(-(Hypergeometric2F1[1, n, 1 + n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]/((1 + E^((2*I)*(e + f*x)))^n*n)) - ((1 + E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(-1 - n)*Hypergeometric2F1[1, 1 + n, 2 + n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/(1 + n) + Hypergeometric2F1[n, n, 1 + n, (1 - E^((2*I)*(e + f*x)))/2]/(2^n*n))*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*(1 + E^((2*I)*e))*((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^n*f*(Cos[f*x] + I*Sin[f*x])^3*Tan[e + f*x]^n)","B",0
312,1,168,75,2.112355,"\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx","Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^2,x]","\frac{e^{-2 i e} 2^{-n} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{n+1} \cos ^2(e+f x) (a+i a \tan (e+f x))^2 \left(-2^n+\left(1+e^{2 i (e+f x)}\right)^{n+1} \, _2F_1\left(n+1,n+1;n+2;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)\right) \tan ^{-n}(e+f x) (d \tan (e+f x))^n}{f (n+1) (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{2 a^2 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)}",1,"((((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(1 + n)*Cos[e + f*x]^2*(-2^n + (1 + E^((2*I)*(e + f*x)))^(1 + n)*Hypergeometric2F1[1 + n, 1 + n, 2 + n, (1 - E^((2*I)*(e + f*x)))/2])*(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^2)/(2^n*E^((2*I)*e)*f*(1 + n)*(Cos[f*x] + I*Sin[f*x])^2*Tan[e + f*x]^n)","B",1
313,1,159,43,0.8821611,"\int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x]),x]","\frac{a e^{-i e} 2^{-n-1} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{n+1} \left(1+e^{2 i (e+f x)}\right)^{n+1} \cos (e+f x) (1+i \tan (e+f x)) \, _2F_1\left(n+1,n+1;n+2;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right) \tan ^{-n}(e+f x) (d \tan (e+f x))^n}{f (n+1) (\cos (f x)+i \sin (f x))}","\frac{a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)}",1,"(2^(-1 - n)*a*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(1 + n)*(1 + E^((2*I)*(e + f*x)))^(1 + n)*Cos[e + f*x]*Hypergeometric2F1[1 + n, 1 + n, 2 + n, (1 - E^((2*I)*(e + f*x)))/2]*(1 + I*Tan[e + f*x])*(d*Tan[e + f*x])^n)/(E^(I*e)*f*(1 + n)*(Cos[f*x] + I*Sin[f*x])*Tan[e + f*x]^n)","B",1
314,0,0,158,18.7563946,"\int \frac{(d \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x]),x]","\int \frac{(d \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx","\frac{i n (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{2 a d^2 f (n+2)}+\frac{(1-n) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{2 a d f (n+1)}+\frac{(d \tan (e+f x))^{n+1}}{2 d f (a+i a \tan (e+f x))}",1,"Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x]), x]","F",-1
315,0,0,209,6.0185341,"\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2,x]","\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","\frac{i (2-n) n (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{4 a^2 d^2 f (n+2)}+\frac{(1-n)^2 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{4 a^2 d f (n+1)}+\frac{(2-n) (d \tan (e+f x))^{n+1}}{4 a^2 d f (1+i \tan (e+f x))}+\frac{(d \tan (e+f x))^{n+1}}{4 d f (a+i a \tan (e+f x))^2}",1,"Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x]","F",-1
316,0,0,274,21.3277247,"\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3,x]","\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","\frac{i (5-2 n) (2-n) n (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{24 a^3 d^2 f (n+2)}+\frac{(1-2 n) (1-n) (3-n) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{24 a^3 d f (n+1)}+\frac{(5-2 n) (2-n) (d \tan (e+f x))^{n+1}}{24 d f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{(7-2 n) (d \tan (e+f x))^{n+1}}{24 a d f (a+i a \tan (e+f x))^2}+\frac{(d \tan (e+f x))^{n+1}}{6 d f (a+i a \tan (e+f x))^3}",1,"Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3, x]","F",-1
317,0,0,326,28.9741405,"\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^4} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^4,x]","\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^4} \, dx","\frac{i (2-n)^2 (4-n) n (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{48 a^4 d^2 f (n+2)}+\frac{(1-n) (3-n) \left(n^2-4 n+1\right) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{48 a^4 d f (n+1)}+\frac{\left(n^2-7 n+13\right) (d \tan (e+f x))^{n+1}}{48 a^4 d f (1+i \tan (e+f x))^2}+\frac{(2-n)^2 (4-n) (d \tan (e+f x))^{n+1}}{48 a^4 d f (1+i \tan (e+f x))}+\frac{(5-n) (d \tan (e+f x))^{n+1}}{24 a d f (a+i a \tan (e+f x))^3}+\frac{(d \tan (e+f x))^{n+1}}{8 d f (a+i a \tan (e+f x))^4}",1,"Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^4, x]","F",-1
318,1,44,43,0.07293,"\int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^n*(a - I*a*Tan[e + f*x]),x]","\frac{a \tan (e+f x) (d \tan (e+f x))^n \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{f (n+1)}","\frac{a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{d f (n+1)}",1,"(a*Hypergeometric2F1[1, 1 + n, 2 + n, (-I)*Tan[e + f*x]]*Tan[e + f*x]*(d*Tan[e + f*x])^n)/(f*(1 + n))","A",1
319,1,123,158,0.710043,"\int \frac{(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^n/(a - I*a*Tan[e + f*x]),x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left(-\frac{(n-1) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{a (n+1)}-\frac{i n \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{a (n+2)}+\frac{1}{a-i a \tan (e+f x)}\right)}{2 f}","-\frac{i n (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{2 a d^2 f (n+2)}+\frac{(1-n) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{2 a d f (n+1)}+\frac{(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*(-(((-1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2])/(a*(1 + n))) - (I*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(a*(2 + n)) + (a - I*a*Tan[e + f*x])^(-1)))/(2*f)","A",1
320,0,0,89,2.1313683,"\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^{3/2} \, dx","Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^(3/2),x]","\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^{3/2} \, dx","\frac{a \sqrt{a+i a \tan (e+f x)} F_1\left(n+1;-\frac{1}{2},1;n+2;-i \tan (e+f x),i \tan (e+f x)\right) (d \tan (e+f x))^{n+1}}{d f (n+1) \sqrt{1+i \tan (e+f x)}}",1,"Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^(3/2), x]","F",-1
321,0,0,89,1.0944141,"\int (d \tan (e+f x))^n \sqrt{a+i a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^n*Sqrt[a + I*a*Tan[e + f*x]],x]","\int (d \tan (e+f x))^n \sqrt{a+i a \tan (e+f x)} \, dx","\frac{a \sqrt{1+i \tan (e+f x)} F_1\left(n+1;\frac{1}{2},1;n+2;-i \tan (e+f x),i \tan (e+f x)\right) (d \tan (e+f x))^{n+1}}{d f (n+1) \sqrt{a+i a \tan (e+f x)}}",1,"Integrate[(d*Tan[e + f*x])^n*Sqrt[a + I*a*Tan[e + f*x]], x]","F",-1
322,-1,0,88,180.0009912,"\int \frac{(d \tan (e+f x))^n}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[(d*Tan[e + f*x])^n/Sqrt[a + I*a*Tan[e + f*x]],x]","\text{\$Aborted}","\frac{\sqrt{1+i \tan (e+f x)} F_1\left(n+1;\frac{3}{2},1;n+2;-i \tan (e+f x),i \tan (e+f x)\right) (d \tan (e+f x))^{n+1}}{d f (n+1) \sqrt{a+i a \tan (e+f x)}}",1,"$Aborted","F",-1
323,0,0,91,34.4470552,"\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(3/2),x]","\int \frac{(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx","\frac{\sqrt{1+i \tan (e+f x)} F_1\left(n+1;\frac{5}{2},1;n+2;-i \tan (e+f x),i \tan (e+f x)\right) (d \tan (e+f x))^{n+1}}{a d f (n+1) \sqrt{a+i a \tan (e+f x)}}",1,"Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(3/2), x]","F",-1
324,0,0,88,5.5766189,"\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx","Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^m,x]","\int (d \tan (e+f x))^n (a+i a \tan (e+f x))^m \, dx","\frac{(1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \tan (e+f x))^{n+1} F_1(n+1;1-m,1;n+2;-i \tan (e+f x),i \tan (e+f x))}{d f (n+1)}",1,"Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^m, x]","F",-1
325,0,0,205,43.4206931,"\int \tan ^4(c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Tan[c + d*x]^4*(a + I*a*Tan[c + d*x])^m,x]","\int \tan ^4(c+d x) (a+i a \tan (c+d x))^m \, dx","-\frac{i (a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}-\frac{i m \tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d \left(m^2+5 m+6\right)}+\frac{2 i (a+i a \tan (c+d x))^m}{d \left(m^2+5 m+6\right)}+\frac{i \left(m^2+3 m+6\right) (a+i a \tan (c+d x))^{m+1}}{a d (m+1) (m+2) (m+3)}+\frac{\tan ^3(c+d x) (a+i a \tan (c+d x))^m}{d (m+3)}",1,"Integrate[Tan[c + d*x]^4*(a + I*a*Tan[c + d*x])^m, x]","F",-1
326,0,0,144,33.1057162,"\int \tan ^3(c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^m,x]","\int \tan ^3(c+d x) (a+i a \tan (c+d x))^m \, dx","\frac{(a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}-\frac{m (a+i a \tan (c+d x))^{m+1}}{a d \left(m^2+3 m+2\right)}+\frac{\tan ^2(c+d x) (a+i a \tan (c+d x))^m}{d (m+2)}-\frac{2 (a+i a \tan (c+d x))^m}{d m (m+2)}",1,"Integrate[Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^m, x]","F",-1
327,0,0,82,17.0666482,"\int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m,x]","\int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx","\frac{i (a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}-\frac{i (a+i a \tan (c+d x))^{m+1}}{a d (m+1)}",1,"Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m, x]","F",-1
328,1,134,70,4.9243711,"\int \tan (c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^m,x]","\frac{2^{m-1} \left(e^{i d x}\right)^m \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \left(m \left(-e^{2 i (c+d x)}\right) \, _2F_1\left(1,1;m+2;-e^{2 i (c+d x)}\right)+m+1\right) \sec ^{-m}(c+d x) (\cos (d x)+i \sin (d x))^{-m} (a+i a \tan (c+d x))^m}{d m (m+1)}","\frac{(a+i a \tan (c+d x))^m}{d m}-\frac{(a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}",1,"(2^(-1 + m)*(E^(I*d*x))^m*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(1 + m - E^((2*I)*(c + d*x))*m*Hypergeometric2F1[1, 1, 2 + m, -E^((2*I)*(c + d*x))])*(a + I*a*Tan[c + d*x])^m)/(d*m*(1 + m)*Sec[c + d*x]^m*(Cos[d*x] + I*Sin[d*x])^m)","A",0
329,1,128,49,0.4782498,"\int (a+i a \tan (c+d x))^m \, dx","Integrate[(a + I*a*Tan[c + d*x])^m,x]","-\frac{i 2^{m-1} \left(1+e^{2 i (c+d x)}\right) \left(e^{i d x}\right)^m \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^m \, _2F_1\left(1,1;m+1;-e^{2 i (c+d x)}\right) \sec ^{-m}(c+d x) (\cos (d x)+i \sin (d x))^{-m} (a+i a \tan (c+d x))^m}{d m}","-\frac{i (a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}",1,"((-I)*2^(-1 + m)*(E^(I*d*x))^m*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^m*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, 1, 1 + m, -E^((2*I)*(c + d*x))]*(a + I*a*Tan[c + d*x])^m)/(d*m*Sec[c + d*x]^m*(Cos[d*x] + I*Sin[d*x])^m)","B",0
330,0,0,89,11.7034027,"\int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^m,x]","\int \cot (c+d x) (a+i a \tan (c+d x))^m \, dx","\frac{(a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}-\frac{(a+i a \tan (c+d x))^m \, _2F_1(1,m;m+1;i \tan (c+d x)+1)}{d m}",1,"Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^m, x]","F",-1
331,0,0,116,22.4940866,"\int \cot ^2(c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^m,x]","\int \cot ^2(c+d x) (a+i a \tan (c+d x))^m \, dx","\frac{i (a+i a \tan (c+d x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (c+d x)+1)\right)}{2 d m}-\frac{i (a+i a \tan (c+d x))^m \, _2F_1(1,m;m+1;i \tan (c+d x)+1)}{d}-\frac{\cot (c+d x) (a+i a \tan (c+d x))^m}{d}",1,"Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^m, x]","F",-1
332,0,0,81,12.0383538,"\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^m \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^m,x]","\int \tan ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^m \, dx","\frac{2 \tan ^{\frac{5}{2}}(c+d x) (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left(\frac{5}{2};1-m,1;\frac{7}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{5 d}",1,"Integrate[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^m, x]","F",-1
333,0,0,81,5.2147444,"\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^m \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^m,x]","\int \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^m \, dx","\frac{2 \tan ^{\frac{3}{2}}(c+d x) (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left(\frac{3}{2};1-m,1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{3 d}",1,"Integrate[Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^m, x]","F",-1
334,0,0,79,5.4523308,"\int \frac{(a+i a \tan (c+d x))^m}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^m/Sqrt[Tan[c + d*x]],x]","\int \frac{(a+i a \tan (c+d x))^m}{\sqrt{\tan (c+d x)}} \, dx","\frac{2 \sqrt{\tan (c+d x)} (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left(\frac{1}{2};1-m,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}",1,"Integrate[(a + I*a*Tan[c + d*x])^m/Sqrt[Tan[c + d*x]], x]","F",-1
335,0,0,79,3.3137518,"\int \frac{(a+i a \tan (c+d x))^m}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^m/Tan[c + d*x]^(3/2),x]","\int \frac{(a+i a \tan (c+d x))^m}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","-\frac{2 (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left(-\frac{1}{2};1-m,1;\frac{1}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\tan (c+d x)}}",1,"Integrate[(a + I*a*Tan[c + d*x])^m/Tan[c + d*x]^(3/2), x]","F",-1
336,1,117,115,0.796282,"\int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x]),x]","\frac{\left(\frac{1}{15}+\frac{i}{15}\right) a (d \tan (e+f x))^{5/2} \left(-15 \sqrt[4]{-1} \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)+(1-i) \sqrt{\tan (e+f x)} \left(3 \tan ^2(e+f x)+5 \tan (e+f x)-15\right)+15 (-1)^{3/4} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)\right)}{f \tan ^{\frac{5}{2}}(e+f x)}","\frac{\sqrt{2} a d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}-\frac{2 a d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a d (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 a (d \tan (e+f x))^{5/2}}{5 f}",1,"((1/15 + I/15)*a*(d*Tan[e + f*x])^(5/2)*(-15*(-1)^(1/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + 15*(-1)^(3/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + (1 - I)*Sqrt[Tan[e + f*x]]*(-15 + 5*Tan[e + f*x] + 3*Tan[e + f*x]^2)))/(f*Tan[e + f*x]^(5/2))","C",1
337,1,105,93,0.4367052,"\int (d \tan (e+f x))^{3/2} (a+a \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x]),x]","\frac{\left(\frac{1}{3}+\frac{i}{3}\right) a (d \tan (e+f x))^{3/2} \left(-3 (-1)^{3/4} \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)+(1-i) \sqrt{\tan (e+f x)} (\tan (e+f x)+3)+3 \sqrt[4]{-1} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)\right)}{f \tan ^{\frac{3}{2}}(e+f x)}","\frac{\sqrt{2} a d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}+\frac{2 a d \sqrt{d \tan (e+f x)}}{f}+\frac{2 a (d \tan (e+f x))^{3/2}}{3 f}",1,"((1/3 + I/3)*a*(d*Tan[e + f*x])^(3/2)*(-3*(-1)^(3/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + 3*(-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + (1 - I)*Sqrt[Tan[e + f*x]]*(3 + Tan[e + f*x])))/(f*Tan[e + f*x]^(3/2))","C",1
338,1,92,72,0.1093007,"\int \sqrt{d \tan (e+f x)} (a+a \tan (e+f x)) \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x]),x]","\frac{(1+i) a \sqrt{d \tan (e+f x)} \left(\sqrt[4]{-1} \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)+(1-i) \sqrt{\tan (e+f x)}-(-1)^{3/4} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)\right)}{f \sqrt{\tan (e+f x)}}","\frac{2 a \sqrt{d \tan (e+f x)}}{f}-\frac{\sqrt{2} a \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}",1,"((1 + I)*a*((-1)^(1/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] - (-1)^(3/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + (1 - I)*Sqrt[Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[Tan[e + f*x]])","C",1
339,1,74,50,0.0874275,"\int \frac{a+a \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a + a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]],x]","-\frac{(1-i) \sqrt[4]{-1} a \sqrt{\tan (e+f x)} \left(\tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)+i \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (e+f x)}\right)\right)}{f \sqrt{d \tan (e+f x)}}","-\frac{\sqrt{2} a \tan ^{-1}\left(\frac{\sqrt{d} (1-\tan (e+f x))}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{d} f}",1,"((-1 + I)*(-1)^(1/4)*a*(ArcTan[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + I*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]])*Sqrt[Tan[e + f*x]])/(f*Sqrt[d*Tan[e + f*x]])","C",1
340,1,64,74,0.1228018,"\int \frac{a+a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2),x]","-\frac{(1+i) a \left(\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-i \tan (e+f x)\right)-i \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};i \tan (e+f x)\right)\right)}{d f \sqrt{d \tan (e+f x)}}","\frac{\sqrt{2} a \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{3/2} f}-\frac{2 a}{d f \sqrt{d \tan (e+f x)}}",1,"((-1 - I)*a*(Hypergeometric2F1[-1/2, 1, 1/2, (-I)*Tan[e + f*x]] - I*Hypergeometric2F1[-1/2, 1, 1/2, I*Tan[e + f*x]]))/(d*f*Sqrt[d*Tan[e + f*x]])","C",1
341,1,68,98,0.148228,"\int \frac{a+a \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2),x]","-\frac{\left(\frac{1}{3}+\frac{i}{3}\right) a \left(\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-i \tan (e+f x)\right)-i \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};i \tan (e+f x)\right)\right)}{d f (d \tan (e+f x))^{3/2}}","\frac{\sqrt{2} a \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{5/2} f}-\frac{2 a}{d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 a}{3 d f (d \tan (e+f x))^{3/2}}",1,"((-1/3 - I/3)*a*(Hypergeometric2F1[-3/2, 1, -1/2, (-I)*Tan[e + f*x]] - I*Hypergeometric2F1[-3/2, 1, -1/2, I*Tan[e + f*x]]))/(d*f*(d*Tan[e + f*x])^(3/2))","C",1
342,1,68,121,0.227767,"\int \frac{a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx","Integrate[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2),x]","-\frac{\left(\frac{1}{5}+\frac{i}{5}\right) a \left(\, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-i \tan (e+f x)\right)-i \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};i \tan (e+f x)\right)\right)}{d f (d \tan (e+f x))^{5/2}}","-\frac{\sqrt{2} a \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{7/2} f}+\frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}",1,"((-1/5 - I/5)*a*(Hypergeometric2F1[-5/2, 1, -3/2, (-I)*Tan[e + f*x]] - I*Hypergeometric2F1[-5/2, 1, -3/2, I*Tan[e + f*x]]))/(d*f*(d*Tan[e + f*x])^(5/2))","C",1
343,1,187,269,1.2743263,"\int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^2,x]","\frac{a^2 (d \tan (e+f x))^{5/2} \left(-70 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)+70 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+20 \tan ^{\frac{7}{2}}(e+f x)+56 \tan ^{\frac{5}{2}}(e+f x)-280 \sqrt{\tan (e+f x)}-35 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+35 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{70 f \tan ^{\frac{5}{2}}(e+f x)}","-\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}+\frac{\sqrt{2} a^2 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{f}-\frac{a^2 d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} f}+\frac{a^2 d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} f}-\frac{4 a^2 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}+\frac{4 a^2 (d \tan (e+f x))^{5/2}}{5 f}",1,"(a^2*(d*Tan[e + f*x])^(5/2)*(-70*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] + 70*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - 35*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 35*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 280*Sqrt[Tan[e + f*x]] + 56*Tan[e + f*x]^(5/2) + 20*Tan[e + f*x]^(7/2)))/(70*f*Tan[e + f*x]^(5/2))","A",1
344,1,52,246,0.3932297,"\int (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2 \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x])^2,x]","\frac{2 a^2 (d \tan (e+f x))^{3/2} \left(-10 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+3 \tan (e+f x)+10\right)}{15 f}","\frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{\sqrt{2} a^2 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{f}-\frac{a^2 d^{3/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} f}+\frac{a^2 d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} f}+\frac{2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac{4 a^2 (d \tan (e+f x))^{3/2}}{3 f}",1,"(2*a^2*(d*Tan[e + f*x])^(3/2)*(10 - 10*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2] + 3*Tan[e + f*x]))/(15*f)","C",1
345,1,175,244,0.4906577,"\int \sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^2 \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^2,x]","\frac{a^2 \sqrt{d \tan (e+f x)} \left(6 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-6 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+4 \tan ^{\frac{3}{2}}(e+f x)+24 \sqrt{\tan (e+f x)}+3 \sqrt{2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-3 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{6 f \sqrt{\tan (e+f x)}}","\frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{f}-\frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{a^2 \sqrt{d} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} f}-\frac{a^2 \sqrt{d} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} f}",1,"(a^2*Sqrt[d*Tan[e + f*x]]*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 24*Sqrt[Tan[e + f*x]] + 4*Tan[e + f*x]^(3/2)))/(6*f*Sqrt[Tan[e + f*x]])","A",1
346,1,53,222,0.2650549,"\int \frac{(a+a \tan (e+f x))^2}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a + a*Tan[e + f*x])^2/Sqrt[d*Tan[e + f*x]],x]","\frac{2 a^2 \sqrt{d \tan (e+f x)} \left(2 \tan (e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+3\right)}{3 d f}","-\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{\sqrt{d} f}+\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{\sqrt{d} f}+\frac{2 a^2 \sqrt{d \tan (e+f x)}}{d f}+\frac{a^2 \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} \sqrt{d} f}-\frac{a^2 \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} \sqrt{d} f}",1,"(2*a^2*Sqrt[d*Tan[e + f*x]]*(3 + 2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Tan[e + f*x]))/(3*d*f)","C",1
347,1,232,222,1.8173122,"\int \frac{(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2),x]","-\frac{a^2 (\tan (e+f x)+1)^2 \left(-4 \sin ^2(e+f x) \tan (e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+6 \sin (2 (e+f x)) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)+3 \sqrt{2} \cos ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x) \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+\log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)}{6 f (d \tan (e+f x))^{3/2} (\sin (e+f x)+\cos (e+f x))^2}","-\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{3/2} f}+\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{d^{3/2} f}-\frac{a^2 \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} d^{3/2} f}+\frac{a^2 \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} d^{3/2} f}-\frac{2 a^2}{d f \sqrt{d \tan (e+f x)}}",1,"-1/6*(a^2*(1 + Tan[e + f*x])^2*(6*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[2*(e + f*x)] - 4*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2*Tan[e + f*x] + 3*Sqrt[2]*Cos[e + f*x]^2*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Tan[e + f*x]^(3/2)))/(f*(Cos[e + f*x] + Sin[e + f*x])^2*(d*Tan[e + f*x])^(3/2))","C",1
348,1,229,247,1.3555258,"\int \frac{(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2),x]","-\frac{a^2 (\cot (e+f x)+1)^2 \left(48 \sin ^2(e+f x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)+4 \sin (2 (e+f x)) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(e+f x)\right)+3 \sqrt{2} \cos ^2(e+f x) \tan ^{\frac{5}{2}}(e+f x) \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+\log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)}{12 d^2 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^2}","\frac{\sqrt{2} a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{d^{5/2} f}-\frac{\sqrt{2} a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{d^{5/2} f}-\frac{a^2 \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} d^{5/2} f}+\frac{a^2 \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{\sqrt{2} d^{5/2} f}-\frac{4 a^2}{d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 a^2}{3 d f (d \tan (e+f x))^{3/2}}",1,"-1/12*(a^2*(1 + Cot[e + f*x])^2*(48*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2 + 4*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[e + f*x]^2]*Sin[2*(e + f*x)] + 3*Sqrt[2]*Cos[e + f*x]^2*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Tan[e + f*x]^(5/2)))/(d^2*f*(Cos[e + f*x] + Sin[e + f*x])^2*Sqrt[d*Tan[e + f*x]])","C",1
349,1,375,210,4.3297527,"\int (d \tan (e+f x))^{7/2} (a+a \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^(7/2)*(a + a*Tan[e + f*x])^3,x]","\frac{a^3 d^3 \cos (e+f x) (\tan (e+f x)+1)^3 \sqrt{d \tan (e+f x)} \left(3080 \cos ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+420 \sin ^2(e+f x) \tan ^{\frac{7}{2}}(e+f x)+770 \sin (2 (e+f x)) \tan ^{\frac{7}{2}}(e+f x)+2310 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2310 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+1320 \cos ^2(e+f x) \tan ^{\frac{7}{2}}(e+f x)-1848 \cos ^2(e+f x) \tan ^{\frac{5}{2}}(e+f x)-3080 \cos ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x)+9240 \cos ^2(e+f x) \sqrt{\tan (e+f x)}+1155 \sqrt{2} \cos ^2(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-1155 \sqrt{2} \cos ^2(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{2310 f \sqrt{\tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}","-\frac{2 \sqrt{2} a^3 d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}+\frac{4 a^3 d^3 \sqrt{d \tan (e+f x)}}{f}-\frac{4 a^3 d^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left(a^3 \tan (e+f x)+a^3\right) (d \tan (e+f x))^{9/2}}{11 d f}+\frac{16 a^3 (d \tan (e+f x))^{9/2}}{33 d f}+\frac{4 a^3 (d \tan (e+f x))^{7/2}}{7 f}-\frac{4 a^3 d (d \tan (e+f x))^{5/2}}{5 f}",1,"(a^3*d^3*Cos[e + f*x]*Sqrt[d*Tan[e + f*x]]*(1 + Tan[e + f*x])^3*(2310*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 - 2310*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 + 1155*Sqrt[2]*Cos[e + f*x]^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 1155*Sqrt[2]*Cos[e + f*x]^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 9240*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] - 3080*Cos[e + f*x]^2*Tan[e + f*x]^(3/2) + 3080*Cos[e + f*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) - 1848*Cos[e + f*x]^2*Tan[e + f*x]^(5/2) + 1320*Cos[e + f*x]^2*Tan[e + f*x]^(7/2) + 420*Sin[e + f*x]^2*Tan[e + f*x]^(7/2) + 770*Sin[2*(e + f*x)]*Tan[e + f*x]^(7/2)))/(2310*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[Tan[e + f*x]])","C",1
350,1,729,186,6.1126481,"\int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3,x]","\frac{4 \cos ^3(e+f x) \cot (e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)}{3 f (\sin (e+f x)+\cos (e+f x))^3}-\frac{\sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{f \tan ^{\frac{5}{2}}(e+f x) (\sin (e+f x)+\cos (e+f x))^3}+\frac{\sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{f \tan ^{\frac{5}{2}}(e+f x) (\sin (e+f x)+\cos (e+f x))^3}+\frac{4 \cos ^3(e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{5 f (\sin (e+f x)+\cos (e+f x))^3}+\frac{6 \sin (e+f x) \cos ^2(e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{7 f (\sin (e+f x)+\cos (e+f x))^3}+\frac{2 \sin ^2(e+f x) \cos (e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{9 f (\sin (e+f x)+\cos (e+f x))^3}-\frac{\cos ^3(e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{\sqrt{2} f \tan ^{\frac{5}{2}}(e+f x) (\sin (e+f x)+\cos (e+f x))^3}+\frac{\cos ^3(e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{\sqrt{2} f \tan ^{\frac{5}{2}}(e+f x) (\sin (e+f x)+\cos (e+f x))^3}-\frac{4 \cos ^3(e+f x) \cot ^2(e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{f (\sin (e+f x)+\cos (e+f x))^3}-\frac{4 \cos ^3(e+f x) \cot (e+f x) (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}{3 f (\sin (e+f x)+\cos (e+f x))^3}","-\frac{2 \sqrt{2} a^3 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}-\frac{4 a^3 d^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 \left(a^3 \tan (e+f x)+a^3\right) (d \tan (e+f x))^{7/2}}{9 d f}+\frac{40 a^3 (d \tan (e+f x))^{7/2}}{63 d f}+\frac{4 a^3 (d \tan (e+f x))^{5/2}}{5 f}-\frac{4 a^3 d (d \tan (e+f x))^{3/2}}{3 f}",1,"(4*Cos[e + f*x]^3*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(5*f*(Cos[e + f*x] + Sin[e + f*x])^3) - (4*Cos[e + f*x]^3*Cot[e + f*x]*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(3*f*(Cos[e + f*x] + Sin[e + f*x])^3) - (4*Cos[e + f*x]^3*Cot[e + f*x]^2*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(f*(Cos[e + f*x] + Sin[e + f*x])^3) + (4*Cos[e + f*x]^3*Cot[e + f*x]*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(3*f*(Cos[e + f*x] + Sin[e + f*x])^3) + (6*Cos[e + f*x]^2*Sin[e + f*x]*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(7*f*(Cos[e + f*x] + Sin[e + f*x])^3) + (2*Cos[e + f*x]*Sin[e + f*x]^2*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(9*f*(Cos[e + f*x] + Sin[e + f*x])^3) - (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(f*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(5/2)) + (Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(f*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(5/2)) - (Cos[e + f*x]^3*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(Sqrt[2]*f*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(5/2)) + (Cos[e + f*x]^3*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)/(Sqrt[2]*f*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(5/2))","C",1
351,1,332,160,2.6598446,"\int (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x])^3,x]","-\frac{a^3 \cos (e+f x) (\tan (e+f x)+1)^3 (d \tan (e+f x))^{3/2} \left(280 \cos ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)-60 \sin ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x)-126 \sin (2 (e+f x)) \tan ^{\frac{3}{2}}(e+f x)+210 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-210 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-280 \cos ^2(e+f x) \tan ^{\frac{3}{2}}(e+f x)+840 \cos ^2(e+f x) \sqrt{\tan (e+f x)}+105 \sqrt{2} \cos ^2(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-105 \sqrt{2} \cos ^2(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{210 f \tan ^{\frac{3}{2}}(e+f x) (\sin (e+f x)+\cos (e+f x))^3}","\frac{2 \sqrt{2} a^3 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}+\frac{32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac{4 a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac{4 a^3 d \sqrt{d \tan (e+f x)}}{f}+\frac{2 \left(a^3 \tan (e+f x)+a^3\right) (d \tan (e+f x))^{5/2}}{7 d f}",1,"-1/210*(a^3*Cos[e + f*x]*(d*Tan[e + f*x])^(3/2)*(1 + Tan[e + f*x])^3*(210*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 - 210*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 + 105*Sqrt[2]*Cos[e + f*x]^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 105*Sqrt[2]*Cos[e + f*x]^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 840*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] - 280*Cos[e + f*x]^2*Tan[e + f*x]^(3/2) + 280*Cos[e + f*x]^2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) - 60*Sin[e + f*x]^2*Tan[e + f*x]^(3/2) - 126*Sin[2*(e + f*x)]*Tan[e + f*x]^(3/2)))/(f*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(3/2))","C",1
352,1,315,138,1.7973651,"\int \sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^3 \, dx","Integrate[Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^3,x]","\frac{a^3 \cos (e+f x) (\tan (e+f x)+1)^3 \sqrt{d \tan (e+f x)} \left(3 \left(4 \sin ^2(e+f x) \sqrt{\tan (e+f x)}+10 \sin (2 (e+f x)) \sqrt{\tan (e+f x)}+10 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-10 \sqrt{2} \cos ^2(e+f x) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+40 \cos ^2(e+f x) \sqrt{\tan (e+f x)}+5 \sqrt{2} \cos ^2(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-5 \sqrt{2} \cos ^2(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)-20 \sin (2 (e+f x)) \sqrt{\tan (e+f x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)\right)}{30 f \sqrt{\tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}","\frac{2 \sqrt{2} a^3 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{f}+\frac{8 a^3 (d \tan (e+f x))^{3/2}}{5 d f}+\frac{4 a^3 \sqrt{d \tan (e+f x)}}{f}+\frac{2 \left(a^3 \tan (e+f x)+a^3\right) (d \tan (e+f x))^{3/2}}{5 d f}",1,"(a^3*Cos[e + f*x]*(3*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 + 5*Sqrt[2]*Cos[e + f*x]^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 5*Sqrt[2]*Cos[e + f*x]^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 40*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] + 4*Sin[e + f*x]^2*Sqrt[Tan[e + f*x]] + 10*Sin[2*(e + f*x)]*Sqrt[Tan[e + f*x]]) - 20*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[2*(e + f*x)]*Sqrt[Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]]*(1 + Tan[e + f*x])^3)/(30*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[Tan[e + f*x]])","C",1
353,1,553,117,6.0894208,"\int \frac{(a+a \tan (e+f x))^3}{\sqrt{d \tan (e+f x)}} \, dx","Integrate[(a + a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]],x]","\frac{4 \sin ^2(e+f x) \cos (e+f x) (a \tan (e+f x)+a)^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)}{3 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}+\frac{\sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{\tan (e+f x)} (a \tan (e+f x)+a)^3}{f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}-\frac{\sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{\tan (e+f x)} (a \tan (e+f x)+a)^3}{f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}+\frac{6 \sin (e+f x) \cos ^2(e+f x) (a \tan (e+f x)+a)^3}{f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}+\frac{2 \sin ^2(e+f x) \cos (e+f x) (a \tan (e+f x)+a)^3}{3 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}+\frac{\cos ^3(e+f x) \sqrt{\tan (e+f x)} (a \tan (e+f x)+a)^3 \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{\sqrt{2} f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}-\frac{\cos ^3(e+f x) \sqrt{\tan (e+f x)} (a \tan (e+f x)+a)^3 \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{\sqrt{2} f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}","\frac{16 a^3 \sqrt{d \tan (e+f x)}}{3 d f}+\frac{2 \left(a^3 \tan (e+f x)+a^3\right) \sqrt{d \tan (e+f x)}}{3 d f}-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{d} f}",1,"(6*Cos[e + f*x]^2*Sin[e + f*x]*(a + a*Tan[e + f*x])^3)/(f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]]) + (2*Cos[e + f*x]*Sin[e + f*x]^2*(a + a*Tan[e + f*x])^3)/(3*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]]) + (4*Cos[e + f*x]*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2*(a + a*Tan[e + f*x])^3)/(3*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]]) + (Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*Sqrt[Tan[e + f*x]]*(a + a*Tan[e + f*x])^3)/(f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]]) - (Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*Sqrt[Tan[e + f*x]]*(a + a*Tan[e + f*x])^3)/(f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]]) + (Cos[e + f*x]^3*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[Tan[e + f*x]]*(a + a*Tan[e + f*x])^3)/(Sqrt[2]*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]]) - (Cos[e + f*x]^3*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[Tan[e + f*x]]*(a + a*Tan[e + f*x])^3)/(Sqrt[2]*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]])","C",1
354,1,314,114,1.7701891,"\int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(3/2),x]","\frac{(a \tan (e+f x)+a)^3 \left(4 \sin ^3(e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)-4 \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)-2 \sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)+2 \sqrt{2} \cos ^3(e+f x) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)+4 \sin ^2(e+f x) \cos (e+f x)-\sqrt{2} \cos ^3(e+f x) \tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+\sqrt{2} \cos ^3(e+f x) \tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{2 f (d \tan (e+f x))^{3/2} (\sin (e+f x)+\cos (e+f x))^3}","-\frac{2 \sqrt{2} a^3 \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{3/2} f}+\frac{4 a^3 \sqrt{d \tan (e+f x)}}{d^2 f}-\frac{2 \left(a^3 \tan (e+f x)+a^3\right)}{d f \sqrt{d \tan (e+f x)}}",1,"((a + a*Tan[e + f*x])^3*(-4*Cos[e + f*x]^2*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[e + f*x] + 4*Cos[e + f*x]*Sin[e + f*x]^2 + 4*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[e + f*x]^3 - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*Tan[e + f*x]^(3/2) + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^3*Tan[e + f*x]^(3/2) - Sqrt[2]*Cos[e + f*x]^3*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2) + Sqrt[2]*Cos[e + f*x]^3*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2)))/(2*f*(Cos[e + f*x] + Sin[e + f*x])^3*(d*Tan[e + f*x])^(3/2))","C",1
355,1,272,117,4.0324385,"\int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(5/2),x]","\frac{a^3 (\tan (e+f x)+1)^3 \left(8 \sin ^3(e+f x) \tan (e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)-8 \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(e+f x)\right)-72 \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)-9 \sqrt{2} \cos ^3(e+f x) \tan ^{\frac{5}{2}}(e+f x) \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+\log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)}{12 f (d \tan (e+f x))^{5/2} (\sin (e+f x)+\cos (e+f x))^3}","\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{5/2} f}-\frac{16 a^3}{3 d^2 f \sqrt{d \tan (e+f x)}}-\frac{2 \left(a^3 \tan (e+f x)+a^3\right)}{3 d f (d \tan (e+f x))^{3/2}}",1,"(a^3*(1 + Tan[e + f*x])^3*(-8*Cos[e + f*x]^2*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[e + f*x]^2]*Sin[e + f*x] - 72*Cos[e + f*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2 + 8*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sin[e + f*x]^3*Tan[e + f*x] - 9*Sqrt[2]*Cos[e + f*x]^3*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Tan[e + f*x]^(5/2)))/(12*f*(Cos[e + f*x] + Sin[e + f*x])^3*(d*Tan[e + f*x])^(5/2))","C",1
356,1,271,141,2.1845206,"\int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx","Integrate[(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(7/2),x]","-\frac{a^3 (\cot (e+f x)+1)^3 \left(8 \sin (e+f x) \cos ^2(e+f x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(e+f x)\right)+5 \left(24 \sin ^3(e+f x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)+8 \sin ^2(e+f x) \cos (e+f x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(e+f x)\right)+\sqrt{2} \cos ^3(e+f x) \tan ^{\frac{7}{2}}(e+f x) \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+\log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)\right)}{20 d^3 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^3}","\frac{2 \sqrt{2} a^3 \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{7/2} f}-\frac{4 a^3}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac{2 \left(a^3 \tan (e+f x)+a^3\right)}{5 d f (d \tan (e+f x))^{5/2}}",1,"-1/20*(a^3*(1 + Cot[e + f*x])^3*(8*Cos[e + f*x]^2*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[e + f*x]^2]*Sin[e + f*x] + 5*(8*Cos[e + f*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2 + 24*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[e + f*x]^3 + Sqrt[2]*Cos[e + f*x]^3*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Tan[e + f*x]^(7/2))))/(d^3*f*(Cos[e + f*x] + Sin[e + f*x])^3*Sqrt[d*Tan[e + f*x]])","C",1
357,1,175,165,0.8544113,"\int \frac{(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{9/2}} \, dx","Integrate[(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(9/2),x]","-\frac{a^3 \cos (e+f x) (\cot (e+f x)+1)^3 \sqrt{d \tan (e+f x)} \left(70 \sin ^2(e+f x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(e+f x)\right)+35 \sin (2 (e+f x)) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(e+f x)\right)+42 \cos ^2(e+f x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(e+f x)\right)+10 \cos ^2(e+f x) \cot (e+f x) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(e+f x)\right)\right)}{35 d^5 f (\sin (e+f x)+\cos (e+f x))^3}","-\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{d^{9/2} f}+\frac{4 a^3}{d^4 f \sqrt{d \tan (e+f x)}}-\frac{4 a^3}{3 d^3 f (d \tan (e+f x))^{3/2}}-\frac{32 a^3}{35 d^2 f (d \tan (e+f x))^{5/2}}-\frac{2 \left(a^3 \tan (e+f x)+a^3\right)}{7 d f (d \tan (e+f x))^{7/2}}",1,"-1/35*(a^3*Cos[e + f*x]*(1 + Cot[e + f*x])^3*(10*Cos[e + f*x]^2*Cot[e + f*x]*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[e + f*x]^2] + 42*Cos[e + f*x]^2*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[e + f*x]^2] + 70*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[e + f*x]^2]*Sin[e + f*x]^2 + 35*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[e + f*x]^2]*Sin[2*(e + f*x)])*Sqrt[d*Tan[e + f*x]])/(d^5*f*(Cos[e + f*x] + Sin[e + f*x])^3)","C",1
358,1,110,111,0.9179711,"\int \frac{(d \tan (e+f x))^{5/2}}{a+a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^(5/2)/(a + a*Tan[e + f*x]),x]","\frac{\left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-2 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+4 \sqrt{\tan (e+f x)}\right) (d \tan (e+f x))^{5/2}}{2 a f \tan ^{\frac{5}{2}}(e+f x)}","-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{a f}+\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{2} a f}+\frac{2 d^2 \sqrt{d \tan (e+f x)}}{a f}",1,"((Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*ArcTan[Sqrt[Tan[e + f*x]]] + 4*Sqrt[Tan[e + f*x]])*(d*Tan[e + f*x])^(5/2))/(2*a*f*Tan[e + f*x]^(5/2))","A",1
359,1,107,87,0.3379973,"\int \frac{(d \tan (e+f x))^{3/2}}{a+a \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(a + a*Tan[e + f*x]),x]","\frac{(d \tan (e+f x))^{3/2} \left(4 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\sqrt{2} \left(\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)}{4 a f \tan ^{\frac{3}{2}}(e+f x)}","\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{a f}-\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{2} a f}",1,"((4*ArcTan[Sqrt[Tan[e + f*x]]] + Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]))*(d*Tan[e + f*x])^(3/2))/(4*a*f*Tan[e + f*x]^(3/2))","A",1
360,1,98,89,0.1816842,"\int \frac{\sqrt{d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(a + a*Tan[e + f*x]),x]","-\frac{\left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)\right) \sqrt{d \tan (e+f x)}}{2 a f \sqrt{\tan (e+f x)}}","-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{a f}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{2} a f}",1,"-1/2*((Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 2*ArcTan[Sqrt[Tan[e + f*x]]])*Sqrt[d*Tan[e + f*x]])/(a*f*Sqrt[Tan[e + f*x]])","A",1
361,1,107,81,0.5066821,"\int \frac{1}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))} \, dx","Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])),x]","\frac{\sqrt{\tan (e+f x)} \left(4 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\sqrt{2} \left(\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)\right)\right)}{4 a f \sqrt{d \tan (e+f x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{a \sqrt{d} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{d} (\tan (e+f x)+1)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{2} a \sqrt{d} f}",1,"((4*ArcTan[Sqrt[Tan[e + f*x]]] + Sqrt[2]*(-Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] + Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]))*Sqrt[Tan[e + f*x]])/(4*a*f*Sqrt[d*Tan[e + f*x]])","A",1
362,1,122,111,1.6773579,"\int \frac{1}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))} \, dx","Integrate[1/((d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x])),x]","-\frac{-\sqrt{2} \sqrt{\tan (e+f x)} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)+\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{\tan (e+f x)}+2 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right) \sqrt{\tan (e+f x)}+4}{2 a d f \sqrt{d \tan (e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{a d^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{2} a d^{3/2} f}-\frac{2}{a d f \sqrt{d \tan (e+f x)}}",1,"-1/2*(4 - Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[Tan[e + f*x]] + Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[Tan[e + f*x]] + 2*ArcTan[Sqrt[Tan[e + f*x]]]*Sqrt[Tan[e + f*x]])/(a*d*f*Sqrt[d*Tan[e + f*x]])","A",1
363,1,130,135,1.4082948,"\int \frac{1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))} \, dx","Integrate[1/((d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])),x]","\frac{12 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)+24 \tan (e+f x)+3 \sqrt{2} \tan ^{\frac{3}{2}}(e+f x) \left(\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)-8}{12 a d f (d \tan (e+f x))^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{a d^{5/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{\sqrt{2} a d^{5/2} f}+\frac{2}{a d^2 f \sqrt{d \tan (e+f x)}}-\frac{2}{3 a d f (d \tan (e+f x))^{3/2}}",1,"(-8 + 24*Tan[e + f*x] + 12*ArcTan[Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2) + 3*Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Tan[e + f*x]^(3/2))/(12*a*d*f*(d*Tan[e + f*x])^(3/2))","A",1
364,1,226,281,3.4093726,"\int \frac{(d \tan (e+f x))^{5/2}}{(a+a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^(5/2)/(a + a*Tan[e + f*x])^2,x]","\frac{\csc (e+f x) (d \tan (e+f x))^{5/2} (\sin (e+f x)+\cos (e+f x))^2 \left(\frac{\sec (e+f x) \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+12 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\sqrt{2} \log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{\tan ^{\frac{3}{2}}(e+f x)}-\frac{4 \cot (e+f x)}{\sin (e+f x)+\cos (e+f x)}\right)}{8 a^2 f (\tan (e+f x)+1)^2}","\frac{3 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a^2 f}+\frac{d^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 \sqrt{2} a^2 f}-\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{2 \sqrt{2} a^2 f}+\frac{d^{5/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 f}-\frac{d^{5/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 f}-\frac{d^2 \sqrt{d \tan (e+f x)}}{2 f \left(a^2 \tan (e+f x)+a^2\right)}",1,"(Csc[e + f*x]*(Cos[e + f*x] + Sin[e + f*x])^2*((-4*Cot[e + f*x])/(Cos[e + f*x] + Sin[e + f*x]) + ((2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 12*ArcTan[Sqrt[Tan[e + f*x]]] + Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Sec[e + f*x])/Tan[e + f*x]^(3/2))*(d*Tan[e + f*x])^(5/2))/(8*a^2*f*(1 + Tan[e + f*x])^2)","A",1
365,1,229,279,2.0484899,"\int \frac{(d \tan (e+f x))^{3/2}}{(a+a \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(a + a*Tan[e + f*x])^2,x]","\frac{\sec (e+f x) (d \tan (e+f x))^{3/2} (\sin (e+f x)+\cos (e+f x))^2 \left(\frac{2 \cot (e+f x)}{\sin (e+f x)+\cos (e+f x)}-\frac{\csc (e+f x) \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+4 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)-\sqrt{2} \log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)+\sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)}{2 \sqrt{\tan (e+f x)}}\right)}{4 a^2 f (\tan (e+f x)+1)^2}","-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a^2 f}-\frac{d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 \sqrt{2} a^2 f}+\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{2 \sqrt{2} a^2 f}+\frac{d^{3/2} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 f}-\frac{d^{3/2} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 f}+\frac{d \sqrt{d \tan (e+f x)}}{2 f \left(a^2 \tan (e+f x)+a^2\right)}",1,"(Sec[e + f*x]*(Cos[e + f*x] + Sin[e + f*x])^2*((2*Cot[e + f*x])/(Cos[e + f*x] + Sin[e + f*x]) - (Csc[e + f*x]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 4*ArcTan[Sqrt[Tan[e + f*x]]] - Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]))/(2*Sqrt[Tan[e + f*x]]))*(d*Tan[e + f*x])^(3/2))/(4*a^2*f*(1 + Tan[e + f*x])^2)","A",1
366,1,192,278,1.3838019,"\int \frac{\sqrt{d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(a + a*Tan[e + f*x])^2,x]","\frac{\sqrt{d \tan (e+f x)} \left(-\frac{2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+4 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\sqrt{2} \log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{\tan (e+f x)}}-\frac{2 \cos (e+f x)}{\sin (e+f x)+\cos (e+f x)}\right)}{4 a^2 f}","-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a^2 f}-\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 \sqrt{2} a^2 f}+\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{2 \sqrt{2} a^2 f}-\frac{\sqrt{d \tan (e+f x)}}{2 f \left(a^2 \tan (e+f x)+a^2\right)}-\frac{\sqrt{d} \log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 f}+\frac{\sqrt{d} \log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 f}",1,"(((-2*Cos[e + f*x])/(Cos[e + f*x] + Sin[e + f*x]) - (2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 4*ArcTan[Sqrt[Tan[e + f*x]]] + Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])/(2*Sqrt[Tan[e + f*x]]))*Sqrt[d*Tan[e + f*x]])/(4*a^2*f)","A",1
367,1,337,281,0.7466088,"\int \frac{1}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^2} \, dx","Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^2),x]","\frac{\sqrt{\tan (e+f x)} \left(12 \sin (e+f x) \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+12 \cos (e+f x) \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+4 \cos (e+f x) \sqrt{\tan (e+f x)}-\sqrt{2} \sin (e+f x) \log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)+\sqrt{2} \sin (e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\sqrt{2} \cos (e+f x) \log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)+\sqrt{2} \cos (e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) (\sin (e+f x)+\cos (e+f x))-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) (\sin (e+f x)+\cos (e+f x))\right)}{8 a^2 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a^2 \sqrt{d} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 \sqrt{2} a^2 \sqrt{d} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{2 \sqrt{2} a^2 \sqrt{d} f}+\frac{\sqrt{d \tan (e+f x)}}{2 d f \left(a^2 \tan (e+f x)+a^2\right)}-\frac{\log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 \sqrt{d} f}+\frac{\log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 \sqrt{d} f}",1,"((12*ArcTan[Sqrt[Tan[e + f*x]]]*Cos[e + f*x] - Sqrt[2]*Cos[e + f*x]*Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] + Sqrt[2]*Cos[e + f*x]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 12*ArcTan[Sqrt[Tan[e + f*x]]]*Sin[e + f*x] - Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]]*Sin[e + f*x] + Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sin[e + f*x] + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*(Cos[e + f*x] + Sin[e + f*x]) - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*(Cos[e + f*x] + Sin[e + f*x]) + 4*Cos[e + f*x]*Sqrt[Tan[e + f*x]])*Sqrt[Tan[e + f*x]])/(8*a^2*f*(Cos[e + f*x] + Sin[e + f*x])*Sqrt[d*Tan[e + f*x]])","A",1
368,1,203,306,1.4704358,"\int \frac{1}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2} \, dx","Integrate[1/((d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x])^2),x]","\frac{\tan ^{\frac{3}{2}}(e+f x) \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-10 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\frac{\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{\sqrt{2}}-\frac{2 (5 \sin (e+f x)+4 \cos (e+f x))}{\sqrt{\tan (e+f x)} (\sin (e+f x)+\cos (e+f x))}\right)}{4 a^2 f (d \tan (e+f x))^{3/2}}","-\frac{5 \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a^2 d^{3/2} f}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 \sqrt{2} a^2 d^{3/2} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{2 \sqrt{2} a^2 d^{3/2} f}+\frac{\log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 d^{3/2} f}-\frac{\log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 d^{3/2} f}-\frac{5}{2 a^2 d f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f \left(a^2 \tan (e+f x)+a^2\right) \sqrt{d \tan (e+f x)}}",1,"((Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - 10*ArcTan[Sqrt[Tan[e + f*x]]] + (Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])/Sqrt[2] - (2*(4*Cos[e + f*x] + 5*Sin[e + f*x]))/((Cos[e + f*x] + Sin[e + f*x])*Sqrt[Tan[e + f*x]]))*Tan[e + f*x]^(3/2))/(4*a^2*f*(d*Tan[e + f*x])^(3/2))","A",1
369,1,203,331,3.9720563,"\int \frac{1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx","Integrate[1/((d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^2),x]","\frac{\tan ^{\frac{5}{2}}(e+f x) \left(-\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)+\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+14 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\frac{\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{\sqrt{2}}+\frac{2 \left(20 \cot (e+f x)-4 \csc ^2(e+f x)+31\right)}{3 \sqrt{\tan (e+f x)} (\cot (e+f x)+1)}\right)}{4 a^2 f (d \tan (e+f x))^{5/2}}","\frac{7 \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 a^2 d^{5/2} f}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{2 \sqrt{2} a^2 d^{5/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right)}{2 \sqrt{2} a^2 d^{5/2} f}+\frac{\log \left(\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 d^{5/2} f}-\frac{\log \left(\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right)}{4 \sqrt{2} a^2 d^{5/2} f}+\frac{9}{2 a^2 d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f \left(a^2 \tan (e+f x)+a^2\right) (d \tan (e+f x))^{3/2}}-\frac{7}{6 a^2 d f (d \tan (e+f x))^{3/2}}",1,"((-(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]) + Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 14*ArcTan[Sqrt[Tan[e + f*x]]] + (Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])/Sqrt[2] + (2*(31 + 20*Cot[e + f*x] - 4*Csc[e + f*x]^2))/(3*(1 + Cot[e + f*x])*Sqrt[Tan[e + f*x]]))*Tan[e + f*x]^(5/2))/(4*a^2*f*(d*Tan[e + f*x])^(5/2))","A",1
370,1,346,189,6.2720847,"\int \frac{(d \tan (e+f x))^{9/2}}{(a+a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(9/2)/(a + a*Tan[e + f*x])^3,x]","\frac{\cot (e+f x) \csc ^3(e+f x) (d \tan (e+f x))^{9/2} (\sin (e+f x)+\cos (e+f x))^3 \left(-\frac{11 \sin (e+f x)}{8 (\sin (e+f x)+\cos (e+f x))}-\frac{1}{8 (\sin (e+f x)+\cos (e+f x))^2}+\frac{7}{2}\right)}{f (a \tan (e+f x)+a)^3}+\frac{\sec ^3(e+f x) (d \tan (e+f x))^{9/2} (\sin (e+f x)+\cos (e+f x))^3 \left(\frac{2 \sqrt{2} \cos (2 (e+f x)) \csc (e+f x) \sec ^3(e+f x) \left(\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)\right)}{(1-\tan (e+f x)) \left(\tan ^2(e+f x)+1\right) (\cot (e+f x)+1)}-\frac{62 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right) (\tan (e+f x)+1) \csc (e+f x) \sec ^3(e+f x)}{\left(\tan ^2(e+f x)+1\right)^2 (\cot (e+f x)+1)}\right)}{16 f \tan ^{\frac{9}{2}}(e+f x) (a \tan (e+f x)+a)^3}","-\frac{31 d^{9/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 f}+\frac{d^{9/2} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 f}+\frac{27 d^4 \sqrt{d \tan (e+f x)}}{8 a^3 f}-\frac{9 d^3 (d \tan (e+f x))^{3/2}}{8 a^3 f (\tan (e+f x)+1)}-\frac{d^2 (d \tan (e+f x))^{5/2}}{4 a f (a \tan (e+f x)+a)^2}",1,"(Cot[e + f*x]*Csc[e + f*x]^3*(Cos[e + f*x] + Sin[e + f*x])^3*(7/2 - 1/(8*(Cos[e + f*x] + Sin[e + f*x])^2) - (11*Sin[e + f*x])/(8*(Cos[e + f*x] + Sin[e + f*x])))*(d*Tan[e + f*x])^(9/2))/(f*(a + a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[e + f*x] + Sin[e + f*x])^3*(d*Tan[e + f*x])^(9/2)*((-62*ArcTan[Sqrt[Tan[e + f*x]]]*Csc[e + f*x]*Sec[e + f*x]^3*(1 + Tan[e + f*x]))/((1 + Cot[e + f*x])*(1 + Tan[e + f*x]^2)^2) + (2*Sqrt[2]*Cos[2*(e + f*x)]*Csc[e + f*x]*(-Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] + Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]])*Sec[e + f*x]^3)/((1 + Cot[e + f*x])*(1 - Tan[e + f*x])*(1 + Tan[e + f*x]^2))))/(16*f*Tan[e + f*x]^(9/2)*(a + a*Tan[e + f*x])^3)","A",1
371,1,183,165,2.8896575,"\int \frac{(d \tan (e+f x))^{7/2}}{(a+a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(7/2)/(a + a*Tan[e + f*x])^3,x]","\frac{(d \tan (e+f x))^{7/2} (\sin (e+f x)+\cos (e+f x))^3 \left(\frac{2 \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)+11 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)\right) \csc (e+f x) \sec ^2(e+f x)}{\tan ^{\frac{5}{2}}(e+f x)}-\frac{\csc ^5(e+f x) (9 \sin (2 (e+f x))+7 \cos (2 (e+f x))+7)}{(\cot (e+f x)+1)^2}\right)}{16 a^3 f (\tan (e+f x)+1)^3}","\frac{11 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 f}+\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 f}-\frac{7 d^3 \sqrt{d \tan (e+f x)}}{8 a^3 f (\tan (e+f x)+1)}-\frac{d^2 (d \tan (e+f x))^{3/2}}{4 a f (a \tan (e+f x)+a)^2}",1,"((Cos[e + f*x] + Sin[e + f*x])^3*(-((Csc[e + f*x]^5*(7 + 7*Cos[2*(e + f*x)] + 9*Sin[2*(e + f*x)]))/(1 + Cot[e + f*x])^2) + (2*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] + 11*ArcTan[Sqrt[Tan[e + f*x]]])*Csc[e + f*x]*Sec[e + f*x]^2)/Tan[e + f*x]^(5/2))*(d*Tan[e + f*x])^(7/2))/(16*a^3*f*(1 + Tan[e + f*x])^3)","A",1
372,1,192,164,3.0503995,"\int \frac{(d \tan (e+f x))^{5/2}}{(a+a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(5/2)/(a + a*Tan[e + f*x])^3,x]","\frac{\sec (e+f x) (d \tan (e+f x))^{5/2} (\sin (e+f x)+\cos (e+f x))^3 \left(\frac{\csc ^4(e+f x) (5 \sin (2 (e+f x))+3 \cos (2 (e+f x))+3)}{(\cot (e+f x)+1)^2}+\frac{2 \csc (e+f x) \sec (e+f x) \left(\tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+\sqrt{2} \left(\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)}{\tan ^{\frac{3}{2}}(e+f x)}\right)}{16 a^3 f (\tan (e+f x)+1)^3}","\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 f}-\frac{d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 f}+\frac{5 d^2 \sqrt{d \tan (e+f x)}}{8 a^3 f (\tan (e+f x)+1)}-\frac{d^2 \sqrt{d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2}",1,"(Sec[e + f*x]*(Cos[e + f*x] + Sin[e + f*x])^3*((Csc[e + f*x]^4*(3 + 3*Cos[2*(e + f*x)] + 5*Sin[2*(e + f*x)]))/(1 + Cot[e + f*x])^2 + (2*Csc[e + f*x]*(ArcTan[Sqrt[Tan[e + f*x]]] + Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]))*Sec[e + f*x])/Tan[e + f*x]^(3/2))*(d*Tan[e + f*x])^(5/2))/(16*a^3*f*(1 + Tan[e + f*x])^3)","A",1
373,1,127,164,2.3686608,"\int \frac{(d \tan (e+f x))^{3/2}}{(a+a \tan (e+f x))^3} \, dx","Integrate[(d*Tan[e + f*x])^(3/2)/(a + a*Tan[e + f*x])^3,x]","\frac{d \sqrt{d \tan (e+f x)} \left(\frac{-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-5 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)}{\sqrt{\tan (e+f x)}}+\frac{(\cot (e+f x)-1) \cot (e+f x)}{(\cot (e+f x)+1)^2}\right)}{8 a^3 f}","-\frac{5 d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 f}-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 f}-\frac{d \sqrt{d \tan (e+f x)}}{8 f \left(a^3 \tan (e+f x)+a^3\right)}+\frac{d \sqrt{d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2}",1,"(d*(((-1 + Cot[e + f*x])*Cot[e + f*x])/(1 + Cot[e + f*x])^2 + (-2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - 5*ArcTan[Sqrt[Tan[e + f*x]]])/Sqrt[Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(8*a^3*f)","A",1
374,1,253,161,0.8618306,"\int \frac{\sqrt{d \tan (e+f x)}}{(a+a \tan (e+f x))^3} \, dx","Integrate[Sqrt[d*Tan[e + f*x]]/(a + a*Tan[e + f*x])^3,x]","-\frac{\sqrt{d \tan (e+f x)} \left(5 \sqrt{\tan (e+f x)}+2 \sqrt{2} \log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-2 \sqrt{2} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-2 (\sin (2 (e+f x))+1) \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+5 \cos (2 (e+f x)) \sqrt{\tan (e+f x)}+\sin (2 (e+f x)) \left(3 \sqrt{\tan (e+f x)}+2 \sqrt{2} \left(\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)\right)\right)}{16 a^3 f \sqrt{\tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^2}","\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 f}-\frac{3 \sqrt{d \tan (e+f x)}}{8 f \left(a^3 \tan (e+f x)+a^3\right)}+\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 f}-\frac{\sqrt{d \tan (e+f x)}}{4 a f (a \tan (e+f x)+a)^2}",1,"-1/16*((2*Sqrt[2]*Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*ArcTan[Sqrt[Tan[e + f*x]]]*(1 + Sin[2*(e + f*x)]) + Sin[2*(e + f*x)]*(2*Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]) + 3*Sqrt[Tan[e + f*x]]) + 5*Sqrt[Tan[e + f*x]] + 5*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]])*Sqrt[d*Tan[e + f*x]])/(a^3*f*(Cos[e + f*x] + Sin[e + f*x])^2*Sqrt[Tan[e + f*x]])","A",1
375,1,217,165,1.0487435,"\int \frac{1}{\sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^3} \, dx","Integrate[1/(Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^3),x]","\frac{\sqrt{\tan (e+f x)} \left(22 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+9 \sqrt{\tan (e+f x)}+22 \sin (2 (e+f x)) \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+7 \sin (2 (e+f x)) \sqrt{\tan (e+f x)}+9 \cos (2 (e+f x)) \sqrt{\tan (e+f x)}+4 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) (\sin (e+f x)+\cos (e+f x))^2-4 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) (\sin (e+f x)+\cos (e+f x))^2\right)}{16 a^3 f \sqrt{d \tan (e+f x)} (\sin (e+f x)+\cos (e+f x))^2}","\frac{11 \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 \sqrt{d} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 \sqrt{d} f}+\frac{7 \sqrt{d \tan (e+f x)}}{8 a^3 d f (\tan (e+f x)+1)}+\frac{\sqrt{d \tan (e+f x)}}{4 a d f (a \tan (e+f x)+a)^2}",1,"((22*ArcTan[Sqrt[Tan[e + f*x]]] + 4*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*(Cos[e + f*x] + Sin[e + f*x])^2 - 4*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*(Cos[e + f*x] + Sin[e + f*x])^2 + 22*ArcTan[Sqrt[Tan[e + f*x]]]*Sin[2*(e + f*x)] + 9*Sqrt[Tan[e + f*x]] + 9*Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]] + 7*Sin[2*(e + f*x)]*Sqrt[Tan[e + f*x]])*Sqrt[Tan[e + f*x]])/(16*a^3*f*(Cos[e + f*x] + Sin[e + f*x])^2*Sqrt[d*Tan[e + f*x]])","A",1
376,1,173,189,1.4257635,"\int \frac{1}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^3} \, dx","Integrate[1/((d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x])^3),x]","\frac{\tan ^{\frac{3}{2}}(e+f x) \left(-62 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)+2 \sqrt{2} \left(\log \left(-\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}-1\right)-\log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)\right)-\frac{\sqrt{\tan (e+f x)} (90 \cos (2 (e+f x))+75 \cot (e+f x)-11 \cos (3 (e+f x)) \csc (e+f x)+90)}{2 (\sin (e+f x)+\cos (e+f x))^2}\right)}{16 a^3 f (d \tan (e+f x))^{3/2}}","-\frac{31 \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 d^{3/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 d^{3/2} f}-\frac{27}{8 a^3 d f \sqrt{d \tan (e+f x)}}+\frac{9}{8 a^3 d f (\tan (e+f x)+1) \sqrt{d \tan (e+f x)}}+\frac{1}{4 a d f (a \tan (e+f x)+a)^2 \sqrt{d \tan (e+f x)}}",1,"((-62*ArcTan[Sqrt[Tan[e + f*x]]] + 2*Sqrt[2]*(Log[-1 + Sqrt[2]*Sqrt[Tan[e + f*x]] - Tan[e + f*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]) - ((90 + 90*Cos[2*(e + f*x)] + 75*Cot[e + f*x] - 11*Cos[3*(e + f*x)]*Csc[e + f*x])*Sqrt[Tan[e + f*x]])/(2*(Cos[e + f*x] + Sin[e + f*x])^2))*Tan[e + f*x]^(3/2))/(16*a^3*f*(d*Tan[e + f*x])^(3/2))","A",1
377,1,368,215,6.3017492,"\int \frac{1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3} \, dx","Integrate[1/((d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3),x]","\frac{\tan ^3(e+f x) \sec ^3(e+f x) (\sin (e+f x)+\cos (e+f x))^3 \left(6 \cot (e+f x)-\frac{2}{3} \csc ^2(e+f x)-\frac{17 \sin (e+f x)}{8 (\sin (e+f x)+\cos (e+f x))}+\frac{1}{8 (\sin (e+f x)+\cos (e+f x))^2}+\frac{8}{3}\right)}{f (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}+\frac{\tan ^{\frac{5}{2}}(e+f x) \sec ^3(e+f x) (\sin (e+f x)+\cos (e+f x))^3 \left(\frac{126 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right) (\tan (e+f x)+1) \csc (e+f x) \sec ^3(e+f x)}{\left(\tan ^2(e+f x)+1\right)^2 (\cot (e+f x)+1)}+\frac{2 \sin (2 (e+f x)) \left(\sqrt{2} \left(\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right)-\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right)\right)-2 \tan ^{-1}\left(\sqrt{\tan (e+f x)}\right)\right) (\tan (e+f x)+1) \csc ^2(e+f x) \sec ^2(e+f x)}{\left(\tan ^2(e+f x)+1\right) (\cot (e+f x)+1)}\right)}{16 f (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}}","\frac{59 \tan ^{-1}\left(\frac{\sqrt{d \tan (e+f x)}}{\sqrt{d}}\right)}{8 a^3 d^{5/2} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{d}-\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right)}{2 \sqrt{2} a^3 d^{5/2} f}+\frac{63}{8 a^3 d^2 f \sqrt{d \tan (e+f x)}}+\frac{11}{8 a^3 d f (\tan (e+f x)+1) (d \tan (e+f x))^{3/2}}-\frac{55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac{1}{4 a d f (a \tan (e+f x)+a)^2 (d \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^3*(Cos[e + f*x] + Sin[e + f*x])^3*(8/3 + 6*Cot[e + f*x] - (2*Csc[e + f*x]^2)/3 + 1/(8*(Cos[e + f*x] + Sin[e + f*x])^2) - (17*Sin[e + f*x])/(8*(Cos[e + f*x] + Sin[e + f*x])))*Tan[e + f*x]^3)/(f*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(5/2)*((126*ArcTan[Sqrt[Tan[e + f*x]]]*Csc[e + f*x]*Sec[e + f*x]^3*(1 + Tan[e + f*x]))/((1 + Cot[e + f*x])*(1 + Tan[e + f*x]^2)^2) + (2*(Sqrt[2]*(-ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] + ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]) - 2*ArcTan[Sqrt[Tan[e + f*x]]])*Csc[e + f*x]^2*Sec[e + f*x]^2*Sin[2*(e + f*x)]*(1 + Tan[e + f*x]))/((1 + Cot[e + f*x])*(1 + Tan[e + f*x]^2))))/(16*f*(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3)","A",1
378,1,122,264,1.026912,"\int \tan ^5(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^5*Sqrt[1 + Tan[e + f*x]],x]","\frac{2 \sqrt{\tan (e+f x)+1} \left(35 \tan ^4(e+f x)+5 \tan ^3(e+f x)-69 \tan ^2(e+f x)-13 \tan (e+f x)+341\right)-315 \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)-315 \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{315 f}","-\frac{\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \tan (e+f x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{2 (\tan (e+f x)+1)^{3/2} \tan ^3(e+f x)}{9 f}-\frac{4 (\tan (e+f x)+1)^{3/2} \tan ^2(e+f x)}{21 f}-\frac{26 (\tan (e+f x)+1)^{3/2} \tan (e+f x)}{105 f}+\frac{52 (\tan (e+f x)+1)^{3/2}}{315 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan (e+f x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"(-315*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] - 315*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + f*x]]*(341 - 13*Tan[e + f*x] - 69*Tan[e + f*x]^2 + 5*Tan[e + f*x]^3 + 35*Tan[e + f*x]^4))/(315*f)","C",1
379,1,100,208,0.3240499,"\int \tan ^3(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^3*Sqrt[1 + Tan[e + f*x]],x]","\frac{2 \sqrt{\tan (e+f x)+1} \left(3 \tan ^2(e+f x)+\tan (e+f x)-17\right)+15 \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+15 \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{15 f}","\frac{\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \tan (e+f x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{2 \tan (e+f x) (\tan (e+f x)+1)^{3/2}}{5 f}-\frac{4 (\tan (e+f x)+1)^{3/2}}{15 f}-\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan (e+f x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"(15*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 15*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + f*x]]*(-17 + Tan[e + f*x] + 3*Tan[e + f*x]^2))/(15*f)","C",1
380,1,78,166,0.0658172,"\int \tan (e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Tan[e + f*x]*Sqrt[1 + Tan[e + f*x]],x]","-\frac{-2 \sqrt{\tan (e+f x)+1}+\sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+\sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{f}","-\frac{\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \tan (e+f x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan (e+f x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"-((Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] - 2*Sqrt[1 + Tan[e + f*x]])/f)","C",1
381,1,78,165,0.076813,"\int \cot (e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]],x]","\frac{-2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+\sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+\sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{f}","\frac{\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \tan (e+f x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}+\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan (e+f x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/f","C",1
382,1,124,221,0.4628883,"\int \cot ^3(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]],x]","-\frac{-9 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+4 \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+4 \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)+2 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+\sqrt{\tan (e+f x)+1} \cot (e+f x)}{4 f}","-\frac{\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \tan (e+f x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{9 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{4 f}-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan (e+f x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac{\sqrt{\tan (e+f x)+1} \cot (e+f x)}{4 f}",1,"-1/4*(-9*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 4*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 4*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 2*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/f","C",1
383,1,169,273,0.8326615,"\int \cot ^5(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Cot[e + f*x]^5*Sqrt[1 + Tan[e + f*x]],x]","\frac{-417 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+192 \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+192 \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)-48 \sqrt{\tan (e+f x)+1} \cot ^4(e+f x)-8 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)+106 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+33 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{192 f}","\frac{\sqrt{\frac{1}{2} \left(\sqrt{2}-1\right)} \tan ^{-1}\left(\frac{\left(2-\sqrt{2}\right) \tan (e+f x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{139 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{64 f}+\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan (e+f x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac{53 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}+\frac{11 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{64 f}",1,"(-417*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 192*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 192*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 33*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 106*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]] - 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 48*Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(192*f)","C",1
384,1,118,318,0.7141287,"\int \tan ^4(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^4*Sqrt[1 + Tan[e + f*x]],x]","\frac{-35 i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+35 i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)+2 \sqrt{\tan (e+f x)+1} \left((5 \tan (e+f x)+1) \sec ^2(e+f x)-2 (9 \tan (e+f x)+5)\right)}{35 f}","-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{2 (\tan (e+f x)+1)^{3/2} \tan ^2(e+f x)}{7 f}-\frac{8 (\tan (e+f x)+1)^{3/2} \tan (e+f x)}{35 f}-\frac{18 (\tan (e+f x)+1)^{3/2}}{35 f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}",1,"((-35*I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (35*I)*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + f*x]]*(Sec[e + f*x]^2*(1 + 5*Tan[e + f*x]) - 2*(5 + 9*Tan[e + f*x])))/(35*f)","C",1
385,1,86,266,0.1626041,"\int \tan ^2(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^2*Sqrt[1 + Tan[e + f*x]],x]","\frac{2 (\tan (e+f x)+1)^{3/2}+3 i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)-3 i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{3 f}","\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}",1,"((3*I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] - (3*I)*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*(1 + Tan[e + f*x])^(3/2))/(3*f)","C",1
386,1,67,247,0.0299934,"\int \sqrt{1+\tan (e+f x)} \, dx","Integrate[Sqrt[1 + Tan[e + f*x]],x]","-\frac{i \left(\sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)-\sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)\right)}{f}","-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}",1,"((-I)*(Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] - Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]]))/f","C",1
387,1,102,288,0.2648012,"\int \cot ^2(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]],x]","-\frac{\tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)-i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)+\sqrt{\tan (e+f x)+1} \cot (e+f x)}{f}","\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}-\frac{\tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}-\frac{\sqrt{\tan (e+f x)+1} \cot (e+f x)}{f}",1,"-((ArcTanh[Sqrt[1 + Tan[e + f*x]]] - I*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + I*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f)","C",1
388,1,151,346,0.4129293,"\int \cot ^4(e+f x) \sqrt{1+\tan (e+f x)} \, dx","Integrate[Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]],x]","\frac{21 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)-24 i \sqrt{1-i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+24 i \sqrt{1+i} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)-8 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)-2 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+27 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{24 f}","-\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{2}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{2 \left(1+\sqrt{2}\right)} f}+\frac{7 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{8 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac{9 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{8 f}",1,"(21*ArcTanh[Sqrt[1 + Tan[e + f*x]]] - (24*I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (24*I)*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 27*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] - 2*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]] - 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(24*f)","C",1
389,1,132,369,1.1061133,"\int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Tan[e + f*x]^5*(1 + Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{\tan (e+f x)+1} \left(21 \tan ^5(e+f x)+28 \tan ^4(e+f x)-32 \tan ^3(e+f x)-54 \tan ^2(e+f x)+72 \tan (e+f x)+318\right)-231 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)-231 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{231 f}","\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}-\frac{4 (\tan (e+f x)+1)^{5/2} \tan ^2(e+f x)}{33 f}-\frac{50 (\tan (e+f x)+1)^{5/2} \tan (e+f x)}{231 f}+\frac{20 (\tan (e+f x)+1)^{5/2}}{231 f}+\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}",1,"(-231*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] - 231*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + f*x]]*(318 + 72*Tan[e + f*x] - 54*Tan[e + f*x]^2 - 32*Tan[e + f*x]^3 + 28*Tan[e + f*x]^4 + 21*Tan[e + f*x]^5))/(231*f)","C",1
390,1,112,315,0.5892234,"\int \tan ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Tan[e + f*x]^3*(1 + Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{\tan (e+f x)+1} \left(15 \tan ^3(e+f x)+24 \tan ^2(e+f x)-32 \tan (e+f x)-146\right)+105 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+105 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{105 f}","-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{2 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}-\frac{4 (\tan (e+f x)+1)^{5/2}}{35 f}-\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}-\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}",1,"(105*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 105*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + f*x]]*(-146 - 32*Tan[e + f*x] + 24*Tan[e + f*x]^2 + 15*Tan[e + f*x]^3))/(105*f)","C",1
391,1,90,271,0.2224832,"\int \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Tan[e + f*x]*(1 + Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{\tan (e+f x)+1} (\tan (e+f x)+4)-3 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)-3 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{3 f}","\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}",1,"(-3*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] - 3*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + f*x]]*(4 + Tan[e + f*x]))/(3*f)","C",1
392,1,78,253,0.1778866,"\int \cot (e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]*(1 + Tan[e + f*x])^(3/2),x]","\frac{-2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+(1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{f}","-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}-\frac{2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}",1,"(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/f","C",1
393,1,125,307,0.6364454,"\int \cot ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^3*(1 + Tan[e + f*x])^(3/2),x]","-\frac{-5 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+4 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+4 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)+2 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+5 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{4 f}","\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}+\frac{5 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{4 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac{5 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{4 f}",1,"-1/4*(-5*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 4*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 4*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 5*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 2*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/f","C",1
394,1,169,361,1.4512926,"\int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^5*(1 + Tan[e + f*x])^(3/2),x]","\frac{-83 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+64 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+64 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)-16 \sqrt{\tan (e+f x)+1} \cot ^4(e+f x)-24 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)+30 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+83 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{64 f}","-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{2 \sqrt{1+\sqrt{2}} f}-\frac{83 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{64 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac{3 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{8 f}+\frac{15 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{32 f}+\frac{83 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{64 f}",1,"(-83*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 64*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 64*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 83*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 30*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]] - 24*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 16*Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(64*f)","C",1
395,1,122,227,1.5886522,"\int \tan ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Tan[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{\tan (e+f x)+1} \left(7 \tan ^4(e+f x)+10 \tan ^3(e+f x)-12 \tan ^2(e+f x)-26 \tan (e+f x)+52\right)-\frac{126 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}-\frac{126 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}}{63 f}","-\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{2 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{9 f}-\frac{8 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{63 f}-\frac{22 (\tan (e+f x)+1)^{5/2}}{63 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"((-126*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] - (126*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 2*Sqrt[1 + Tan[e + f*x]]*(52 - 26*Tan[e + f*x] - 12*Tan[e + f*x]^2 + 10*Tan[e + f*x]^3 + 7*Tan[e + f*x]^4))/(63*f)","C",1
396,1,100,173,0.3764935,"\int \tan ^2(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Tan[e + f*x]^2*(1 + Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{\tan (e+f x)+1} \left(\tan ^2(e+f x)+2 \tan (e+f x)-4\right)+\frac{10 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{10 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}}{5 f}","\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{2 (\tan (e+f x)+1)^{5/2}}{5 f}-\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"((10*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (10*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 2*Sqrt[1 + Tan[e + f*x]]*(-4 + 2*Tan[e + f*x] + Tan[e + f*x]^2))/(5*f)","C",1
397,1,79,156,0.0700074,"\int (1+\tan (e+f x))^{3/2} \, dx","Integrate[(1 + Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{\tan (e+f x)+1}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}}{f}","-\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}",1,"((-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] - (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 2*Sqrt[1 + Tan[e + f*x]])/f","C",1
398,1,100,178,0.1995773,"\int \cot ^2(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^2*(1 + Tan[e + f*x])^(3/2),x]","\frac{-3 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}-\sqrt{\tan (e+f x)+1} \cot (e+f x)}{f}","\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{3 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{\sqrt{\tan (e+f x)+1} \cot (e+f x)}{f}",1,"(-3*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] - Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f","C",1
399,1,147,238,1.0427917,"\int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx","Integrate[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]","-\frac{-75 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+\frac{48 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{48 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}+8 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)+14 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)-21 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{24 f}","-\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}+\frac{25 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{8 f}-\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}-\frac{7 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac{7 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{8 f}",1,"-1/24*(-75*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (48*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (48*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] - 21*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 14*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]] + 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/f","C",1
400,1,112,241,0.8596023,"\int \frac{\tan ^5(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Tan[e + f*x]^5/Sqrt[1 + Tan[e + f*x]],x]","\frac{4 \sqrt{\tan (e+f x)+1} \left(15 \tan ^3(e+f x)-18 \tan ^2(e+f x)-11 \tan (e+f x)+22\right)-\frac{210 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}-\frac{210 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}}{210 f}","-\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}+\frac{2 \sqrt{\tan (e+f x)+1} \tan ^3(e+f x)}{7 f}-\frac{12 \sqrt{\tan (e+f x)+1} \tan ^2(e+f x)}{35 f}-\frac{22 \sqrt{\tan (e+f x)+1} \tan (e+f x)}{105 f}+\frac{44 \sqrt{\tan (e+f x)+1}}{105 f}-\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}",1,"((-210*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] - (210*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 4*Sqrt[1 + Tan[e + f*x]]*(22 - 11*Tan[e + f*x] - 18*Tan[e + f*x]^2 + 15*Tan[e + f*x]^3))/(210*f)","C",1
401,1,90,187,0.208008,"\int \frac{\tan ^3(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Tan[e + f*x]^3/Sqrt[1 + Tan[e + f*x]],x]","\frac{4 (\tan (e+f x)-2) \sqrt{\tan (e+f x)+1}+\frac{6 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{6 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}}{6 f}","\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}+\frac{2 \tan (e+f x) \sqrt{\tan (e+f x)+1}}{3 f}-\frac{4 \sqrt{\tan (e+f x)+1}}{3 f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}",1,"((6*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (6*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 4*(-2 + Tan[e + f*x])*Sqrt[1 + Tan[e + f*x]])/(6*f)","C",1
402,1,67,143,0.032543,"\int \frac{\tan (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Tan[e + f*x]/Sqrt[1 + Tan[e + f*x]],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i} f}","-\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}-\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}",1,"-(ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]]/(Sqrt[1 - I]*f)) - ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]]/(Sqrt[1 + I]*f)","C",1
403,1,83,161,0.0741386,"\int \frac{\cot (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Cot[e + f*x]/Sqrt[1 + Tan[e + f*x]],x]","-\frac{2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i} f}","\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}-\frac{2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}",1,"(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f + ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]]/(Sqrt[1 - I]*f) + ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]]/(Sqrt[1 + I]*f)","C",1
404,1,125,215,0.3123176,"\int \frac{\cot ^3(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Cot[e + f*x]^3/Sqrt[1 + Tan[e + f*x]],x]","\frac{5 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}-\frac{4 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}-2 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+3 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{4 f}","-\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}+\frac{5 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{4 f}-\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}+\frac{3 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{4 f}",1,"(5*ArcTanh[Sqrt[1 + Tan[e + f*x]]] - (4*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] - (4*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] + 3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] - 2*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(4*f)","C",1
405,1,169,269,1.3604283,"\int \frac{\cot ^5(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Cot[e + f*x]^5/Sqrt[1 + Tan[e + f*x]],x]","\frac{-345 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+\frac{192 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{192 \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{\sqrt{1+i}}-48 \sqrt{\tan (e+f x)+1} \cot ^4(e+f x)+56 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)+26 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)-39 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{192 f}","\frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left(\frac{\left(1-\sqrt{2}\right) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left(5 \sqrt{2}-7\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}-\frac{115 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{64 f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left(\frac{\left(1+\sqrt{2}\right) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left(7+5 \sqrt{2}\right)} \sqrt{\tan (e+f x)+1}}\right)}{2 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}+\frac{7 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac{13 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}-\frac{13 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{64 f}",1,"(-345*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (192*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (192*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] - 39*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 26*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]] + 56*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 48*Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(192*f)","C",1
406,1,112,311,0.8281622,"\int \frac{\tan ^4(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Tan[e + f*x]^4/Sqrt[1 + Tan[e + f*x]],x]","\frac{15 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+15 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)-4 \sqrt{\tan (e+f x)+1} \sec ^2(e+f x) (2 \sin (2 (e+f x))+5 \cos (2 (e+f x))+2)}{30 f}","-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}+\frac{2 \sqrt{\tan (e+f x)+1} \tan ^2(e+f x)}{5 f}-\frac{8 \sqrt{\tan (e+f x)+1} \tan (e+f x)}{15 f}-\frac{14 \sqrt{\tan (e+f x)+1}}{15 f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}",1,"(15*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 15*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] - 4*Sec[e + f*x]^2*(2 + 5*Cos[2*(e + f*x)] + 2*Sin[2*(e + f*x)])*Sqrt[1 + Tan[e + f*x]])/(30*f)","C",1
407,1,80,257,0.0965558,"\int \frac{\tan ^2(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Tan[e + f*x]^2/Sqrt[1 + Tan[e + f*x]],x]","-\frac{-4 \sqrt{\tan (e+f x)+1}+(1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{2 f}","\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}",1,"-1/2*((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] - 4*Sqrt[1 + Tan[e + f*x]])/f","C",1
408,1,66,240,0.0230173,"\int \frac{1}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[1/Sqrt[1 + Tan[e + f*x]],x]","\frac{(1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)}{2 f}","-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}",1,"((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/(2*f)","C",1
409,1,101,280,0.2385423,"\int \frac{\cot ^2(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Cot[e + f*x]^2/Sqrt[1 + Tan[e + f*x]],x]","-\frac{-2 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+(1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+(1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)+2 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{2 f}","\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}+\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}-\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}+\frac{\tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{f}-\frac{\sqrt{\tan (e+f x)+1} \cot (e+f x)}{f}",1,"-1/2*(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f","C",1
410,1,147,339,0.6171196,"\int \frac{\cot ^4(e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx","Integrate[Cot[e + f*x]^4/Sqrt[1 + Tan[e + f*x]],x]","\frac{-9 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)+12 (1-i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right)+12 (1+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right)-8 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)+10 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+9 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{24 f}","-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2 \left(1+\sqrt{2}\right)}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left(\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left(1+\sqrt{2}\right)}}{\sqrt{2 \left(\sqrt{2}-1\right)}}\right)}{2 f}-\frac{\log \left(\tan (e+f x)-\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}+\frac{\log \left(\tan (e+f x)+\sqrt{2 \left(1+\sqrt{2}\right)} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right)}{4 \sqrt{1+\sqrt{2}} f}-\frac{3 \tanh ^{-1}\left(\sqrt{\tan (e+f x)+1}\right)}{8 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}+\frac{5 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac{3 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{8 f}",1,"(-9*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 12*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 12*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 9*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 10*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]] - 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(24*f)","C",1
411,0,0,161,0.7721532,"\int (d \tan (e+f x))^n (a+a \tan (e+f x))^m \, dx","Integrate[(d*Tan[e + f*x])^n*(a + a*Tan[e + f*x])^m,x]","\int (d \tan (e+f x))^n (a+a \tan (e+f x))^m \, dx","\frac{(\tan (e+f x)+1)^{-m} (a \tan (e+f x)+a)^m (d \tan (e+f x))^{n+1} F_1(n+1;-m,1;n+2;-\tan (e+f x),-i \tan (e+f x))}{2 d f (n+1)}+\frac{(\tan (e+f x)+1)^{-m} (a \tan (e+f x)+a)^m (d \tan (e+f x))^{n+1} F_1(n+1;-m,1;n+2;-\tan (e+f x),i \tan (e+f x))}{2 d f (n+1)}",1,"Integrate[(d*Tan[e + f*x])^n*(a + a*Tan[e + f*x])^m, x]","F",-1
412,1,95,93,0.3093503,"\int \tan ^5(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^5*(a + b*Tan[c + d*x]),x]","-\frac{a \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}-\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan ^5(c+d x)}{5 d}-\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan (c+d x)}{d}","\frac{a \tan ^4(c+d x)}{4 d}-\frac{a \tan ^2(c+d x)}{2 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan ^5(c+d x)}{5 d}-\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan (c+d x)}{d}-b x",1,"-((b*ArcTan[Tan[c + d*x]])/d) + (b*Tan[c + d*x])/d - (b*Tan[c + d*x]^3)/(3*d) + (b*Tan[c + d*x]^5)/(5*d) - (a*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)","A",1
413,1,79,77,0.1987027,"\int \tan ^4(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^4*(a + b*Tan[c + d*x]),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 \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}","\frac{a \tan ^3(c+d x)}{3 d}-\frac{a \tan (c+d x)}{d}+a x+\frac{b \tan ^4(c+d x)}{4 d}-\frac{b \tan ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}",1,"(a*ArcTan[Tan[c + d*x]])/d - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d) - (b*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)","A",1
414,1,67,60,0.136065,"\int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x]),x]","\frac{a \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}+\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}","\frac{a \tan ^2(c+d x)}{2 d}+\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x",1,"(b*ArcTan[Tan[c + d*x]])/d - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d) + (a*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
415,1,51,44,0.1043649,"\int \tan ^2(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x]),x]","-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{b \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{a \tan (c+d x)}{d}-a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d}",1,"-((a*ArcTan[Tan[c + d*x]])/d) + (a*Tan[c + d*x])/d + (b*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d)","A",1
416,1,38,29,0.020435,"\int \tan (c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x]),x]","-\frac{a \log (\cos (c+d x))}{d}-\frac{b \tan ^{-1}(\tan (c+d x))}{d}+\frac{b \tan (c+d x)}{d}","-\frac{a \log (\cos (c+d x))}{d}+\frac{b \tan (c+d x)}{d}-b x",1,"-((b*ArcTan[Tan[c + d*x]])/d) - (a*Log[Cos[c + d*x]])/d + (b*Tan[c + d*x])/d","A",1
417,1,17,17,0.004679,"\int (a+b \tan (c+d x)) \, dx","Integrate[a + b*Tan[c + d*x],x]","a x-\frac{b \log (\cos (c+d x))}{d}","a x-\frac{b \log (\cos (c+d x))}{d}",1,"a*x - (b*Log[Cos[c + d*x]])/d","A",1
418,1,24,16,0.0253032,"\int \cot (c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x]),x]","\frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b x","\frac{a \log (\sin (c+d x))}{d}+b x",1,"b*x + (a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
419,1,51,29,0.108661,"\int \cot ^2(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x]),x]","\frac{b (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}-\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 \cot (c+d x)}{d}-a x+\frac{b \log (\sin (c+d x))}{d}",1,"-((a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d) + (b*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
420,1,66,46,0.2912875,"\int \cot ^3(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[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 \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}","-\frac{a \cot ^2(c+d x)}{2 d}-\frac{a \log (\sin (c+d x))}{d}-\frac{b \cot (c+d x)}{d}-b x",1,"-((b*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d) - (a*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d)","C",1
421,1,70,60,0.2868579,"\int \cot ^4(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[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 \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}+\frac{a \cot (c+d x)}{d}+a x-\frac{b \cot ^2(c+d x)}{2 d}-\frac{b \log (\sin (c+d x))}{d}",1,"-1/3*(a*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d - (b*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d)","C",1
422,1,82,75,0.3975558,"\int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[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 \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 \cot ^4(c+d x)}{4 d}+\frac{a \cot ^2(c+d x)}{2 d}+\frac{a \log (\sin (c+d x))}{d}-\frac{b \cot ^3(c+d x)}{3 d}+\frac{b \cot (c+d x)}{d}+b x",1,"-1/3*(b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/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)","C",1
423,1,82,93,0.3563415,"\int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x]),x]","\frac{b \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 \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 \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{b \cot ^4(c+d x)}{4 d}+\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\sin (c+d x))}{d}",1,"-1/5*(a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/d + (b*(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)","C",1
424,1,110,120,0.6153293,"\int \tan ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","\frac{30 \left(a^2-b^2\right) \tan ^{-1}(\tan (c+d x))+10 \left(a^2-b^2\right) \tan ^3(c+d x)-30 \left(a^2-b^2\right) \tan (c+d x)+15 a b \tan ^4(c+d x)-30 a b \tan ^2(c+d x)-60 a b \log (\cos (c+d x))+6 b^2 \tan ^5(c+d x)}{30 d}","\frac{\left(a^2-b^2\right) \tan ^3(c+d x)}{3 d}-\frac{\left(a^2-b^2\right) \tan (c+d x)}{d}+x \left(a^2-b^2\right)+\frac{a b \tan ^4(c+d x)}{2 d}-\frac{a b \tan ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(30*(a^2 - b^2)*ArcTan[Tan[c + d*x]] - 60*a*b*Log[Cos[c + d*x]] - 30*(a^2 - b^2)*Tan[c + d*x] - 30*a*b*Tan[c + d*x]^2 + 10*(a^2 - b^2)*Tan[c + d*x]^3 + 15*a*b*Tan[c + d*x]^4 + 6*b^2*Tan[c + d*x]^5)/(30*d)","A",1
425,1,113,98,0.4438368,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}+\frac{2 a b \tan ^{-1}(\tan (c+d x))}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}-\frac{2 a b \tan (c+d x)}{d}-\frac{b^2 \left(-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right)}{4 d}","\frac{\left(a^2-b^2\right) \tan ^2(c+d x)}{2 d}+\frac{\left(a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}-\frac{2 a b \tan (c+d x)}{d}+2 a b x+\frac{b^2 \tan ^4(c+d x)}{4 d}",1,"(2*a*b*ArcTan[Tan[c + d*x]])/d - (2*a*b*Tan[c + d*x])/d + (2*a*b*Tan[c + d*x]^3)/(3*d) + (a^2*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/(2*d) - (b^2*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4))/(4*d)","A",1
426,1,99,63,0.3220857,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \tan ^{-1}(\tan (c+d x))}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \left(\tan ^2(c+d x)+2 \log (\cos (c+d x))\right)}{d}+\frac{b^2 \tan ^{-1}(\tan (c+d x))}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d}-\frac{b^2 \tan (c+d x)}{d}","-x \left(a^2-b^2\right)+\frac{(a+b \tan (c+d x))^3}{3 b d}+\frac{2 a b \log (\cos (c+d x))}{d}-\frac{b^2 \tan (c+d x)}{d}",1,"-((a^2*ArcTan[Tan[c + d*x]])/d) + (b^2*ArcTan[Tan[c + d*x]])/d + (a^2*Tan[c + d*x])/d - (b^2*Tan[c + d*x])/d + (b^2*Tan[c + d*x]^3)/(3*d) + (a*b*(2*Log[Cos[c + d*x]] + Tan[c + d*x]^2))/d","A",1
427,1,74,58,0.2774939,"\int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^2,x]","\frac{4 a b \tan (c+d x)+(a-i b)^2 \log (\tan (c+d x)+i)+(a+i b)^2 \log (-\tan (c+d x)+i)+b^2 \tan ^2(c+d x)}{2 d}","-\frac{\left(a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{(a+b \tan (c+d x))^2}{2 d}+\frac{a b \tan (c+d x)}{d}-2 a b x",1,"((a + I*b)^2*Log[I - Tan[c + d*x]] + (a - I*b)^2*Log[I + Tan[c + d*x]] + 4*a*b*Tan[c + d*x] + b^2*Tan[c + d*x]^2)/(2*d)","C",1
428,1,69,39,0.1088687,"\int (a+b \tan (c+d x))^2 \, dx","Integrate[(a + b*Tan[c + d*x])^2,x]","\frac{2 b^2 \tan (c+d x)-i \left((a+i b)^2 \log (-\tan (c+d x)+i)-(a-i b)^2 \log (\tan (c+d x)+i)\right)}{2 d}","x \left(a^2-b^2\right)-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"((-I)*((a + I*b)^2*Log[I - Tan[c + d*x]] - (a - I*b)^2*Log[I + Tan[c + d*x]]) + 2*b^2*Tan[c + d*x])/(2*d)","C",1
429,1,43,35,0.05328,"\int \cot (c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+2 a b x-\frac{b^2 \log (\cos (c+d x))}{d}","\frac{a^2 \log (\sin (c+d x))}{d}+2 a b x-\frac{b^2 \log (\cos (c+d x))}{d}",1,"2*a*b*x - (b^2*Log[Cos[c + d*x]])/d + (a^2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
430,1,82,41,0.1519447,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","\frac{-2 a^2 \cot (c+d x)+i \left((a-i b)^2 (-\log (\tan (c+d x)+i))+(a+i b)^2 \log (-\tan (c+d x)+i)-4 i a b \log (\tan (c+d x))\right)}{2 d}","-x \left(a^2-b^2\right)-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}",1,"(-2*a^2*Cot[c + d*x] + I*((a + I*b)^2*Log[I - Tan[c + d*x]] - (4*I)*a*b*Log[Tan[c + d*x]] - (a - I*b)^2*Log[I + Tan[c + d*x]]))/(2*d)","C",1
431,1,92,58,0.2598483,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{-a^2 \cot ^2(c+d x)-4 a b \cot (c+d x)+(a-i b)^2 \log (\tan (c+d x)+i)+(a+i b)^2 \log (-\tan (c+d x)+i)-2 (a-b) (a+b) \log (\tan (c+d x))}{2 d}","-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^2(c+d x)}{2 d}-\frac{2 a b \cot (c+d x)}{d}-2 a b x",1,"(-4*a*b*Cot[c + d*x] - a^2*Cot[c + d*x]^2 + (a + I*b)^2*Log[I - Tan[c + d*x]] - 2*(a - b)*(a + b)*Log[Tan[c + d*x]] + (a - I*b)^2*Log[I + Tan[c + d*x]])/(2*d)","C",1
432,1,103,78,0.6926586,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \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 b \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{d}-\frac{b^2 \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}","\frac{\left(a^2-b^2\right) \cot (c+d x)}{d}+x \left(a^2-b^2\right)-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^2(c+d x)}{d}-\frac{2 a b \log (\sin (c+d x))}{d}",1,"-1/3*(a^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/d - (b^2*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d - (a*b*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/d","C",1
433,1,122,98,0.5414456,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]","\frac{a^2 \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{2 a b \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^2 \left(\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","\frac{\left(a^2-b^2\right) \cot ^2(c+d x)}{2 d}+\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^4(c+d x)}{4 d}-\frac{2 a b \cot ^3(c+d x)}{3 d}+\frac{2 a b \cot (c+d x)}{d}+2 a b x",1,"(-2*a*b*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) - (b^2*(Cot[c + d*x]^2 + 2*Log[Cos[c + d*x]] + 2*Log[Tan[c + d*x]]))/(2*d) + (a^2*(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)","C",1
434,1,121,120,1.0293198,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \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 b \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)}{2 d}-\frac{b^2 \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)}{3 d}","\frac{\left(a^2-b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2-b^2\right) \cot (c+d x)}{d}-x \left(a^2-b^2\right)-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a b \cot ^4(c+d x)}{2 d}+\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\sin (c+d x))}{d}",1,"-1/5*(a^2*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/d - (b^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (a*b*(2*Cot[c + d*x]^2 - Cot[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/(2*d)","C",1
435,1,161,147,0.7907355,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","\frac{-3 \left(a^5+10 b^2 (a+i b)^3 \log (-\tan (c+d x)+i)+10 b^2 (a-i b)^3 \log (\tan (c+d x)+i)\right)+30 a b^2 \left(a^2-3 b^2\right) \tan ^2(c+d x)-20 b^3 \left(b^2-3 a^2\right) \tan ^3(c+d x)+60 b^3 \left(b^2-3 a^2\right) \tan (c+d x)+45 a b^4 \tan ^4(c+d x)+12 b^5 \tan ^5(c+d x)}{60 b^2 d}","-\frac{b \left(a^2-b^2\right) \tan (c+d x)}{d}+\frac{a \left(a^2-3 b^2\right) \log (\cos (c+d x))}{d}+b x \left(3 a^2-b^2\right)-\frac{a (a+b \tan (c+d x))^4}{20 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}-\frac{(a+b \tan (c+d x))^3}{3 d}-\frac{a (a+b \tan (c+d x))^2}{2 d}",1,"(-3*(a^5 + 10*(a + I*b)^3*b^2*Log[I - Tan[c + d*x]] + 10*(a - I*b)^3*b^2*Log[I + Tan[c + d*x]]) + 60*b^3*(-3*a^2 + b^2)*Tan[c + d*x] + 30*a*b^2*(a^2 - 3*b^2)*Tan[c + d*x]^2 - 20*b^3*(-3*a^2 + b^2)*Tan[c + d*x]^3 + 45*a*b^4*Tan[c + d*x]^4 + 12*b^5*Tan[c + d*x]^5)/(60*b^2*d)","C",1
436,1,97,94,0.9973313,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","\frac{-12 a b^2 \tan (c+d x)+\frac{(a+b \tan (c+d x))^4}{b}+2 i (a+i b)^3 \log (-\tan (c+d x)+i)+2 (b+i a)^3 \log (\tan (c+d x)+i)-2 b^3 \tan ^2(c+d x)}{4 d}","\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}-a x \left(a^2-3 b^2\right)-\frac{2 a b^2 \tan (c+d x)}{d}+\frac{(a+b \tan (c+d x))^4}{4 b d}-\frac{b (a+b \tan (c+d x))^2}{2 d}",1,"((2*I)*(a + I*b)^3*Log[I - Tan[c + d*x]] + 2*(I*a + b)^3*Log[I + Tan[c + d*x]] - 12*a*b^2*Tan[c + d*x] - 2*b^3*Tan[c + d*x]^2 + (a + b*Tan[c + d*x])^4/b)/(4*d)","C",1
437,1,100,97,0.5596026,"\int \tan (c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^3,x]","\frac{-6 b \left(b^2-3 a^2\right) \tan (c+d x)+9 a b^2 \tan ^2(c+d x)+3 \left((a-i b)^3 \log (\tan (c+d x)+i)+(a+i b)^3 \log (-\tan (c+d x)+i)\right)+2 b^3 \tan ^3(c+d x)}{6 d}","\frac{b \left(a^2-b^2\right) \tan (c+d x)}{d}-\frac{a \left(a^2-3 b^2\right) \log (\cos (c+d x))}{d}-b x \left(3 a^2-b^2\right)+\frac{(a+b \tan (c+d x))^3}{3 d}+\frac{a (a+b \tan (c+d x))^2}{2 d}",1,"(3*((a + I*b)^3*Log[I - Tan[c + d*x]] + (a - I*b)^3*Log[I + Tan[c + d*x]]) - 6*b*(-3*a^2 + b^2)*Tan[c + d*x] + 9*a*b^2*Tan[c + d*x]^2 + 2*b^3*Tan[c + d*x]^3)/(6*d)","C",1
438,1,79,72,0.2359545,"\int (a+b \tan (c+d x))^3 \, dx","Integrate[(a + b*Tan[c + d*x])^3,x]","\frac{6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)}{2 d}","-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+a x \left(a^2-3 b^2\right)+\frac{2 a b^2 \tan (c+d x)}{d}+\frac{b (a+b \tan (c+d x))^2}{2 d}",1,"((I*a - b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2)/(2*d)","C",1
439,1,79,62,0.1960894,"\int \cot (c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^3,x]","-\frac{-2 a^3 \log (\tan (c+d x))-2 b^2 (a+b \tan (c+d x))+(a+i b)^3 \log (-\tan (c+d x)+i)+(a-i b)^3 \log (\tan (c+d x)+i)}{2 d}","\frac{a^3 \log (\sin (c+d x))}{d}+b x \left(3 a^2-b^2\right)+\frac{b^2 (a+b \tan (c+d x))}{d}-\frac{3 a b^2 \log (\cos (c+d x))}{d}",1,"-1/2*((a + I*b)^3*Log[I - Tan[c + d*x]] - 2*a^3*Log[Tan[c + d*x]] + (a - I*b)^3*Log[I + Tan[c + d*x]] - 2*b^2*(a + b*Tan[c + d*x]))/d","C",1
440,1,80,69,0.1340573,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","-\frac{a^3 \cot (c+d x)-\frac{1}{2} (b+i a)^3 \log (-\cot (c+d x)+i)+\frac{1}{2} (-b+i a)^3 \log (\cot (c+d x)+i)-b^3 \log (\tan (c+d x))}{d}","-a x \left(a^2-3 b^2\right)+\frac{3 a^2 b \log (\sin (c+d x))}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}-\frac{b^3 \log (\cos (c+d x))}{d}",1,"-((a^3*Cot[c + d*x] - ((I*a + b)^3*Log[I - Cot[c + d*x]])/2 + ((I*a - b)^3*Log[I + Cot[c + d*x]])/2 - b^3*Log[Tan[c + d*x]])/d)","C",1
441,1,96,83,0.3275021,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","\frac{a^3 \left(-\cot ^2(c+d x)\right)-2 a \left(a^2-3 b^2\right) \log (\tan (c+d x))-6 a^2 b \cot (c+d x)+(a+i b)^3 \log (-\tan (c+d x)+i)+(a-i b)^3 \log (\tan (c+d x)+i)}{2 d}","-\frac{a \left(a^2-3 b^2\right) \log (\sin (c+d x))}{d}-b x \left(3 a^2-b^2\right)-\frac{5 a^2 b \cot (c+d x)}{2 d}-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}",1,"(-6*a^2*b*Cot[c + d*x] - a^3*Cot[c + d*x]^2 + (a + I*b)^3*Log[I - Tan[c + d*x]] - 2*a*(a^2 - 3*b^2)*Log[Tan[c + d*x]] + (a - I*b)^3*Log[I + Tan[c + d*x]])/(2*d)","C",1
442,1,120,104,1.1143158,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]","\frac{-2 a^3 \cot ^3(c+d x)+6 a \left(a^2-3 b^2\right) \cot (c+d x)+6 b \left(b^2-3 a^2\right) \log (\tan (c+d x))-9 a^2 b \cot ^2(c+d x)+3 (-b+i a)^3 \log (-\tan (c+d x)+i)-3 (b+i a)^3 \log (\tan (c+d x)+i)}{6 d}","\frac{a \left(a^2-3 b^2\right) \cot (c+d x)}{d}-\frac{b \left(3 a^2-b^2\right) \log (\sin (c+d x))}{d}+a x \left(a^2-3 b^2\right)-\frac{7 a^2 b \cot ^2(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))}{3 d}",1,"(6*a*(a^2 - 3*b^2)*Cot[c + d*x] - 9*a^2*b*Cot[c + d*x]^2 - 2*a^3*Cot[c + d*x]^3 + 3*(I*a - b)^3*Log[I - Tan[c + d*x]] + 6*b*(-3*a^2 + b^2)*Log[Tan[c + d*x]] - 3*(I*a + b)^3*Log[I + Tan[c + d*x]])/(6*d)","C",1
443,1,118,130,1.5986811,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]","-\frac{a^3 \cot ^4(c+d x)-2 a \left(a^2-3 b^2\right) \cot ^2(c+d x)+4 b \left(b^2-3 a^2\right) \cot (c+d x)+4 a^2 b \cot ^3(c+d x)+2 (a-i b)^3 \log (-\cot (c+d x)+i)+2 (a+i b)^3 \log (\cot (c+d x)+i)}{4 d}","\frac{a \left(a^2-3 b^2\right) \cot ^2(c+d x)}{2 d}+\frac{b \left(3 a^2-b^2\right) \cot (c+d x)}{d}+\frac{a \left(a^2-3 b^2\right) \log (\sin (c+d x))}{d}+b x \left(3 a^2-b^2\right)-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}",1,"-1/4*(4*b*(-3*a^2 + b^2)*Cot[c + d*x] - 2*a*(a^2 - 3*b^2)*Cot[c + d*x]^2 + 4*a^2*b*Cot[c + d*x]^3 + a^3*Cot[c + d*x]^4 + 2*(a - I*b)^3*Log[I - Cot[c + d*x]] + 2*(a + I*b)^3*Log[I + Cot[c + d*x]])/d","C",1
444,1,152,157,0.9678282,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{1}{5} a^3 \cot ^5(c+d x)-\frac{1}{3} a \left(a^2-3 b^2\right) \cot ^3(c+d x)-\frac{1}{2} b \left(3 a^2-b^2\right) \cot ^2(c+d x)+a \left(a^2-3 b^2\right) \cot (c+d x)+\frac{3}{4} a^2 b \cot ^4(c+d x)-\frac{1}{2} (b+i a)^3 \log (-\cot (c+d x)+i)+\frac{1}{2} (-b+i a)^3 \log (\cot (c+d x)+i)}{d}","\frac{a \left(a^2-3 b^2\right) \cot ^3(c+d x)}{3 d}+\frac{b \left(3 a^2-b^2\right) \cot ^2(c+d x)}{2 d}-\frac{a \left(a^2-3 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2-b^2\right) \log (\sin (c+d x))}{d}-a x \left(a^2-3 b^2\right)-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}",1,"-((a*(a^2 - 3*b^2)*Cot[c + d*x] - (b*(3*a^2 - b^2)*Cot[c + d*x]^2)/2 - (a*(a^2 - 3*b^2)*Cot[c + d*x]^3)/3 + (3*a^2*b*Cot[c + d*x]^4)/4 + (a^3*Cot[c + d*x]^5)/5 - ((I*a + b)^3*Log[I - Cot[c + d*x]])/2 + ((I*a - b)^3*Log[I + Cot[c + d*x]])/2)/d)","C",1
445,1,190,181,1.1881021,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]","\frac{-2 \left(a^6+15 b^2 (a+i b)^4 \log (-\tan (c+d x)+i)+15 b^2 (a-i b)^4 \log (\tan (c+d x)+i)\right)-15 b^4 \left(b^2-6 a^2\right) \tan ^4(c+d x)+80 a b^3 \left(a^2-b^2\right) \tan ^3(c+d x)-240 a b^3 \left(a^2-b^2\right) \tan (c+d x)+30 b^2 \left(a^4-6 a^2 b^2+b^4\right) \tan ^2(c+d x)+48 a b^5 \tan ^5(c+d x)+10 b^6 \tan ^6(c+d x)}{60 b^2 d}","-\frac{\left(a^2-b^2\right) (a+b \tan (c+d x))^2}{2 d}-\frac{a b \left(a^2-3 b^2\right) \tan (c+d x)}{d}+4 a b x \left(a^2-b^2\right)+\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (\cos (c+d x))}{d}-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^3}{3 d}",1,"(-2*(a^6 + 15*(a + I*b)^4*b^2*Log[I - Tan[c + d*x]] + 15*(a - I*b)^4*b^2*Log[I + Tan[c + d*x]]) - 240*a*b^3*(a^2 - b^2)*Tan[c + d*x] + 30*b^2*(a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]^2 + 80*a*b^3*(a^2 - b^2)*Tan[c + d*x]^3 - 15*b^4*(-6*a^2 + b^2)*Tan[c + d*x]^4 + 48*a*b^5*Tan[c + d*x]^5 + 10*b^6*Tan[c + d*x]^6)/(60*b^2*d)","C",1
446,1,122,128,0.7478886,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]","\frac{30 b^2 \left(b^2-6 a^2\right) \tan (c+d x)-60 a b^3 \tan ^2(c+d x)+\frac{6 (a+b \tan (c+d x))^5}{b}+15 i (a+i b)^4 \log (-\tan (c+d x)+i)-15 i (a-i b)^4 \log (\tan (c+d x)+i)-10 b^4 \tan ^3(c+d x)}{30 d}","-\frac{b^2 \left(3 a^2-b^2\right) \tan (c+d x)}{d}+\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}-x \left(a^4-6 a^2 b^2+b^4\right)+\frac{(a+b \tan (c+d x))^5}{5 b d}-\frac{b (a+b \tan (c+d x))^3}{3 d}-\frac{a b (a+b \tan (c+d x))^2}{d}",1,"((15*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (15*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 30*b^2*(-6*a^2 + b^2)*Tan[c + d*x] - 60*a*b^3*Tan[c + d*x]^2 - 10*b^4*Tan[c + d*x]^3 + (6*(a + b*Tan[c + d*x])^5)/b)/(30*d)","C",1
447,1,123,130,1.6574173,"\int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^4,x]","\frac{-6 b^2 \left(b^2-6 a^2\right) \tan ^2(c+d x)+48 a b \left(a^2-b^2\right) \tan (c+d x)+16 a b^3 \tan ^3(c+d x)+6 \left((a-i b)^4 \log (\tan (c+d x)+i)+(a+i b)^4 \log (-\tan (c+d x)+i)\right)+3 b^4 \tan ^4(c+d x)}{12 d}","\frac{\left(a^2-b^2\right) (a+b \tan (c+d x))^2}{2 d}+\frac{a b \left(a^2-3 b^2\right) \tan (c+d x)}{d}-4 a b x \left(a^2-b^2\right)-\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (\cos (c+d x))}{d}+\frac{(a+b \tan (c+d x))^4}{4 d}+\frac{a (a+b \tan (c+d x))^3}{3 d}",1,"(6*((a + I*b)^4*Log[I - Tan[c + d*x]] + (a - I*b)^4*Log[I + Tan[c + d*x]]) + 48*a*b*(a^2 - b^2)*Tan[c + d*x] - 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^2 + 16*a*b^3*Tan[c + d*x]^3 + 3*b^4*Tan[c + d*x]^4)/(12*d)","C",1
448,1,105,103,0.3742975,"\int (a+b \tan (c+d x))^4 \, dx","Integrate[(a + b*Tan[c + d*x])^4,x]","\frac{-6 b^2 \left(b^2-6 a^2\right) \tan (c+d x)+12 a b^3 \tan ^2(c+d x)+3 i (a-i b)^4 \log (\tan (c+d x)+i)-3 i (a+i b)^4 \log (-\tan (c+d x)+i)+2 b^4 \tan ^3(c+d x)}{6 d}","\frac{b^2 \left(3 a^2-b^2\right) \tan (c+d x)}{d}-\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}+x \left(a^4-6 a^2 b^2+b^4\right)+\frac{b (a+b \tan (c+d x))^3}{3 d}+\frac{a b (a+b \tan (c+d x))^2}{d}",1,"((-3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] + (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] - 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] + 12*a*b^3*Tan[c + d*x]^2 + 2*b^4*Tan[c + d*x]^3)/(6*d)","C",1
449,1,94,92,0.4142479,"\int \cot (c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^4,x]","\frac{2 a^4 \log (\tan (c+d x))+6 a b^3 \tan (c+d x)+b^2 (a+b \tan (c+d x))^2-(a+i b)^4 \log (-\tan (c+d x)+i)-(a-i b)^4 \log (\tan (c+d x)+i)}{2 d}","\frac{a^4 \log (\sin (c+d x))}{d}-\frac{b^2 \left(6 a^2-b^2\right) \log (\cos (c+d x))}{d}+4 a b x \left(a^2-b^2\right)+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 (a+b \tan (c+d x))^2}{2 d}",1,"(-((a + I*b)^4*Log[I - Tan[c + d*x]]) + 2*a^4*Log[Tan[c + d*x]] - (a - I*b)^4*Log[I + Tan[c + d*x]] + 6*a*b^3*Tan[c + d*x] + b^2*(a + b*Tan[c + d*x])^2)/(2*d)","C",1
450,1,94,97,0.2707462,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]","-\frac{a^4 \cot (c+d x)-4 a b^3 \log (\tan (c+d x))+\frac{1}{2} i (a-i b)^4 \log (-\cot (c+d x)+i)-\frac{1}{2} i (a+i b)^4 \log (\cot (c+d x)+i)-b^4 \tan (c+d x)}{d}","\frac{4 a^3 b \log (\sin (c+d x))}{d}+\frac{b^2 \left(a^2+b^2\right) \tan (c+d x)}{d}-\frac{a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-x \left(a^4-6 a^2 b^2+b^4\right)-\frac{4 a b^3 \log (\cos (c+d x))}{d}",1,"-((a^4*Cot[c + d*x] + (I/2)*(a - I*b)^4*Log[I - Cot[c + d*x]] - (I/2)*(a + I*b)^4*Log[I + Cot[c + d*x]] - 4*a*b^3*Log[Tan[c + d*x]] - b^4*Tan[c + d*x])/d)","C",1
451,1,90,99,0.3151234,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]","-\frac{a^4 \cot ^2(c+d x)+8 a^3 b \cot (c+d x)-(a-i b)^4 \log (-\cot (c+d x)+i)-(a+i b)^4 \log (\cot (c+d x)+i)-2 b^4 \log (\tan (c+d x))}{2 d}","-\frac{3 a^3 b \cot (c+d x)}{d}-\frac{a^2 \left(a^2-6 b^2\right) \log (\sin (c+d x))}{d}-4 a b x \left(a^2-b^2\right)-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{b^4 \log (\cos (c+d x))}{d}",1,"-1/2*(8*a^3*b*Cot[c + d*x] + a^4*Cot[c + d*x]^2 - (a - I*b)^4*Log[I - Cot[c + d*x]] - (a + I*b)^4*Log[I + Cot[c + d*x]] - 2*b^4*Log[Tan[c + d*x]])/d","C",1
452,1,125,117,1.2739289,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^4,x]","-\frac{2 a^4 \cot ^3(c+d x)+12 a^3 b \cot ^2(c+d x)-6 a^2 \left(a^2-6 b^2\right) \cot (c+d x)+24 a b \left(a^2-b^2\right) \log (\tan (c+d x))+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-3 i (a-i b)^4 \log (\tan (c+d x)+i)}{6 d}","-\frac{4 a^3 b \cot ^2(c+d x)}{3 d}+\frac{a^2 \left(3 a^2-17 b^2\right) \cot (c+d x)}{3 d}-\frac{4 a b \left(a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+x \left(a^4-6 a^2 b^2+b^4\right)",1,"-1/6*(-6*a^2*(a^2 - 6*b^2)*Cot[c + d*x] + 12*a^3*b*Cot[c + d*x]^2 + 2*a^4*Cot[c + d*x]^3 + (3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] + 24*a*b*(a^2 - b^2)*Log[Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]])/d","C",1
453,1,147,141,2.3705537,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]","\frac{-3 a^4 \cot ^4(c+d x)-16 a^3 b \cot ^3(c+d x)+6 a^2 \left(a^2-6 b^2\right) \cot ^2(c+d x)+48 a b \left(a^2-b^2\right) \cot (c+d x)-6 \left(-2 \left(a^4-6 a^2 b^2+b^4\right) \log (\tan (c+d x))+(a-i b)^4 \log (\tan (c+d x)+i)+(a+i b)^4 \log (-\tan (c+d x)+i)\right)}{12 d}","-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}+\frac{a^2 \left(2 a^2-11 b^2\right) \cot ^2(c+d x)}{4 d}+\frac{4 a b \left(a^2-b^2\right) \cot (c+d x)}{d}+4 a b x \left(a^2-b^2\right)-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (\sin (c+d x))}{d}",1,"(48*a*b*(a^2 - b^2)*Cot[c + d*x] + 6*a^2*(a^2 - 6*b^2)*Cot[c + d*x]^2 - 16*a^3*b*Cot[c + d*x]^3 - 3*a^4*Cot[c + d*x]^4 - 6*((a + I*b)^4*Log[I - Tan[c + d*x]] - 2*(a^4 - 6*a^2*b^2 + b^4)*Log[Tan[c + d*x]] + (a - I*b)^4*Log[I + Tan[c + d*x]]))/(12*d)","C",1
454,1,154,170,0.4192516,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^4,x]","-\frac{\frac{1}{5} a^4 \cot ^5(c+d x)+a^3 b \cot ^4(c+d x)-\frac{1}{3} a^2 \left(a^2-6 b^2\right) \cot ^3(c+d x)+\left(a^4-6 a^2 b^2+b^4\right) \cot (c+d x)-2 a b (a-b) (a+b) \cot ^2(c+d x)+\frac{1}{2} i (a-i b)^4 \log (-\cot (c+d x)+i)-\frac{1}{2} i (a+i b)^4 \log (\cot (c+d x)+i)}{d}","-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}+\frac{a^2 \left(5 a^2-27 b^2\right) \cot ^3(c+d x)}{15 d}+\frac{2 a b \left(a^2-b^2\right) \cot ^2(c+d x)}{d}+\frac{4 a b \left(a^2-b^2\right) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}-\frac{\left(a^4-6 a^2 b^2+b^4\right) \cot (c+d x)}{d}-x \left(a^4-6 a^2 b^2+b^4\right)",1,"-(((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x] - 2*a*(a - b)*b*(a + b)*Cot[c + d*x]^2 - (a^2*(a^2 - 6*b^2)*Cot[c + d*x]^3)/3 + a^3*b*Cot[c + d*x]^4 + (a^4*Cot[c + d*x]^5)/5 + (I/2)*(a - I*b)^4*Log[I - Cot[c + d*x]] - (I/2)*(a + I*b)^4*Log[I + Cot[c + d*x]])/d)","C",1
455,1,178,198,0.4975858,"\int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]","-\frac{\frac{1}{6} a^4 \cot ^6(c+d x)+\frac{4}{5} a^3 b \cot ^5(c+d x)-\frac{1}{4} a^2 \left(a^2-6 b^2\right) \cot ^4(c+d x)+\frac{1}{2} \left(a^4-6 a^2 b^2+b^4\right) \cot ^2(c+d x)-\frac{4}{3} a b (a-b) (a+b) \cot ^3(c+d x)+4 a b (a-b) (a+b) \cot (c+d x)-\frac{1}{2} (a-i b)^4 \log (-\cot (c+d x)+i)-\frac{1}{2} (a+i b)^4 \log (\cot (c+d x)+i)}{d}","-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}+\frac{a^2 \left(3 a^2-16 b^2\right) \cot ^4(c+d x)}{12 d}+\frac{4 a b \left(a^2-b^2\right) \cot ^3(c+d x)}{3 d}-\frac{4 a b \left(a^2-b^2\right) \cot (c+d x)}{d}-4 a b x \left(a^2-b^2\right)-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}-\frac{\left(a^4-6 a^2 b^2+b^4\right) \cot ^2(c+d x)}{2 d}-\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (\sin (c+d x))}{d}",1,"-((4*a*(a - b)*b*(a + b)*Cot[c + d*x] + ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/2 - (4*a*(a - b)*b*(a + b)*Cot[c + d*x]^3)/3 - (a^2*(a^2 - 6*b^2)*Cot[c + d*x]^4)/4 + (4*a^3*b*Cot[c + d*x]^5)/5 + (a^4*Cot[c + d*x]^6)/6 - ((a - I*b)^4*Log[I - Cot[c + d*x]])/2 - ((a + I*b)^4*Log[I + Cot[c + d*x]])/2)/d)","C",1
456,1,167,154,1.7261494,"\int \frac{\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x]),x]","\frac{6 \left(2 a^6 \log (a+b \tan (c+d x))+b^5 (b+i a) \log (-\tan (c+d x)+i)+b^5 (b-i a) \log (\tan (c+d x)+i)\right)-12 a b \left(a^4-b^4\right) \tan (c+d x)+6 b^2 \left(a^4-b^4\right) \tan ^2(c+d x)+3 b^4 \left(a^2+b^2\right) \tan ^4(c+d x)-4 a b^3 \left(a^2+b^2\right) \tan ^3(c+d x)}{12 b^5 d \left(a^2+b^2\right)}","-\frac{b \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{a x}{a^2+b^2}-\frac{a \left(a^2-b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2-b^2\right) \tan ^2(c+d x)}{2 b^3 d}+\frac{a^6 \log (a+b \tan (c+d x))}{b^5 d \left(a^2+b^2\right)}-\frac{a \tan ^3(c+d x)}{3 b^2 d}+\frac{\tan ^4(c+d x)}{4 b d}",1,"(6*(b^5*(I*a + b)*Log[I - Tan[c + d*x]] + b^5*((-I)*a + b)*Log[I + Tan[c + d*x]] + 2*a^6*Log[a + b*Tan[c + d*x]]) - 12*a*b*(a^4 - b^4)*Tan[c + d*x] + 6*b^2*(a^4 - b^4)*Tan[c + d*x]^2 - 4*a*b^3*(a^2 + b^2)*Tan[c + d*x]^3 + 3*b^4*(a^2 + b^2)*Tan[c + d*x]^4)/(12*b^5*(a^2 + b^2)*d)","C",1
457,1,155,125,0.6112036,"\int \frac{\tan ^5(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x]),x]","\frac{a^2 \tan (c+d x)}{b^3 d}-\frac{a^5 \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)}-\frac{a \tan ^2(c+d x)}{2 b^2 d}+\frac{\log (-\tan (c+d x)+i)}{2 d (a+i b)}+\frac{\log (\tan (c+d x)+i)}{2 d (a-i b)}+\frac{\tan ^3(c+d x)}{3 b d}-\frac{\tan (c+d x)}{b d}","-\frac{a \log (\cos (c+d x))}{d \left(a^2+b^2\right)}+\frac{b x}{a^2+b^2}+\frac{\left(a^2-b^2\right) \tan (c+d x)}{b^3 d}-\frac{a^5 \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)}-\frac{a \tan ^2(c+d x)}{2 b^2 d}+\frac{\tan ^3(c+d x)}{3 b d}",1,"Log[I - Tan[c + d*x]]/(2*(a + I*b)*d) + Log[I + Tan[c + d*x]]/(2*(a - I*b)*d) - (a^5*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)*d) + (a^2*Tan[c + d*x])/(b^3*d) - Tan[c + d*x]/(b*d) - (a*Tan[c + d*x]^2)/(2*b^2*d) + Tan[c + d*x]^3/(3*b*d)","C",1
458,1,107,97,0.4122906,"\int \frac{\tan ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 a^4 \log (a+b \tan (c+d x))}{b^3 \left(a^2+b^2\right)}-\frac{2 a \tan (c+d x)}{b^2}+\frac{\log (-\tan (c+d x)+i)}{-b+i a}-\frac{\log (\tan (c+d x)+i)}{b+i a}+\frac{\tan ^2(c+d x)}{b}}{2 d}","\frac{b \log (\cos (c+d x))}{d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}+\frac{a^4 \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan ^2(c+d x)}{2 b d}",1,"(Log[I - Tan[c + d*x]]/(I*a - b) - Log[I + Tan[c + d*x]]/(I*a + b) + (2*a^4*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)) - (2*a*Tan[c + d*x])/b^2 + Tan[c + d*x]^2/b)/(2*d)","C",1
459,1,91,79,0.4072969,"\int \frac{\tan ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x]),x]","-\frac{\frac{2 a^3 \log (a+b \tan (c+d x))}{b^2 \left(a^2+b^2\right)}+\frac{\log (-\tan (c+d x)+i)}{a+i b}+\frac{\log (\tan (c+d x)+i)}{a-i b}-\frac{2 \tan (c+d x)}{b}}{2 d}","\frac{a \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{b x}{a^2+b^2}-\frac{a^3 \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)}+\frac{\tan (c+d x)}{b d}",1,"-1/2*(Log[I - Tan[c + d*x]]/(a + I*b) + Log[I + Tan[c + d*x]]/(a - I*b) + (2*a^3*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)) - (2*Tan[c + d*x])/b)/d","C",1
460,1,78,66,0.1021086,"\int \frac{\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x]),x]","\frac{2 a^2 \log (a+b \tan (c+d x))+b (b+i a) \log (-\tan (c+d x)+i)+b (b-i a) \log (\tan (c+d x)+i)}{2 b d \left(a^2+b^2\right)}","\frac{a^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b d \left(a^2+b^2\right)}-\frac{a x}{a^2+b^2}-\frac{\log (\cos (c+d x))}{b d}",1,"(b*(I*a + b)*Log[I - Tan[c + d*x]] + b*((-I)*a + b)*Log[I + Tan[c + d*x]] + 2*a^2*Log[a + b*Tan[c + d*x]])/(2*b*(a^2 + b^2)*d)","C",1
461,1,66,46,0.131861,"\int \frac{\tan (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{2 (b-i a) (c+d x)-a \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 i a \tan ^{-1}(\tan (c+d x))}{2 d \left(a^2+b^2\right)}","\frac{b x}{a^2+b^2}-\frac{a \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}",1,"(2*((-I)*a + b)*(c + d*x) + (2*I)*a*ArcTan[Tan[c + d*x]] - a*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])/(2*(a^2 + b^2)*d)","C",1
462,1,76,45,0.0646381,"\int \frac{1}{a+b \tan (c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(-1),x]","\frac{(-b-i a) \log (-\tan (c+d x)+i)+i (a+i b) \log (\tan (c+d x)+i)+2 b \log (a+b \tan (c+d x))}{2 d \left(a^2+b^2\right)}","\frac{b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}",1,"(((-I)*a - b)*Log[I - Tan[c + d*x]] + I*(a + I*b)*Log[I + Tan[c + d*x]] + 2*b*Log[a + b*Tan[c + d*x]])/(2*(a^2 + b^2)*d)","C",1
463,1,91,66,0.1868337,"\int \frac{\cot (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x]),x]","-\frac{\frac{2 b^2 \log (a+b \tan (c+d x))}{a^3+a b^2}+\frac{\log (-\tan (c+d x)+i)}{a+i b}+\frac{\log (\tan (c+d x)+i)}{a-i b}-\frac{2 \log (\tan (c+d x))}{a}}{2 d}","-\frac{b^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left(a^2+b^2\right)}-\frac{b x}{a^2+b^2}+\frac{\log (\sin (c+d x))}{a d}",1,"-1/2*(Log[I - Tan[c + d*x]]/(a + I*b) - (2*Log[Tan[c + d*x]])/a + Log[I + Tan[c + d*x]]/(a - I*b) + (2*b^2*Log[a + b*Tan[c + d*x]])/(a^3 + a*b^2))/d","C",1
464,1,96,81,0.4934219,"\int \frac{\cot ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x]),x]","-\frac{-\frac{b^3 \log (a \cot (c+d x)+b)}{a^2 \left(a^2+b^2\right)}-\frac{\log (-\cot (c+d x)+i)}{2 (b+i a)}+\frac{\log (\cot (c+d x)+i)}{2 (-b+i a)}+\frac{\cot (c+d x)}{a}}{d}","-\frac{a x}{a^2+b^2}+\frac{b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)}-\frac{b \log (\sin (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"-((Cot[c + d*x]/a - Log[I - Cot[c + d*x]]/(2*(I*a + b)) + Log[I + Cot[c + d*x]]/(2*(I*a - b)) - (b^3*Log[b + a*Cot[c + d*x]])/(a^2*(a^2 + b^2)))/d)","C",1
465,1,106,107,0.6798799,"\int \frac{\cot ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Tan[c + d*x]),x]","-\frac{-\frac{2 b \cot (c+d x)}{a^2}+\frac{2 b^4 \log (a \cot (c+d x)+b)}{a^3 \left(a^2+b^2\right)}-\frac{\log (-\cot (c+d x)+i)}{a-i b}-\frac{\log (\cot (c+d x)+i)}{a+i b}+\frac{\cot ^2(c+d x)}{a}}{2 d}","\frac{b x}{a^2+b^2}+\frac{b \cot (c+d x)}{a^2 d}-\frac{\left(a^2-b^2\right) \log (\sin (c+d x))}{a^3 d}-\frac{b^4 \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)}-\frac{\cot ^2(c+d x)}{2 a d}",1,"-1/2*((-2*b*Cot[c + d*x])/a^2 + Cot[c + d*x]^2/a - Log[I - Cot[c + d*x]]/(a - I*b) - Log[I + Cot[c + d*x]]/(a + I*b) + (2*b^4*Log[b + a*Cot[c + d*x]])/(a^3*(a^2 + b^2)))/d","C",1
466,1,131,133,1.9215538,"\int \frac{\cot ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^4/(a + b*Tan[c + d*x]),x]","-\frac{-\frac{3 b \cot ^2(c+d x)}{a^2}-\frac{6 b^5 \log (a \cot (c+d x)+b)}{a^4 \left(a^2+b^2\right)}-\frac{6 \left(a^2-b^2\right) \cot (c+d x)}{a^3}+\frac{3 \log (-\cot (c+d x)+i)}{b+i a}+\frac{3 i \log (\cot (c+d x)+i)}{a+i b}+\frac{2 \cot ^3(c+d x)}{a}}{6 d}","\frac{a x}{a^2+b^2}+\frac{b \cot ^2(c+d x)}{2 a^2 d}+\frac{b \left(a^2-b^2\right) \log (\sin (c+d x))}{a^4 d}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)}+\frac{\left(a^2-b^2\right) \cot (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a d}",1,"-1/6*((-6*(a^2 - b^2)*Cot[c + d*x])/a^3 - (3*b*Cot[c + d*x]^2)/a^2 + (2*Cot[c + d*x]^3)/a + (3*Log[I - Cot[c + d*x]])/(I*a + b) + ((3*I)*Log[I + Cot[c + d*x]])/(a + I*b) - (6*b^5*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)))/d","C",1
467,1,242,239,6.2336136,"\int \frac{\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]","\frac{\tan ^4(c+d x)}{3 b d (a+b \tan (c+d x))}+\frac{-\frac{2 a \tan ^3(c+d x)}{b d (a+b \tan (c+d x))}+\frac{-\frac{6 \left(1-\frac{2 a^2}{b^2}\right) \tan (c+d x)}{d}-\frac{12 a^5 \left(2 a^2+3 b^2\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^2}-\frac{6 a^4 \left(2 a^2+b^2\right)}{b^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{3 i b^2 \log (-\tan (c+d x)+i)}{d (a+i b)^2}-\frac{3 i b^2 \log (\tan (c+d x)+i)}{d (a-i b)^2}}{2 b}}{3 b}","-\frac{a^2 \tan ^4(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(4 a^2+b^2\right) \tan ^3(c+d x)}{3 b^2 d \left(a^2+b^2\right)}-\frac{2 a b \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}-\frac{a \left(2 a^2+b^2\right) \tan ^2(c+d x)}{b^3 d \left(a^2+b^2\right)}-\frac{2 a^5 \left(2 a^2+3 b^2\right) \log (a+b \tan (c+d x))}{b^5 d \left(a^2+b^2\right)^2}+\frac{\left(4 a^4+2 a^2 b^2-b^4\right) \tan (c+d x)}{b^4 d \left(a^2+b^2\right)}",1,"Tan[c + d*x]^4/(3*b*d*(a + b*Tan[c + d*x])) + ((-2*a*Tan[c + d*x]^3)/(b*d*(a + b*Tan[c + d*x])) + (((3*I)*b^2*Log[I - Tan[c + d*x]])/((a + I*b)^2*d) - ((3*I)*b^2*Log[I + Tan[c + d*x]])/((a - I*b)^2*d) - (12*a^5*(2*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d) - (6*(1 - (2*a^2)/b^2)*Tan[c + d*x])/d - (6*a^4*(2*a^2 + b^2))/(b^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(2*b))/(3*b)","C",1
468,1,182,197,2.9331425,"\int \frac{\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 a^4 \left(3 a^2+5 b^2\right) \log (a+b \tan (c+d x))}{b^3 \left(a^2+b^2\right)^2}+\frac{6 a^5+4 a^3 b^2}{b^3 \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\tan ^3(c+d x)}{a+b \tan (c+d x)}-\frac{3 a \tan ^2(c+d x)}{b (a+b \tan (c+d x))}+\frac{b \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{b \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 b d}","-\frac{a^2 \tan ^3(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(3 a^2+b^2\right) \tan ^2(c+d x)}{2 b^2 d \left(a^2+b^2\right)}-\frac{\left(a^2-b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{2 a b x}{\left(a^2+b^2\right)^2}-\frac{a \left(3 a^2+2 b^2\right) \tan (c+d x)}{b^3 d \left(a^2+b^2\right)}+\frac{a^4 \left(3 a^2+5 b^2\right) \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)^2}",1,"((b*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (b*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*a^4*(3*a^2 + 5*b^2)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2) + (6*a^5 + 4*a^3*b^2)/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])) - (3*a*Tan[c + d*x]^2)/(b*(a + b*Tan[c + d*x])) + Tan[c + d*x]^3/(a + b*Tan[c + d*x]))/(2*b*d)","C",1
469,1,329,155,2.6759499,"\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","\frac{b^2 \left(a^2+b^2\right)^2 \tan ^2(c+d x)+2 i a^3 \left(a^2+2 b^2\right) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left(2 a \left(a^2+b^2\right)^2 \log (\cos (c+d x))-\left(a^3 \left(a^2+2 b^2\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)\right)+(a+i b)^2 \left(-2 i a^3-4 a^2 b+2 i a b^2+b^3\right) (c+d x)\right)+b \tan (c+d x) \left(-2 i a^5 c-2 i a^5 d x+2 a^5-4 i a^3 b^2 c-4 i a^3 b^2 d x+3 a^3 b^2+a^2 b^3 c+a^2 b^3 d x+2 a \left(a^2+b^2\right)^2 \log (\cos (c+d x))-a^3 \left(a^2+2 b^2\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+a b^4-b^5 c-b^5 d x\right)}{b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","-\frac{a^2 \tan ^2(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(2 a^2+b^2\right) \tan (c+d x)}{b^2 d \left(a^2+b^2\right)}+\frac{2 a b \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}-\frac{2 a^3 \left(a^2+2 b^2\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^2}",1,"(a*((a + I*b)^2*((-2*I)*a^3 - 4*a^2*b + (2*I)*a*b^2 + b^3)*(c + d*x) + 2*a*(a^2 + b^2)^2*Log[Cos[c + d*x]] - a^3*(a^2 + 2*b^2)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*(2*a^5 + 3*a^3*b^2 + a*b^4 - (2*I)*a^5*c - (4*I)*a^3*b^2*c + a^2*b^3*c - b^5*c - (2*I)*a^5*d*x - (4*I)*a^3*b^2*d*x + a^2*b^3*d*x - b^5*d*x + 2*a*(a^2 + b^2)^2*Log[Cos[c + d*x]] - a^3*(a^2 + 2*b^2)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] + b^2*(a^2 + b^2)^2*Tan[c + d*x]^2 + (2*I)*a^3*(a^2 + 2*b^2)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
470,1,251,114,1.0680692,"\int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]","\frac{-2 i a^2 \left(a^2+3 b^2\right) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left(-2 \left(a^2+b^2\right)^2 \log (\cos (c+d x))+a \left(a \left(a^2+3 b^2\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 (2 b+i a) (a+i b)^2 (c+d x)\right)\right)+b \tan (c+d x) \left(-2 \left(a^2+b^2\right)^2 \log (\cos (c+d x))+a \left(a \left(a^2+3 b^2\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 i \left(a^3 (c+d x+i)+a b^2 (3 c+3 d x+i)+2 i b^3 (c+d x)\right)\right)\right)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","\frac{a^2 \left(a^2+3 b^2\right) \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)^2}+\frac{\left(a^2-b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{2 a b x}{\left(a^2+b^2\right)^2}+\frac{a^3}{b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}",1,"(a*(-2*(a^2 + b^2)^2*Log[Cos[c + d*x]] + a*(2*(a + I*b)^2*(I*a + 2*b)*(c + d*x) + a*(a^2 + 3*b^2)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])) + b*(-2*(a^2 + b^2)^2*Log[Cos[c + d*x]] + a*((2*I)*((2*I)*b^3*(c + d*x) + a^3*(I + c + d*x) + a*b^2*(I + 3*c + 3*d*x)) + a*(a^2 + 3*b^2)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]))*Tan[c + d*x] - (2*I)*a^2*(a^2 + 3*b^2)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
471,1,161,88,0.6933143,"\int \frac{\tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","\frac{\tan (c+d x) \left(-a b^2 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+(a+i b) \left(a^2-a b (c+d x+i)-i b^2 (c+d x)\right)\right)+2 i a b \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))-a \left(a b \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+(a+i b)^2 (c+d x)\right)}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","-\frac{a^2}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}",1,"(-(a*((a + I*b)^2*(c + d*x) + a*b*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])) + ((a + I*b)*(a^2 - I*b^2*(c + d*x) - a*b*(I + c + d*x)) - a*b^2*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] + (2*I)*a*b*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
472,1,181,82,0.4756382,"\int \frac{\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^2,x]","\frac{a \left(2 \left(\left(b^2-a^2\right) \log (a+b \tan (c+d x))+a^2+b^2\right)+(a-i b)^2 \log (-\tan (c+d x)+i)+(a+i b)^2 \log (\tan (c+d x)+i)\right)+b \tan (c+d x) \left(2 \left(b^2-a^2\right) \log (a+b \tan (c+d x))+(a-i b)^2 \log (-\tan (c+d x)+i)+(a+i b)^2 \log (\tan (c+d x)+i)\right)}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","\frac{a}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{2 a b x}{\left(a^2+b^2\right)^2}",1,"(a*((a - I*b)^2*Log[I - Tan[c + d*x]] + (a + I*b)^2*Log[I + Tan[c + d*x]] + 2*(a^2 + b^2 + (-a^2 + b^2)*Log[a + b*Tan[c + d*x]])) + b*((a - I*b)^2*Log[I - Tan[c + d*x]] + (a + I*b)^2*Log[I + Tan[c + d*x]] + 2*(-a^2 + b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
473,1,106,82,1.3154968,"\int \frac{1}{(a+b \tan (c+d x))^2} \, dx","Integrate[(a + b*Tan[c + d*x])^(-2),x]","\frac{\frac{2 b \left(2 a \log (a+b \tan (c+d x))-\frac{a^2+b^2}{a+b \tan (c+d x)}\right)}{\left(a^2+b^2\right)^2}-\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","-\frac{b}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}",1,"(((-I)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*(2*a*Log[a + b*Tan[c + d*x]] - (a^2 + b^2)/(a + b*Tan[c + d*x])))/(a^2 + b^2)^2)/(2*d)","C",1
474,1,154,107,0.6848485,"\int \frac{\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^2,x]","\frac{-\frac{b^2 \left(3 a^2+b^2\right) \log (a+b \tan (c+d x))}{a \left(a^2+b^2\right)}+\frac{\left(a^2+b^2\right) \log (\tan (c+d x))}{a}+\frac{b^2}{a+b \tan (c+d x)}-\frac{a (a-i b) \log (-\tan (c+d x)+i)}{2 (a+i b)}-\frac{a (a+i b) \log (\tan (c+d x)+i)}{2 (a-i b)}}{a d \left(a^2+b^2\right)}","\frac{b^2}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{b^2 \left(3 a^2+b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)^2}-\frac{2 a b x}{\left(a^2+b^2\right)^2}+\frac{\log (\sin (c+d x))}{a^2 d}",1,"(-1/2*(a*(a - I*b)*Log[I - Tan[c + d*x]])/(a + I*b) + ((a^2 + b^2)*Log[Tan[c + d*x]])/a - (a*(a + I*b)*Log[I + Tan[c + d*x]])/(2*(a - I*b)) - (b^2*(3*a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)) + b^2/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)","C",1
475,1,136,150,0.7520827,"\int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{\cot (c+d x)}{a^2}-\frac{b^4}{a^3 \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{2 b^3 \left(2 a^2+b^2\right) \log (a \cot (c+d x)+b)}{a^3 \left(a^2+b^2\right)^2}+\frac{i \log (-\cot (c+d x)+i)}{2 (a-i b)^2}-\frac{i \log (\cot (c+d x)+i)}{2 (a+i b)^2}}{d}","-\frac{2 b \log (\sin (c+d x))}{a^3 d}-\frac{b \left(a^2+2 b^2\right)}{a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}+\frac{2 b^3 \left(2 a^2+b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)^2}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))}",1,"-((Cot[c + d*x]/a^2 - b^4/(a^3*(a^2 + b^2)*(b + a*Cot[c + d*x])) + ((I/2)*Log[I - Cot[c + d*x]])/(a - I*b)^2 - ((I/2)*Log[I + Cot[c + d*x]])/(a + I*b)^2 - (2*b^3*(2*a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(a^3*(a^2 + b^2)^2))/d)","C",1
476,1,146,189,1.2580339,"\int \frac{\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]","-\frac{-\frac{4 b \cot (c+d x)}{a^3}+\frac{\cot ^2(c+d x)}{a^2}+\frac{2 b^5}{a^4 \left(a^2+b^2\right) (a \cot (c+d x)+b)}+\frac{2 b^4 \left(5 a^2+3 b^2\right) \log (a \cot (c+d x)+b)}{a^4 \left(a^2+b^2\right)^2}-\frac{\log (-\cot (c+d x)+i)}{(a-i b)^2}-\frac{\log (\cot (c+d x)+i)}{(a+i b)^2}}{2 d}","\frac{2 a b x}{\left(a^2+b^2\right)^2}+\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\left(a^2-3 b^2\right) \log (\sin (c+d x))}{a^4 d}-\frac{b^4 \left(5 a^2+3 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^2}+\frac{b^2 \left(2 a^2+3 b^2\right)}{a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}",1,"-1/2*((-4*b*Cot[c + d*x])/a^3 + Cot[c + d*x]^2/a^2 + (2*b^5)/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])) - Log[I - Cot[c + d*x]]/(a - I*b)^2 - Log[I + Cot[c + d*x]]/(a + I*b)^2 + (2*b^4*(5*a^2 + 3*b^2)*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)^2))/d","C",1
477,1,243,283,4.5120055,"\int \frac{\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]","\frac{-\frac{a^4 \left(6 a^2+5 b^2\right)}{b^4 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{2 a^4 \left(6 a^4+17 a^2 b^2+15 b^4\right) \log (a+b \tan (c+d x))}{b^4 \left(a^2+b^2\right)^3}+\frac{4 a^3 \left(6 a^4+11 a^2 b^2+4 b^4\right)}{b^4 \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^2}-\frac{4 a \tan ^3(c+d x)}{b (a+b \tan (c+d x))^2}+\frac{i b \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{b \log (\tan (c+d x)+i)}{(b+i a)^3}}{2 b d}","-\frac{a^2 \tan ^4(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{2 a^2 \left(a^2+2 b^2\right) \tan ^3(c+d x)}{b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}-\frac{a \left(6 a^4+11 a^2 b^2+3 b^4\right) \tan (c+d x)}{b^4 d \left(a^2+b^2\right)^2}+\frac{a^4 \left(6 a^4+17 a^2 b^2+15 b^4\right) \log (a+b \tan (c+d x))}{b^5 d \left(a^2+b^2\right)^3}+\frac{\left(6 a^4+11 a^2 b^2+b^4\right) \tan ^2(c+d x)}{2 b^3 d \left(a^2+b^2\right)^2}",1,"((I*b*Log[I - Tan[c + d*x]])/(a + I*b)^3 - (b*Log[I + Tan[c + d*x]])/(I*a + b)^3 + (2*a^4*(6*a^4 + 17*a^2*b^2 + 15*b^4)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^3) - (a^4*(6*a^2 + 5*b^2))/(b^4*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (4*a*Tan[c + d*x]^3)/(b*(a + b*Tan[c + d*x])^2) + Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2 + (4*a^3*(6*a^4 + 11*a^2*b^2 + 4*b^4))/(b^4*(a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*b*d)","C",1
478,1,694,239,6.5256026,"\int \frac{\tan ^5(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^3,x]","\frac{a^5 \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{2 b^2 d (a-i b)^2 (a+i b)^2 (a+b \tan (c+d x))^3}+\frac{b \left(3 a^2-b^2\right) (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{d (a-i b)^3 (a+i b)^3 (a+b \tan (c+d x))^3}+\frac{\sec ^3(c+d x) \left(2 a^5 \sin (c+d x)+5 a^3 b^2 \sin (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d (a-i b)^2 (a+i b)^2 (a+b \tan (c+d x))^3}-\frac{i \left(-3 a^7-9 a^5 b^2-10 a^3 b^4\right) \tan ^{-1}(\tan (c+d x)) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{b^4 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3}+\frac{\left(-3 a^7-9 a^5 b^2-10 a^3 b^4\right) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)}{2 b^4 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3}+\frac{3 a \sec ^3(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{b^4 d (a+b \tan (c+d x))^3}+\frac{\tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{b^3 d (a+b \tan (c+d x))^3}-\frac{i \left(3 a^{12} b^3-3 i a^{11} b^4+15 a^{10} b^5-15 i a^9 b^6+31 a^8 b^7-31 i a^7 b^8+29 a^6 b^9-29 i a^5 b^{10}+10 a^4 b^{11}-10 i a^3 b^{12}\right) (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{b^7 d (a-i b)^6 (a+i b)^5 (a+b \tan (c+d x))^3}","-\frac{a^2 \tan ^3(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 \left(3 a^2+7 b^2\right) \tan ^2(c+d x)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{a \left(a^2-3 b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{b x \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3}+\frac{\left(3 a^4+6 a^2 b^2+b^4\right) \tan (c+d x)}{b^3 d \left(a^2+b^2\right)^2}-\frac{a^3 \left(3 a^4+9 a^2 b^2+10 b^4\right) \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)^3}",1,"(a^5*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(2*(a - I*b)^2*(a + I*b)^2*b^2*d*(a + b*Tan[c + d*x])^3) + (b*(3*a^2 - b^2)*(c + d*x)*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/((a - I*b)^3*(a + I*b)^3*d*(a + b*Tan[c + d*x])^3) - (I*(3*a^12*b^3 - (3*I)*a^11*b^4 + 15*a^10*b^5 - (15*I)*a^9*b^6 + 31*a^8*b^7 - (31*I)*a^7*b^8 + 29*a^6*b^9 - (29*I)*a^5*b^10 + 10*a^4*b^11 - (10*I)*a^3*b^12)*(c + d*x)*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/((a - I*b)^6*(a + I*b)^5*b^7*d*(a + b*Tan[c + d*x])^3) - (I*(-3*a^7 - 9*a^5*b^2 - 10*a^3*b^4)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(b^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3) + (3*a*Log[Cos[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(b^4*d*(a + b*Tan[c + d*x])^3) + ((-3*a^7 - 9*a^5*b^2 - 10*a^3*b^4)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(2*b^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*(2*a^5*Sin[c + d*x] + 5*a^3*b^2*Sin[c + d*x]))/((a - I*b)^2*(a + I*b)^2*b^3*d*(a + b*Tan[c + d*x])^3) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*Tan[c + d*x])/(b^3*d*(a + b*Tan[c + d*x])^3)","C",1
479,1,351,183,2.162196,"\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(-2 a^2 b \left(a^2+b^2\right) \left(a^2+4 b^2\right) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))-2 \left(a^2+b^2\right)^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+2 a b^3 \left(a^2-3 b^2\right) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2+a^4 \left(-b^2\right) \left(a^2+b^2\right)+2 i a^2 \left(a^4+3 a^2 b^2+6 b^4\right) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2+a^2 \left(a^4+3 a^2 b^2+6 b^4\right) (a \cos (c+d x)+b \sin (c+d x))^2 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)-2 i a^2 \left(a^4+3 a^2 b^2+6 b^4\right) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2\right)}{2 b^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3}","-\frac{a^2 \tan ^2(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}+\frac{a^2 \left(a^4+3 a^2 b^2+6 b^4\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^3}+\frac{a^3 \left(a^2+3 b^2\right)}{b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}",1,"(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-(a^4*b^2*(a^2 + b^2)) - 2*a^2*b*(a^2 + b^2)*(a^2 + 4*b^2)*Sin[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + 2*a*b^3*(a^2 - 3*b^2)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (2*I)*a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - (2*I)*a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*ArcTan[Tan[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 2*(a^2 + b^2)^3*Log[Cos[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2))/(2*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3)","C",1
480,1,269,149,5.2898035,"\int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{2 b}{\left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{4 a b \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^2}-\frac{a b \left(\frac{\left(a^2+b^2\right) \left(5 a^2+4 a b \tan (c+d x)+b^2\right)}{(a+b \tan (c+d x))^2}+\left(2 b^2-6 a^2\right) \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^3}-\frac{a}{b (a+b \tan (c+d x))^2}-\frac{2 \tan (c+d x)}{(a+b \tan (c+d x))^2}+\frac{a \log (-\tan (c+d x)+i)}{(b-i a)^3}+\frac{a \log (\tan (c+d x)+i)}{(b+i a)^3}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 b d}","-\frac{a^2 \left(a^2+5 b^2\right)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{a^2 \tan (c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a \left(a^2-3 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{b x \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3}",1,"((I*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (a*Log[I - Tan[c + d*x]])/((-I)*a + b)^3 - (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (a*Log[I + Tan[c + d*x]])/(I*a + b)^3 - (4*a*b*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 - a/(b*(a + b*Tan[c + d*x])^2) - (2*Tan[c + d*x])/(a + b*Tan[c + d*x])^2 + (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])) - (a*b*((-6*a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]] + ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3)/(2*b*d)","C",1
481,1,200,129,3.9328348,"\int \frac{\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","-\frac{a \left(\frac{2 a (a-b) (a+b)}{b \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{(b+i a)^3 \log (-\tan (c+d x)+i)}{\left(a^2+b^2\right)^2}-\frac{2 b \left(b^2-3 a^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^2}+\frac{i (a+i b) \log (\tan (c+d x)+i)}{(a-i b)^2}\right)-\frac{b^2 \tan ^3(c+d x)}{(a+b \tan (c+d x))^2}+\frac{b \tan ^2(c+d x)}{a+b \tan (c+d x)}}{2 a d \left(a^2+b^2\right)}","-\frac{a^2}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{2 a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b \left(3 a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}",1,"-1/2*(-((b^2*Tan[c + d*x]^3)/(a + b*Tan[c + d*x])^2) + (b*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]) + a*(((I*a + b)^3*Log[I - Tan[c + d*x]])/(a^2 + b^2)^2 + (I*(a + I*b)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b*(-3*a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 + (2*a*(a - b)*(a + b))/(b*(a^2 + b^2)*(a + b*Tan[c + d*x]))))/(a*(a^2 + b^2)*d)","C",1
482,1,234,129,3.8362812,"\int \frac{\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^3,x]","\frac{-\frac{2 b}{\left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{4 a b \log (a+b \tan (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 \tan (c+d x)+b^2\right)}{(a+b \tan (c+d x))^2}+\left(2 b^2-6 a^2\right) \log (a+b \tan (c+d x))\right)}{\left(a^2+b^2\right)^3}+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right)-\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 b d}","\frac{a}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a^2-b^2}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{a \left(a^2-3 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{b x \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3}",1,"(((-I)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (4*a*b*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 - (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])) + a*((I*Log[I - Tan[c + d*x]])/(a + I*b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 + (b*((-6*a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]] + ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3))/(2*b*d)","C",1
483,1,127,122,1.8078436,"\int \frac{1}{(a+b \tan (c+d x))^3} \, dx","Integrate[(a + b*Tan[c + d*x])^(-3),x]","\frac{\frac{b \left(\left(6 a^2-2 b^2\right) \log (a+b \tan (c+d x))-\frac{\left(a^2+b^2\right) \left(5 a^2+4 a b \tan (c+d x)+b^2\right)}{(a+b \tan (c+d x))^2}\right)}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (c+d x)+i)}{(b-i a)^3}+\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}}{2 d}","-\frac{2 a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}",1,"(Log[I - Tan[c + d*x]]/((-I)*a + b)^3 + Log[I + Tan[c + d*x]]/(I*a + b)^3 + (b*((6*a^2 - 2*b^2)*Log[a + b*Tan[c + d*x]] - ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3)/(2*d)","C",1
484,1,209,168,4.2119428,"\int \frac{\cot (c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{4 a b^2}{\left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 b^2}{a^2+a b \tan (c+d x)}+\frac{2 \left(a^2+b^2\right) \log (\tan (c+d x))}{a^2}-\frac{2 b^2 \left(6 a^4+3 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))}{a^2 \left(a^2+b^2\right)^2}+\frac{b^2}{(a+b \tan (c+d x))^2}-\frac{a (a-i b) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{a (a+i b) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 a d \left(a^2+b^2\right)}","\frac{\log (\sin (c+d x))}{a^3 d}+\frac{b^2 \left(3 a^2+b^2\right)}{a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b^2}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{b x \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3}-\frac{b^2 \left(6 a^4+3 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)^3}",1,"(-((a*(a - I*b)*Log[I - Tan[c + d*x]])/(a + I*b)^2) + (2*(a^2 + b^2)*Log[Tan[c + d*x]])/a^2 - (a*(a + I*b)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b^2*(6*a^4 + 3*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) + b^2/(a + b*Tan[c + d*x])^2 + (4*a*b^2)/((a^2 + b^2)*(a + b*Tan[c + d*x])) + (2*b^2)/(a^2 + a*b*Tan[c + d*x]))/(2*a*(a^2 + b^2)*d)","C",1
485,1,178,211,4.7572346,"\int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{2 \cot (c+d x)}{a^3}+\frac{b^5}{a^4 \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{2 b^4 \left(5 a^2+3 b^2\right)}{a^4 \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{2 b^3 \left(10 a^4+9 a^2 b^2+3 b^4\right) \log (a \cot (c+d x)+b)}{a^4 \left(a^2+b^2\right)^3}+\frac{\log (-\cot (c+d x)+i)}{(b+i a)^3}+\frac{\log (\cot (c+d x)+i)}{(b-i a)^3}}{2 d}","-\frac{3 b \log (\sin (c+d x))}{a^4 d}-\frac{b \left(2 a^2+3 b^2\right)}{2 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}+\frac{b^3 \left(10 a^4+9 a^2 b^2+3 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^3}-\frac{b \left(a^4+6 a^2 b^2+3 b^4\right)}{a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2}",1,"-1/2*((2*Cot[c + d*x])/a^3 + b^5/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])^2) - (2*b^4*(5*a^2 + 3*b^2))/(a^4*(a^2 + b^2)^2*(b + a*Cot[c + d*x])) + Log[I - Cot[c + d*x]]/(I*a + b)^3 + Log[I + Cot[c + d*x]]/((-I)*a + b)^3 - (2*b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)^3))/d","C",1
486,1,1281,315,6.502712,"\int \frac{\tan ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]","\frac{(a \cos (c+d x)+b \sin (c+d x)) \left(24 \sin (2 (c+d x)) a^{11}+12 \sin (4 (c+d x)) a^{11}+39 b a^{10}+12 b \cos (2 (c+d x)) a^{10}-27 b \cos (4 (c+d x)) a^{10}+158 b^2 \sin (2 (c+d x)) a^9+35 b^2 \sin (4 (c+d x)) a^9+171 b^3 a^8+32 b^3 \cos (2 (c+d x)) a^8-115 b^3 \cos (4 (c+d x)) a^8-9 b^4 (c+d x) a^7-12 b^4 (c+d x) \cos (2 (c+d x)) a^7-3 b^4 (c+d x) \cos (4 (c+d x)) a^7+396 b^4 \sin (2 (c+d x)) a^7+18 b^4 \sin (4 (c+d x)) a^7+276 b^5 a^6-16 b^5 \cos (2 (c+d x)) a^6-196 b^5 \cos (4 (c+d x)) a^6-18 b^5 (c+d x) \sin (2 (c+d x)) a^6-9 b^5 (c+d x) \sin (4 (c+d x)) a^6+45 b^6 (c+d x) a^5+72 b^6 (c+d x) \cos (2 (c+d x)) a^5+27 b^6 (c+d x) \cos (4 (c+d x)) a^5+412 b^6 \sin (2 (c+d x)) a^5-74 b^6 \sin (4 (c+d x)) a^5+180 b^7 a^4-72 b^7 \cos (2 (c+d x)) a^4-108 b^7 \cos (4 (c+d x)) a^4+102 b^7 (c+d x) \sin (2 (c+d x)) a^4+57 b^7 (c+d x) \sin (4 (c+d x)) a^4+45 b^8 (c+d x) a^3-12 b^8 (c+d x) \cos (2 (c+d x)) a^3-57 b^8 (c+d x) \cos (4 (c+d x)) a^3+168 b^8 \sin (2 (c+d x)) a^3-78 b^8 \sin (4 (c+d x)) a^3+45 b^9 a^2-48 b^9 \cos (2 (c+d x)) a^2+3 b^9 \cos (4 (c+d x)) a^2+18 b^9 (c+d x) \sin (2 (c+d x)) a^2-27 b^9 (c+d x) \sin (4 (c+d x)) a^2-9 b^{10} (c+d x) a+9 b^{10} (c+d x) \cos (4 (c+d x)) a+18 b^{10} \sin (2 (c+d x)) a-9 b^{10} \sin (4 (c+d x)) a+9 b^{11}-12 b^{11} \cos (2 (c+d x))+3 b^{11} \cos (4 (c+d x))-6 b^{11} (c+d x) \sin (2 (c+d x))+3 b^{11} (c+d x) \sin (4 (c+d x))\right) \sec ^5(c+d x)}{24 b^4 (b-i a)^4 (i a+b)^4 d (a+b \tan (c+d x))^4}-\frac{4 i \left(-5 i a^3 b^{17}+5 a^4 b^{16}-21 i a^5 b^{15}+21 a^6 b^{14}-37 i a^7 b^{13}+37 a^8 b^{12}-36 i a^9 b^{11}+36 a^{10} b^{10}-21 i a^{11} b^9+21 a^{12} b^8-7 i a^{13} b^7+7 a^{14} b^6-i a^{15} b^5+a^{16} b^4\right) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{(a-i b)^8 (a+i b)^7 b^9 d (a+b \tan (c+d x))^4}+\frac{4 i \left(a^9+4 b^2 a^7+6 b^4 a^5+5 b^6 a^3\right) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{b^5 \left(a^2+b^2\right)^4 d (a+b \tan (c+d x))^4}+\frac{4 a \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{b^5 d (a+b \tan (c+d x))^4}-\frac{2 \left(a^9+4 b^2 a^7+6 b^4 a^5+5 b^6 a^3\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right) (a \cos (c+d x)+b \sin (c+d x))^4 \sec ^4(c+d x)}{b^5 \left(a^2+b^2\right)^4 d (a+b \tan (c+d x))^4}","-\frac{a^2 \tan ^4(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{a^2 \left(2 a^2+5 b^2\right) \tan ^3(c+d x)}{3 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^4}-\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{\left(a^2+b^2\right)^4}-\frac{2 a^2 \left(a^4+3 a^2 b^2+4 b^4\right) \tan ^2(c+d x)}{b^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{\left(4 a^6+12 a^4 b^2+13 a^2 b^4+b^6\right) \tan (c+d x)}{b^4 d \left(a^2+b^2\right)^3}-\frac{4 a^3 \left(a^6+4 a^4 b^2+6 a^2 b^4+5 b^6\right) \log (a+b \tan (c+d x))}{b^5 d \left(a^2+b^2\right)^4}",1,"((-4*I)*(a^16*b^4 - I*a^15*b^5 + 7*a^14*b^6 - (7*I)*a^13*b^7 + 21*a^12*b^8 - (21*I)*a^11*b^9 + 36*a^10*b^10 - (36*I)*a^9*b^11 + 37*a^8*b^12 - (37*I)*a^7*b^13 + 21*a^6*b^14 - (21*I)*a^5*b^15 + 5*a^4*b^16 - (5*I)*a^3*b^17)*(c + d*x)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/((a - I*b)^8*(a + I*b)^7*b^9*d*(a + b*Tan[c + d*x])^4) + ((4*I)*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 5*a^3*b^6)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^5*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) + (4*a*Log[Cos[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^5*d*(a + b*Tan[c + d*x])^4) - (2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 5*a^3*b^6)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^5*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) + (Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])*(39*a^10*b + 171*a^8*b^3 + 276*a^6*b^5 + 180*a^4*b^7 + 45*a^2*b^9 + 9*b^11 - 9*a^7*b^4*(c + d*x) + 45*a^5*b^6*(c + d*x) + 45*a^3*b^8*(c + d*x) - 9*a*b^10*(c + d*x) + 12*a^10*b*Cos[2*(c + d*x)] + 32*a^8*b^3*Cos[2*(c + d*x)] - 16*a^6*b^5*Cos[2*(c + d*x)] - 72*a^4*b^7*Cos[2*(c + d*x)] - 48*a^2*b^9*Cos[2*(c + d*x)] - 12*b^11*Cos[2*(c + d*x)] - 12*a^7*b^4*(c + d*x)*Cos[2*(c + d*x)] + 72*a^5*b^6*(c + d*x)*Cos[2*(c + d*x)] - 12*a^3*b^8*(c + d*x)*Cos[2*(c + d*x)] - 27*a^10*b*Cos[4*(c + d*x)] - 115*a^8*b^3*Cos[4*(c + d*x)] - 196*a^6*b^5*Cos[4*(c + d*x)] - 108*a^4*b^7*Cos[4*(c + d*x)] + 3*a^2*b^9*Cos[4*(c + d*x)] + 3*b^11*Cos[4*(c + d*x)] - 3*a^7*b^4*(c + d*x)*Cos[4*(c + d*x)] + 27*a^5*b^6*(c + d*x)*Cos[4*(c + d*x)] - 57*a^3*b^8*(c + d*x)*Cos[4*(c + d*x)] + 9*a*b^10*(c + d*x)*Cos[4*(c + d*x)] + 24*a^11*Sin[2*(c + d*x)] + 158*a^9*b^2*Sin[2*(c + d*x)] + 396*a^7*b^4*Sin[2*(c + d*x)] + 412*a^5*b^6*Sin[2*(c + d*x)] + 168*a^3*b^8*Sin[2*(c + d*x)] + 18*a*b^10*Sin[2*(c + d*x)] - 18*a^6*b^5*(c + d*x)*Sin[2*(c + d*x)] + 102*a^4*b^7*(c + d*x)*Sin[2*(c + d*x)] + 18*a^2*b^9*(c + d*x)*Sin[2*(c + d*x)] - 6*b^11*(c + d*x)*Sin[2*(c + d*x)] + 12*a^11*Sin[4*(c + d*x)] + 35*a^9*b^2*Sin[4*(c + d*x)] + 18*a^7*b^4*Sin[4*(c + d*x)] - 74*a^5*b^6*Sin[4*(c + d*x)] - 78*a^3*b^8*Sin[4*(c + d*x)] - 9*a*b^10*Sin[4*(c + d*x)] - 9*a^6*b^5*(c + d*x)*Sin[4*(c + d*x)] + 57*a^4*b^7*(c + d*x)*Sin[4*(c + d*x)] - 27*a^2*b^9*(c + d*x)*Sin[4*(c + d*x)] + 3*b^11*(c + d*x)*Sin[4*(c + d*x)]))/(24*b^4*((-I)*a + b)^4*(I*a + b)^4*d*(a + b*Tan[c + d*x])^4)","C",1
487,1,788,256,6.3977122,"\int \frac{\tan ^5(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^4,x]","-\frac{a^4 \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{3 b d (a-i b)^2 (a+i b)^2 (a+b \tan (c+d x))^4}-\frac{a^4 \left(3 a^2+13 b^2\right) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{6 b^2 d (a-i b)^3 (a+i b)^3 (a+b \tan (c+d x))^4}+\frac{\sec ^4(c+d x) \left(-3 a^6 \sin (c+d x)-11 a^4 b^2 \sin (c+d x)-30 a^2 b^4 \sin (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^3}{3 b^3 d (a-i b)^3 (a+i b)^3 (a+b \tan (c+d x))^4}-\frac{i \left(a^8+4 a^6 b^2+5 a^4 b^4+10 a^2 b^6\right) \tan ^{-1}(\tan (c+d x)) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^4 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^4}+\frac{\left(a^8+4 a^6 b^2+5 a^4 b^4+10 a^2 b^6\right) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)}{2 b^4 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^4}-\frac{\sec ^4(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^4}{b^4 d (a+b \tan (c+d x))^4}+\frac{4 a b (a-b) (a+b) (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{d (a-i b)^4 (a+i b)^4 (a+b \tan (c+d x))^4}+\frac{\left(i a^{15} b^3+a^{14} b^4+7 i a^{13} b^5+7 a^{12} b^6+20 i a^{11} b^7+20 a^{10} b^8+38 i a^9 b^9+38 a^8 b^{10}+49 i a^7 b^{11}+49 a^6 b^{12}+35 i a^5 b^{13}+35 a^4 b^{14}+10 i a^3 b^{15}+10 a^2 b^{16}\right) (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}{b^7 d (a-i b)^8 (a+i b)^7 (a+b \tan (c+d x))^4}","-\frac{a^2 \tan ^3(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{a^2 \left(a^2+3 b^2\right) \tan ^2(c+d x)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{4 a b x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^4}-\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{a^2 \left(a^6+4 a^4 b^2+5 a^2 b^4+10 b^6\right) \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)^4}+\frac{a^3 \left(a^4+3 a^2 b^2+6 b^4\right)}{b^4 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}",1,"-1/6*(a^4*(3*a^2 + 13*b^2)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/((a - I*b)^3*(a + I*b)^3*b^2*d*(a + b*Tan[c + d*x])^4) + (4*a*(a - b)*b*(a + b)*(c + d*x)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/((a - I*b)^4*(a + I*b)^4*d*(a + b*Tan[c + d*x])^4) + ((I*a^15*b^3 + a^14*b^4 + (7*I)*a^13*b^5 + 7*a^12*b^6 + (20*I)*a^11*b^7 + 20*a^10*b^8 + (38*I)*a^9*b^9 + 38*a^8*b^10 + (49*I)*a^7*b^11 + 49*a^6*b^12 + (35*I)*a^5*b^13 + 35*a^4*b^14 + (10*I)*a^3*b^15 + 10*a^2*b^16)*(c + d*x)*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/((a - I*b)^8*(a + I*b)^7*b^7*d*(a + b*Tan[c + d*x])^4) - (I*(a^8 + 4*a^6*b^2 + 5*a^4*b^4 + 10*a^2*b^6)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^4*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) - (Log[Cos[c + d*x]]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(b^4*d*(a + b*Tan[c + d*x])^4) + ((a^8 + 4*a^6*b^2 + 5*a^4*b^4 + 10*a^2*b^6)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)/(2*b^4*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^4) + (Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(-3*a^6*Sin[c + d*x] - 11*a^4*b^2*Sin[c + d*x] - 30*a^2*b^4*Sin[c + d*x]))/(3*(a - I*b)^3*(a + I*b)^3*b^3*d*(a + b*Tan[c + d*x])^4) - (a^4*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*Tan[c + d*x])/(3*(a - I*b)^2*(a + I*b)^2*b*d*(a + b*Tan[c + d*x])^4)","C",1
488,1,236,208,2.5254891,"\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^4,x]","-\frac{\frac{6 a b^3}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{6 b^3 \left(b^2-3 a^2\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{24 a b^3 (a-b) (a+b) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}+\frac{2 a^4}{b \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{3 i b^2 \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 i b^2 \log (\tan (c+d x)+i)}{(b+i a)^4}+\frac{6 b \tan ^2(c+d x)}{(a+b \tan (c+d x))^3}+\frac{6 a \tan (c+d x)}{(a+b \tan (c+d x))^3}}{6 b^2 d}","-\frac{a^2 \tan ^2(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{\left(a^2+b^2\right)^4}-\frac{a^2 \left(2 a^4+7 a^2 b^2+17 b^4\right)}{3 b^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{a^3 \left(a^2+4 b^2\right)}{3 b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}",1,"-1/6*(((3*I)*b^2*Log[I - Tan[c + d*x]])/(a + I*b)^4 - ((3*I)*b^2*Log[I + Tan[c + d*x]])/(I*a + b)^4 - (24*a*(a - b)*b^3*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 + (2*a^4)/(b*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (6*a*Tan[c + d*x])/(a + b*Tan[c + d*x])^3 + (6*b*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3 + (6*a*b^3)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) - (6*b^3*(-3*a^2 + b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x])))/(b^2*d)","C",1
489,1,387,189,6.2355661,"\int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^4,x]","-\frac{\tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac{\frac{a}{3 b d (a+b \tan (c+d x))^3}+\frac{2 b \left(\frac{-\frac{2 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b}{2 \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}-\frac{\log (-\tan (c+d x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (c+d x)+i)}{2 (b+i a)^3}}{b}-\frac{a \left(-\frac{b \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{b}{3 \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}-\frac{i \log (-\tan (c+d x)+i)}{2 (a+i b)^4}+\frac{i \log (\tan (c+d x)+i)}{2 (a-i b)^4}\right)}{b}\right)}{d}}{2 b}","-\frac{a^2 \left(a^2+7 b^2\right)}{6 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{a^2 \tan (c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{a \left(a^2-3 b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{4 a b x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^4}+\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}",1,"-1/2*Tan[c + d*x]/(b*d*(a + b*Tan[c + d*x])^3) - (a/(3*b*d*(a + b*Tan[c + d*x])^3) + (2*b*(-((a*(((-1/2*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + ((I/2)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (4*a*(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 - b/(3*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/b) + (-1/2*Log[I - Tan[c + d*x]]/(I*a - b)^3 + Log[I + Tan[c + d*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/b))/d)/(2*b)","C",1
490,1,324,169,6.2198958,"\int \frac{\tan ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]","\frac{b^2 \tan ^3(c+d x)}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{\frac{3 a \tan (c+d x)}{d (a+b \tan (c+d x))^2}-\frac{3 a \left(\frac{2 b}{\left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{4 a b \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^2}-2 a \left(\frac{4 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b}{\left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{2 b \left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (c+d x)+i)}{(-b+i a)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right)+\frac{i \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{i \log (\tan (c+d x)+i)}{(a-i b)^2}\right)}{2 d}}{3 a \left(a^2+b^2\right)}","-\frac{a^2}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{4 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}-\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{\left(a^2+b^2\right)^4}",1,"(b^2*Tan[c + d*x]^3)/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((3*a*Tan[c + d*x])/(d*(a + b*Tan[c + d*x])^2) - (3*a*((I*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (4*a*b*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 + (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])) - 2*a*(Log[I - Tan[c + d*x]]/(I*a - b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 - (2*b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 + b/((a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (4*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))))/(2*d))/(3*a*(a^2 + b^2))","C",1
491,1,319,172,6.2464553,"\int \frac{\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^4,x]","\frac{a \left(\frac{6 b \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{6 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{2 b}{\left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{24 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^4}+\frac{3 i \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 i \log (\tan (c+d x)+i)}{(a-i b)^4}\right)}{6 b d}-\frac{\frac{4 a b}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b}{\left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{2 b \left(3 a^2-b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (c+d x)+i)}{(-b+i a)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}}{2 b d}","\frac{a \left(a^2-3 b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{a}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{a^2-b^2}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{4 a b x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^4}-\frac{\left(a^4-6 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}",1,"(a*(((3*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 - ((3*I)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (24*a*(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 + (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (6*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/(6*b*d) - (Log[I - Tan[c + d*x]]/(I*a - b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 - (2*b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 + b/((a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (4*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*b*d)","C",1
492,1,176,165,3.022795,"\int \frac{1}{(a+b \tan (c+d x))^4} \, dx","Integrate[(a + b*Tan[c + d*x])^(-4),x]","\frac{\frac{2 b \left(12 a \left(a^2-b^2\right) \log (a+b \tan (c+d x))-\frac{\left(a^2+b^2\right) \left(13 a^4+3 a b \left(7 a^2-b^2\right) \tan (c+d x)+2 a^2 b^2+\left(9 a^2 b^2-3 b^4\right) \tan ^2(c+d x)+b^4\right)}{(a+b \tan (c+d x))^3}\right)}{\left(a^2+b^2\right)^4}-\frac{3 i \log (-\tan (c+d x)+i)}{(a+i b)^4}+\frac{3 i \log (\tan (c+d x)+i)}{(a-i b)^4}}{6 d}","-\frac{b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{b}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{\left(a^2+b^2\right)^4}",1,"(((-3*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + ((3*I)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (2*b*(12*a*(a^2 - b^2)*Log[a + b*Tan[c + d*x]] - ((a^2 + b^2)*(13*a^4 + 2*a^2*b^2 + b^4 + 3*a*b*(7*a^2 - b^2)*Tan[c + d*x] + (9*a^2*b^2 - 3*b^4)*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3))/(a^2 + b^2)^4)/(6*d)","C",1
493,1,243,226,2.0466076,"\int \frac{\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^4,x]","\frac{\frac{2 a b^2 \left(a^2+b^2\right)}{(a+b \tan (c+d x))^3}+\frac{3 \left(3 a^2 b^2+b^4\right)}{(a+b \tan (c+d x))^2}+\frac{6 \left(6 a^4 b^2+3 a^2 b^4+b^6\right)}{a \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{3 \left(-a^4 (a-i b)^4 \log (-\tan (c+d x)+i)-a^4 (a+i b)^4 \log (\tan (c+d x)+i)+2 \left(a^2+b^2\right)^4 \log (\tan (c+d x))-2 b^2 \left(10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right) \log (a+b \tan (c+d x))\right)}{a^2 \left(a^2+b^2\right)^2}}{6 a^2 d \left(a^2+b^2\right)^2}","\frac{\log (\sin (c+d x))}{a^4 d}+\frac{b^2 \left(3 a^2+b^2\right)}{2 a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{b^2}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{4 a b x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^4}-\frac{b^2 \left(10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^4}+\frac{b^2 \left(6 a^4+3 a^2 b^2+b^4\right)}{a^3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}",1,"((3*(-(a^4*(a - I*b)^4*Log[I - Tan[c + d*x]]) + 2*(a^2 + b^2)^4*Log[Tan[c + d*x]] - a^4*(a + I*b)^4*Log[I + Tan[c + d*x]] - 2*b^2*(10*a^6 + 5*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[a + b*Tan[c + d*x]]))/(a^2*(a^2 + b^2)^2) + (2*a*b^2*(a^2 + b^2))/(a + b*Tan[c + d*x])^3 + (3*(3*a^2*b^2 + b^4))/(a + b*Tan[c + d*x])^2 + (6*(6*a^4*b^2 + 3*a^2*b^4 + b^6))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(6*a^2*(a^2 + b^2)^2*d)","C",1
494,1,241,278,3.5458416,"\int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]","-\frac{\frac{\cot (c+d x)}{a^4}-\frac{b^6}{3 a^5 \left(a^2+b^2\right) (a \cot (c+d x)+b)^3}+\frac{b^5 \left(3 a^2+2 b^2\right)}{a^5 \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)^2}-\frac{b^4 \left(15 a^4+17 a^2 b^2+6 b^4\right)}{a^5 \left(a^2+b^2\right)^3 (a \cot (c+d x)+b)}-\frac{4 b^3 \left(5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right) \log (a \cot (c+d x)+b)}{a^5 \left(a^2+b^2\right)^4}+\frac{i \log (-\cot (c+d x)+i)}{2 (a-i b)^4}-\frac{i \log (\cot (c+d x)+i)}{2 (a+i b)^4}}{d}","-\frac{4 b \log (\sin (c+d x))}{a^5 d}-\frac{b \left(3 a^2+4 b^2\right)}{3 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}-\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{\left(a^2+b^2\right)^4}-\frac{b \left(a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right)}{a^4 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{b \left(a^4+4 a^2 b^2+2 b^4\right)}{a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}+\frac{4 b^3 \left(5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left(a^2+b^2\right)^4}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^3}",1,"-((Cot[c + d*x]/a^4 - b^6/(3*a^5*(a^2 + b^2)*(b + a*Cot[c + d*x])^3) + (b^5*(3*a^2 + 2*b^2))/(a^5*(a^2 + b^2)^2*(b + a*Cot[c + d*x])^2) - (b^4*(15*a^4 + 17*a^2*b^2 + 6*b^4))/(a^5*(a^2 + b^2)^3*(b + a*Cot[c + d*x])) + ((I/2)*Log[I - Cot[c + d*x]])/(a - I*b)^4 - ((I/2)*Log[I + Cot[c + d*x]])/(a + I*b)^4 - (4*b^3*(5*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[b + a*Cot[c + d*x]])/(a^5*(a^2 + b^2)^4))/d)","C",1
495,1,65,31,0.0378361,"\int \frac{1}{3+5 \tan (c+d x)} \, dx","Integrate[(3 + 5*Tan[c + d*x])^(-1),x]","-\frac{\left(\frac{5}{68}+\frac{3 i}{68}\right) \log (-\tan (c+d x)+i)}{d}-\frac{\left(\frac{5}{68}-\frac{3 i}{68}\right) \log (\tan (c+d x)+i)}{d}+\frac{5 \log (5 \tan (c+d x)+3)}{34 d}","\frac{5 \log (5 \sin (c+d x)+3 \cos (c+d x))}{34 d}+\frac{3 x}{34}",1,"((-5/68 - (3*I)/68)*Log[I - Tan[c + d*x]])/d - ((5/68 - (3*I)/68)*Log[I + Tan[c + d*x]])/d + (5*Log[3 + 5*Tan[c + d*x]])/(34*d)","C",1
496,1,67,50,0.259407,"\int \frac{1}{(3+5 \tan (c+d x))^2} \, dx","Integrate[(3 + 5*Tan[c + d*x])^(-2),x]","-\frac{\frac{170}{5 \tan (c+d x)+3}+(15-8 i) \log (-\tan (c+d x)+i)+(15+8 i) \log (\tan (c+d x)+i)-30 \log (5 \tan (c+d x)+3)}{1156 d}","-\frac{5}{34 d (5 \tan (c+d x)+3)}+\frac{15 \log (5 \sin (c+d x)+3 \cos (c+d x))}{578 d}-\frac{4 x}{289}",1,"-1/1156*((15 - 8*I)*Log[I - Tan[c + d*x]] + (15 + 8*I)*Log[I + Tan[c + d*x]] - 30*Log[3 + 5*Tan[c + d*x]] + 170/(3 + 5*Tan[c + d*x]))/d","C",1
497,1,84,69,0.580384,"\int \frac{1}{(3+5 \tan (c+d x))^3} \, dx","Integrate[(3 + 5*Tan[c + d*x])^(-3),x]","\frac{\left(\frac{1}{39304}+\frac{i}{39304}\right) \left((47+52 i) \log (-\tan (c+d x)+i)-(52+47 i) \log (\tan (c+d x)+i)+(5-5 i) \left(\log (5 \tan (c+d x)+3)-\frac{85 (6 \tan (c+d x)+7)}{(5 \tan (c+d x)+3)^2}\right)\right)}{d}","-\frac{15}{578 d (5 \tan (c+d x)+3)}-\frac{5}{68 d (5 \tan (c+d x)+3)^2}+\frac{5 \log (5 \sin (c+d x)+3 \cos (c+d x))}{19652 d}-\frac{99 x}{19652}",1,"((1/39304 + I/39304)*((47 + 52*I)*Log[I - Tan[c + d*x]] - (52 + 47*I)*Log[I + Tan[c + d*x]] + (5 - 5*I)*(Log[3 + 5*Tan[c + d*x]] - (85*(7 + 6*Tan[c + d*x]))/(3 + 5*Tan[c + d*x])^2)))/d","C",1
498,1,87,88,1.1543254,"\int \frac{1}{(3+5 \tan (c+d x))^4} \, dx","Integrate[(3 + 5*Tan[c + d*x])^(-4),x]","\frac{-\frac{170 \left(75 \tan ^2(c+d x)+855 \tan (c+d x)+1064\right)}{(5 \tan (c+d x)+3)^3}+(720+483 i) \log (-\tan (c+d x)+i)+(720-483 i) \log (\tan (c+d x)+i)-1440 \log (5 \tan (c+d x)+3)}{2004504 d}","-\frac{5}{19652 d (5 \tan (c+d x)+3)}-\frac{15}{1156 d (5 \tan (c+d x)+3)^2}-\frac{5}{102 d (5 \tan (c+d x)+3)^3}-\frac{60 \log (5 \sin (c+d x)+3 \cos (c+d x))}{83521 d}-\frac{161 x}{334084}",1,"((720 + 483*I)*Log[I - Tan[c + d*x]] + (720 - 483*I)*Log[I + Tan[c + d*x]] - 1440*Log[3 + 5*Tan[c + d*x]] - (170*(1064 + 855*Tan[c + d*x] + 75*Tan[c + d*x]^2))/(3 + 5*Tan[c + d*x])^3)/(2004504*d)","C",1
499,1,65,31,0.0375664,"\int \frac{1}{5+3 \tan (c+d x)} \, dx","Integrate[(5 + 3*Tan[c + d*x])^(-1),x]","-\frac{\left(\frac{3}{68}+\frac{5 i}{68}\right) \log (-\tan (c+d x)+i)}{d}-\frac{\left(\frac{3}{68}-\frac{5 i}{68}\right) \log (\tan (c+d x)+i)}{d}+\frac{3 \log (3 \tan (c+d x)+5)}{34 d}","\frac{3 \log (3 \sin (c+d x)+5 \cos (c+d x))}{34 d}+\frac{5 x}{34}",1,"((-3/68 - (5*I)/68)*Log[I - Tan[c + d*x]])/d - ((3/68 - (5*I)/68)*Log[I + Tan[c + d*x]])/d + (3*Log[5 + 3*Tan[c + d*x]])/(34*d)","C",1
500,1,67,50,0.2710222,"\int \frac{1}{(5+3 \tan (c+d x))^2} \, dx","Integrate[(5 + 3*Tan[c + d*x])^(-2),x]","-\frac{\frac{102}{3 \tan (c+d x)+5}+(15+8 i) \log (-\tan (c+d x)+i)+(15-8 i) \log (\tan (c+d x)+i)-30 \log (3 \tan (c+d x)+5)}{1156 d}","-\frac{3}{34 d (3 \tan (c+d x)+5)}+\frac{15 \log (3 \sin (c+d x)+5 \cos (c+d x))}{578 d}+\frac{4 x}{289}",1,"-1/1156*((15 + 8*I)*Log[I - Tan[c + d*x]] + (15 - 8*I)*Log[I + Tan[c + d*x]] - 30*Log[5 + 3*Tan[c + d*x]] + 102/(5 + 3*Tan[c + d*x]))/d","C",1
501,1,86,69,0.6856592,"\int \frac{1}{(5+3 \tan (c+d x))^3} \, dx","Integrate[(5 + 3*Tan[c + d*x])^(-3),x]","\frac{\left(\frac{1}{39304}+\frac{i}{39304}\right) \left((-47+52 i) \log (-\tan (c+d x)+i)-(52-47 i) \log (\tan (c+d x)+i)+(3-3 i) \left(33 \log (3 \tan (c+d x)+5)-\frac{17 (30 \tan (c+d x)+67)}{(3 \tan (c+d x)+5)^2}\right)\right)}{d}","-\frac{15}{578 d (3 \tan (c+d x)+5)}-\frac{3}{68 d (3 \tan (c+d x)+5)^2}+\frac{99 \log (3 \sin (c+d x)+5 \cos (c+d x))}{19652 d}-\frac{5 x}{19652}",1,"((1/39304 + I/39304)*((-47 + 52*I)*Log[I - Tan[c + d*x]] - (52 - 47*I)*Log[I + Tan[c + d*x]] + (3 - 3*I)*(33*Log[5 + 3*Tan[c + d*x]] - (17*(67 + 30*Tan[c + d*x]))/(5 + 3*Tan[c + d*x])^2)))/d","C",1
502,1,95,88,0.6601322,"\int \frac{1}{(5+3 \tan (c+d x))^4} \, dx","Integrate[(5 + 3*Tan[c + d*x])^(-4),x]","-\frac{\frac{3366}{3 \tan (c+d x)+5}+\frac{8670}{(3 \tan (c+d x)+5)^2}+\frac{19652}{(3 \tan (c+d x)+5)^3}+(240-161 i) \log (-\tan (c+d x)+i)+(240+161 i) \log (\tan (c+d x)+i)-480 \log (3 \tan (c+d x)+5)}{668168 d}","-\frac{99}{19652 d (3 \tan (c+d x)+5)}-\frac{15}{1156 d (3 \tan (c+d x)+5)^2}-\frac{1}{34 d (3 \tan (c+d x)+5)^3}+\frac{60 \log (3 \sin (c+d x)+5 \cos (c+d x))}{83521 d}-\frac{161 x}{334084}",1,"-1/668168*((240 - 161*I)*Log[I - Tan[c + d*x]] + (240 + 161*I)*Log[I + Tan[c + d*x]] - 480*Log[5 + 3*Tan[c + d*x]] + 19652/(5 + 3*Tan[c + d*x])^3 + 8670/(5 + 3*Tan[c + d*x])^2 + 3366/(5 + 3*Tan[c + d*x]))/d","C",1
503,1,167,456,19.6319957,"\int \tan ^4(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(8 a^3-2 b \left(2 a^2+25 b^2\right) \tan (c+d x)+3 b^2 \sec ^2(c+d x) (a+5 b \tan (c+d x))-38 a b^2\right)}{b^3}-105 i \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+105 i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{105 d}","\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 \left(8 a^2-35 b^2\right) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}",1,"((-105*I)*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (105*I)*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2*Sqrt[a + b*Tan[c + d*x]]*(8*a^3 - 38*a*b^2 - 2*b*(2*a^2 + 25*b^2)*Tan[c + d*x] + 3*b^2*Sec[c + d*x]^2*(a + 5*b*Tan[c + d*x])))/b^3)/(105*d)","C",1
504,1,140,159,1.0511817,"\int \tan ^3(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(-2 a^2+a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)-15 b^2\right)}{b^2}+15 \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+15 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{15 d}","-\frac{4 a (a+b \tan (c+d x))^{3/2}}{15 b^2 d}+\frac{2 \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{5 b d}-\frac{2 \sqrt{a+b \tan (c+d x)}}{d}+\frac{\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(15*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 15*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-2*a^2 - 15*b^2 + a*b*Tan[c + d*x] + 3*b^2*Tan[c + d*x]^2))/b^2)/(15*d)","A",1
505,1,115,382,0.2464751,"\int \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]],x]","\frac{2 (a+b \tan (c+d x))^{3/2}}{3 b d}+\frac{i \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 (a+b \tan (c+d x))^{3/2}}{3 b d}",1,"(I*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(a + b*Tan[c + d*x])^(3/2))/(3*b*d)","C",1
506,1,100,106,0.1080758,"\int \tan (c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]],x]","-\frac{-2 \sqrt{a+b \tan (c+d x)}+\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","\frac{2 \sqrt{a+b \tan (c+d x)}}{d}-\frac{\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"-((Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - 2*Sqrt[a + b*Tan[c + d*x]])/d)","A",1
507,1,87,358,0.0328831,"\int \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]],x]","-\frac{i \left(\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{d}","\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}",1,"((-I)*(Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]))/d","C",1
508,1,111,116,0.1206756,"\int \cot (c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]],x]","\frac{-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d","A",1
509,1,139,415,0.773021,"\int \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]],x]","-\frac{\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}-i \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}","-\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}",1,"-(((b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] - I*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + I*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d)","C",1
510,1,166,189,1.1253552,"\int \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\left(8 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)-\sqrt{a} \left(4 a \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+4 a \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+\cot (c+d x) \sqrt{a+b \tan (c+d x)} (2 a \cot (c+d x)+b)\right)}{4 a^{3/2} d}","\frac{\left(8 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d}-\frac{\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{\cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}",1,"((8*a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*a*Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*a*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(b + 2*a*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*a^(3/2)*d)","A",1
511,1,268,209,1.5930437,"\int \tan ^4(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{2 \left(8 a^2-63 b^2\right) (a+b \tan (c+d x))^{5/2}}{b^2}-\frac{315 \left(a^2 \sqrt{-b^2}+2 a b^2+\left(-b^2\right)^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}+\frac{315 \left(a^2 \sqrt{-b^2}-2 a b^2+\left(-b^2\right)^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}+630 b^2 \sqrt{a+b \tan (c+d x)}+70 \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2}-\frac{40 a \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{b}}{315 b d}","\frac{2 \left(8 a^2-63 b^2\right) (a+b \tan (c+d x))^{5/2}}{315 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2}}{9 b d}+\frac{2 b \sqrt{a+b \tan (c+d x)}}{d}-\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((-315*(2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] + (315*(-2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + 630*b^2*Sqrt[a + b*Tan[c + d*x]] + (2*(8*a^2 - 63*b^2)*(a + b*Tan[c + d*x])^(5/2))/b^2 - (40*a*Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2))/b + 70*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2))/(315*b*d)","A",1
512,1,170,181,1.6082893,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2),x]","\frac{2 \sqrt{a+b \tan (c+d x)} \left(b \left(3 a^2-35 b^2\right) \tan (c+d x)-2 a \left(3 a^2+70 b^2\right)+24 a b^2 \tan ^2(c+d x)+15 b^3 \tan ^3(c+d x)\right)}{105 b^2 d}+\frac{(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{4 a (a+b \tan (c+d x))^{5/2}}{35 b^2 d}+\frac{2 \tan (c+d x) (a+b \tan (c+d x))^{5/2}}{7 b d}-\frac{2 (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{2 a \sqrt{a+b \tan (c+d x)}}{d}+\frac{(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*Sqrt[a + b*Tan[c + d*x]]*(-2*a*(3*a^2 + 70*b^2) + b*(3*a^2 - 35*b^2)*Tan[c + d*x] + 24*a*b^2*Tan[c + d*x]^2 + 15*b^3*Tan[c + d*x]^3))/(105*b^2*d)","A",1
513,1,158,135,1.0747977,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{2 (a+b \tan (c+d x))^{5/2}}{b}+5 (b+i a) \left(-\sqrt{a+b \tan (c+d x)}+\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)\right)+5 i (a+i b) \left(\sqrt{a+b \tan (c+d x)}-\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{5 d}","\frac{2 (a+b \tan (c+d x))^{5/2}}{5 b d}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{d}+\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((2*(a + b*Tan[c + d*x])^(5/2))/b + 5*(I*a + b)*(Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - Sqrt[a + b*Tan[c + d*x]]) + (5*I)*(a + I*b)*(-(Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]) + Sqrt[a + b*Tan[c + d*x]]))/(5*d)","A",1
514,1,116,128,0.3196256,"\int \tan (c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2),x]","\frac{2 \sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x))-3 (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-3 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{3 d}","\frac{2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 a \sqrt{a+b \tan (c+d x)}}{d}-\frac{(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(-3*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - 3*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(4*a + b*Tan[c + d*x]))/(3*d)","A",1
515,1,106,111,0.1042024,"\int (a+b \tan (c+d x))^{3/2} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2),x]","\frac{2 b \sqrt{a+b \tan (c+d x)}-i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","\frac{2 b \sqrt{a+b \tan (c+d x)}}{d}-\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((-I)*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + I*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]])/d","A",1
516,1,111,116,0.1978716,"\int \cot (c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2),x]","\frac{-2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+(a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(-2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d","A",1
517,1,186,149,0.3731409,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2),x]","\frac{-3 \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{a-i b} (b+i a) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+b \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)-i a \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)-a \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}","\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{3 \sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}",1,"(-3*Sqrt[a]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + Sqrt[a - I*b]*(I*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*a*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + I*b]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - a*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d","A",1
518,1,168,189,1.4557852,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\left(8 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)-\sqrt{a} \left(4 (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+4 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+\cot (c+d x) \sqrt{a+b \tan (c+d x)} (2 a \cot (c+d x)+5 b)\right)}{4 \sqrt{a} d}","\frac{\left(8 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} d}-\frac{(a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{a \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{5 b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}",1,"((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(5*b + 2*a*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*Sqrt[a]*d)","A",1
519,1,198,211,3.6477234,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \sqrt{a+b \tan (c+d x)} \left(-10 a^4+a b \left(5 a^2-231 b^2\right) \tan (c+d x)-483 a^2 b^2+\left(75 a^2 b^2-63 b^4\right) \tan ^2(c+d x)+95 a b^3 \tan ^3(c+d x)+35 b^4 \tan ^4(c+d x)+315 b^4\right)}{315 b^2 d}+\frac{(a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{2 \left(a^2-b^2\right) \sqrt{a+b \tan (c+d x)}}{d}-\frac{4 a (a+b \tan (c+d x))^{7/2}}{63 b^2 d}+\frac{2 \tan (c+d x) (a+b \tan (c+d x))^{7/2}}{9 b d}-\frac{2 (a+b \tan (c+d x))^{5/2}}{5 d}-\frac{2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{(a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*Sqrt[a + b*Tan[c + d*x]]*(-10*a^4 - 483*a^2*b^2 + 315*b^4 + a*b*(5*a^2 - 231*b^2)*Tan[c + d*x] + (75*a^2*b^2 - 63*b^4)*Tan[c + d*x]^2 + 95*a*b^3*Tan[c + d*x]^3 + 35*b^4*Tan[c + d*x]^4))/(315*b^2*d)","A",1
520,1,205,158,2.1305497,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{1}{2} \sec ^3(c+d x) \sqrt{a+b \tan (c+d x)} \left(\left(3 a^3-58 a b^2\right) \cos (3 (c+d x))+3 a \left(3 a^2-46 b^2\right) \cos (c+d x)-2 b \sin (c+d x) \left(\left(10 b^2-9 a^2\right) \cos (2 (c+d x))-9 a^2+4 b^2\right)\right)+21 i b (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-21 i b (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{21 b d}","\frac{2 (a+b \tan (c+d x))^{7/2}}{7 b d}-\frac{2 b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{4 a b \sqrt{a+b \tan (c+d x)}}{d}+\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((21*I)*(a - I*b)^(5/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (21*I)*(a + I*b)^(5/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (Sec[c + d*x]^3*(3*a*(3*a^2 - 46*b^2)*Cos[c + d*x] + (3*a^3 - 58*a*b^2)*Cos[3*(c + d*x)] - 2*b*(-9*a^2 + 4*b^2 + (-9*a^2 + 10*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/2)/(21*b*d)","A",1
521,1,138,158,0.7136029,"\int \tan (c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \sqrt{a+b \tan (c+d x)} \left(23 a^2+11 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)-15 b^2\right)-15 (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-15 (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{15 d}","\frac{2 \left(a^2-b^2\right) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{2 a (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{(a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(-15*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - 15*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(23*a^2 - 15*b^2 + 11*a*b*Tan[c + d*x] + 3*b^2*Tan[c + d*x]^2))/(15*d)","A",1
522,1,121,134,0.4069045,"\int (a+b \tan (c+d x))^{5/2} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 b \sqrt{a+b \tan (c+d x)} (7 a+b \tan (c+d x))-3 i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+3 i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{3 d}","\frac{2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{4 a b \sqrt{a+b \tan (c+d x)}}{d}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((-3*I)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (3*I)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]*(7*a + b*Tan[c + d*x]))/(3*d)","A",1
523,1,220,138,0.1918959,"\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2),x]","\frac{-2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+a^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 b^2 \sqrt{a+b \tan (c+d x)}-b^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 i a b \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+(a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}","-\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}+\frac{2 b^2 \sqrt{a+b \tan (c+d x)}}{d}+\frac{(a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{(a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(-2*a^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + a^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2*I)*a*Sqrt[a + I*b]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b^2*Sqrt[a + b*Tan[c + d*x]])/d","A",1
524,1,233,151,0.5620502,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2),x]","\frac{-5 a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)-i a^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)-a^2 \cot (c+d x) \sqrt{a+b \tan (c+d x)}+i b^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+2 a b \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}","-\frac{5 a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{d}-\frac{a^2 \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{d}+\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"(-5*a^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*a^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*a*Sqrt[a + I*b]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + I*Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - a^2*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d","A",1
525,1,268,192,1.4229302,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2),x]","-\frac{-\sqrt{a} \left(8 a^2-15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)+4 a^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+2 a^2 \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}-4 b^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+4 (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+8 i a b \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)+9 a b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}","\frac{\sqrt{a} \left(8 a^2-15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 d}-\frac{a^2 \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{(a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}-\frac{(a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{9 a b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}",1,"-1/4*(-(Sqrt[a]*(8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]) + 4*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*a^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (8*I)*a*Sqrt[a + I*b]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - 4*Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 9*a*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]] + 2*a^2*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d","A",1
526,1,185,237,3.6343373,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2),x]","-\frac{\frac{15 b \left(b^2-8 a^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\cot (c+d x) \sqrt{a+b \tan (c+d x)} \left(8 a^2 \cot ^2(c+d x)-24 a^2+26 a b \cot (c+d x)+33 b^2\right)+24 i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-24 i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{24 d}","\frac{5 b \left(8 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{8 \sqrt{a} d}+\frac{\left(8 a^2-11 b^2\right) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}",1,"-1/24*((15*b*(-8*a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (24*I)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (24*I)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(-24*a^2 + 33*b^2 + 26*a*b*Cot[c + d*x] + 8*a^2*Cot[c + d*x]^2)*Sqrt[a + b*Tan[c + d*x]])/d","A",1
527,1,143,167,0.9476969,"\int (a+b \tan (c+d x))^{7/2} \, dx","Integrate[(a + b*Tan[c + d*x])^(7/2),x]","\frac{2 b \sqrt{a+b \tan (c+d x)} \left(58 a^2+16 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)-15 b^2\right)-15 i (a-i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)+15 i (a+i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{15 d}","\frac{2 b \left(3 a^2-b^2\right) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{i (a-i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d}+\frac{i (a+i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d}",1,"((-15*I)*(a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (15*I)*(a + I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]*(58*a^2 - 15*b^2 + 16*a*b*Tan[c + d*x] + 3*b^2*Tan[c + d*x]^2))/(15*d)","A",1
528,1,185,229,3.4663598,"\int \frac{\tan ^5(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^5/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(-48 a^3+b \left(24 a^2-35 b^2\right) \tan (c+d x)-18 a b^2 \tan ^2(c+d x)+70 a b^2+15 b^3 \tan ^3(c+d x)\right)}{b^4}-\frac{105 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}-\frac{105 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}}{105 d}","-\frac{4 a \left(24 a^2-35 b^2\right) \sqrt{a+b \tan (c+d x)}}{105 b^4 d}+\frac{2 \left(24 a^2-35 b^2\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{105 b^3 d}-\frac{12 a \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{35 b^2 d}+\frac{2 \tan ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{7 b d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"((-105*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] - (105*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-48*a^3 + 70*a*b^2 + b*(24*a^2 - 35*b^2)*Tan[c + d*x] - 18*a*b^2*Tan[c + d*x]^2 + 15*b^3*Tan[c + d*x]^3))/b^4)/(105*d)","A",1
529,1,184,500,3.3844984,"\int \frac{\tan ^4(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^4/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(8 a^2-4 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)-15 b^2\right)-\frac{15 \left(-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}}{b^2}-\frac{15 \sqrt{-b^2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}}{15 b d}","-\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 \left(8 a^2-15 b^2\right) \sqrt{a+b \tan (c+d x)}}{15 b^3 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{15 b^2 d}+\frac{2 \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{5 b d}",1,"((-15*Sqrt[-b^2]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] + ((-15*(-b^2)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + 2*Sqrt[a + b*Tan[c + d*x]]*(8*a^2 - 15*b^2 - 4*a*b*Tan[c + d*x] + 3*b^2*Tan[c + d*x]^2))/b^2)/(15*b*d)","A",1
530,1,159,140,0.7527178,"\int \frac{\tan ^3(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^3/Sqrt[a + b*Tan[c + d*x]],x]","\frac{3 b^2 \sqrt{a-i b} (a+i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)-(a-i b) \left(2 (a+i b) (2 a-b \tan (c+d x)) \sqrt{a+b \tan (c+d x)}-3 b^2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)\right)}{3 b^2 d \left(a^2+b^2\right)}","-\frac{4 a \sqrt{a+b \tan (c+d x)}}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"(3*Sqrt[a - I*b]*(a + I*b)*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (a - I*b)*(-3*Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*(a + I*b)*(2*a - b*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(3*b^2*(a^2 + b^2)*d)","A",1
531,1,108,424,0.1958755,"\int \frac{\tan ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^2/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)}}{b}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{d}","\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 \sqrt{a+b \tan (c+d x)}}{b d}",1,"((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]])/b)/d","C",1
532,1,107,87,0.0683673,"\int \frac{\tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (-a+i b)}+\frac{\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (-a-i b)}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"(Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) + (Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((-a - I*b)*d)","A",1
533,1,87,402,0.0483337,"\int \frac{1}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Tan[c + d*x]],x]","-\frac{i \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}\right)}{d}","-\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}",1,"((-I)*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]))/d","C",1
534,1,111,116,0.1441436,"\int \frac{\cot (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]/Sqrt[a + b*Tan[c + d*x]],x]","\frac{-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{d}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}",1,"((-2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b])/d","A",1
535,1,142,461,0.7974666,"\int \frac{\cot ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^2/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a}}{d}","\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{b \log \left(-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left(\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right)}{2 \sqrt{2} d \sqrt{a^2+b^2} \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right)}{\sqrt{2} d \sqrt{a^2+b^2} \sqrt{a-\sqrt{a^2+b^2}}}-\frac{\cot (c+d x) \sqrt{a+b \tan (c+d x)}}{a d}",1,"((b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/a)/d","C",1
536,1,203,194,3.0990896,"\int \frac{\cot ^3(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^3/Sqrt[a + b*Tan[c + d*x]],x]","\frac{-\frac{4 a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}-\frac{4 a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}+\frac{\left(8 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}-2 a \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}+3 b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a^2 d}","\frac{3 b \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a^2 d}+\frac{\left(8 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{5/2} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d \sqrt{a-i b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d \sqrt{a+i b}}-\frac{\cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 a d}",1,"(((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] - (4*a^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] - (4*a^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + 3*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]] - 2*a*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(4*a^2*d)","A",1
537,1,311,282,6.4635414,"\int \frac{\tan ^5(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^(3/2),x]","\frac{2 \left(\frac{2 \left(8 a^2-5 b^2\right) \sqrt{a+b \tan (c+d x)}+\frac{5 b^4 \left(b^2-a \sqrt{-b^2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{-b^2} \left(a^2+b^2\right) \sqrt{a-\sqrt{-b^2}}}-\frac{5 b^4 \left(a-\sqrt{-b^2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\left(a^2+b^2\right) \sqrt{a+\sqrt{-b^2}}}+\frac{2 a^3 \left(8 a^2+3 b^2\right)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}}{2 b^3 d}-\frac{2 a \tan ^2(c+d x)}{b d \sqrt{a+b \tan (c+d x)}}\right)}{5 b}+\frac{2 \tan ^3(c+d x)}{5 b d \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2 \tan ^3(c+d x)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left(6 a^2+b^2\right) \tan ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{5 b^2 d \left(a^2+b^2\right)}-\frac{2 a \left(8 a^2+3 b^2\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{5 b^3 d \left(a^2+b^2\right)}+\frac{2 \left(16 a^4+6 a^2 b^2-5 b^4\right) \sqrt{a+b \tan (c+d x)}}{5 b^4 d \left(a^2+b^2\right)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"(2*Tan[c + d*x]^3)/(5*b*d*Sqrt[a + b*Tan[c + d*x]]) + (2*((-2*a*Tan[c + d*x]^2)/(b*d*Sqrt[a + b*Tan[c + d*x]]) + ((5*b^4*(b^2 - a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(Sqrt[-b^2]*(a^2 + b^2)*Sqrt[a - Sqrt[-b^2]]) - (5*b^4*(a - Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/((a^2 + b^2)*Sqrt[a + Sqrt[-b^2]]) + (2*a^3*(8*a^2 + 3*b^2))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + 2*(8*a^2 - 5*b^2)*Sqrt[a + b*Tan[c + d*x]])/(2*b^3*d)))/(5*b)","A",1
538,1,252,226,3.2525554,"\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{2 a^2 \left(8 a^2+5 b^2\right)}{b^2 \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{3 b^2 \left(\frac{a}{\sqrt{-b^2}}+1\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\left(a^2+b^2\right) \sqrt{a-\sqrt{-b^2}}}+\frac{3 \left(a \sqrt{-b^2}+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\left(a^2+b^2\right) \sqrt{a+\sqrt{-b^2}}}+\frac{2 \tan ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}}-\frac{8 a \tan (c+d x)}{b \sqrt{a+b \tan (c+d x)}}}{3 b d}","-\frac{2 a^2 \tan ^2(c+d x)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left(4 a^2+b^2\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 d \left(a^2+b^2\right)}-\frac{2 a \left(8 a^2+5 b^2\right) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left(a^2+b^2\right)}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((3*b^2*(1 + a/Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/((a^2 + b^2)*Sqrt[a - Sqrt[-b^2]]) + (3*(b^2 + a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/((a^2 + b^2)*Sqrt[a + Sqrt[-b^2]]) - (2*a^2*(8*a^2 + 5*b^2))/(b^2*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) - (8*a*Tan[c + d*x])/(b*Sqrt[a + b*Tan[c + d*x]]) + (2*Tan[c + d*x]^2)/Sqrt[a + b*Tan[c + d*x]])/(3*b*d)","A",1
539,1,243,165,1.4894419,"\int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{(b+i a) \sqrt{a+b \tan (c+d x)}}+\frac{a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{(-b+i a) \sqrt{a+b \tan (c+d x)}}+\frac{4 a}{b \sqrt{a+b \tan (c+d x)}}+\frac{2 \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{b d}","-\frac{2 a^2 \tan (c+d x)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left(2 a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}{b^2 d \left(a^2+b^2\right)}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (4*a)/(b*Sqrt[a + b*Tan[c + d*x]]) - (a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)])/((I*a + b)*Sqrt[a + b*Tan[c + d*x]]) + (a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])/((I*a - b)*Sqrt[a + b*Tan[c + d*x]]) + (2*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]])/(b*d)","C",1
540,1,119,125,0.1631611,"\int \frac{\tan ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^(3/2),x]","\frac{b (b-i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \left(-i b \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)+2 a+2 i b\right)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"(b*((-I)*a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*(2*a + (2*I)*b - I*b*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]))/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])","C",1
541,1,100,116,0.1789198,"\int \frac{\tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^(3/2),x]","\frac{(a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}","\frac{2 a}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])","C",1
542,1,105,120,0.1104534,"\int \frac{1}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-3/2),x]","\frac{i (a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(-b-i a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}","-\frac{2 b}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"(I*(a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])","C",1
543,1,165,150,1.1875046,"\int \frac{\cot (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{2 \left(a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{2 b^2}{\sqrt{a+b \tan (c+d x)}}+\frac{a (a+i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{\sqrt{a-i b}}+\frac{a (a-i b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{\sqrt{a+i b}}}{a d \left(a^2+b^2\right)}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d}+\frac{2 b^2}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}",1,"((-2*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (a*(a + I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (a*(a - I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*b^2)/Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)*d)","A",1
544,1,184,192,3.4163127,"\int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{\frac{b \left(a^2+3 b^2\right)}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{i a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{(a-i b)^{3/2}}+\frac{i a^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{(a+i b)^{3/2}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{a \cot (c+d x)}{\sqrt{a+b \tan (c+d x)}}}{a^2 d}","\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}-\frac{b \left(a^2+3 b^2\right)}{a^2 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}-\frac{\cot (c+d x)}{a d \sqrt{a+b \tan (c+d x)}}",1,"-(((-3*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] - (I*a^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(a - I*b)^(3/2) + (I*a^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(a + I*b)^(3/2) + (b*(a^2 + 3*b^2))/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) + (a*Cot[c + d*x])/Sqrt[a + b*Tan[c + d*x]])/(a^2*d))","A",1
545,1,259,241,5.5093058,"\int \frac{\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{\left(8 a^2-15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{-\frac{4 a^3 \left(a+\sqrt{-b^2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\sqrt{a-\sqrt{-b^2}}}+\frac{4 a^3 \left(\sqrt{-b^2}-a\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\sqrt{a+\sqrt{-b^2}}}+\frac{-2 a^2 \left(a^2+b^2\right) \cot ^2(c+d x)+5 a b \left(a^2+b^2\right) \cot (c+d x)+7 a^2 b^2+15 b^4}{\sqrt{a+b \tan (c+d x)}}}{a \left(a^2+b^2\right)}}{4 a^2 d}","\frac{5 b \cot (c+d x)}{4 a^2 d \sqrt{a+b \tan (c+d x)}}+\frac{\left(8 a^2-15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{7/2} d}+\frac{b^2 \left(7 a^2+15 b^2\right)}{4 a^3 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{3/2}}-\frac{\cot ^2(c+d x)}{2 a d \sqrt{a+b \tan (c+d x)}}",1,"(((8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + ((-4*a^3*(a + Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] + (4*a^3*(-a + Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + (7*a^2*b^2 + 15*b^4 + 5*a*b*(a^2 + b^2)*Cot[c + d*x] - 2*a^2*(a^2 + b^2)*Cot[c + d*x]^2)/Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)))/(4*a^2*d)","A",1
546,1,353,291,5.1524853,"\int \frac{\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \left(-\frac{3 b \left(a^2 \sqrt{-b^2}-2 a b^2+\left(-b^2\right)^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{2 \sqrt{-b^2} \left(a^2+b^2\right)^2 \sqrt{a-\sqrt{-b^2}}}-\frac{3 b \left(a^2 \sqrt{-b^2}+2 a b^2+\left(-b^2\right)^{3/2}\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{2 \sqrt{-b^2} \left(a^2+b^2\right)^2 \sqrt{a+\sqrt{-b^2}}}-\frac{3 a^2 \left(8 a^4+15 a^2 b^2+5 b^4\right)}{b^3 \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{a^3 \left(8 a^2+7 b^2\right)}{b^3 \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}}-\frac{6 a \tan ^2(c+d x)}{b (a+b \tan (c+d x))^{3/2}}\right)}{3 b d}","-\frac{2 a^2 \tan ^3(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{4 a^2 \left(a^2+2 b^2\right) \tan ^2(c+d x)}{b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{4 a \left(8 a^4+15 a^2 b^2+4 b^4\right) \sqrt{a+b \tan (c+d x)}}{3 b^4 d \left(a^2+b^2\right)^2}+\frac{2 \left(8 a^4+15 a^2 b^2+b^4\right) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left(a^2+b^2\right)^2}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"(2*((-3*b*(-2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*Sqrt[-b^2]*(a^2 + b^2)^2*Sqrt[a - Sqrt[-b^2]]) - (3*b*(2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*Sqrt[-b^2]*(a^2 + b^2)^2*Sqrt[a + Sqrt[-b^2]]) + (a^3*(8*a^2 + 7*b^2))/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) - (6*a*Tan[c + d*x]^2)/(b*(a + b*Tan[c + d*x])^(3/2)) + Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2) - (3*a^2*(8*a^4 + 15*a^2*b^2 + 5*b^4))/(b^3*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]])))/(3*b*d)","A",1
547,1,308,226,3.0297947,"\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{2 a^2 \left(8 a^2+9 b^2\right)}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{3 \left(-b^2\right)^{3/2} \left(a^2+2 a \sqrt{-b^2}-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right)}{\left(a^2+b^2\right)^2 \sqrt{a-\sqrt{-b^2}}}+\frac{3 \left(-b^2\right)^{3/2} \left(-a^2+2 a \sqrt{-b^2}+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right)}{\left(a^2+b^2\right)^2 \sqrt{a+\sqrt{-b^2}}}-\frac{12 a b^4}{\left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{6 b^2 \tan ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}}+\frac{24 a b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 b^3 d}","-\frac{2 a^2 \tan ^2(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(4 a^2+3 b^2\right) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left(a^2+b^2\right)}+\frac{4 a^3 \left(2 a^2+5 b^2\right)}{3 b^3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((3*(-b^2)^(3/2)*(a^2 - b^2 + 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/((a^2 + b^2)^2*Sqrt[a - Sqrt[-b^2]]) + (3*(-b^2)^(3/2)*(-a^2 + b^2 + 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/((a^2 + b^2)^2*Sqrt[a + Sqrt[-b^2]]) + (2*a^2*(8*a^2 + 9*b^2))/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (24*a*b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (6*b^2*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^(3/2) - (12*a*b^4)/((a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]]))/(3*b^3*d)","A",1
548,1,220,172,1.4960522,"\int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(5/2),x]","\frac{-\frac{a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{b+i a}+\frac{a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{-b+i a}+\frac{3 (a+b \tan (c+d x)) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)}{b+i a}+\frac{3 i (a+b \tan (c+d x)) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{a+i b}-\frac{4 a}{b}-6 \tan (c+d x)}{3 b d (a+b \tan (c+d x))^{3/2}}","-\frac{4 a^2 \left(a^2+4 b^2\right)}{3 b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{2 a^2 \tan (c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((-4*a)/b - (a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)])/(I*a + b) + (a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(I*a - b) - 6*Tan[c + d*x] + (3*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x]))/(I*a + b) + ((3*I)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x]))/(a + I*b))/(3*b*d*(a + b*Tan[c + d*x])^(3/2))","C",1
549,1,122,157,0.1921147,"\int \frac{\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2),x]","\frac{b (b-i a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \left(-i b \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)+2 a+2 i b\right)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}","-\frac{2 a^2}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{4 a b}{d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"(b*((-I)*a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*(2*a + (2*I)*b - I*b*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)]))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))","C",1
550,1,103,155,0.1827619,"\int \frac{\tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^(5/2),x]","\frac{(a+i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}","\frac{2 a}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"((a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))","C",1
551,1,108,152,0.0909866,"\int \frac{1}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-5/2),x]","\frac{i (a+i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(-b-i a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}","-\frac{4 a b}{d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{2 b}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"(I*(a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))","C",1
552,1,237,195,2.2399881,"\int \frac{\cot (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]/(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \left(\frac{3 b^2 \left(3 a^2+b^2\right)}{a \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{-\frac{3 \left(a^2+b^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a}}+\frac{3 a^2 (a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{2 \sqrt{a+i b}}+\frac{3 a^2 (a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{2 \sqrt{a-i b}}}{a \left(a^2+b^2\right)}+\frac{b^2}{(a+b \tan (c+d x))^{3/2}}\right)}{3 a d \left(a^2+b^2\right)}","-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d}+\frac{2 b^2 \left(3 a^2+b^2\right)}{a^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}",1,"(2*(((-3*(a^2 + b^2)^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (3*a^2*(a + I*b)^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(2*Sqrt[a - I*b]) + (3*a^2*(a - I*b)^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(2*Sqrt[a + I*b]))/(a*(a^2 + b^2)) + b^2/(a + b*Tan[c + d*x])^(3/2) + (3*b^2*(3*a^2 + b^2))/(a*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(3*a*(a^2 + b^2)*d)","A",1
553,1,232,245,4.7660067,"\int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{-\frac{15 b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b \left(3 a^2+5 b^2\right)}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-3 i a^2 \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{(a-i b)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{(a+i b)^{5/2}}\right)+\frac{3 b \left(a^4+10 a^2 b^2+5 b^4\right)}{a \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{3 a \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 a^2 d}","\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right)}{a^{7/2} d}-\frac{b \left(3 a^2+5 b^2\right)}{3 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{b \left(a^4+10 a^2 b^2+5 b^4\right)}{a^3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{5/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{5/2}}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}",1,"-1/3*((-15*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) - (3*I)*a^2*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/(a - I*b)^(5/2) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(a + I*b)^(5/2)) + (b*(3*a^2 + 5*b^2))/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (3*a*Cot[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (3*b*(a^4 + 10*a^2*b^2 + 5*b^4))/(a*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]]))/(a^2*d)","A",1
554,1,108,194,0.2454261,"\int \frac{1}{(a+b \tan (c+d x))^{7/2}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-7/2),x]","\frac{i (a+i b) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{a+b \tan (c+d x)}{a-i b}\right)+(-b-i a) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{a+b \tan (c+d x)}{a+i b}\right)}{5 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{5/2}}","-\frac{2 b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^3 \sqrt{a+b \tan (c+d x)}}-\frac{4 a b}{3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}-\frac{2 b}{5 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{5/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right)}{d (a-i b)^{7/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right)}{d (a+i b)^{7/2}}",1,"(I*(a + I*b)*Hypergeometric2F1[-5/2, 1, -3/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1[-5/2, 1, -3/2, (a + b*Tan[c + d*x])/(a + I*b)])/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2))","C",1
555,1,106,202,0.5889835,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]),x]","\frac{-15 \sqrt[4]{-1} (b+i a) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 \sqrt{\tan (c+d x)} \left(5 a \tan (c+d x)+3 b \tan ^2(c+d x)-15 b\right)+15 (-1)^{3/4} (a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{15 d}","\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{2 b \sqrt{\tan (c+d x)}}{d}",1,"(-15*(-1)^(1/4)*(I*a + b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 15*(-1)^(3/4)*(a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(-15*b + 5*a*Tan[c + d*x] + 3*b*Tan[c + d*x]^2))/(15*d)","C",1
556,1,94,184,0.2448472,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]),x]","\frac{3 \sqrt[4]{-1} (a-i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 \sqrt{\tan (c+d x)} (3 a+b \tan (c+d x))+3 \sqrt[4]{-1} (a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{3 d}","\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 a \sqrt{\tan (c+d x)}}{d}+\frac{2 b \tan ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(3*(-1)^(1/4)*(a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 3*(-1)^(1/4)*(a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(3*a + b*Tan[c + d*x]))/(3*d)","C",1
557,1,79,166,0.092266,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x)) \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]),x]","\frac{\sqrt[4]{-1} (b+i a) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-(-1)^{3/4} (a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 b \sqrt{\tan (c+d x)}}{d}","-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b \sqrt{\tan (c+d x)}}{d}",1,"((-1)^(1/4)*(I*a + b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (-1)^(3/4)*(a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*b*Sqrt[Tan[c + d*x]])/d","C",1
558,1,61,150,0.0429467,"\int \frac{a+b \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])/Sqrt[Tan[c + d*x]],x]","-\frac{\sqrt[4]{-1} \left((a-i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{d}","-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"-(((-1)^(1/4)*((a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/d)","C",1
559,1,69,166,0.1054981,"\int \frac{a+b \tan (c+d x)}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])/Tan[c + d*x]^(3/2),x]","\frac{-\left((a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-i \tan (c+d x)\right)\right)-(a-i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};i \tan (c+d x)\right)}{d \sqrt{\tan (c+d x)}}","\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a}{d \sqrt{\tan (c+d x)}}",1,"(-((a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (-I)*Tan[c + d*x]]) - (a - I*b)*Hypergeometric2F1[-1/2, 1, 1/2, I*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])","C",1
560,1,72,184,0.1093779,"\int \frac{a+b \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])/Tan[c + d*x]^(5/2),x]","\frac{-\left((a+i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-i \tan (c+d x)\right)\right)-(a-i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};i \tan (c+d x)\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b}{d \sqrt{\tan (c+d x)}}",1,"(-((a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (-I)*Tan[c + d*x]]) - (a - I*b)*Hypergeometric2F1[-3/2, 1, -1/2, I*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2))","C",1
561,1,72,202,0.1186982,"\int \frac{a+b \tan (c+d x)}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])/Tan[c + d*x]^(7/2),x]","\frac{-\left((a+i b) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-i \tan (c+d x)\right)\right)-(a-i b) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};i \tan (c+d x)\right)}{5 d \tan ^{\frac{5}{2}}(c+d x)}","-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a}{5 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\tan (c+d x)}}-\frac{2 b}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-((a + I*b)*Hypergeometric2F1[-5/2, 1, -3/2, (-I)*Tan[c + d*x]]) - (a - I*b)*Hypergeometric2F1[-5/2, 1, -3/2, I*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2))","C",1
562,1,133,268,1.1294282,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2,x]","\frac{2 \sqrt{\tan (c+d x)} \left(35 \left(a^2-b^2\right) \tan (c+d x)+42 a b \tan ^2(c+d x)-210 a b+15 b^2 \tan ^3(c+d x)\right)-105 (-1)^{3/4} (a-i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+105 (-1)^{3/4} (a+i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{105 d}","\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2-b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{4 a b \tan ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{4 a b \sqrt{\tan (c+d x)}}{d}+\frac{2 b^2 \tan ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(-105*(-1)^(3/4)*(a - I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(3/4)*(a + I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(-210*a*b + 35*(a^2 - b^2)*Tan[c + d*x] + 42*a*b*Tan[c + d*x]^2 + 15*b^2*Tan[c + d*x]^3))/(105*d)","C",1
563,1,120,249,0.7355044,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2,x]","\frac{2 \sqrt{\tan (c+d x)} \left(15 a^2+10 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)-15 b^2\right)+15 \sqrt[4]{-1} (a-i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+15 \sqrt[4]{-1} (a+i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{15 d}","\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2-b^2\right) \sqrt{\tan (c+d x)}}{d}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{4 a b \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b^2 \tan ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(15*(-1)^(1/4)*(a - I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 15*(-1)^(1/4)*(a + I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(15*a^2 - 15*b^2 + 10*a*b*Tan[c + d*x] + 3*b^2*Tan[c + d*x]^2))/(15*d)","C",1
564,1,99,223,0.2796195,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2 \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2,x]","\frac{3 (-1)^{3/4} (a-i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 b \sqrt{\tan (c+d x)} (6 a+b \tan (c+d x))-3 (-1)^{3/4} (a+i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{3 d}","-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{4 a b \sqrt{\tan (c+d x)}}{d}+\frac{2 b^2 \tan ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(3*(-1)^(3/4)*(a - I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 3*(-1)^(3/4)*(a + I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*b*Sqrt[Tan[c + d*x]]*(6*a + b*Tan[c + d*x]))/(3*d)","C",1
565,1,85,204,0.1175673,"\int \frac{(a+b \tan (c+d x))^2}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Sqrt[Tan[c + d*x]],x]","-\frac{\sqrt[4]{-1} (a-i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} (a+i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-2 b^2 \sqrt{\tan (c+d x)}}{d}","-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 \sqrt{\tan (c+d x)}}{d}",1,"-(((-1)^(1/4)*(a - I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (-1)^(1/4)*(a + I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 2*b^2*Sqrt[Tan[c + d*x]])/d)","C",1
566,1,166,204,0.9642548,"\int \frac{(a+b \tan (c+d x))^2}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(3/2),x]","-\frac{\frac{4 (a-b) (a+b) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)}{\sqrt{\tan (c+d x)}}+\sqrt{2} a b \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\frac{4 b^2}{\sqrt{\tan (c+d x)}}}{2 d}","\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2}{d \sqrt{\tan (c+d x)}}",1,"-1/2*(Sqrt[2]*a*b*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + (4*b^2)/Sqrt[Tan[c + d*x]] + (4*(a - b)*(a + b)*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2])/Sqrt[Tan[c + d*x]])/d","C",1
567,1,77,223,0.2221405,"\int \frac{(a+b \tan (c+d x))^2}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(5/2),x]","-\frac{2 \left(\left(a^2-b^2\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+b \left(6 a \tan (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+b\right)\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\tan (c+d x)}}",1,"(-2*((a^2 - b^2)*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + b*(b + 6*a*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]*Tan[c + d*x])))/(3*d*Tan[c + d*x]^(3/2))","C",1
568,1,81,249,0.2392867,"\int \frac{(a+b \tan (c+d x))^2}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(7/2),x]","\frac{-6 \left(a^2-b^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)-2 b \left(10 a \tan (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+3 b\right)}{15 d \tan ^{\frac{5}{2}}(c+d x)}","-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 \left(a^2-b^2\right)}{d \sqrt{\tan (c+d x)}}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{4 a b}{3 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-6*(a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] - 2*b*(3*b + 10*a*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]*Tan[c + d*x]))/(15*d*Tan[c + d*x]^(5/2))","C",1
569,1,164,328,3.081547,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3,x]","\frac{2 \sqrt{\tan (c+d x)} \left(-63 b \left(b^2-3 a^2\right) \tan ^2(c+d x)+105 a \left(a^2-3 b^2\right) \tan (c+d x)+315 b \left(b^2-3 a^2\right)+135 a b^2 \tan ^3(c+d x)+35 b^3 \tan ^4(c+d x)\right)-315 (-1)^{3/4} (a-i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+315 (-1)^{3/4} (a+i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{315 d}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b \left(3 a^2-b^2\right) \tan ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a \left(a^2-3 b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 b \left(3 a^2-b^2\right) \sqrt{\tan (c+d x)}}{d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 \tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac{40 a b^2 \tan ^{\frac{7}{2}}(c+d x)}{63 d}",1,"(-315*(-1)^(3/4)*(a - I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 315*(-1)^(3/4)*(a + I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(315*b*(-3*a^2 + b^2) + 105*a*(a^2 - 3*b^2)*Tan[c + d*x] - 63*b*(-3*a^2 + b^2)*Tan[c + d*x]^2 + 135*a*b^2*Tan[c + d*x]^3 + 35*b^3*Tan[c + d*x]^4))/(315*d)","C",1
570,1,144,299,1.4041907,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3,x]","\frac{2 \sqrt{\tan (c+d x)} \left(105 \left(a^3-3 a b^2\right)-35 b \left(b^2-3 a^2\right) \tan (c+d x)+63 a b^2 \tan ^2(c+d x)+15 b^3 \tan ^3(c+d x)\right)+105 \sqrt[4]{-1} (a-i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+105 \sqrt[4]{-1} (a+i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{105 d}","\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b \left(3 a^2-b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a \left(a^2-3 b^2\right) \sqrt{\tan (c+d x)}}{d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))}{7 d}+\frac{32 a b^2 \tan ^{\frac{5}{2}}(c+d x)}{35 d}",1,"(105*(-1)^(1/4)*(a - I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 105*(-1)^(1/4)*(a + I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(105*(a^3 - 3*a*b^2) - 35*b*(-3*a^2 + b^2)*Tan[c + d*x] + 63*a*b^2*Tan[c + d*x]^2 + 15*b^3*Tan[c + d*x]^3))/(105*d)","C",1
571,1,120,272,0.7995743,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^3 \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3,x]","\frac{2 b \sqrt{\tan (c+d x)} \left(15 a^2+5 a b \tan (c+d x)+b^2 \tan ^2(c+d x)-5 b^2\right)-5 \sqrt[4]{-1} (b+i a)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-5 (-1)^{3/4} (a+i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{5 d}","-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b \left(3 a^2-b^2\right) \sqrt{\tan (c+d x)}}{d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{8 a b^2 \tan ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d}",1,"(-5*(-1)^(1/4)*(I*a + b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 5*(-1)^(3/4)*(a + I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*b*Sqrt[Tan[c + d*x]]*(15*a^2 - 5*b^2 + 5*a*b*Tan[c + d*x] + b^2*Tan[c + d*x]^2))/(5*d)","C",1
572,1,101,245,0.3423349,"\int \frac{(a+b \tan (c+d x))^3}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Sqrt[Tan[c + d*x]],x]","\frac{2 b^2 \sqrt{\tan (c+d x)} (9 a+b \tan (c+d x))-3 \sqrt[4]{-1} (a-i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-3 \sqrt[4]{-1} (a+i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{3 d}","-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{2 b^2 \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}{3 d}+\frac{16 a b^2 \sqrt{\tan (c+d x)}}{3 d}",1,"(-3*(-1)^(1/4)*(a - I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 3*(-1)^(1/4)*(a + I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*b^2*Sqrt[Tan[c + d*x]]*(9*a + b*Tan[c + d*x]))/(3*d)","C",1
573,1,192,245,1.9110017,"\int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(3/2),x]","\frac{-8 a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+\sqrt{2} b \left(b^2-3 a^2\right) \sqrt{\tan (c+d x)} \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+8 b^2 (a+b \tan (c+d x))-32 a b^2}{4 d \sqrt{\tan (c+d x)}}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}{d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{d \sqrt{\tan (c+d x)}}",1,"(-32*a*b^2 - 8*a*(a^2 - 3*b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2] + Sqrt[2]*b*(-3*a^2 + b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]] + 8*b^2*(a + b*Tan[c + d*x]))/(4*d*Sqrt[Tan[c + d*x]])","C",1
574,1,91,245,0.2806402,"\int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(5/2),x]","\frac{-6 b^2 (a+b \tan (c+d x))+(a-i b)^3 \left(-\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};i \tan (c+d x)\right)\right)-(a+i b)^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-i \tan (c+d x)\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}","\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{16 a^2 b}{3 d \sqrt{\tan (c+d x)}}",1,"(-((a + I*b)^3*Hypergeometric2F1[-3/2, 1, -1/2, (-I)*Tan[c + d*x]]) - (a - I*b)^3*Hypergeometric2F1[-3/2, 1, -1/2, I*Tan[c + d*x]] - 6*b^2*(a + b*Tan[c + d*x]))/(3*d*Tan[c + d*x]^(3/2))","C",1
575,1,103,270,0.4887147,"\int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(7/2),x]","-\frac{2 \left(3 a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)+b \left(5 \left(3 a^2-b^2\right) \tan (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+b (9 a+5 b \tan (c+d x))\right)\right)}{15 d \tan ^{\frac{5}{2}}(c+d x)}","-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 a \left(a^2-3 b^2\right)}{d \sqrt{\tan (c+d x)}}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{8 a^2 b}{5 d \tan ^{\frac{3}{2}}(c+d x)}",1,"(-2*(3*a*(a^2 - 3*b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + b*(5*(3*a^2 - b^2)*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(9*a + 5*b*Tan[c + d*x]))))/(15*d*Tan[c + d*x]^(5/2))","C",1
576,1,103,299,0.5885065,"\int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(9/2),x]","-\frac{2 \left(5 a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(c+d x)\right)+b \left(7 \left(3 a^2-b^2\right) \tan (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)+b (15 a+7 b \tan (c+d x))\right)\right)}{35 d \tan ^{\frac{7}{2}}(c+d x)}","-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 a \left(a^2-3 b^2\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(3 a^2-b^2\right)}{d \sqrt{\tan (c+d x)}}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{32 a^2 b}{35 d \tan ^{\frac{5}{2}}(c+d x)}",1,"(-2*(5*a*(a^2 - 3*b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[c + d*x]^2] + b*(7*(3*a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(15*a + 7*b*Tan[c + d*x]))))/(35*d*Tan[c + d*x]^(7/2))","C",1
577,1,104,326,0.6711813,"\int \frac{(a+b \tan (c+d x))^3}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(11/2),x]","-\frac{2 \left(7 a \left(a^2-3 b^2\right) \, _2F_1\left(-\frac{9}{4},1;-\frac{5}{4};-\tan ^2(c+d x)\right)+3 b \left(3 \left(3 a^2-b^2\right) \tan (c+d x) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(c+d x)\right)+b (7 a+3 b \tan (c+d x))\right)\right)}{63 d \tan ^{\frac{9}{2}}(c+d x)}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 a \left(a^2-3 b^2\right)}{5 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(3 a^2-b^2\right)}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a \left(a^2-3 b^2\right)}{d \sqrt{\tan (c+d x)}}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{40 a^2 b}{63 d \tan ^{\frac{7}{2}}(c+d x)}",1,"(-2*(7*a*(a^2 - 3*b^2)*Hypergeometric2F1[-9/4, 1, -5/4, -Tan[c + d*x]^2] + 3*b*(3*(3*a^2 - b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[c + d*x]^2]*Tan[c + d*x] + b*(7*a + 3*b*Tan[c + d*x]))))/(63*d*Tan[c + d*x]^(9/2))","C",1
578,1,61,150,0.058678,"\int \frac{a+b \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])/Sqrt[Tan[c + d*x]],x]","-\frac{\sqrt[4]{-1} \left((a-i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{d}","-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d}-\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"-(((-1)^(1/4)*((a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/d)","C",1
579,1,82,162,0.1461144,"\int \frac{a+b \tan (c+d x)}{\sqrt{-\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])/Sqrt[-Tan[c + d*x]],x]","\frac{\sqrt[4]{-1} \tan ^{\frac{3}{2}}(c+d x) \left((a-i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{d (-\tan (c+d x))^{3/2}}","\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{-\tan (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{-\tan (c+d x)}+1\right)}{\sqrt{2} d}+\frac{(a+b) \log \left(-\tan (c+d x)-\sqrt{2} \sqrt{-\tan (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \log \left(-\tan (c+d x)+\sqrt{2} \sqrt{-\tan (c+d x)}+1\right)}{2 \sqrt{2} d}",1,"((-1)^(1/4)*((a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*Tan[c + d*x]^(3/2))/(d*(-Tan[c + d*x])^(3/2))","C",1
580,1,83,208,0.0811739,"\int \frac{a+b \tan (c+d x)}{\sqrt{e \tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])/Sqrt[e*Tan[c + d*x]],x]","-\frac{\sqrt[4]{-1} \sqrt{\tan (c+d x)} \left((a-i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{d \sqrt{e \tan (c+d x)}}","-\frac{(a+b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}+\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}-\frac{(a-b) \log \left(\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}+\frac{(a-b) \log \left(\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}",1,"-(((-1)^(1/4)*((a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*Sqrt[Tan[c + d*x]])/(d*Sqrt[e*Tan[c + d*x]]))","C",1
581,1,84,214,0.1665478,"\int \frac{a+b \tan (c+d x)}{\sqrt{-e \tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])/Sqrt[-(e*Tan[c + d*x])],x]","\frac{\sqrt[4]{-1} e \tan ^{\frac{3}{2}}(c+d x) \left((a-i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a+i b) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{d (-e \tan (c+d x))^{3/2}}","\frac{(a-b) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{-e \tan (c+d x)}}{\sqrt{e}}\right)}{\sqrt{2} d \sqrt{e}}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{-e \tan (c+d x)}}{\sqrt{e}}+1\right)}{\sqrt{2} d \sqrt{e}}+\frac{(a+b) \log \left(-\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{-e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}-\frac{(a+b) \log \left(-\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{-e \tan (c+d x)}+\sqrt{e}\right)}{2 \sqrt{2} d \sqrt{e}}",1,"((-1)^(1/4)*e*((a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*Tan[c + d*x]^(3/2))/(d*(-(e*Tan[c + d*x]))^(3/2))","C",1
582,1,243,300,1.8303528,"\int \frac{\tan ^{\frac{9}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x]),x]","\frac{-\frac{15 \left(8 a^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+8 \sqrt{b} \left(b^2-a^2\right) \left(a^2+b^2\right) \sqrt{\tan (c+d x)}+2 \sqrt{2} b^{7/2} (a+b) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+\sqrt{2} b^{7/2} (b-a) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{b^{3/2} \left(a^2+b^2\right)}-40 a \tan ^{\frac{3}{2}}(c+d x)+24 b \tan ^{\frac{5}{2}}(c+d x)}{60 b^2 d}","-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 \left(a^2-b^2\right) \sqrt{\tan (c+d x)}}{b^3 d}-\frac{2 a^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{7/2} d \left(a^2+b^2\right)}-\frac{2 a \tan ^{\frac{3}{2}}(c+d x)}{3 b^2 d}+\frac{2 \tan ^{\frac{5}{2}}(c+d x)}{5 b d}",1,"((-15*(2*Sqrt[2]*b^(7/2)*(a + b)*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + 8*a^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*b^(7/2)*(-a + b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*Sqrt[b]*(-a^2 + b^2)*(a^2 + b^2)*Sqrt[Tan[c + d*x]]))/(b^(3/2)*(a^2 + b^2)) - 40*a*Tan[c + d*x]^(3/2) + 24*b*Tan[c + d*x]^(5/2))/(60*b^2*d)","A",1
583,1,222,271,0.872581,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x]),x]","\frac{-\frac{6 \sqrt{2} b^2 (a-b) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}-\frac{3 \sqrt{2} b^2 (a+b) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{24 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \left(a^2+b^2\right)}-24 a \sqrt{\tan (c+d x)}+8 b \tan ^{\frac{3}{2}}(c+d x)}{12 b^2 d}","-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} d \left(a^2+b^2\right)}-\frac{2 a \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 \tan ^{\frac{3}{2}}(c+d x)}{3 b d}",1,"((-6*Sqrt[2]*(a - b)*b^2*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (24*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)) - (3*Sqrt[2]*b^2*(a + b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) - 24*a*Sqrt[Tan[c + d*x]] + 8*b*Tan[c + d*x]^(3/2))/(12*b^2*d)","A",1
584,1,155,250,0.1722643,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x]),x]","\frac{-2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+2 a^2 \sqrt{b} \sqrt{\tan (c+d x)}+\sqrt[4]{-1} b^{3/2} (b-i a) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} b^{3/2} (b+i a) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+2 b^{5/2} \sqrt{\tan (c+d x)}}{b^{3/2} d \left(a^2+b^2\right)}","\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2} d \left(a^2+b^2\right)}+\frac{2 \sqrt{\tan (c+d x)}}{b d}",1,"((-1)^(1/4)*b^(3/2)*((-I)*a + b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 2*a^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*b^(3/2)*(I*a + b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*a^2*Sqrt[b]*Sqrt[Tan[c + d*x]] + 2*b^(5/2)*Sqrt[Tan[c + d*x]])/(b^(3/2)*(a^2 + b^2)*d)","C",1
585,1,227,232,0.2811871,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x]),x]","\frac{3 a \left(8 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \sqrt{2} \sqrt{b} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\sqrt{2} \sqrt{b} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \sqrt{b} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+8 b^{3/2} \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)}{12 \sqrt{b} d \left(a^2+b^2\right)}","\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} d \left(a^2+b^2\right)}",1,"(3*a*(2*Sqrt[2]*Sqrt[b]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*Sqrt[b]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 8*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*Sqrt[b]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Sqrt[b]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*b^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(12*Sqrt[b]*(a^2 + b^2)*d)","C",1
586,1,204,232,0.1787106,"\int \frac{\sqrt{\tan (c+d x)}}{a+b \tan (c+d x)} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x]),x]","\frac{-24 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+8 a \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-6 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+6 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} b \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+3 \sqrt{2} b \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{12 d \left(a^2+b^2\right)}","-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}",1,"(-6*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - 24*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 3*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*a*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(12*(a^2 + b^2)*d)","C",1
587,1,225,232,0.193842,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])),x]","\frac{-6 \sqrt{2} a^{3/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+6 \sqrt{2} a^{3/2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-3 \sqrt{2} a^{3/2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+3 \sqrt{2} a^{3/2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+24 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-8 \sqrt{a} b \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)}{12 \sqrt{a} d \left(a^2+b^2\right)}","-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d \left(a^2+b^2\right)}",1,"(-6*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + 24*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 3*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 3*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 8*Sqrt[a]*b*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(12*Sqrt[a]*(a^2 + b^2)*d)","C",1
588,1,131,250,0.5183974,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]","\frac{-\frac{2 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2}}-\frac{2 \left(a^2+b^2\right)}{a \sqrt{\tan (c+d x)}}-(-1)^{3/4} (a+i b) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} (b+i a) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)}","\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{2 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d \left(a^2+b^2\right)}-\frac{2}{a d \sqrt{\tan (c+d x)}}",1,"(-((-1)^(3/4)*(a + I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]) - (2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/a^(3/2) + (-1)^(1/4)*(I*a + b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (2*(a^2 + b^2))/(a*Sqrt[Tan[c + d*x]]))/((a^2 + b^2)*d)","C",1
589,1,222,271,1.8229304,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]","\frac{\frac{6 \sqrt{2} a^2 (a-b) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{3 \sqrt{2} a^2 (a+b) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{24 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)}-\frac{8 a}{\tan ^{\frac{3}{2}}(c+d x)}+\frac{24 b}{\sqrt{\tan (c+d x)}}}{12 a^2 d}","\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b}{a^2 d \sqrt{\tan (c+d x)}}+\frac{2 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d \left(a^2+b^2\right)}-\frac{2}{3 a d \tan ^{\frac{3}{2}}(c+d x)}",1,"((6*Sqrt[2]*a^2*(a - b)*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (24*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) + (3*Sqrt[2]*a^2*(a + b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) - (8*a)/Tan[c + d*x]^(3/2) + (24*b)/Sqrt[Tan[c + d*x]])/(12*a^2*d)","A",1
590,1,248,300,3.6813707,"\int \frac{1}{\tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x])),x]","\frac{-15 \left(\frac{2 \sqrt{2} a^2 (a+b) \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}-\frac{\sqrt{2} a^2 (a-b) \left(\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)}{a^2+b^2}+\frac{8 b^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \left(a^2+b^2\right)}-\frac{8 (a-b) (a+b)}{a \sqrt{\tan (c+d x)}}\right)-\frac{24 a}{\tan ^{\frac{5}{2}}(c+d x)}+\frac{40 b}{\tan ^{\frac{3}{2}}(c+d x)}}{60 a^2 d}","-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b^{9/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{7/2} d \left(a^2+b^2\right)}+\frac{2 \left(a^2-b^2\right)}{a^3 d \sqrt{\tan (c+d x)}}-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}",1,"(-15*((2*Sqrt[2]*a^2*(a + b)*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (8*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)) - (Sqrt[2]*a^2*(a - b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(a^2 + b^2) - (8*(a - b)*(a + b))/(a*Sqrt[Tan[c + d*x]])) - (24*a)/Tan[c + d*x]^(5/2) + (40*b)/Tan[c + d*x]^(3/2))/(60*a^2*d)","A",1
591,1,442,399,1.8894547,"\int \frac{\tan ^{\frac{9}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x])^2,x]","\frac{b^2 \tan ^{\frac{11}{2}}(c+d x)}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{-\frac{b \tan ^{\frac{9}{2}}(c+d x)}{d}+\frac{2 \left(\frac{9 a b \tan ^{\frac{7}{2}}(c+d x)}{2 d}+\frac{2 \left(-\frac{63 a^2 b \tan ^{\frac{5}{2}}(c+d x)}{4 d}+\frac{2 \left(\frac{105 a b \left(5 a^2+2 b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{8 d}+\frac{2 \left(-\frac{945 a^2 b \left(5 a^2+4 b^2\right) \sqrt{\tan (c+d x)}}{16 d}+\frac{2 \left(\frac{2 \left(-\frac{945}{32} a^3 b^6+\frac{945}{64} a^3 b^4 \left(5 a^2+4 b^2\right)+\frac{945}{64} a^3 b^2 \left(5 a^4+4 a^2 b^2-2 b^4\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)}+\frac{-\frac{\sqrt[4]{-1} \left(\frac{945 a^2 b^6}{16}-\frac{945}{32} i a b^5 (a-b) (a+b)\right) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{\sqrt[4]{-1} \left(\frac{945 a^2 b^6}{16}+\frac{945}{32} i a b^5 (a-b) (a+b)\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}}{a^2+b^2}\right)}{b}\right)}{3 b}\right)}{5 b}\right)}{7 b}\right)}{9 b}}{a \left(a^2+b^2\right)}","-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(5 a^2+2 b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2+b^2\right)}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a \left(5 a^2+4 b^2\right) \sqrt{\tan (c+d x)}}{b^3 d \left(a^2+b^2\right)}+\frac{a^{7/2} \left(5 a^2+9 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{7/2} d \left(a^2+b^2\right)^2}",1,"(b^2*Tan[c + d*x]^(11/2))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (-((b*Tan[c + d*x]^(9/2))/d) + (2*((9*a*b*Tan[c + d*x]^(7/2))/(2*d) + (2*((-63*a^2*b*Tan[c + d*x]^(5/2))/(4*d) + (2*((2*((2*((2*((-945*a^3*b^6)/32 + (945*a^3*b^4*(5*a^2 + 4*b^2))/64 + (945*a^3*b^2*(5*a^4 + 4*a^2*b^2 - 2*b^4))/64)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*((945*a^2*b^6)/16 - ((945*I)/32)*a*(a - b)*b^5*(a + b))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((945*a^2*b^6)/16 + ((945*I)/32)*a*(a - b)*b^5*(a + b))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2)))/b - (945*a^2*b*(5*a^2 + 4*b^2)*Sqrt[Tan[c + d*x]])/(16*d)))/(3*b) + (105*a*b*(5*a^2 + 2*b^2)*Tan[c + d*x]^(3/2))/(8*d)))/(5*b)))/(7*b)))/(9*b))/(a*(a^2 + b^2))","C",1
592,1,375,358,1.2072274,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^2,x]","\frac{-7 a^{5/2} b^3 \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-7 a^{7/2} b^2 \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-3 a^{11/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-3 a^{9/2} b \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+3 a^5 \sqrt{b} \sqrt{\tan (c+d x)}+2 a^4 b^{3/2} \tan ^{\frac{3}{2}}(c+d x)+5 a^3 b^{5/2} \sqrt{\tan (c+d x)}+4 a^2 b^{7/2} \tan ^{\frac{3}{2}}(c+d x)+\sqrt[4]{-1} b^{5/2} (b-i a)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) (a+b \tan (c+d x))+2 a b^{9/2} \sqrt{\tan (c+d x)}+\sqrt[4]{-1} b^{5/2} (b+i a)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) (a+b \tan (c+d x))+2 b^{11/2} \tan ^{\frac{3}{2}}(c+d x)}{b^{5/2} d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(3 a^2+2 b^2\right) \sqrt{\tan (c+d x)}}{b^2 d \left(a^2+b^2\right)}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a^{5/2} \left(3 a^2+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{5/2} d \left(a^2+b^2\right)^2}",1,"(-3*a^(11/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 7*a^(7/2)*b^2*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + 3*a^5*Sqrt[b]*Sqrt[Tan[c + d*x]] + 5*a^3*b^(5/2)*Sqrt[Tan[c + d*x]] + 2*a*b^(9/2)*Sqrt[Tan[c + d*x]] - 3*a^(9/2)*b*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Tan[c + d*x] - 7*a^(5/2)*b^3*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Tan[c + d*x] + 2*a^4*b^(3/2)*Tan[c + d*x]^(3/2) + 4*a^2*b^(7/2)*Tan[c + d*x]^(3/2) + 2*b^(11/2)*Tan[c + d*x]^(3/2) + (-1)^(1/4)*b^(5/2)*((-I)*a + b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*(a + b*Tan[c + d*x]) + (-1)^(1/4)*b^(5/2)*(I*a + b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*(a + b*Tan[c + d*x]))/(b^(5/2)*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
593,1,157,318,1.6692673,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^2,x]","\frac{-\frac{a^2 \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}{b (a+b \tan (c+d x))}+\frac{a^{3/2} \left(a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2}}+(-1)^{3/4} (b-i a)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(-1)^{3/4} (a-i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)^2}","\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a^2 \sqrt{\tan (c+d x)}}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a^{3/2} \left(a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2} d \left(a^2+b^2\right)^2}",1,"((-1)^(3/4)*((-I)*a + b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a^(3/2)*(a^2 + 5*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/b^(3/2) + (-1)^(3/4)*(a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (a^2*(a^2 + b^2)*Sqrt[Tan[c + d*x]])/(b*(a + b*Tan[c + d*x])))/((a^2 + b^2)^2*d)","C",1
594,1,151,312,1.2621856,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{\sqrt{a} \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b}}+\frac{a \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}+\sqrt[4]{-1} (a+i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} (a-i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)^2}","\frac{\sqrt{a} \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a \sqrt{\tan (c+d x)}}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}",1,"((-1)^(1/4)*(a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (Sqrt[a]*(a^2 - 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[b] + (-1)^(1/4)*(a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a*(a^2 + b^2)*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x]))/((a^2 + b^2)^2*d)","C",1
595,1,182,316,0.7081714,"\int \frac{\sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^2,x]","\frac{-\frac{\sqrt{a} \sqrt{b} \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^2+b^2}+\frac{(-1)^{3/4} a \left((a+i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-(a-i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{b^2 \tan ^{\frac{3}{2}}(c+d x)}{a+b \tan (c+d x)}-b \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right)}","-\frac{\sqrt{b} \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{b \sqrt{\tan (c+d x)}}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}",1,"(-((Sqrt[a]*Sqrt[b]*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^2 + b^2)) + ((-1)^(3/4)*a*((a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) - b*Sqrt[Tan[c + d*x]] + (b^2*Tan[c + d*x]^(3/2))/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)","C",1
596,1,166,317,0.620309,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2),x]","\frac{-\frac{\sqrt[4]{-1} a \left((a+i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+(a-i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{b^{3/2} \left(5 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)}+\frac{b^2 \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}}{a d \left(a^2+b^2\right)}","-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^2 \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^{3/2} \left(5 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} d \left(a^2+b^2\right)^2}",1,"((b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) - ((-1)^(1/4)*a*((a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (b^2*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)","C",1
597,1,195,358,1.7902011,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]","-\frac{\frac{2 a^2+3 b^2}{a \sqrt{\tan (c+d x)}}+\frac{(-1)^{3/4} a \left((a+i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-(a-i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{a^2+b^2}+\frac{b^{5/2} \left(7 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \left(a^2+b^2\right)}-\frac{b^2}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))}}{a d \left(a^2+b^2\right)}","\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^2}{a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}-\frac{2 a^2+3 b^2}{a^2 d \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{b^{5/2} \left(7 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{5/2} d \left(a^2+b^2\right)^2}",1,"-(((b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)) + ((-1)^(3/4)*a*((a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (2*a^2 + 3*b^2)/(a*Sqrt[Tan[c + d*x]]) - b^2/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)*d))","C",1
598,1,230,397,4.405876,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2),x]","\frac{-\frac{2 a^2+5 b^2}{a \tan ^{\frac{3}{2}}(c+d x)}+\frac{3 b \left(4 a^2+5 b^2\right)}{a^2 \sqrt{\tan (c+d x)}}+\frac{3 \left(\sqrt[4]{-1} a^{7/2} (a+i b)^2 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+\sqrt[4]{-1} a^{7/2} (a-i b)^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+b^{7/2} \left(9 a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)\right)}{a^{5/2} \left(a^2+b^2\right)}+\frac{3 b^2}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}}{3 a d \left(a^2+b^2\right)}","\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^2}{a d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac{2 a^2+5 b^2}{3 a^2 d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x)}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^{7/2} \left(9 a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{7/2} d \left(a^2+b^2\right)^2}+\frac{b \left(4 a^2+5 b^2\right)}{a^3 d \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}",1,"((3*((-1)^(1/4)*a^(7/2)*(a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + b^(7/2)*(9*a^2 + 5*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + (-1)^(1/4)*a^(7/2)*(a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^(5/2)*(a^2 + b^2)) - (2*a^2 + 5*b^2)/(a*Tan[c + d*x]^(3/2)) + (3*b*(4*a^2 + 5*b^2))/(a^2*Sqrt[Tan[c + d*x]]) + (3*b^2)/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])))/(3*a*(a^2 + b^2)*d)","C",1
599,1,723,493,6.3961907,"\int \frac{\tan ^{\frac{11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^(11/2)/(a + b*Tan[c + d*x])^3,x]","\frac{b^2 \tan ^{\frac{13}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{-\frac{b \tan ^{\frac{11}{2}}(c+d x)}{d (a+b \tan (c+d x))}+\frac{2 \left(\frac{9 a b \tan ^{\frac{9}{2}}(c+d x)}{2 d (a+b \tan (c+d x))}+\frac{2 \left(-\frac{63 a^2 b \tan ^{\frac{7}{2}}(c+d x)}{4 d (a+b \tan (c+d x))}+\frac{2 \left(\frac{105 a b \left(7 a^2+4 b^2\right) \tan ^{\frac{5}{2}}(c+d x)}{8 d (a+b \tan (c+d x))}+\frac{2 \left(-\frac{315 a^2 b \left(35 a^2+32 b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{16 d (a+b \tan (c+d x))}+\frac{2 \left(-\frac{945 a b \left(35 a^4+32 a^2 b^2-4 b^4\right) \sqrt{\tan (c+d x)}}{32 d (a+b \tan (c+d x))}-\frac{2 \left(\frac{\left(-a \left(\frac{945}{128} a^5 b^2 \left(35 a^2+32 b^2\right)-\frac{945 a b^8}{32}\right)-\frac{945}{128} a^2 b^4 \left(35 a^4+32 a^2 b^2-4 b^4\right)\right) \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\frac{2 \left(\frac{945 a^4 b^8}{16}-\frac{945}{256} a^4 b^4 \left(35 a^4+67 a^2 b^2+24 b^4\right)-\frac{945}{256} a^4 b^2 \left(35 a^6+67 a^4 b^2+32 a^2 b^4-8 b^6\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)}+\frac{-\frac{\sqrt[4]{-1} \left(\frac{945}{32} a^3 b^6 \left(a^2-3 b^2\right)+\frac{945}{32} i a^2 b^7 \left(3 a^2-b^2\right)\right) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{\sqrt[4]{-1} \left(\frac{945}{32} a^3 b^6 \left(a^2-3 b^2\right)-\frac{945}{32} i a^2 b^7 \left(3 a^2-b^2\right)\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}}{a^2+b^2}}{a \left(a^2+b^2\right)}\right)}{b}\right)}{b}\right)}{3 b}\right)}{5 b}\right)}{7 b}\right)}{9 b}}{2 a \left(a^2+b^2\right)}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a^2 \left(7 a^2+15 b^2\right) \tan ^{\frac{5}{2}}(c+d x)}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{a^2 \tan ^{\frac{7}{2}}(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a \left(35 a^4+67 a^2 b^2+24 b^4\right) \sqrt{\tan (c+d x)}}{4 b^4 d \left(a^2+b^2\right)^2}+\frac{\left(35 a^4+67 a^2 b^2+8 b^4\right) \tan ^{\frac{3}{2}}(c+d x)}{12 b^3 d \left(a^2+b^2\right)^2}+\frac{a^{7/2} \left(35 a^4+102 a^2 b^2+99 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 b^{9/2} d \left(a^2+b^2\right)^3}",1,"(b^2*Tan[c + d*x]^(13/2))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (-((b*Tan[c + d*x]^(11/2))/(d*(a + b*Tan[c + d*x]))) + (2*((9*a*b*Tan[c + d*x]^(9/2))/(2*d*(a + b*Tan[c + d*x])) + (2*((-63*a^2*b*Tan[c + d*x]^(7/2))/(4*d*(a + b*Tan[c + d*x])) + (2*((105*a*b*(7*a^2 + 4*b^2)*Tan[c + d*x]^(5/2))/(8*d*(a + b*Tan[c + d*x])) + (2*((-315*a^2*b*(35*a^2 + 32*b^2)*Tan[c + d*x]^(3/2))/(16*d*(a + b*Tan[c + d*x])) + (2*((-945*a*b*(35*a^4 + 32*a^2*b^2 - 4*b^4)*Sqrt[Tan[c + d*x]])/(32*d*(a + b*Tan[c + d*x])) - (2*(((2*((945*a^4*b^8)/16 - (945*a^4*b^4*(35*a^4 + 67*a^2*b^2 + 24*b^4))/256 - (945*a^4*b^2*(35*a^6 + 67*a^4*b^2 + 32*a^2*b^4 - 8*b^6))/256)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*((945*a^3*b^6*(a^2 - 3*b^2))/32 + ((945*I)/32)*a^2*b^7*(3*a^2 - b^2))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((945*a^3*b^6*(a^2 - 3*b^2))/32 - ((945*I)/32)*a^2*b^7*(3*a^2 - b^2))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2))/(a*(a^2 + b^2)) + (((-945*a^2*b^4*(35*a^4 + 32*a^2*b^2 - 4*b^4))/128 - a*((-945*a*b^8)/32 + (945*a^5*b^2*(35*a^2 + 32*b^2))/128))*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))))/b))/b))/(3*b)))/(5*b)))/(7*b)))/(9*b))/(2*a*(a^2 + b^2))","C",1
600,1,403,444,5.3710299,"\int \frac{\tan ^{\frac{9}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{2 a \left(5 a^2+4 b^2\right) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}{b^2}+\frac{2 a^2 \left(15 a^2+16 b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}{b^3}-\frac{2 a^2 \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))}{b}-\frac{a (a+b \tan (c+d x)) \left(a \sqrt{b} \left(a^2+b^2\right) \left(15 a^4+31 a^2 b^2+24 b^4\right) \sqrt{\tan (c+d x)}+(a+b \tan (c+d x)) \left(a^{5/2} \left(15 a^4+46 a^2 b^2+63 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-4 (-1)^{3/4} b^{7/2} (a+i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-4 \sqrt[4]{-1} b^{7/2} (b+i a)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)\right)}{b^{7/2} \left(a^2+b^2\right)^2}-2 b \tan ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))+2 a \tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))+2 b^2 \tan ^{\frac{11}{2}}(c+d x)}{4 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}","-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a^2 \left(5 a^2+13 b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{a^2 \tan ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(15 a^4+31 a^2 b^2+8 b^4\right) \sqrt{\tan (c+d x)}}{4 b^3 d \left(a^2+b^2\right)^2}-\frac{a^{5/2} \left(15 a^4+46 a^2 b^2+63 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 b^{7/2} d \left(a^2+b^2\right)^3}",1,"(2*b^2*Tan[c + d*x]^(11/2) + (2*a^2*(15*a^2 + 16*b^2)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/b^3 + (2*a*(5*a^2 + 4*b^2)*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))/b^2 - (2*a^2*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]))/b + 2*a*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]) - 2*b*Tan[c + d*x]^(9/2)*(a + b*Tan[c + d*x]) - (a*(a + b*Tan[c + d*x])*(a*Sqrt[b]*(a^2 + b^2)*(15*a^4 + 31*a^2*b^2 + 24*b^4)*Sqrt[Tan[c + d*x]] + (-4*(-1)^(3/4)*(a + I*b)^3*b^(7/2)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + a^(5/2)*(15*a^4 + 46*a^2*b^2 + 63*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 4*(-1)^(1/4)*b^(7/2)*(I*a + b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*(a + b*Tan[c + d*x])))/(b^(7/2)*(a^2 + b^2)^2))/(4*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)","C",1
601,1,373,396,4.8848774,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^3,x]","\frac{-\frac{a (a+b \tan (c+d x)) \left(2 \sqrt{b} \left(a^2+b^2\right)^2 \left(3 a^2+4 b^2\right) \sqrt{\tan (c+d x)}-2 b^{5/2} \left(a^2+b^2\right)^2 \tan ^{\frac{5}{2}}(c+d x)+2 a b^{3/2} \left(a^2+b^2\right)^2 \tan ^{\frac{3}{2}}(c+d x)-\sqrt{b} \left(a^2+b^2\right) \left(3 a^4+3 a^2 b^2+8 b^4\right) \sqrt{\tan (c+d x)}-(a+b \tan (c+d x)) \left(a^{3/2} \left(3 a^4+6 a^2 b^2+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-4 \sqrt[4]{-1} b^{5/2} (a+i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-4 \sqrt[4]{-1} b^{5/2} (a-i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)\right)}{\left(a^2+b^2\right)^2}-2 b^{7/2} \tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))+2 b^{9/2} \tan ^{\frac{9}{2}}(c+d x)}{4 a b^{5/2} d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}","-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a^2 \tan ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 \left(3 a^2+11 b^2\right) \sqrt{\tan (c+d x)}}{4 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{a^{3/2} \left(3 a^4+6 a^2 b^2+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 b^{5/2} d \left(a^2+b^2\right)^3}",1,"(2*b^(9/2)*Tan[c + d*x]^(9/2) - 2*b^(7/2)*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]) - (a*(a + b*Tan[c + d*x])*(2*Sqrt[b]*(a^2 + b^2)^2*(3*a^2 + 4*b^2)*Sqrt[Tan[c + d*x]] - Sqrt[b]*(a^2 + b^2)*(3*a^4 + 3*a^2*b^2 + 8*b^4)*Sqrt[Tan[c + d*x]] + 2*a*b^(3/2)*(a^2 + b^2)^2*Tan[c + d*x]^(3/2) - 2*b^(5/2)*(a^2 + b^2)^2*Tan[c + d*x]^(5/2) - (-4*(-1)^(1/4)*(a + I*b)^3*b^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 4*(-1)^(1/4)*(a - I*b)^3*b^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*(a + b*Tan[c + d*x])))/(a^2 + b^2)^2)/(4*a*b^(5/2)*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)","C",1
602,1,325,390,4.2236973,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^3,x]","\frac{-\frac{a (a+b \tan (c+d x)) \left(2 a \sqrt{b} \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)}-a \sqrt{b} \left(a^2+9 b^2\right) \left(a^2+b^2\right) \sqrt{\tan (c+d x)}-2 b^{3/2} \left(a^2+b^2\right)^2 \tan ^{\frac{3}{2}}(c+d x)-(a+b \tan (c+d x)) \left(\sqrt{a} \left(a^4+18 a^2 b^2-15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-4 (-1)^{3/4} b^{3/2} (a+i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-4 \sqrt[4]{-1} b^{3/2} (b+i a)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)\right)}{\left(a^2+b^2\right)^2}-2 b^{5/2} \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))+2 b^{7/2} \tan ^{\frac{7}{2}}(c+d x)}{4 a b^{3/2} d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}","\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a^2 \sqrt{\tan (c+d x)}}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a \left(a^2+9 b^2\right) \sqrt{\tan (c+d x)}}{4 b d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{a} \left(a^4+18 a^2 b^2-15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 b^{3/2} d \left(a^2+b^2\right)^3}",1,"(2*b^(7/2)*Tan[c + d*x]^(7/2) - 2*b^(5/2)*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]) - (a*(a + b*Tan[c + d*x])*(2*a*Sqrt[b]*(a^2 + b^2)^2*Sqrt[Tan[c + d*x]] - a*Sqrt[b]*(a^2 + b^2)*(a^2 + 9*b^2)*Sqrt[Tan[c + d*x]] - 2*b^(3/2)*(a^2 + b^2)^2*Tan[c + d*x]^(3/2) - (-4*(-1)^(3/4)*(a + I*b)^3*b^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[a]*(a^4 + 18*a^2*b^2 - 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 4*(-1)^(1/4)*b^(3/2)*(I*a + b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*(a + b*Tan[c + d*x])))/(a^2 + b^2)^2)/(4*a*b^(3/2)*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)","C",1
603,1,326,385,3.7334003,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{(a+b \tan (c+d x)) \left(a^{5/2} b^{5/2} \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)}+\frac{1}{2} \left(a^{5/2} b^{5/2} \left(3 a^2-5 b^2\right) \left(a^2+b^2\right) \sqrt{\tan (c+d x)}+(a+b \tan (c+d x)) \left(4 \sqrt[4]{-1} a^{5/2} b^{5/2} (a+i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+4 \sqrt[4]{-1} a^{5/2} b^{5/2} (a-i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+a^2 b^2 \left(3 a^4-26 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)\right)\right)\right)}{a^{3/2} \left(a^2+b^2\right)^2}-b^{7/2} \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))+b^{9/2} \tan ^{\frac{5}{2}}(c+d x)}{2 a b^{5/2} d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{a \sqrt{\tan (c+d x)}}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{\left(3 a^2-5 b^2\right) \sqrt{\tan (c+d x)}}{4 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\left(3 a^4-26 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 \sqrt{a} \sqrt{b} d \left(a^2+b^2\right)^3}",1,"(b^(9/2)*Tan[c + d*x]^(5/2) - b^(7/2)*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]) + ((a + b*Tan[c + d*x])*(a^(5/2)*b^(5/2)*(a^2 + b^2)^2*Sqrt[Tan[c + d*x]] + (a^(5/2)*b^(5/2)*(3*a^2 - 5*b^2)*(a^2 + b^2)*Sqrt[Tan[c + d*x]] + (4*(-1)^(1/4)*a^(5/2)*(a + I*b)^3*b^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + a^2*b^2*(3*a^4 - 26*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + 4*(-1)^(1/4)*a^(5/2)*(a - I*b)^3*b^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*(a + b*Tan[c + d*x]))/2))/(a^(3/2)*(a^2 + b^2)^2))/(2*a*b^(5/2)*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)","C",1
604,1,259,389,5.5861664,"\int \frac{\sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{4 \sqrt[4]{-1} a b \left((-b+i a)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-(b+i a)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)\right)}{\left(a^2+b^2\right)^2}+\frac{\left(7 a^2 b^2-b^4\right) \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{b^{3/2} \left(-15 a^4+18 a^2 b^2+b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \left(a^2+b^2\right)^2}-\frac{2 b^3 \tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^2}+\frac{2 b^2 \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}}{4 a b d \left(a^2+b^2\right)}","-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{b \left(7 a^2-b^2\right) \sqrt{\tan (c+d x)}}{4 a d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b \sqrt{\tan (c+d x)}}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\sqrt{b} \left(15 a^4-18 a^2 b^2-b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{3/2} d \left(a^2+b^2\right)^3}",1,"-1/4*(-((b^(3/2)*(-15*a^4 + 18*a^2*b^2 + b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^2)) + (4*(-1)^(1/4)*a*b*((I*a - b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (I*a + b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^2 - (2*b^3*Tan[c + d*x]^(3/2))/(a + b*Tan[c + d*x])^2 + (2*b^2*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x]) + ((7*a^2*b^2 - b^4)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])))/(a*b*(a^2 + b^2)*d)","C",1
605,1,235,396,2.754937,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3),x]","\frac{\frac{\left(11 a^2 b^2+3 b^4\right) \sqrt{\tan (c+d x)}}{a \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{-4 \sqrt[4]{-1} a^{5/2} (a+i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-4 \sqrt[4]{-1} a^{5/2} (a-i b)^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)+b^{3/2} \left(35 a^4+6 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{a^{3/2} \left(a^2+b^2\right)^2}+\frac{2 b^2 \sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^2}}{4 a d \left(a^2+b^2\right)}","-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b^2 \left(11 a^2+3 b^2\right) \sqrt{\tan (c+d x)}}{4 a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b^2 \sqrt{\tan (c+d x)}}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b^{3/2} \left(35 a^4+6 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{5/2} d \left(a^2+b^2\right)^3}",1,"((-4*(-1)^(1/4)*a^(5/2)*(a + I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 4*(-1)^(1/4)*a^(5/2)*(a - I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/(a^(3/2)*(a^2 + b^2)^2) + (2*b^2*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x])^2 + ((11*a^2*b^2 + 3*b^4)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d)","C",1
606,1,358,444,4.1745246,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]","\frac{\frac{13 a^2 b^2+5 b^4}{a \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{-46 a^{5/2} b^4-39 a^{9/2} b^2-4 (-1)^{3/4} a^{7/2} (a+i b)^3 \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right) \sqrt{\tan (c+d x)}-4 \sqrt[4]{-1} a^{7/2} (b+i a)^3 \sqrt{\tan (c+d x)} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)-8 a^{13/2}-63 a^4 b^{5/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-46 a^2 b^{9/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-15 b^{13/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)-15 \sqrt{a} b^6}{a^{5/2} \left(a^2+b^2\right)^2}+\frac{2 b^2}{(a+b \tan (c+d x))^2}}{4 a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}","\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b^2 \left(13 a^2+5 b^2\right)}{4 a^2 d \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)} (a+b \tan (c+d x))}+\frac{b^2}{2 a d \left(a^2+b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{b^{5/2} \left(63 a^4+46 a^2 b^2+15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{7/2} d \left(a^2+b^2\right)^3}-\frac{8 a^4+31 a^2 b^2+15 b^4}{4 a^3 d \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)}}",1,"((-8*a^(13/2) - 39*a^(9/2)*b^2 - 46*a^(5/2)*b^4 - 15*Sqrt[a]*b^6 - 4*(-1)^(3/4)*a^(7/2)*(a + I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 63*a^4*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 46*a^2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 15*b^(13/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 4*(-1)^(1/4)*a^(7/2)*(I*a + b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*(a^2 + b^2)^2) + (2*b^2)/(a + b*Tan[c + d*x])^2 + (13*a^2*b^2 + 5*b^4)/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]])","C",1
607,1,495,493,6.1988788,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3),x]","\frac{b^2}{2 a d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac{\frac{\frac{11 a^2 b^2}{2}+\frac{1}{2} b^2 \left(4 a^2+7 b^2\right)}{a d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{-\frac{8 a^4+67 a^2 b^2+35 b^4}{6 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(-\frac{3 b \left(24 a^4+67 a^2 b^2+35 b^4\right)}{4 a d \sqrt{\tan (c+d x)}}-\frac{2 \left(\frac{2 \left(-3 a^6 b^2+\frac{3}{16} a^2 b^2 \left(24 a^4+67 a^2 b^2+35 b^4\right)-\frac{3}{16} b^2 \left(8 a^6-32 a^4 b^2-67 a^2 b^4-35 b^6\right)\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d \left(a^2+b^2\right)}+\frac{-\frac{\sqrt[4]{-1} \left(-\frac{3}{2} a^5 \left(a^2-3 b^2\right)-\frac{3}{2} i a^4 b \left(3 a^2-b^2\right)\right) \tan ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}-\frac{\sqrt[4]{-1} \left(-\frac{3}{2} a^5 \left(a^2-3 b^2\right)+\frac{3}{2} i a^4 b \left(3 a^2-b^2\right)\right) \tanh ^{-1}\left((-1)^{3/4} \sqrt{\tan (c+d x)}\right)}{d}}{a^2+b^2}\right)}{a}\right)}{3 a}}{a \left(a^2+b^2\right)}}{2 a \left(a^2+b^2\right)}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b^2 \left(15 a^2+7 b^2\right)}{4 a^2 d \left(a^2+b^2\right)^2 \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{b^2}{2 a d \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b \left(24 a^4+67 a^2 b^2+35 b^4\right)}{4 a^4 d \left(a^2+b^2\right)^2 \sqrt{\tan (c+d x)}}+\frac{b^{7/2} \left(99 a^4+102 a^2 b^2+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{4 a^{9/2} d \left(a^2+b^2\right)^3}-\frac{8 a^4+67 a^2 b^2+35 b^4}{12 a^3 d \left(a^2+b^2\right)^2 \tan ^{\frac{3}{2}}(c+d x)}",1,"b^2/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2) + (((-2*((-2*((2*(-3*a^6*b^2 + (3*a^2*b^2*(24*a^4 + 67*a^2*b^2 + 35*b^4))/16 - (3*b^2*(8*a^6 - 32*a^4*b^2 - 67*a^2*b^4 - 35*b^6))/16)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*((-3*a^5*(a^2 - 3*b^2))/2 - ((3*I)/2)*a^4*b*(3*a^2 - b^2))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((-3*a^5*(a^2 - 3*b^2))/2 + ((3*I)/2)*a^4*b*(3*a^2 - b^2))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2)))/a - (3*b*(24*a^4 + 67*a^2*b^2 + 35*b^4))/(4*a*d*Sqrt[Tan[c + d*x]])))/(3*a) - (8*a^4 + 67*a^2*b^2 + 35*b^4)/(6*a*d*Tan[c + d*x]^(3/2)))/(a*(a^2 + b^2)) + ((11*a^2*b^2)/2 + (b^2*(4*a^2 + 7*b^2))/2)/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])))/(2*a*(a^2 + b^2))","C",1
608,1,282,231,1.833502,"\int \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}+\frac{-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{-\frac{\sqrt{a} \left(a^2+8 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{b} \sqrt{a+b \tan (c+d x)}}-2 \sqrt[4]{-1} b \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-2 \sqrt[4]{-1} b \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}}{2 b}","-\frac{\left(a^2+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 b^{3/2} d}-\frac{\sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 b d}-\frac{a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b d}+\frac{\sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(2*b*d) + (-1/2*(a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d + (-2*(-1)^(1/4)*Sqrt[-a + I*b]*b*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 2*(-1)^(1/4)*Sqrt[a + I*b]*b*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (Sqrt[a]*(a^2 + 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/d)/(2*b)","A",1
609,1,217,184,1.7558516,"\int \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{a^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}-(-1)^{3/4} \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(-1)^{3/4} \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}","\frac{i \sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}+\frac{i \sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-((-1)^(3/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (-1)^(3/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (a^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
610,1,189,151,0.5896634,"\int \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+\sqrt[4]{-1} \left(\sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","\frac{\sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{2 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(1/4)*(Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (2*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/d","A",1
611,1,123,115,0.0952231,"\int \frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]/Sqrt[Tan[c + d*x]],x]","\frac{(-1)^{3/4} \left(\sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","-\frac{i \sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{i \sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(3/4)*(Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/d","A",1
612,1,154,139,0.3826914,"\int \frac{\sqrt{a+b \tan (c+d x)}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]/Tan[c + d*x]^(3/2),x]","-\frac{\sqrt[4]{-1} \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\frac{2 \sqrt{a+b \tan (c+d x)}}{\sqrt{\tan (c+d x)}}}{d}","-\frac{\sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}+\frac{\sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"-(((-1)^(1/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (-1)^(1/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]])/Sqrt[Tan[c + d*x]])/d)","A",1
613,1,161,181,0.5983874,"\int \frac{\sqrt{a+b \tan (c+d x)}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]/Tan[c + d*x]^(5/2),x]","\frac{-3 (-1)^{3/4} \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+3 (-1)^{3/4} \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\frac{2 (a+b \tan (c+d x))^{3/2}}{a \tan ^{\frac{3}{2}}(c+d x)}}{3 d}","\frac{i \sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{3 a d \sqrt{\tan (c+d x)}}+\frac{i \sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-3*(-1)^(3/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(-1)^(3/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (2*(a + b*Tan[c + d*x])^(3/2))/(a*Tan[c + d*x]^(3/2)))/(3*d)","A",1
614,1,197,221,1.4742876,"\int \frac{\sqrt{a+b \tan (c+d x)}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]/Tan[c + d*x]^(7/2),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2+2 b^2\right) \tan ^2(c+d x)-3 a^2-a b \tan (c+d x)\right)}{a^2 \tan ^{\frac{5}{2}}(c+d x)}+15 \sqrt[4]{-1} \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+15 \sqrt[4]{-1} \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{15 d}","\frac{2 \left(15 a^2+2 b^2\right) \sqrt{a+b \tan (c+d x)}}{15 a^2 d \sqrt{\tan (c+d x)}}+\frac{\sqrt{-b+i a} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 b \sqrt{a+b \tan (c+d x)}}{15 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{\sqrt{b+i a} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(15*(-1)^(1/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 15*(-1)^(1/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2 - a*b*Tan[c + d*x] + (15*a^2 + 2*b^2)*Tan[c + d*x]^2))/(a^2*Tan[c + d*x]^(5/2)))/(15*d)","A",1
615,1,313,280,3.8580529,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2),x]","\frac{-3 a^{3/2} \left(a^2+24 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{b} \left(\sqrt{\tan (c+d x)} \left(3 \left(a^3-8 a b^2\right)+b \left(17 a^2-24 b^2\right) \tan (c+d x)+22 a b^2 \tan ^2(c+d x)+8 b^3 \tan ^3(c+d x)\right)+24 \sqrt[4]{-1} b (-a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}-24 \sqrt[4]{-1} b (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}\right)}{24 b^{3/2} d \sqrt{a+b \tan (c+d x)}}","-\frac{\left(a^2+8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{8 b d}-\frac{a \left(a^2+24 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 b^{3/2} d}+\frac{i (-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}-\frac{a \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}-\frac{i (b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-3*a^(3/2)*(a^2 + 24*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a] + Sqrt[b]*(24*(-1)^(1/4)*(-a + I*b)^(3/2)*b*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] - 24*(-1)^(1/4)*(a + I*b)^(3/2)*b*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[Tan[c + d*x]]*(3*(a^3 - 8*a*b^2) + b*(17*a^2 - 24*b^2)*Tan[c + d*x] + 22*a*b^2*Tan[c + d*x]^2 + 8*b^3*Tan[c + d*x]^3)))/(24*b^(3/2)*d*Sqrt[a + b*Tan[c + d*x]])","A",1
616,1,850,226,6.1258874,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2),x]","\frac{2 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right)^2 \left(\frac{3 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{8 \sqrt{b} \sqrt{\tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right)^{5/2}}+\frac{1}{4} \left(\frac{1}{\frac{b \tan (c+d x)}{a}+1}+\frac{3}{2 \left(\frac{b \tan (c+d x)}{a}+1\right)^2}\right)\right)-i \left(-2 b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right) \left(\frac{\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{b} \sqrt{\tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{b \tan (c+d x)}{a}+1\right)}\right)-(-a-i b) \left(\frac{2 \sqrt[4]{-1} (-a-i b) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{2 \sqrt{a} \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right) \sqrt{\frac{b \tan (c+d x)}{a}+1}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{2 d}+\frac{2 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right)^2 \left(\frac{3 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{8 \sqrt{b} \sqrt{\tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right)^{5/2}}+\frac{1}{4} \left(\frac{1}{\frac{b \tan (c+d x)}{a}+1}+\frac{3}{2 \left(\frac{b \tan (c+d x)}{a}+1\right)^2}\right)\right)-i \left(2 b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right) \left(\frac{\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{2 \sqrt{b} \sqrt{\tan (c+d x)} \left(\frac{b \tan (c+d x)}{a}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{b \tan (c+d x)}{a}+1\right)}\right)-(i b-a) \left(\frac{2 \sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+2 \sqrt[4]{-1} \sqrt{i b-a} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{i b-a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)\right)}{2 d}","\frac{\left(3 a^2-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 \sqrt{b} d}+\frac{(-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 d}+\frac{(b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(2*a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)^2*((3*Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(8*Sqrt[b]*Sqrt[Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)^(5/2)) + (3/(2*(1 + (b*Tan[c + d*x])/a)^2) + (1 + (b*Tan[c + d*x])/a)^(-1))/4) - I*(-2*b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)*((Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*Sqrt[b]*Sqrt[Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)^(3/2)) + 1/(2*(1 + (b*Tan[c + d*x])/a))) - (-a - I*b)*((2*(-1)^(1/4)*(-a - I*b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])))/(2*d) + (2*a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)^2*((3*Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(8*Sqrt[b]*Sqrt[Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)^(5/2)) + (3/(2*(1 + (b*Tan[c + d*x])/a)^2) + (1 + (b*Tan[c + d*x])/a)^(-1))/4) - I*(2*b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)*((Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*Sqrt[b]*Sqrt[Tan[c + d*x]]*(1 + (b*Tan[c + d*x])/a)^(3/2)) + 1/(2*(1 + (b*Tan[c + d*x])/a))) - (-a + I*b)*(2*(-1)^(1/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])))/(2*d)","B",1
617,1,219,186,2.5876215,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\sqrt[4]{-1} (-a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \tan (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{\frac{b \tan (c+d x)}{a}+1}}}{d}","-\frac{i (-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{i (b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-((-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (-1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[1 + (b*Tan[c + d*x])/a])/d","A",1
618,1,203,152,0.8021426,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Sqrt[Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a} b^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+\sqrt[4]{-1} \left(\sqrt{-a+i b} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{a+i b} (b-i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(1/4)*(Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[a + I*b]*((-I)*a + b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (2*Sqrt[a]*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/d","A",1
619,1,175,145,0.3410598,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(3/2),x]","\frac{\sqrt[4]{-1} (-a+i b)^{3/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt[4]{-1} (a+i b)^{3/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-2 a \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}","\frac{i (-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}-\frac{i (b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - (-1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 2*a*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])","A",1
620,1,176,173,0.8640575,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(5/2),x]","\frac{3 (-1)^{3/4} (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-3 \sqrt[4]{-1} \sqrt{-a+i b} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\frac{2 \sqrt{a+b \tan (c+d x)} (a+4 b \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)}}{3 d}","\frac{(-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{8 b \sqrt{a+b \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}+\frac{(b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-3*(-1)^(1/4)*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(-1)^(3/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (2*Sqrt[a + b*Tan[c + d*x]]*(a + 4*b*Tan[c + d*x]))/Tan[c + d*x]^(3/2))/(3*d)","A",1
621,1,197,224,1.66643,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(7/2),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(\left(5 a^2-b^2\right) \tan ^2(c+d x)-a^2-2 a b \tan (c+d x)\right)}{a \tan ^{\frac{5}{2}}(c+d x)}-5 \sqrt[4]{-1} (-a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+5 \sqrt[4]{-1} (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{5 d}","\frac{2 \left(5 a^2-b^2\right) \sqrt{a+b \tan (c+d x)}}{5 a d \sqrt{\tan (c+d x)}}-\frac{i (-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{4 b \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 a \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{i (b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-5*(-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 5*(-1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-a^2 - 2*a*b*Tan[c + d*x] + (5*a^2 - b^2)*Tan[c + d*x]^2))/(a*Tan[c + d*x]^(5/2)))/(5*d)","A",1
622,1,229,266,2.9290187,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(9/2),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(-15 a^3+2 b \left(70 a^2+3 b^2\right) \tan ^3(c+d x)+a \left(35 a^2-3 b^2\right) \tan ^2(c+d x)-24 a^2 b \tan (c+d x)\right)}{a^2 \tan ^{\frac{7}{2}}(c+d x)}-105 (-1)^{3/4} (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+105 \sqrt[4]{-1} \sqrt{-a+i b} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{105 d}","\frac{2 \left(35 a^2-3 b^2\right) \sqrt{a+b \tan (c+d x)}}{105 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 b \left(70 a^2+3 b^2\right) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \sqrt{\tan (c+d x)}}-\frac{(-b+i a)^{3/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{16 b \sqrt{a+b \tan (c+d x)}}{35 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{(b+i a)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(105*(-1)^(1/4)*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 105*(-1)^(3/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-15*a^3 - 24*a^2*b*Tan[c + d*x] + a*(35*a^2 - 3*b^2)*Tan[c + d*x]^2 + 2*b*(70*a^2 + 3*b^2)*Tan[c + d*x]^3))/(a^2*Tan[c + d*x]^(7/2)))/(105*d)","A",1
623,1,349,332,3.4039688,"\int \tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2),x]","-\frac{2 \left(5 a^2+48 b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+3 a \left(5 a^2+112 b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{3 \sqrt{a} \left(5 a^4+240 a^2 b^2-128 b^4\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+192 \sqrt[4]{-1} b (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+192 \sqrt[4]{-1} b (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-48 \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{7/2}+8 a \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{192 b d}","-\frac{\left(5 a^2+48 b^2\right) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac{a \left(5 a^2+112 b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{64 b d}-\frac{\left(5 a^4+240 a^2 b^2-128 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{64 b^{3/2} d}+\frac{(-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac{a \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}-\frac{(b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"-1/192*(192*(-1)^(1/4)*(-a + I*b)^(5/2)*b*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 192*(-1)^(1/4)*(a + I*b)^(5/2)*b*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*a*(5*a^2 + 112*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*(5*a^2 + 48*b^2)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*a*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) - 48*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2) + (3*Sqrt[a]*(5*a^4 + 240*a^2*b^2 - 128*b^4)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(b*d)","A",1
624,1,300,277,3.198541,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2),x]","\frac{3 \left(11 a^2-8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{15 a^{3/2} \left(a^2-8 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+8 b^2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}-24 (-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+24 (-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+26 a b \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}","\frac{\left(11 a^2-8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{8 d}+\frac{5 a \left(a^2-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 \sqrt{b} d}+\frac{b^2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{i (-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{13 a b \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{i (b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-24*(-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 24*(-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(11*a^2 - 8*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 26*a*b*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + 8*b^2*Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]] + (15*a^(3/2)*(a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(24*d)","A",1
625,1,264,231,2.0382788,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{\sqrt{a} \sqrt{b} \left(15 a^2-8 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+2 b^2 \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}+4 \sqrt[4]{-1} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+4 \sqrt[4]{-1} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+9 a b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 d}","\frac{\sqrt{b} \left(15 a^2-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 d}+\frac{b^2 \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{(-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{9 a b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 d}+\frac{(b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(4*(-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 4*(-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 9*a*b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b^2*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*Sqrt[b]*(15*a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/(4*d)","A",1
626,1,221,188,0.8137186,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Sqrt[Tan[c + d*x]],x]","\frac{\frac{5 a^{3/2} b^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+b^2 \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+(-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}","\frac{5 a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b^2 \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{i (-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{i (b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b^2*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (5*a^(3/2)*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]])/d","A",1
627,1,244,183,2.2244268,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(3/2),x]","\frac{\frac{2 \sqrt{a} b^{5/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}-\frac{2 a^2 \sqrt{a+b \tan (c+d x)}+\sqrt[4]{-1} (-a+i b)^{5/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} (a+i b)^{5/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{\tan (c+d x)}}}{d}","-\frac{2 a^2 \sqrt{a+b \tan (c+d x)}}{d \sqrt{\tan (c+d x)}}+\frac{2 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((2*Sqrt[a]*b^(5/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]] - ((-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + (-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + 2*a^2*Sqrt[a + b*Tan[c + d*x]])/Sqrt[Tan[c + d*x]])/d","A",1
628,1,170,182,0.8996674,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(5/2),x]","\frac{-3 (-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+3 (-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\frac{2 a \sqrt{a+b \tan (c+d x)} (a+7 b \tan (c+d x))}{\tan ^{\frac{3}{2}}(c+d x)}}{3 d}","-\frac{2 a^2 \sqrt{a+b \tan (c+d x)}}{3 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{i (-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{14 a b \sqrt{a+b \tan (c+d x)}}{3 d \sqrt{\tan (c+d x)}}-\frac{i (b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-3*(-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (2*a*Sqrt[a + b*Tan[c + d*x]]*(a + 7*b*Tan[c + d*x]))/Tan[c + d*x]^(3/2))/(3*d)","A",1
629,1,194,219,1.6532569,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(7/2),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2-23 b^2\right) \tan ^2(c+d x)-3 a^2-11 a b \tan (c+d x)\right)}{\tan ^{\frac{5}{2}}(c+d x)}+15 \sqrt[4]{-1} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+15 \sqrt[4]{-1} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{15 d}","\frac{2 \left(15 a^2-23 b^2\right) \sqrt{a+b \tan (c+d x)}}{15 d \sqrt{\tan (c+d x)}}-\frac{2 a^2 \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{(-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{22 a b \sqrt{a+b \tan (c+d x)}}{15 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{(b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(15*(-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 15*(-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2 - 11*a*b*Tan[c + d*x] + (15*a^2 - 23*b^2)*Tan[c + d*x]^2))/Tan[c + d*x]^(5/2))/(15*d)","A",1
630,1,249,270,3.6224028,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(9/2),x]","-\frac{\frac{\sec ^3(c+d x) \sqrt{a+b \tan (c+d x)} \left(\left(10 a^3-9 a b^2\right) \cos (3 (c+d x))+a \left(2 a^2+9 b^2\right) \cos (c+d x)+2 b \sin (c+d x) \left(\left(58 a^2-3 b^2\right) \cos (2 (c+d x))-40 a^2+3 b^2\right)\right)}{a \tan ^{\frac{7}{2}}(c+d x)}-42 (-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+42 (-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{42 d}","\frac{2 \left(7 a^2-9 b^2\right) \sqrt{a+b \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(49 a^2-3 b^2\right) \sqrt{a+b \tan (c+d x)}}{21 a d \sqrt{\tan (c+d x)}}-\frac{2 a^2 \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{i (-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{6 a b \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{i (b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"-1/42*(-42*(-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 42*(-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (Sec[c + d*x]^3*(a*(2*a^2 + 9*b^2)*Cos[c + d*x] + (10*a^3 - 9*a*b^2)*Cos[3*(c + d*x)] + 2*b*(-40*a^2 + 3*b^2 + (58*a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/(a*Tan[c + d*x]^(7/2)))/d","A",1
631,1,300,318,4.216443,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\tan ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(11/2),x]","\frac{\frac{\sec ^4(c+d x) \sqrt{a+b \tan (c+d x)} \left(-987 a^4+272 a^3 b \sin (2 (c+d x))-326 a^3 b \sin (4 (c+d x))+1374 a^2 b^2+4 \left(280 a^4-483 a^2 b^2-10 b^4\right) \cos (2 (c+d x))+\left(-413 a^4+558 a^2 b^2+10 b^4\right) \cos (4 (c+d x))-10 a b^3 \sin (2 (c+d x))+5 a b^3 \sin (4 (c+d x))+30 b^4\right)}{a^2 \tan ^{\frac{9}{2}}(c+d x)}-1260 \sqrt[4]{-1} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-1260 \sqrt[4]{-1} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{1260 d}","\frac{2 b \left(231 a^2-5 b^2\right) \sqrt{a+b \tan (c+d x)}}{315 a d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(21 a^2-25 b^2\right) \sqrt{a+b \tan (c+d x)}}{105 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a^2 \sqrt{a+b \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}-\frac{2 \left(315 a^4-483 a^2 b^2-10 b^4\right) \sqrt{a+b \tan (c+d x)}}{315 a^2 d \sqrt{\tan (c+d x)}}+\frac{(-b+i a)^{5/2} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{38 a b \sqrt{a+b \tan (c+d x)}}{63 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{(b+i a)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(-1260*(-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 1260*(-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + (Sec[c + d*x]^4*(-987*a^4 + 1374*a^2*b^2 + 30*b^4 + 4*(280*a^4 - 483*a^2*b^2 - 10*b^4)*Cos[2*(c + d*x)] + (-413*a^4 + 558*a^2*b^2 + 10*b^4)*Cos[4*(c + d*x)] + 272*a^3*b*Sin[2*(c + d*x)] - 10*a*b^3*Sin[2*(c + d*x)] - 326*a^3*b*Sin[4*(c + d*x)] + 5*a*b^3*Sin[4*(c + d*x)])*Sqrt[a + b*Tan[c + d*x]])/(a^2*Tan[c + d*x]^(9/2)))/(1260*d)","A",1
632,1,270,232,4.0877006,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(7/2)/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{\sqrt{a} \left(3 a^2-8 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}-\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+2 b \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}-3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}","\frac{\left(3 a^2-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 b^{5/2} d}-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-4*(-1)^(3/4)*b^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - (4*(-1)^(3/4)*b^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - 3*a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*(3*a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/(4*b^2*d)","A",1
633,1,221,188,2.0152422,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(5/2)/Sqrt[a + b*Tan[c + d*x]],x]","\frac{-\frac{a^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{b^{3/2} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{b}}{d}","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{b d}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/b - (a^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(b^(3/2)*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
634,1,189,152,0.8203006,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(3/2)/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\frac{2 \sqrt{a} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+(-1)^{3/4} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{d}","-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(3/4)*(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b] + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]) + (2*Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
635,1,123,115,0.1234628,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt[4]{-1} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(1/4)*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]))/d","A",1
636,1,124,109,0.123959,"\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{(-1)^{3/4} \left(-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{d}","\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(3/4)*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) - ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]))/d","A",1
637,1,157,147,0.4215806,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{a \sqrt{\tan (c+d x)}}}{d}","-\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{a d \sqrt{\tan (c+d x)}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*Sqrt[a + b*Tan[c + d*x]])/(a*Sqrt[Tan[c + d*x]]))/d","A",1
638,1,172,180,1.6047865,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{-\frac{2 (a-2 b \tan (c+d x)) \sqrt{a+b \tan (c+d x)}}{a^2 \tan ^{\frac{3}{2}}(c+d x)}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{3 d}","\frac{4 b \sqrt{a+b \tan (c+d x)}}{3 a^2 d \sqrt{\tan (c+d x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*(a - 2*b*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/(a^2*Tan[c + d*x]^(3/2)))/(3*d)","A",1
639,1,197,229,2.6815995,"\int \frac{1}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2-8 b^2\right) \tan ^2(c+d x)-3 a^2+4 a b \tan (c+d x)\right)}{a^3 \tan ^{\frac{5}{2}}(c+d x)}-\frac{15 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{15 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{15 d}","\frac{8 b \sqrt{a+b \tan (c+d x)}}{15 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(15 a^2-8 b^2\right) \sqrt{a+b \tan (c+d x)}}{15 a^3 d \sqrt{\tan (c+d x)}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{5 a d \tan ^{\frac{5}{2}}(c+d x)}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-15*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + (15*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2 + 4*a*b*Tan[c + d*x] + (15*a^2 - 8*b^2)*Tan[c + d*x]^2))/(a^3*Tan[c + d*x]^(5/2)))/(15*d)","A",1
640,1,270,250,3.222243,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b \tan (c+d x)}{a}\right)}{a \sqrt{a+b \tan (c+d x)}}+\frac{5 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}-\frac{5 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}-\frac{5 \sqrt{\tan (c+d x)}}{(a-i b) \sqrt{a+b \tan (c+d x)}}-\frac{5 \sqrt{\tan (c+d x)}}{(a+i b) \sqrt{a+b \tan (c+d x)}}}{5 d}","-\frac{2 a^2 \tan ^{\frac{3}{2}}(c+d x)}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{\left(3 a^2+b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{b^2 d \left(a^2+b^2\right)}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{5/2} d}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"((5*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) - (5*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) - (5*Sqrt[Tan[c + d*x]])/((a - I*b)*Sqrt[a + b*Tan[c + d*x]]) - (5*Sqrt[Tan[c + d*x]])/((a + I*b)*Sqrt[a + b*Tan[c + d*x]]) + (2*Hypergeometric2F1[3/2, 5/2, 7/2, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*Sqrt[1 + (b*Tan[c + d*x])/a])/(a*Sqrt[a + b*Tan[c + d*x]]))/(5*d)","C",1
641,1,321,195,1.307223,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{2 \sqrt{a} \sqrt{-a+i b} \sqrt{a+i b} \left(a^2+b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{b} \left(\sqrt{-a+i b} \left(-2 a^2 \sqrt{a+i b} \sqrt{\tan (c+d x)}-\sqrt[4]{-1} b (a-i b) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}\right)+\sqrt[4]{-1} b (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}\right)}{b^{3/2} d (-a+i b)^{3/2} (a+i b)^{3/2} \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2 \sqrt{\tan (c+d x)}}{b d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"-((2*Sqrt[a]*Sqrt[-a + I*b]*Sqrt[a + I*b]*(a^2 + b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a] + Sqrt[b]*((-1)^(1/4)*(a + I*b)^(3/2)*b*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[-a + I*b]*(-2*a^2*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]] - (-1)^(1/4)*(a - I*b)*b*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]])))/((-a + I*b)^(3/2)*(a + I*b)^(3/2)*b^(3/2)*d*Sqrt[a + b*Tan[c + d*x]]))","A",1
642,1,182,154,1.4444479,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\frac{\frac{\sqrt[4]{-1} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}+\frac{2 a \sqrt{\tan (c+d x)}}{(a+i b) \sqrt{a+b \tan (c+d x)}}}{a-i b}-\frac{(-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}}{d}","\frac{2 a \sqrt{\tan (c+d x)}}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(-(((-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2)) + (((-1)^(1/4)*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) + (2*a*Sqrt[Tan[c + d*x]])/((a + I*b)*Sqrt[a + b*Tan[c + d*x]]))/(a - I*b))/d","A",1
643,1,163,149,1.2238061,"\int \frac{\sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^(3/2),x]","\frac{-\frac{2 b \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}}{d}","-\frac{2 b \sqrt{\tan (c+d x)}}{d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) + ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) - (2*b*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
644,1,183,159,0.929685,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]","-\frac{-\frac{2 b^2 \sqrt{\tan (c+d x)}}{a \sqrt{a+b \tan (c+d x)}}+\frac{(-1)^{3/4} (a+i b) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{d \left(a^2+b^2\right)}","\frac{2 b^2 \sqrt{\tan (c+d x)}}{a d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"-((((-1)^(3/4)*(a + I*b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((-1)^(1/4)*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*b^2*Sqrt[Tan[c + d*x]])/(a*Sqrt[a + b*Tan[c + d*x]]))/((a^2 + b^2)*d))","A",1
645,1,202,193,4.4134006,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)),x]","-\frac{\frac{\frac{2 \left(b \left(a^2+2 b^2\right) \tan (c+d x)+a \left(a^2+b^2\right)\right)}{a^2 \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} (a-i b) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{a^2+b^2}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}}{d}","-\frac{2 b \left(a^2+2 b^2\right) \sqrt{\tan (c+d x)}}{a^2 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{2}{a d \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"-((((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) + (((-1)^(1/4)*(a - I*b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*(a*(a^2 + b^2) + b*(a^2 + 2*b^2)*Tan[c + d*x]))/(a^2*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]))/(a^2 + b^2))/d)","A",1
646,1,223,241,5.6165441,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\frac{2 b^2 \left(5 a^2+8 b^2\right) \tan ^2(c+d x)+8 a b \left(a^2+b^2\right) \tan (c+d x)-2 a^2 \left(a^2+b^2\right)}{a^3 \left(a^2+b^2\right) \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}-\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}}{3 d}","\frac{8 b}{3 a^2 d \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left(5 a^2+8 b^2\right) \sqrt{\tan (c+d x)}}{3 a^3 d \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{2}{3 a d \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"((-3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) + (-2*a^2*(a^2 + b^2) + 8*a*b*(a^2 + b^2)*Tan[c + d*x] + 2*b^2*(5*a^2 + 8*b^2)*Tan[c + d*x]^2)/(a^3*(a^2 + b^2)*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]))/(3*d)","A",1
647,1,407,317,6.163565,"\int \frac{\tan ^{\frac{9}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \tan ^{\frac{7}{2}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} \, _2F_1\left(\frac{5}{2},\frac{7}{2};\frac{9}{2};-\frac{b \tan (c+d x)}{a}\right)}{7 a^2 d \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (a-i b) (-a+i b)^{3/2}}-\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-a-i b) (a+i b)^{3/2}}+\frac{\tan ^{\frac{3}{2}}(c+d x)}{3 d (-a+i b) (a+b \tan (c+d x))^{3/2}}-\frac{\tan ^{\frac{3}{2}}(c+d x)}{3 d (a+i b) (a+b \tan (c+d x))^{3/2}}-\frac{i \sqrt{\tan (c+d x)}}{d (a-i b) (-a+i b) \sqrt{a+b \tan (c+d x)}}+\frac{i \sqrt{\tan (c+d x)}}{d (-a-i b) (a+i b) \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2 \tan ^{\frac{5}{2}}(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 a^2 \left(5 a^2+11 b^2\right) \tan ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{\left(5 a^4+10 a^2 b^2+b^4\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{b^3 d \left(a^2+b^2\right)^2}-\frac{5 a \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{7/2} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((a - I*b)*(-a + I*b)^(3/2)*d) - ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((-a - I*b)*(a + I*b)^(3/2)*d) + Tan[c + d*x]^(3/2)/(3*(-a + I*b)*d*(a + b*Tan[c + d*x])^(3/2)) - Tan[c + d*x]^(3/2)/(3*(a + I*b)*d*(a + b*Tan[c + d*x])^(3/2)) - (I*Sqrt[Tan[c + d*x]])/((a - I*b)*(-a + I*b)*d*Sqrt[a + b*Tan[c + d*x]]) + (I*Sqrt[Tan[c + d*x]])/((-a - I*b)*(a + I*b)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*Hypergeometric2F1[5/2, 7/2, 9/2, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(7/2)*Sqrt[1 + (b*Tan[c + d*x])/a])/(7*a^2*d*Sqrt[a + b*Tan[c + d*x]])","C",1
648,1,468,251,6.234619,"\int \frac{\tan ^{\frac{7}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^(5/2),x]","\frac{2 \sqrt{a+b \tan (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d \sqrt{\frac{b \tan (c+d x)}{a}+1}}-\frac{2 \sqrt{\tan (c+d x)}}{b^2 d \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-a+i b)^{3/2} (b+i a)}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a) (a+i b)^{3/2}}-\frac{i \tan ^{\frac{3}{2}}(c+d x)}{3 d (a-i b) (a+b \tan (c+d x))^{3/2}}+\frac{i \tan ^{\frac{3}{2}}(c+d x)}{3 d (a+i b) (a+b \tan (c+d x))^{3/2}}-\frac{2 \tan ^{\frac{3}{2}}(c+d x)}{3 b d (a+b \tan (c+d x))^{3/2}}-\frac{i \sqrt{\tan (c+d x)}}{d (-b+i a) (a+i b) \sqrt{a+b \tan (c+d x)}}-\frac{i \sqrt{\tan (c+d x)}}{d (a-i b) (b+i a) \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2 \tan ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 a^2 \left(a^2+3 b^2\right) \sqrt{\tan (c+d x)}}{b^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{5/2} d}-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"-(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((-a + I*b)^(3/2)*(I*a + b)*d)) + ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)*(a + I*b)^(3/2)*d) - ((I/3)*Tan[c + d*x]^(3/2))/((a - I*b)*d*(a + b*Tan[c + d*x])^(3/2)) + ((I/3)*Tan[c + d*x]^(3/2))/((a + I*b)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*Tan[c + d*x]^(3/2))/(3*b*d*(a + b*Tan[c + d*x])^(3/2)) - (I*Sqrt[Tan[c + d*x]])/((I*a - b)*(a + I*b)*d*Sqrt[a + b*Tan[c + d*x]]) - (2*Sqrt[Tan[c + d*x]])/(b^2*d*Sqrt[a + b*Tan[c + d*x]]) - (I*Sqrt[Tan[c + d*x]])/((a - I*b)*(I*a + b)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[a]*b^(5/2)*d*Sqrt[1 + (b*Tan[c + d*x])/a])","A",1
649,1,189,214,3.4037057,"\int \frac{\tan ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{2 a \sqrt{\tan (c+d x)} \left(\left(a^2+7 b^2\right) \tan (c+d x)+6 a b\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}+\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{5/2}}-\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{5/2}}}{3 d}","-\frac{2 a^2 \sqrt{\tan (c+d x)}}{3 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left(a^2+7 b^2\right) \sqrt{\tan (c+d x)}}{3 b d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"((3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(5/2) - (3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(5/2) + (2*a*Sqrt[Tan[c + d*x]]*(6*a*b + (a^2 + 7*b^2)*Tan[c + d*x]))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^(3/2)))/(3*d)","A",1
650,1,198,199,3.0780025,"\int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{2 \sqrt{\tan (c+d x)} \left(2 b \left(a^2-2 b^2\right) \tan (c+d x)+3 a \left(a^2-b^2\right)\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{5/2}}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{5/2}}}{3 d}","\frac{2 a \sqrt{\tan (c+d x)}}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{4 \left(a^2-2 b^2\right) \sqrt{\tan (c+d x)}}{3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"((3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(5/2) + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(5/2) + (2*Sqrt[Tan[c + d*x]]*(3*a*(a^2 - b^2) + 2*b*(a^2 - 2*b^2)*Tan[c + d*x]))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^(3/2)))/(3*d)","A",1
651,1,194,211,3.8806903,"\int \frac{\sqrt{\tan (c+d x)}}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\frac{2 b \sqrt{\tan (c+d x)} \left(\left(b^3-5 a^2 b\right) \tan (c+d x)-6 a^3\right)}{a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}-\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{5/2}}+\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{5/2}}}{3 d}","-\frac{2 b \left(5 a^2-b^2\right) \sqrt{\tan (c+d x)}}{3 a d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}-\frac{2 b \sqrt{\tan (c+d x)}}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"((-3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(5/2) + (3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(5/2) + (2*b*Sqrt[Tan[c + d*x]]*(-6*a^3 + (-5*a^2*b + b^3)*Tan[c + d*x]))/(a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^(3/2)))/(3*d)","A",1
652,1,235,212,1.4304761,"\int \frac{1}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\frac{4 b^2 \left(4 a^2+b^2\right) \sqrt{\tan (c+d x)}}{a^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))^{3/2}}-3 (-1)^{3/4} \left(\frac{(a-i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{3 d \left(a^2+b^2\right)^2}","\frac{4 b^2 \left(4 a^2+b^2\right) \sqrt{\tan (c+d x)}}{3 a^2 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \sqrt{\tan (c+d x)}}{3 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(-3*(-1)^(3/4)*(((a + I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((a - I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*b^2*(a^2 + b^2)*Sqrt[Tan[c + d*x]])/(a*(a + b*Tan[c + d*x])^(3/2)) + (4*b^2*(4*a^2 + b^2)*Sqrt[Tan[c + d*x]])/(a^2*Sqrt[a + b*Tan[c + d*x]]))/(3*(a^2 + b^2)^2*d)","A",1
653,1,294,265,2.7246501,"\int \frac{1}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)),x]","-\frac{-\frac{3 \sqrt[4]{-1} a^2 \left(\frac{(a+i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{(a-i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{\left(a^2+b^2\right)^2}+\frac{2 b \left(3 a^2+4 b^2\right) \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(3 a^4+17 a^2 b^2+8 b^4\right) \sqrt{\tan (c+d x)}}{a \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{6 a}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{3 a^2 d}","-\frac{2 b \left(3 a^2+4 b^2\right) \sqrt{\tan (c+d x)}}{3 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}-\frac{2 b \left(3 a^4+17 a^2 b^2+8 b^4\right) \sqrt{\tan (c+d x)}}{3 a^3 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{i \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{2}{a d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"-1/3*((-3*(-1)^(1/4)*a^2*(((a + I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((a - I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]))/(a^2 + b^2)^2 + (6*a)/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(3*a^2 + 4*b^2)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(3*a^4 + 17*a^2*b^2 + 8*b^4)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]]))/(a^2*d)","A",1
654,1,483,298,6.3199176,"\int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)),x]","-\frac{2}{3 a d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac{2 \left(-\frac{6 b}{a d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{3 \left(\frac{32 b^2 \sqrt{\tan (c+d x)}}{3 a^2 \sqrt{a+b \tan (c+d x)}}-\frac{\frac{b (5 a-2 i b) \sqrt{\tan (c+d x)}}{(a-i b) \sqrt{a+b \tan (c+d x)}}-\frac{3 \sqrt[4]{-1} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}}{3 (b+i a)}+\frac{\frac{b (5 a+2 i b) \sqrt{\tan (c+d x)}}{(a+i b) \sqrt{a+b \tan (c+d x)}}-\frac{3 \sqrt[4]{-1} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}}{3 (-b+i a)}+\frac{16 b^2 \sqrt{\tan (c+d x)}}{3 a (a+b \tan (c+d x))^{3/2}}+\frac{a b \sqrt{\tan (c+d x)}}{3 (-b+i a) (a+b \tan (c+d x))^{3/2}}-\frac{a b \sqrt{\tan (c+d x)}}{3 (b+i a) (a+b \tan (c+d x))^{3/2}}\right)}{2 a d}\right)}{3 a}","\frac{4 b}{a^2 d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left(4 a^4+15 a^2 b^2+8 b^4\right) \sqrt{\tan (c+d x)}}{3 a^4 d \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left(7 a^2+8 b^2\right) \sqrt{\tan (c+d x)}}{3 a^3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{2}{3 a d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"-2/(3*a*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)) - (2*((-6*b)/(a*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (3*((a*b*Sqrt[Tan[c + d*x]])/(3*(I*a - b)*(a + b*Tan[c + d*x])^(3/2)) + (16*b^2*Sqrt[Tan[c + d*x]])/(3*a*(a + b*Tan[c + d*x])^(3/2)) - (a*b*Sqrt[Tan[c + d*x]])/(3*(I*a + b)*(a + b*Tan[c + d*x])^(3/2)) + (32*b^2*Sqrt[Tan[c + d*x]])/(3*a^2*Sqrt[a + b*Tan[c + d*x]]) - ((-3*(-1)^(1/4)*a^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) + ((5*a - (2*I)*b)*b*Sqrt[Tan[c + d*x]])/((a - I*b)*Sqrt[a + b*Tan[c + d*x]]))/(3*(I*a + b)) + ((-3*(-1)^(1/4)*a^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) + ((5*a + (2*I)*b)*b*Sqrt[Tan[c + d*x]])/((a + I*b)*Sqrt[a + b*Tan[c + d*x]]))/(3*(I*a - b))))/(2*a*d)))/(3*a)","A",1
655,1,89,89,0.1799705,"\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{2+3 \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[2 + 3*Tan[c + d*x]]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{-3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{-3+2 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{3+2 i} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{3-2 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{3+2 i} d}",1,"ArcTan[(Sqrt[-3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]]/(Sqrt[-3 + 2*I]*d) + ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]]/(Sqrt[3 + 2*I]*d)","A",1
656,1,89,89,0.1360006,"\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{-2+3 \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[-2 + 3*Tan[c + d*x]]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{-3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{-3+2 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{3+2 i} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{3-2 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{3+2 i} d}",1,"ArcTan[(Sqrt[-3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]]/(Sqrt[-3 + 2*I]*d) + ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]]/(Sqrt[3 + 2*I]*d)","A",1
657,1,101,89,0.1685402,"\int \frac{1}{\sqrt{2-3 \tan (c+d x)} \sqrt{\tan (c+d x)}} \, dx","Integrate[1/(Sqrt[2 - 3*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]","\frac{\sqrt{-3+2 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{3}{13}+\frac{2 i}{13}} \sqrt{2-3 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)-\sqrt{3+2 i} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{13}+\frac{2 i}{13}} \sqrt{2-3 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)}{\sqrt{13} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{2-3 \tan (c+d x)}}\right)}{\sqrt{3-2 i} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{2-3 \tan (c+d x)}}\right)}{\sqrt{3+2 i} d}",1,"(-(Sqrt[3 + 2*I]*ArcTan[(Sqrt[3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]) + Sqrt[-3 + 2*I]*ArcTanh[(Sqrt[-3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]])/(Sqrt[13]*d)","A",1
658,1,101,89,0.1451229,"\int \frac{1}{\sqrt{-2-3 \tan (c+d x)} \sqrt{\tan (c+d x)}} \, dx","Integrate[1/(Sqrt[-2 - 3*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]","\frac{\sqrt{-3+2 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{3}{13}+\frac{2 i}{13}} \sqrt{-3 \tan (c+d x)-2}}{\sqrt{\tan (c+d x)}}\right)-\sqrt{3+2 i} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{13}+\frac{2 i}{13}} \sqrt{-3 \tan (c+d x)-2}}{\sqrt{\tan (c+d x)}}\right)}{\sqrt{13} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right)}{\sqrt{3-2 i} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right)}{\sqrt{3+2 i} d}",1,"(-(Sqrt[3 + 2*I]*ArcTan[(Sqrt[3/13 + (2*I)/13]*Sqrt[-2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]) + Sqrt[-3 + 2*I]*ArcTanh[(Sqrt[-3/13 + (2*I)/13]*Sqrt[-2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]])/(Sqrt[13]*d)","A",1
659,1,89,89,0.1743496,"\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{3+2 \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[3 + 2*Tan[c + d*x]]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{-2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{-2+3 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{2+3 i} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{2-3 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{2+3 i} d}",1,"ArcTan[(Sqrt[-2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]]/(Sqrt[-2 + 3*I]*d) + ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]]/(Sqrt[2 + 3*I]*d)","A",1
660,1,101,89,0.1673518,"\int \frac{1}{\sqrt{3-2 \tan (c+d x)} \sqrt{\tan (c+d x)}} \, dx","Integrate[1/(Sqrt[3 - 2*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]","\frac{\sqrt{-2+3 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{2}{13}+\frac{3 i}{13}} \sqrt{3-2 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)-\sqrt{2+3 i} \tan ^{-1}\left(\frac{\sqrt{\frac{2}{13}+\frac{3 i}{13}} \sqrt{3-2 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)}{\sqrt{13} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{3-2 \tan (c+d x)}}\right)}{\sqrt{2-3 i} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{3-2 \tan (c+d x)}}\right)}{\sqrt{2+3 i} d}",1,"(-(Sqrt[2 + 3*I]*ArcTan[(Sqrt[2/13 + (3*I)/13]*Sqrt[3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]) + Sqrt[-2 + 3*I]*ArcTanh[(Sqrt[-2/13 + (3*I)/13]*Sqrt[3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]])/(Sqrt[13]*d)","A",1
661,1,89,89,0.1444168,"\int \frac{1}{\sqrt{\tan (c+d x)} \sqrt{-3+2 \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[-3 + 2*Tan[c + d*x]]),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{-2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{-2+3 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{2+3 i} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{2-3 i} d}+\frac{\tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{2+3 i} d}",1,"ArcTan[(Sqrt[-2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/(Sqrt[-2 + 3*I]*d) + ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/(Sqrt[2 + 3*I]*d)","A",1
662,1,101,89,0.1524752,"\int \frac{1}{\sqrt{-3-2 \tan (c+d x)} \sqrt{\tan (c+d x)}} \, dx","Integrate[1/(Sqrt[-3 - 2*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]","\frac{\sqrt{-2+3 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{2}{13}+\frac{3 i}{13}} \sqrt{-2 \tan (c+d x)-3}}{\sqrt{\tan (c+d x)}}\right)-\sqrt{2+3 i} \tan ^{-1}\left(\frac{\sqrt{\frac{2}{13}+\frac{3 i}{13}} \sqrt{-2 \tan (c+d x)-3}}{\sqrt{\tan (c+d x)}}\right)}{\sqrt{13} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right)}{\sqrt{2-3 i} d}+\frac{\tan ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right)}{\sqrt{2+3 i} d}",1,"(-(Sqrt[2 + 3*I]*ArcTan[(Sqrt[2/13 + (3*I)/13]*Sqrt[-3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]) + Sqrt[-2 + 3*I]*ArcTanh[(Sqrt[-2/13 + (3*I)/13]*Sqrt[-3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]])/(Sqrt[13]*d)","A",1
663,1,95,95,0.138787,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{2+3 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[2 + 3*Tan[c + d*x]],x]","\frac{i \tan ^{-1}\left(\frac{\sqrt{-3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{-3+2 i} d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{3+2 i} d}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{3-2 i} d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)+2}}\right)}{\sqrt{3+2 i} d}",1,"(I*ArcTan[(Sqrt[-3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/(Sqrt[-3 + 2*I]*d) - (I*ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)","A",1
664,1,95,95,0.1039514,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{-2+3 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[-2 + 3*Tan[c + d*x]],x]","\frac{i \tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{3+2 i} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{-3+2 i} d}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{3+2 i} d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{3 \tan (c+d x)-2}}\right)}{\sqrt{3-2 i} d}",1,"((-I)*ArcTan[(Sqrt[-3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]])/(Sqrt[-3 + 2*I]*d) + (I*ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)","A",1
665,1,103,95,0.0662447,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{2-3 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[2 - 3*Tan[c + d*x]],x]","\frac{i \left(\sqrt{3+2 i} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{13}+\frac{2 i}{13}} \sqrt{2-3 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)+\sqrt{-3+2 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{3}{13}+\frac{2 i}{13}} \sqrt{2-3 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)\right)}{\sqrt{13} d}","\frac{i \tan ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{2-3 \tan (c+d x)}}\right)}{\sqrt{3+2 i} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{2-3 \tan (c+d x)}}\right)}{\sqrt{3-2 i} d}",1,"(I*(Sqrt[3 + 2*I]*ArcTan[(Sqrt[3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]] + Sqrt[-3 + 2*I]*ArcTanh[(Sqrt[-3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)","A",1
666,1,103,95,0.1146021,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{-2-3 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[-2 - 3*Tan[c + d*x]],x]","-\frac{i \left(\sqrt{3+2 i} \tan ^{-1}\left(\frac{\sqrt{\frac{3}{13}+\frac{2 i}{13}} \sqrt{-3 \tan (c+d x)-2}}{\sqrt{\tan (c+d x)}}\right)+\sqrt{-3+2 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{3}{13}+\frac{2 i}{13}} \sqrt{-3 \tan (c+d x)-2}}{\sqrt{\tan (c+d x)}}\right)\right)}{\sqrt{13} d}","\frac{i \tan ^{-1}\left(\frac{\sqrt{3-2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right)}{\sqrt{3-2 i} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{3+2 i} \sqrt{\tan (c+d x)}}{\sqrt{-3 \tan (c+d x)-2}}\right)}{\sqrt{3+2 i} d}",1,"((-I)*(Sqrt[3 + 2*I]*ArcTan[(Sqrt[3/13 + (2*I)/13]*Sqrt[-2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]] + Sqrt[-3 + 2*I]*ArcTanh[(Sqrt[-3/13 + (2*I)/13]*Sqrt[-2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)","A",1
667,1,95,95,0.1362359,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{3+2 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[3 + 2*Tan[c + d*x]],x]","\frac{i \tan ^{-1}\left(\frac{\sqrt{-2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{-2+3 i} d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{2+3 i} d}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{2-3 i} d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)+3}}\right)}{\sqrt{2+3 i} d}",1,"(I*ArcTan[(Sqrt[-2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]])/(Sqrt[-2 + 3*I]*d) - (I*ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*d)","A",1
668,1,103,95,0.1283515,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{3-2 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[3 - 2*Tan[c + d*x]],x]","\frac{i \left(\sqrt{2+3 i} \tan ^{-1}\left(\frac{\sqrt{\frac{2}{13}+\frac{3 i}{13}} \sqrt{3-2 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)+\sqrt{-2+3 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{2}{13}+\frac{3 i}{13}} \sqrt{3-2 \tan (c+d x)}}{\sqrt{\tan (c+d x)}}\right)\right)}{\sqrt{13} d}","\frac{i \tan ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{3-2 \tan (c+d x)}}\right)}{\sqrt{2+3 i} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{3-2 \tan (c+d x)}}\right)}{\sqrt{2-3 i} d}",1,"(I*(Sqrt[2 + 3*I]*ArcTan[(Sqrt[2/13 + (3*I)/13]*Sqrt[3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]] + Sqrt[-2 + 3*I]*ArcTanh[(Sqrt[-2/13 + (3*I)/13]*Sqrt[3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)","A",1
669,1,95,95,0.1170504,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{-3+2 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[-3 + 2*Tan[c + d*x]],x]","\frac{i \tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{2+3 i} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{-2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{-2+3 i} d}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{2+3 i} d}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{2 \tan (c+d x)-3}}\right)}{\sqrt{2-3 i} d}",1,"((-I)*ArcTan[(Sqrt[-2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]])/(Sqrt[-2 + 3*I]*d) + (I*ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*d)","A",1
670,1,103,95,0.091493,"\int \frac{\sqrt{\tan (c+d x)}}{\sqrt{-3-2 \tan (c+d x)}} \, dx","Integrate[Sqrt[Tan[c + d*x]]/Sqrt[-3 - 2*Tan[c + d*x]],x]","-\frac{i \left(\sqrt{2+3 i} \tan ^{-1}\left(\frac{\sqrt{\frac{2}{13}+\frac{3 i}{13}} \sqrt{-2 \tan (c+d x)-3}}{\sqrt{\tan (c+d x)}}\right)+\sqrt{-2+3 i} \tanh ^{-1}\left(\frac{\sqrt{-\frac{2}{13}+\frac{3 i}{13}} \sqrt{-2 \tan (c+d x)-3}}{\sqrt{\tan (c+d x)}}\right)\right)}{\sqrt{13} d}","\frac{i \tan ^{-1}\left(\frac{\sqrt{2-3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right)}{\sqrt{2-3 i} d}-\frac{i \tan ^{-1}\left(\frac{\sqrt{2+3 i} \sqrt{\tan (c+d x)}}{\sqrt{-2 \tan (c+d x)-3}}\right)}{\sqrt{2+3 i} d}",1,"((-I)*(Sqrt[2 + 3*I]*ArcTan[(Sqrt[2/13 + (3*I)/13]*Sqrt[-3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]] + Sqrt[-2 + 3*I]*ArcTanh[(Sqrt[-2/13 + (3*I)/13]*Sqrt[-3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)","A",1
671,1,153,466,0.3948294,"\int \frac{\tan ^{\frac{5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(5/3)/(a + b*Tan[c + d*x]),x]","\frac{30 a \tan ^{\frac{2}{3}}(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \tan (c+d x)}{a}\right)+5 a \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)-2 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)+\log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)\right)+12 b \tan ^{\frac{5}{3}}(c+d x) \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\tan ^2(c+d x)\right)}{20 d \left(a^2+b^2\right)}","-\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{2 d \left(a^2+b^2\right)}+\frac{b \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{2 d \left(a^2+b^2\right)}+\frac{b \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)}+\frac{\sqrt{3} a \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 d \left(a^2+b^2\right)}-\frac{a \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{2 d \left(a^2+b^2\right)}+\frac{a \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{4 d \left(a^2+b^2\right)}+\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}-\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}-\frac{\sqrt{3} a^{5/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt[3]{\tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{b^{2/3} d \left(a^2+b^2\right)}-\frac{3 a^{5/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt[3]{\tan (c+d x)}\right)}{2 b^{2/3} d \left(a^2+b^2\right)}+\frac{a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} d \left(a^2+b^2\right)}",1,"(5*a*(2*Sqrt[3]*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]] - 2*Log[1 + Tan[c + d*x]^(2/3)] + Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)]) + 30*a*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(2/3) + 12*b*Hypergeometric2F1[5/6, 1, 11/6, -Tan[c + d*x]^2]*Tan[c + d*x]^(5/3))/(20*(a^2 + b^2)*d)","C",1
672,1,204,465,0.4129617,"\int \frac{\sqrt[3]{\tan (c+d x)}}{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^(1/3)/(a + b*Tan[c + d*x]),x]","\frac{2 \sqrt[3]{a} b^{2/3} \left(\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sqrt[3]{\tan (c+d x)}+b^{2/3} \tan ^{\frac{2}{3}}(c+d x)\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt[3]{\tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)-2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt[3]{\tan (c+d x)}\right)\right)+3 a \tan ^{\frac{4}{3}}(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\tan ^2(c+d x)\right)+12 b \sqrt[3]{\tan (c+d x)} \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\tan ^2(c+d x)\right)}{4 d \left(a^2+b^2\right)}","-\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{2 d \left(a^2+b^2\right)}+\frac{b \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{2 d \left(a^2+b^2\right)}+\frac{b \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)}-\frac{\sqrt{3} a \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 d \left(a^2+b^2\right)}-\frac{a \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{2 d \left(a^2+b^2\right)}-\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}+\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}+\frac{a \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{4 d \left(a^2+b^2\right)}+\frac{\sqrt{3} \sqrt[3]{a} b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt[3]{\tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{d \left(a^2+b^2\right)}-\frac{3 \sqrt[3]{a} b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt[3]{\tan (c+d x)}\right)}{2 d \left(a^2+b^2\right)}+\frac{\sqrt[3]{a} b^{2/3} \log (a+b \tan (c+d x))}{2 d \left(a^2+b^2\right)}",1,"(2*a^(1/3)*b^(2/3)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x]^(1/3))/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]^(1/3)] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x]^(1/3) + b^(2/3)*Tan[c + d*x]^(2/3)]) + 12*b*Hypergeometric2F1[1/6, 1, 7/6, -Tan[c + d*x]^2]*Tan[c + d*x]^(1/3) + 3*a*Hypergeometric2F1[2/3, 1, 5/3, -Tan[c + d*x]^2]*Tan[c + d*x]^(4/3))/(4*(a^2 + b^2)*d)","C",1
673,1,162,467,0.3288785,"\int \frac{1}{\sqrt[3]{\tan (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(1/3)*(a + b*Tan[c + d*x])),x]","\frac{30 b^2 \tan ^{\frac{2}{3}}(c+d x) \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};-\frac{b \tan (c+d x)}{a}\right)-a \left(5 a \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)-2 \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)+\log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)\right)+12 b \tan ^{\frac{5}{3}}(c+d x) \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};-\tan ^2(c+d x)\right)\right)}{20 a d \left(a^2+b^2\right)}","\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{2 d \left(a^2+b^2\right)}-\frac{b \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{2 d \left(a^2+b^2\right)}-\frac{b \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)}-\frac{\sqrt{3} a \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 d \left(a^2+b^2\right)}-\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}+\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}+\frac{a \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{2 d \left(a^2+b^2\right)}-\frac{a \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{4 d \left(a^2+b^2\right)}-\frac{\sqrt{3} b^{4/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt[3]{\tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt[3]{a} d \left(a^2+b^2\right)}-\frac{3 b^{4/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt[3]{\tan (c+d x)}\right)}{2 \sqrt[3]{a} d \left(a^2+b^2\right)}+\frac{b^{4/3} \log (a+b \tan (c+d x))}{2 \sqrt[3]{a} d \left(a^2+b^2\right)}",1,"(30*b^2*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(2/3) - a*(5*a*(2*Sqrt[3]*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]] - 2*Log[1 + Tan[c + d*x]^(2/3)] + Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)]) + 12*b*Hypergeometric2F1[5/6, 1, 11/6, -Tan[c + d*x]^2]*Tan[c + d*x]^(5/3)))/(20*a*(a^2 + b^2)*d)","C",1
674,1,104,525,0.233741,"\int \frac{1}{\tan ^{\frac{5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[1/(Tan[c + d*x]^(5/3)*(a + b*Tan[c + d*x])),x]","-\frac{3 \left(b^2 \, _2F_1\left(-\frac{2}{3},1;\frac{1}{3};-\frac{b \tan (c+d x)}{a}\right)+a \left(a \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};-\tan ^2(c+d x)\right)+2 b \tan (c+d x) \, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-\tan ^2(c+d x)\right)\right)\right)}{2 a d \left(a^2+b^2\right) \tan ^{\frac{2}{3}}(c+d x)}","\frac{b \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{\tan (c+d x)}\right)}{2 d \left(a^2+b^2\right)}-\frac{b \tan ^{-1}\left(2 \sqrt[3]{\tan (c+d x)}+\sqrt{3}\right)}{2 d \left(a^2+b^2\right)}-\frac{b \tan ^{-1}\left(\sqrt[3]{\tan (c+d x)}\right)}{d \left(a^2+b^2\right)}-\frac{3 b^2}{2 a d \left(a^2+b^2\right) \tan ^{\frac{2}{3}}(c+d x)}-\frac{3 a}{2 d \left(a^2+b^2\right) \tan ^{\frac{2}{3}}(c+d x)}+\frac{\sqrt{3} a \tan ^{-1}\left(\frac{1-2 \tan ^{\frac{2}{3}}(c+d x)}{\sqrt{3}}\right)}{2 d \left(a^2+b^2\right)}+\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)-\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}-\frac{\sqrt{3} b \log \left(\tan ^{\frac{2}{3}}(c+d x)+\sqrt{3} \sqrt[3]{\tan (c+d x)}+1\right)}{4 d \left(a^2+b^2\right)}+\frac{a \log \left(\tan ^{\frac{2}{3}}(c+d x)+1\right)}{2 d \left(a^2+b^2\right)}-\frac{a \log \left(\tan ^{\frac{4}{3}}(c+d x)-\tan ^{\frac{2}{3}}(c+d x)+1\right)}{4 d \left(a^2+b^2\right)}+\frac{\sqrt{3} b^{8/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt[3]{\tan (c+d x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{a^{5/3} d \left(a^2+b^2\right)}-\frac{3 b^{8/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt[3]{\tan (c+d x)}\right)}{2 a^{5/3} d \left(a^2+b^2\right)}+\frac{b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} d \left(a^2+b^2\right)}",1,"(-3*(b^2*Hypergeometric2F1[-2/3, 1, 1/3, -((b*Tan[c + d*x])/a)] + a*(a*Hypergeometric2F1[-1/3, 1, 2/3, -Tan[c + d*x]^2] + 2*b*Hypergeometric2F1[1/6, 1, 7/6, -Tan[c + d*x]^2]*Tan[c + d*x])))/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(2/3))","C",1
675,1,21046,163,26.8161325,"\int \frac{\tan ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(4/3)/Sqrt[a + b*Tan[c + d*x]],x]","\text{Result too large to show}","\frac{3 \tan ^{\frac{7}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{7}{3};1,\frac{1}{2};\frac{10}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{14 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \tan ^{\frac{7}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{7}{3};1,\frac{1}{2};\frac{10}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{14 d \sqrt{a+b \tan (c+d x)}}",1,"Result too large to show","B",0
676,1,7362,163,63.5890644,"\int \frac{\tan ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(2/3)/Sqrt[a + b*Tan[c + d*x]],x]","\text{Result too large to show}","\frac{3 \tan ^{\frac{5}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{5}{3};1,\frac{1}{2};\frac{8}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{10 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \tan ^{\frac{5}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{5}{3};1,\frac{1}{2};\frac{8}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{10 d \sqrt{a+b \tan (c+d x)}}",1,"Result too large to show","B",0
677,1,6316,163,24.1125694,"\int \frac{\sqrt[3]{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^(1/3)/Sqrt[a + b*Tan[c + d*x]],x]","\text{Result too large to show}","\frac{3 \tan ^{\frac{4}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{4}{3};1,\frac{1}{2};\frac{7}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{8 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \tan ^{\frac{4}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{4}{3};1,\frac{1}{2};\frac{7}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{8 d \sqrt{a+b \tan (c+d x)}}",1,"Result too large to show","B",0
678,1,32685,163,71.3315703,"\int \frac{1}{\sqrt[3]{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(1/3)*Sqrt[a + b*Tan[c + d*x]]),x]","\text{Result too large to show}","\frac{3 \tan ^{\frac{2}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{2}{3};1,\frac{1}{2};\frac{5}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{4 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \tan ^{\frac{2}{3}}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{2}{3};1,\frac{1}{2};\frac{5}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{4 d \sqrt{a+b \tan (c+d x)}}",1,"Result too large to show","B",0
679,1,6198,163,34.0860409,"\int \frac{1}{\tan ^{\frac{2}{3}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(2/3)*Sqrt[a + b*Tan[c + d*x]]),x]","\text{Result too large to show}","\frac{3 \sqrt[3]{\tan (c+d x)} \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{2 d \sqrt{a+b \tan (c+d x)}}+\frac{3 \sqrt[3]{\tan (c+d x)} \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(\frac{1}{3};1,\frac{1}{2};\frac{4}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{2 d \sqrt{a+b \tan (c+d x)}}",1,"Result too large to show","B",0
680,1,23355,163,52.9299613,"\int \frac{1}{\tan ^{\frac{4}{3}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Tan[c + d*x]^(4/3)*Sqrt[a + b*Tan[c + d*x]]),x]","\text{Result too large to show}","-\frac{3 \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{2 d \sqrt[3]{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{3 \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{2 d \sqrt[3]{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}",1,"Result too large to show","B",0
681,1,371,525,11.3827499,"\int \tan ^4(e+f x) \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^4*(c + d*Tan[e + f*x])^(1/3),x]","\frac{\frac{3 \sqrt[3]{c+d \tan (e+f x)} \left(9 c^3-d \left(3 c^2+49 d^2\right) \tan (e+f x)+2 d^2 \sec ^2(e+f x) (c+7 d \tan (e+f x))-37 c d^2\right)}{35 d^3}-i \sqrt[3]{c-i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)+\log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)\right)+i \sqrt[3]{c+i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+\log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)\right)}{4 f}","\frac{3 \left(9 c^2-35 d^2\right) (c+d \tan (e+f x))^{4/3}}{140 d^3 f}-\frac{\sqrt{3} \sqrt{-d^2} \sqrt[3]{c-\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 d f}+\frac{\sqrt{3} \sqrt{-d^2} \sqrt[3]{c+\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 d f}-\frac{9 c \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{35 d^2 f}+\frac{3 \sqrt{-d^2} \sqrt[3]{c-\sqrt{-d^2}} \log \left(\sqrt[3]{c-\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 d f}-\frac{3 \sqrt{-d^2} \sqrt[3]{c+\sqrt{-d^2}} \log \left(\sqrt[3]{c+\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 d f}+\frac{\sqrt{-d^2} \sqrt[3]{c-\sqrt{-d^2}} \log (\cos (e+f x))}{4 d f}-\frac{\sqrt{-d^2} \sqrt[3]{c+\sqrt{-d^2}} \log (\cos (e+f x))}{4 d f}-\frac{1}{4} x \sqrt[3]{c-\sqrt{-d^2}}-\frac{1}{4} x \sqrt[3]{c+\sqrt{-d^2}}+\frac{3 \tan ^2(e+f x) (c+d \tan (e+f x))^{4/3}}{10 d f}",1,"((-I)*(c - I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] - 2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) + I*(c + I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) + (3*(c + d*Tan[e + f*x])^(1/3)*(9*c^3 - 37*c*d^2 - d*(3*c^2 + 49*d^2)*Tan[e + f*x] + 2*d^2*Sec[e + f*x]^2*(c + 7*d*Tan[e + f*x])))/(35*d^3))/(4*f)","C",1
682,1,442,373,1.0999465,"\int \tan ^3(e+f x) \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^3*(c + d*Tan[e + f*x])^(1/3),x]","\frac{-9 c^2 \sqrt[3]{c+d \tan (e+f x)}+14 \sqrt{3} d^2 \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)+14 \sqrt{3} d^2 \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)+12 d^2 \tan ^2(e+f x) \sqrt[3]{c+d \tan (e+f x)}-84 d^2 \sqrt[3]{c+d \tan (e+f x)}-14 d^2 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)-14 d^2 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+7 d^2 \sqrt[3]{c-i d} \log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)+7 d^2 \sqrt[3]{c+i d} \log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)+3 c d \tan (e+f x) \sqrt[3]{c+d \tan (e+f x)}}{28 d^2 f}","-\frac{9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac{\sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)}{2 f}+\frac{\sqrt{3} \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)}{2 f}+\frac{3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac{3 \sqrt[3]{c+d \tan (e+f x)}}{f}-\frac{3 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{4 f}-\frac{3 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)}{4 f}-\frac{\sqrt[3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt[3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt[3]{c-i d}+\frac{1}{4} i x \sqrt[3]{c+i d}",1,"(14*Sqrt[3]*(c - I*d)^(1/3)*d^2*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + 14*Sqrt[3]*(c + I*d)^(1/3)*d^2*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 14*(c - I*d)^(1/3)*d^2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 14*(c + I*d)^(1/3)*d^2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + 7*(c - I*d)^(1/3)*d^2*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] + 7*(c + I*d)^(1/3)*d^2*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] - 9*c^2*(c + d*Tan[e + f*x])^(1/3) - 84*d^2*(c + d*Tan[e + f*x])^(1/3) + 3*c*d*Tan[e + f*x]*(c + d*Tan[e + f*x])^(1/3) + 12*d^2*Tan[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3))/(28*d^2*f)","A",1
683,1,313,439,0.8326501,"\int \tan ^2(e+f x) \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[Tan[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3),x]","\frac{\frac{3 (c+d \tan (e+f x))^{4/3}}{d}+i \sqrt[3]{c-i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)+\log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)\right)-i \sqrt[3]{c+i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+\log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)\right)}{4 f}","-\frac{\sqrt{3} d \sqrt[3]{c-\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-d^2} f}+\frac{\sqrt{3} d \sqrt[3]{c+\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-d^2} f}+\frac{3 d \sqrt[3]{c-\sqrt{-d^2}} \log \left(\sqrt[3]{c-\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 \sqrt{-d^2} f}-\frac{3 d \sqrt[3]{c+\sqrt{-d^2}} \log \left(\sqrt[3]{c+\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 \sqrt{-d^2} f}+\frac{d \sqrt[3]{c-\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}-\frac{d \sqrt[3]{c+\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}+\frac{1}{4} x \sqrt[3]{c-\sqrt{-d^2}}+\frac{1}{4} x \sqrt[3]{c+\sqrt{-d^2}}+\frac{3 (c+d \tan (e+f x))^{4/3}}{4 d f}",1,"(I*(c - I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] - 2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) - I*(c + I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) + (3*(c + d*Tan[e + f*x])^(4/3))/d)/(4*f)","C",1
684,1,346,318,0.2433488,"\int \tan (e+f x) \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[Tan[e + f*x]*(c + d*Tan[e + f*x])^(1/3),x]","-\frac{2 \sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)+2 \sqrt{3} \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)-12 \sqrt[3]{c+d \tan (e+f x)}-2 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)-2 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+\sqrt[3]{c-i d} \log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)+\sqrt[3]{c+i d} \log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)}{4 f}","-\frac{\sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)}{2 f}-\frac{\sqrt{3} \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)}{2 f}+\frac{3 \sqrt[3]{c+d \tan (e+f x)}}{f}+\frac{3 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{4 f}+\frac{3 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)}{4 f}+\frac{\sqrt[3]{c-i d} \log (\cos (e+f x))}{4 f}+\frac{\sqrt[3]{c+i d} \log (\cos (e+f x))}{4 f}+\frac{1}{4} i x \sqrt[3]{c-i d}-\frac{1}{4} i x \sqrt[3]{c+i d}",1,"-1/4*(2*Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 2*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + (c - I*d)^(1/3)*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] + (c + I*d)^(1/3)*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] - 12*(c + d*Tan[e + f*x])^(1/3))/f","A",1
685,1,294,415,0.2402862,"\int \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^(1/3),x]","\frac{i \sqrt[3]{c+i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+\log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)\right)-i \sqrt[3]{c-i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)+\log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)\right)}{4 f}","\frac{\sqrt{3} d \sqrt[3]{c-\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-d^2} f}-\frac{\sqrt{3} d \sqrt[3]{c+\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-d^2} f}-\frac{3 d \sqrt[3]{c-\sqrt{-d^2}} \log \left(\sqrt[3]{c-\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 \sqrt{-d^2} f}+\frac{3 d \sqrt[3]{c+\sqrt{-d^2}} \log \left(\sqrt[3]{c+\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 \sqrt{-d^2} f}-\frac{d \sqrt[3]{c-\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}+\frac{d \sqrt[3]{c+\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}-\frac{1}{4} x \sqrt[3]{c-\sqrt{-d^2}}-\frac{1}{4} x \sqrt[3]{c+\sqrt{-d^2}}",1,"((-I)*(c - I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] - 2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) + I*(c + I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]))/(4*f)","C",1
686,1,744,402,0.9007969,"\int \cot (e+f x) \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[Cot[e + f*x]*(c + d*Tan[e + f*x])^(1/3),x]","\frac{-2 \sqrt[3]{c} \log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right)-4 \sqrt{3} \sqrt[3]{c} \tan ^{-1}\left(\frac{2 \sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c}}{\sqrt{3} \sqrt[3]{c}}\right)+2 \sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)+\frac{2 i \sqrt{3} d \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)}{(c+i d)^{2/3}}+\frac{2 \sqrt{3} c \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)}{(c+i d)^{2/3}}+4 \sqrt[3]{c} \log \left(\sqrt[3]{c}-\sqrt[3]{c+d \tan (e+f x)}\right)+\frac{2 i d \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{(c-i d)^{2/3}}-\frac{2 c \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{(c-i d)^{2/3}}-\frac{2 i d \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)}{(c+i d)^{2/3}}-\frac{2 c \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)}{(c+i d)^{2/3}}-\frac{i d \log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)}{(c-i d)^{2/3}}+\frac{c \log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)}{(c-i d)^{2/3}}+\frac{i d \log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)}{(c+i d)^{2/3}}+\frac{c \log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)}{(c+i d)^{2/3}}}{4 f}","-\frac{\sqrt{3} \sqrt[3]{c} \tan ^{-1}\left(\frac{2 \sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c}}{\sqrt{3} \sqrt[3]{c}}\right)}{f}+\frac{\sqrt{3} \sqrt[3]{c-i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)}{2 f}+\frac{\sqrt{3} \sqrt[3]{c+i d} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)}{2 f}+\frac{3 \sqrt[3]{c} \log \left(\sqrt[3]{c}-\sqrt[3]{c+d \tan (e+f x)}\right)}{2 f}-\frac{3 \sqrt[3]{c-i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)}{4 f}-\frac{3 \sqrt[3]{c+i d} \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)}{4 f}-\frac{\sqrt[3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac{\sqrt[3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac{1}{4} i x \sqrt[3]{c-i d}+\frac{1}{4} i x \sqrt[3]{c+i d}-\frac{\sqrt[3]{c} \log (\tan (e+f x))}{2 f}",1,"(-4*Sqrt[3]*c^(1/3)*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3))/(Sqrt[3]*c^(1/3))] + 2*Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + (2*Sqrt[3]*c*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(c + I*d)^(2/3) + ((2*I)*Sqrt[3]*d*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(c + I*d)^(2/3) + 4*c^(1/3)*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - (2*c*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c - I*d)^(2/3) + ((2*I)*d*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c - I*d)^(2/3) - (2*c*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c + I*d)^(2/3) - ((2*I)*d*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c + I*d)^(2/3) - 2*c^(1/3)*Log[c^(2/3) + c^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] + (c*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c - I*d)^(2/3) - (I*d*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c - I*d)^(2/3) + (c*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c + I*d)^(2/3) + (I*d*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c + I*d)^(2/3))/(4*f)","A",1
687,1,464,546,3.3727328,"\int \cot ^2(e+f x) \sqrt[3]{c+d \tan (e+f x)} \, dx","Integrate[Cot[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3),x]","\frac{-\frac{1}{6} \sqrt[3]{c} d \left(\log \left(c^{2/3}+\sqrt[3]{c} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c}}{\sqrt{3} \sqrt[3]{c}}\right)\right)+d \sqrt[3]{c+d \tan (e+f x)}+\frac{1}{4} i c \sqrt[3]{c-i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c-i d}\right)+\log \left(\sqrt[3]{c-i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c-i d)^{2/3}\right)\right)-\frac{1}{4} i c \sqrt[3]{c+i d} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+i d}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c+i d}\right)+\log \left(\sqrt[3]{c+i d} \sqrt[3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}+(c+i d)^{2/3}\right)\right)+\frac{1}{3} \sqrt[3]{c} d \log \left(\sqrt[3]{c}-\sqrt[3]{c+d \tan (e+f x)}\right)-\cot (e+f x) (c+d \tan (e+f x))^{4/3}}{c f}","-\frac{d \tan ^{-1}\left(\frac{2 \sqrt[3]{c+d \tan (e+f x)}+\sqrt[3]{c}}{\sqrt{3} \sqrt[3]{c}}\right)}{\sqrt{3} c^{2/3} f}-\frac{d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac{d \log \left(\sqrt[3]{c}-\sqrt[3]{c+d \tan (e+f x)}\right)}{2 c^{2/3} f}-\frac{\sqrt{3} d \sqrt[3]{c-\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c-\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-d^2} f}+\frac{\sqrt{3} d \sqrt[3]{c+\sqrt{-d^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{c+d \tan (e+f x)}}{\sqrt[3]{c+\sqrt{-d^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-d^2} f}+\frac{3 d \sqrt[3]{c-\sqrt{-d^2}} \log \left(\sqrt[3]{c-\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 \sqrt{-d^2} f}-\frac{3 d \sqrt[3]{c+\sqrt{-d^2}} \log \left(\sqrt[3]{c+\sqrt{-d^2}}-\sqrt[3]{c+d \tan (e+f x)}\right)}{4 \sqrt{-d^2} f}+\frac{d \sqrt[3]{c-\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}-\frac{d \sqrt[3]{c+\sqrt{-d^2}} \log (\cos (e+f x))}{4 \sqrt{-d^2} f}+\frac{1}{4} x \sqrt[3]{c-\sqrt{-d^2}}+\frac{1}{4} x \sqrt[3]{c+\sqrt{-d^2}}-\frac{\cot (e+f x) \sqrt[3]{c+d \tan (e+f x)}}{f}",1,"((c^(1/3)*d*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/3 - (c^(1/3)*d*(2*Sqrt[3]*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3))/(Sqrt[3]*c^(1/3))] + Log[c^(2/3) + c^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]))/6 + (I/4)*c*(c - I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] - 2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) - (I/4)*c*(c + I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)]) + d*(c + d*Tan[e + f*x])^(1/3) - Cot[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(c*f)","C",1
688,1,300,329,0.9781088,"\int (a+b \tan (c+d x))^{5/3} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/3),x]","\frac{(b+i a) \left(2 \sqrt{3} (a-i b)^{2/3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)-(a-i b)^{2/3} \log (\tan (c+d x)+i)+3 \left((a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)\right)\right)+(b-i a) \left(2 \sqrt{3} (a+i b)^{2/3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)-(a+i b)^{2/3} \log (-\tan (c+d x)+i)+3 \left((a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)\right)\right)}{4 d}","\frac{i \sqrt{3} (a-i b)^{5/3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d}-\frac{i \sqrt{3} (a+i b)^{5/3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d}+\frac{3 b (a+b \tan (c+d x))^{2/3}}{2 d}+\frac{3 i (a-i b)^{5/3} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d}-\frac{3 i (a+i b)^{5/3} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d}+\frac{i (a-i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac{i (a+i b)^{5/3} \log (\cos (c+d x))}{4 d}-\frac{1}{4} x (a-i b)^{5/3}-\frac{1}{4} x (a+i b)^{5/3}",1,"((I*a + b)*(2*Sqrt[3]*(a - I*b)^(2/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - (a - I*b)^(2/3)*Log[I + Tan[c + d*x]] + 3*((a - I*b)^(2/3)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + (a + b*Tan[c + d*x])^(2/3))) + ((-I)*a + b)*(2*Sqrt[3]*(a + I*b)^(2/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - (a + I*b)^(2/3)*Log[I - Tan[c + d*x]] + 3*((a + I*b)^(2/3)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + (a + b*Tan[c + d*x])^(2/3))))/(4*d)","A",1
689,1,365,327,0.7953974,"\int (a+b \tan (c+d x))^{4/3} \, dx","Integrate[(a + b*Tan[c + d*x])^(4/3),x]","\frac{(b+i a) \left(6 \sqrt[3]{a+b \tan (c+d x)}-\sqrt[3]{a-i b} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)+\log \left(\sqrt[3]{a-i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3}\right)\right)+2 \sqrt[3]{a-i b} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)\right)-(-b+i a) \left(6 \sqrt[3]{a+b \tan (c+d x)}-\sqrt[3]{a+i b} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)+\log \left(\sqrt[3]{a+i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3}\right)\right)+2 \sqrt[3]{a+i b} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)\right)}{4 d}","-\frac{i \sqrt{3} (a-i b)^{4/3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d}+\frac{i \sqrt{3} (a+i b)^{4/3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d}+\frac{3 b \sqrt[3]{a+b \tan (c+d x)}}{d}+\frac{3 i (a-i b)^{4/3} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d}-\frac{3 i (a+i b)^{4/3} \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d}+\frac{i (a-i b)^{4/3} \log (\cos (c+d x))}{4 d}-\frac{i (a+i b)^{4/3} \log (\cos (c+d x))}{4 d}-\frac{1}{4} x (a-i b)^{4/3}-\frac{1}{4} x (a+i b)^{4/3}",1,"((I*a + b)*(2*(a - I*b)^(1/3)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] - (a - I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] + Log[(a - I*b)^(2/3) + (a - I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]) + 6*(a + b*Tan[c + d*x])^(1/3)) - (I*a - b)*(2*(a + I*b)^(1/3)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] - (a + I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] + Log[(a + I*b)^(2/3) + (a + I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]) + 6*(a + b*Tan[c + d*x])^(1/3)))/(4*d)","A",1
690,1,224,415,0.3445616,"\int (a+b \tan (c+d x))^{2/3} \, dx","Integrate[(a + b*Tan[c + d*x])^(2/3),x]","\frac{\frac{(b+i a) \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)+3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)-\log (\tan (c+d x)+i)\right)}{\sqrt[3]{a-i b}}+\frac{(b-i a) \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)+3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)-\log (-\tan (c+d x)+i)\right)}{\sqrt[3]{a+i b}}}{4 d}","-\frac{\sqrt{3} b \left(a-\sqrt{-b^2}\right)^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d}+\frac{\sqrt{3} b \left(a+\sqrt{-b^2}\right)^{2/3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d}-\frac{3 b \left(a-\sqrt{-b^2}\right)^{2/3} \log \left(\sqrt[3]{a-\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d}+\frac{3 b \left(a+\sqrt{-b^2}\right)^{2/3} \log \left(\sqrt[3]{a+\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d}-\frac{b \left(a-\sqrt{-b^2}\right)^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}+\frac{b \left(a+\sqrt{-b^2}\right)^{2/3} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}-\frac{1}{4} x \left(a-\sqrt{-b^2}\right)^{2/3}-\frac{1}{4} x \left(a+\sqrt{-b^2}\right)^{2/3}",1,"(((I*a + b)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - Log[I + Tan[c + d*x]] + 3*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]))/(a - I*b)^(1/3) + (((-I)*a + b)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - Log[I - Tan[c + d*x]] + 3*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]))/(a + I*b)^(1/3))/(4*d)","C",1
691,1,294,415,0.3940557,"\int \sqrt[3]{a+b \tan (c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(1/3),x]","\frac{i \sqrt[3]{a+i b} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)+\log \left(\sqrt[3]{a+i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3}\right)\right)-i \sqrt[3]{a-i b} \left(2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)-2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)+\log \left(\sqrt[3]{a-i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3}\right)\right)}{4 d}","\frac{\sqrt{3} b \sqrt[3]{a-\sqrt{-b^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d}-\frac{\sqrt{3} b \sqrt[3]{a+\sqrt{-b^2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d}-\frac{3 b \sqrt[3]{a-\sqrt{-b^2}} \log \left(\sqrt[3]{a-\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d}+\frac{3 b \sqrt[3]{a+\sqrt{-b^2}} \log \left(\sqrt[3]{a+\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d}-\frac{b \sqrt[3]{a-\sqrt{-b^2}} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}+\frac{b \sqrt[3]{a+\sqrt{-b^2}} \log (\cos (c+d x))}{4 \sqrt{-b^2} d}-\frac{1}{4} x \sqrt[3]{a-\sqrt{-b^2}}-\frac{1}{4} x \sqrt[3]{a+\sqrt{-b^2}}",1,"((-I)*(a - I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] - 2*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a - I*b)^(2/3) + (a - I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]) + I*(a + I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - 2*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a + I*b)^(2/3) + (a + I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]))/(4*d)","C",1
692,1,251,415,0.3311266,"\int \frac{1}{\sqrt[3]{a+b \tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-1/3),x]","\frac{i \left(\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{\sqrt[3]{a-i b}}-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{\sqrt[3]{a+i b}}+\frac{\log (-\tan (c+d x)+i)}{\sqrt[3]{a+i b}}-\frac{\log (\tan (c+d x)+i)}{\sqrt[3]{a-i b}}+\frac{3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{\sqrt[3]{a-i b}}-\frac{3 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{\sqrt[3]{a+i b}}\right)}{4 d}","-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d \sqrt[3]{a-\sqrt{-b^2}}}+\frac{\sqrt{3} b \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d \sqrt[3]{a+\sqrt{-b^2}}}-\frac{3 b \log \left(\sqrt[3]{a-\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d \sqrt[3]{a-\sqrt{-b^2}}}+\frac{3 b \log \left(\sqrt[3]{a+\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d \sqrt[3]{a+\sqrt{-b^2}}}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \sqrt[3]{a-\sqrt{-b^2}}}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \sqrt[3]{a+\sqrt{-b^2}}}-\frac{x}{4 \sqrt[3]{a-\sqrt{-b^2}}}-\frac{x}{4 \sqrt[3]{a+\sqrt{-b^2}}}",1,"((I/4)*((2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(a - I*b)^(1/3) - (2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(a + I*b)^(1/3) + Log[I - Tan[c + d*x]]/(a + I*b)^(1/3) - Log[I + Tan[c + d*x]]/(a - I*b)^(1/3) + (3*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(a - I*b)^(1/3) - (3*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(a + I*b)^(1/3)))/d","C",1
693,1,313,415,0.5740719,"\int \frac{1}{(a+b \tan (c+d x))^{2/3}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-2/3),x]","\frac{i \left(-\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)+\log \left(\sqrt[3]{a-i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a-i b)^{2/3}\right)}{(a-i b)^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)+\log \left(\sqrt[3]{a+i b} \sqrt[3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}+(a+i b)^{2/3}\right)}{(a+i b)^{2/3}}+\frac{2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{(a-i b)^{2/3}}-\frac{2 \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{(a+i b)^{2/3}}\right)}{4 d}","\frac{\sqrt{3} b \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d \left(a-\sqrt{-b^2}\right)^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left(\frac{\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+\sqrt{-b^2}}}+1}{\sqrt{3}}\right)}{2 \sqrt{-b^2} d \left(a+\sqrt{-b^2}\right)^{2/3}}-\frac{3 b \log \left(\sqrt[3]{a-\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d \left(a-\sqrt{-b^2}\right)^{2/3}}+\frac{3 b \log \left(\sqrt[3]{a+\sqrt{-b^2}}-\sqrt[3]{a+b \tan (c+d x)}\right)}{4 \sqrt{-b^2} d \left(a+\sqrt{-b^2}\right)^{2/3}}-\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \left(a-\sqrt{-b^2}\right)^{2/3}}+\frac{b \log (\cos (c+d x))}{4 \sqrt{-b^2} d \left(a+\sqrt{-b^2}\right)^{2/3}}-\frac{x}{4 \left(a-\sqrt{-b^2}\right)^{2/3}}-\frac{x}{4 \left(a+\sqrt{-b^2}\right)^{2/3}}",1,"((I/4)*((2*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(a - I*b)^(2/3) - (2*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(a + I*b)^(2/3) - (2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]] + Log[(a - I*b)^(2/3) + (a - I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)])/(a - I*b)^(2/3) + (2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] + Log[(a + I*b)^(2/3) + (a + I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)])/(a + I*b)^(2/3)))/d","C",1
694,1,106,336,0.2208079,"\int \frac{1}{(a+b \tan (c+d x))^{4/3}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-4/3),x]","\frac{3 i \left((a+i b) \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \, _2F_1\left(-\frac{1}{3},1;\frac{2}{3};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{2 d \left(a^2+b^2\right) \sqrt[3]{a+b \tan (c+d x)}}","-\frac{3 b}{d \left(a^2+b^2\right) \sqrt[3]{a+b \tan (c+d x)}}+\frac{i \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d (a-i b)^{4/3}}-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d (a+i b)^{4/3}}+\frac{3 i \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d (a-i b)^{4/3}}-\frac{3 i \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d (a+i b)^{4/3}}+\frac{i \log (\cos (c+d x))}{4 d (a-i b)^{4/3}}-\frac{i \log (\cos (c+d x))}{4 d (a+i b)^{4/3}}-\frac{x}{4 (a-i b)^{4/3}}-\frac{x}{4 (a+i b)^{4/3}}",1,"(((3*I)/2)*((a + I*b)*Hypergeometric2F1[-1/3, 1, 2/3, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometric2F1[-1/3, 1, 2/3, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^(1/3))","C",1
695,1,106,338,0.1564403,"\int \frac{1}{(a+b \tan (c+d x))^{5/3}} \, dx","Integrate[(a + b*Tan[c + d*x])^(-5/3),x]","\frac{3 i \left((a+i b) \, _2F_1\left(-\frac{2}{3},1;\frac{1}{3};\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \, _2F_1\left(-\frac{2}{3},1;\frac{1}{3};\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{4 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{2/3}}","-\frac{3 b}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^{2/3}}-\frac{i \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a-i b}}}{\sqrt{3}}\right)}{2 d (a-i b)^{5/3}}+\frac{i \sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[3]{a+b \tan (c+d x)}}{\sqrt[3]{a+i b}}}{\sqrt{3}}\right)}{2 d (a+i b)^{5/3}}+\frac{3 i \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a-i b}\right)}{4 d (a-i b)^{5/3}}-\frac{3 i \log \left(-\sqrt[3]{a+b \tan (c+d x)}+\sqrt[3]{a+i b}\right)}{4 d (a+i b)^{5/3}}+\frac{i \log (\cos (c+d x))}{4 d (a-i b)^{5/3}}-\frac{i \log (\cos (c+d x))}{4 d (a+i b)^{5/3}}-\frac{x}{4 (a-i b)^{5/3}}-\frac{x}{4 (a+i b)^{5/3}}",1,"(((3*I)/4)*((a + I*b)*Hypergeometric2F1[-2/3, 1, 1/3, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometric2F1[-2/3, 1, 1/3, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^(2/3))","C",1
696,1,191,261,1.6269817,"\int (d \tan (e+f x))^n (a+b \tan (e+f x))^4 \, dx","Integrate[(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^4,x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left(\frac{a^2 b^2 (5 n+17)-b^4 (n+3)}{n+1}+\frac{(n+3) \left(a^4-6 a^2 b^2+b^4\right) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{n+1}+\frac{2 a b^3 (n+4) \tan (e+f x)}{n+2}+b^2 (a+b \tan (e+f x))^2+\frac{4 a b (n+3) (a-b) (a+b) \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{n+2}\right)}{f (n+3)}","\frac{4 a b \left(a^2-b^2\right) (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{d^2 f (n+2)}-\frac{b^2 \left(b^2 (n+3)-a^2 (5 n+17)\right) (d \tan (e+f x))^{n+1}}{d f (n+1) (n+3)}+\frac{\left(a^4-6 a^2 b^2+b^4\right) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{2 a b^3 (n+4) \tan (e+f x) (d \tan (e+f x))^{n+1}}{d f (n+2) (n+3)}+\frac{b^2 (a+b \tan (e+f x))^2 (d \tan (e+f x))^{n+1}}{d f (n+3)}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*((-(b^4*(3 + n)) + a^2*b^2*(17 + 5*n))/(1 + n) + ((a^4 - 6*a^2*b^2 + b^4)*(3 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2])/(1 + n) + (2*a*b^3*(4 + n)*Tan[e + f*x])/(2 + n) + (4*a*(a - b)*b*(a + b)*(3 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(2 + n) + b^2*(a + b*Tan[e + f*x])^2))/(f*(3 + n))","A",1
697,1,141,198,0.8084042,"\int (d \tan (e+f x))^n (a+b \tan (e+f x))^3 \, dx","Integrate[(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^3,x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left(a (n+2) \left(a^2-3 b^2\right) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)+b \left((n+1) \left(3 a^2-b^2\right) \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)+b (3 a (n+2)+b (n+1) \tan (e+f x))\right)\right)}{f (n+1) (n+2)}","\frac{b \left(3 a^2-b^2\right) (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{d^2 f (n+2)}+\frac{a \left(a^2-3 b^2\right) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{a b^2 (2 n+5) (d \tan (e+f x))^{n+1}}{d f (n+1) (n+2)}+\frac{b^2 (a+b \tan (e+f x)) (d \tan (e+f x))^{n+1}}{d f (n+2)}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*(a*(a^2 - 3*b^2)*(2 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2] + b*((3*a^2 - b^2)*(1 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x] + b*(3*a*(2 + n) + b*(1 + n)*Tan[e + f*x]))))/(f*(1 + n)*(2 + n))","A",1
698,1,116,140,0.3913799,"\int (d \tan (e+f x))^n (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^2,x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left((n+2) \left(a^2-b^2\right) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)+b \left(2 a (n+1) \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)+b (n+2)\right)\right)}{f (n+1) (n+2)}","\frac{\left(a^2-b^2\right) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{2 a b (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{d^2 f (n+2)}+\frac{b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*((a^2 - b^2)*(2 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2] + b*(b*(2 + n) + 2*a*(1 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(f*(1 + n)*(2 + n))","A",1
699,1,99,103,0.2503404,"\int (d \tan (e+f x))^n (a+b \tan (e+f x)) \, dx","Integrate[(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x]),x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left(a (n+2) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)+b (n+1) \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)\right)}{f (n+1) (n+2)}","\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1)}+\frac{b (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{d^2 f (n+2)}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*(a*(2 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2] + b*(1 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]))/(f*(1 + n)*(2 + n))","A",1
700,1,142,181,0.4909648,"\int \frac{(d \tan (e+f x))^n}{a+b \tan (e+f x)} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + b*Tan[e + f*x]),x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left(a^2 (n+2) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)+b \left(b (n+2) \, _2F_1\left(1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right)-a (n+1) \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)\right)\right)}{a f (n+1) (n+2) \left(a^2+b^2\right)}","-\frac{b (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{d^2 f (n+2) \left(a^2+b^2\right)}+\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1) \left(a^2+b^2\right)}+\frac{b^2 (d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right)}{a d f (n+1) \left(a^2+b^2\right)}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*(a^2*(2 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2] + b*(b*(2 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Tan[e + f*x])/a)] - a*(1 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a*(a^2 + b^2)*f*(1 + n)*(2 + n))","A",1
701,1,198,252,2.5450421,"\int \frac{(d \tan (e+f x))^n}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Tan[e + f*x])^n/(a + b*Tan[e + f*x])^2,x]","\frac{\tan (e+f x) (d \tan (e+f x))^n \left(\frac{a \left(\frac{\left(a^2-b^2\right) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{n+1}-\frac{2 a b \tan (e+f x) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{n+2}\right)}{a^2+b^2}-\frac{b^2 \left(a^2 (n-2)+b^2 n\right) \, _2F_1\left(1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right)}{a (n+1) \left(a^2+b^2\right)}+\frac{b^2}{a+b \tan (e+f x)}\right)}{a f \left(a^2+b^2\right)}","-\frac{2 a b (d \tan (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right)}{d^2 f (n+2) \left(a^2+b^2\right)^2}+\frac{\left(a^2-b^2\right) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right)}{d f (n+1) \left(a^2+b^2\right)^2}+\frac{b^2 \left(a^2 (2-n)-b^2 n\right) (d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right)}{a^2 d f (n+1) \left(a^2+b^2\right)^2}+\frac{b^2 (d \tan (e+f x))^{n+1}}{a d f \left(a^2+b^2\right) (a+b \tan (e+f x))}",1,"(Tan[e + f*x]*(d*Tan[e + f*x])^n*(-((b^2*(a^2*(-2 + n) + b^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Tan[e + f*x])/a)])/(a*(a^2 + b^2)*(1 + n))) + b^2/(a + b*Tan[e + f*x]) + (a*(((a^2 - b^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2])/(1 + n) - (2*a*b*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(2 + n)))/(a^2 + b^2)))/(a*(a^2 + b^2)*f)","A",1
702,0,0,175,9.0617456,"\int \tan ^m(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(3/2),x]","\int \tan ^m(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","\frac{a \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}+\frac{a \tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}",1,"Integrate[Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(3/2), x]","F",-1
703,0,0,173,0.6881738,"\int \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]],x]","\int \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","\frac{\tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}+\frac{\tan ^{m+1}(c+d x) \sqrt{a+b \tan (c+d x)} F_1\left(m+1;-\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{\frac{b \tan (c+d x)}{a}+1}}",1,"Integrate[Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]], x]","F",-1
704,0,0,173,4.4255608,"\int \frac{\tan ^m(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Tan[c + d*x]^m/Sqrt[a + b*Tan[c + d*x]],x]","\int \frac{\tan ^m(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","\frac{\tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{\tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{1}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 d (m+1) \sqrt{a+b \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^m/Sqrt[a + b*Tan[c + d*x]], x]","F",-1
705,0,0,179,19.1241913,"\int \frac{\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(3/2),x]","\int \frac{\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","\frac{\tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},-i \tan (c+d x)\right)}{2 a d (m+1) \sqrt{a+b \tan (c+d x)}}+\frac{\tan ^{m+1}(c+d x) \sqrt{\frac{b \tan (c+d x)}{a}+1} F_1\left(m+1;\frac{3}{2},1;m+2;-\frac{b \tan (c+d x)}{a},i \tan (c+d x)\right)}{2 a d (m+1) \sqrt{a+b \tan (c+d x)}}",1,"Integrate[Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(3/2), x]","F",-1
706,0,0,179,1.5115894,"\int (d \tan (e+f x))^n (a+b \tan (e+f x))^m \, dx","Integrate[(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^m,x]","\int (d \tan (e+f x))^n (a+b \tan (e+f x))^m \, dx","\frac{(d \tan (e+f x))^{n+1} (a+b \tan (e+f x))^m \left(\frac{b \tan (e+f x)}{a}+1\right)^{-m} F_1\left(n+1;-m,1;n+2;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right)}{2 d f (n+1)}+\frac{(d \tan (e+f x))^{n+1} (a+b \tan (e+f x))^m \left(\frac{b \tan (e+f x)}{a}+1\right)^{-m} F_1\left(n+1;-m,1;n+2;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right)}{2 d f (n+1)}",1,"Integrate[(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^m, x]","F",-1
707,1,249,297,2.208308,"\int \tan ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Tan[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","-\frac{(a+b \tan (c+d x))^{n+1} \left(2 (-b+i a) (b+i a) \left(2 a^2-b^2 \left(n^2+5 n+6\right)\right)+i b^3 (n+2) (n+3) (a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)-b^3 (n+2) (n+3) (b+i a) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)-2 b^2 (n+1) (n+2) (a-i b) (a+i b) \tan ^2(c+d x)+4 a b (n+1) (a-i b) (a+i b) \tan (c+d x)\right)}{2 b^3 d (n+1) (n+2) (n+3) (a-i b) (a+i b)}","\frac{\left(2 a^2-b^2 (n+2) (n+3)\right) (a+b \tan (c+d x))^{n+1}}{b^3 d (n+1) (n+2) (n+3)}-\frac{\sqrt{-b^2} (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{2 b d (n+1) \left(a-\sqrt{-b^2}\right)}+\frac{\sqrt{-b^2} (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{2 b d (n+1) \left(a+\sqrt{-b^2}\right)}-\frac{2 a \tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+2) (n+3)}+\frac{\tan ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+3)}",1,"-1/2*((a + b*Tan[c + d*x])^(1 + n)*(2*(I*a - b)*(I*a + b)*(2*a^2 - b^2*(6 + 5*n + n^2)) + I*(a + I*b)*b^3*(2 + n)*(3 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] - b^3*(I*a + b)*(2 + n)*(3 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] + 4*a*(a - I*b)*(a + I*b)*b*(1 + n)*Tan[c + d*x] - 2*(a - I*b)*(a + I*b)*b^2*(1 + n)*(2 + n)*Tan[c + d*x]^2))/((a - I*b)*(a + I*b)*b^3*d*(1 + n)*(2 + n)*(3 + n))","C",1
708,1,135,192,1.0454726,"\int \tan ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(\frac{2 \left(b \tan (c+d x)-\frac{a}{n+1}\right)}{b^2 (n+2)}+\frac{\, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{(n+1) (a-i b)}+\frac{\, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{(n+1) (a+i b)}\right)}{2 d}","-\frac{a (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (a-i b)}+\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}+\frac{\tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)}",1,"((a + b*Tan[c + d*x])^(1 + n)*(Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]/((a - I*b)*(1 + n)) + Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]/((a + I*b)*(1 + n)) + (2*(-(a/(1 + n)) + b*Tan[c + d*x]))/(b^2*(2 + n))))/(2*d)","A",1
709,1,138,193,0.2000859,"\int \tan ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(i b (a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \left(-i b \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)+2 a+2 i b\right)\right)}{2 b d (n+1) (b-i a) (b+i a)}","-\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a-\sqrt{-b^2}\right)}+\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a+\sqrt{-b^2}\right)}+\frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)}",1,"((I*(a + I*b)*b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*(2*a + (2*I)*b - I*b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]))*(a + b*Tan[c + d*x])^(1 + n))/(2*b*((-I)*a + b)*(I*a + b)*d*(1 + n))","C",1
710,1,117,127,0.1818523,"\int \tan (c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^n,x]","-\frac{(a+b \tan (c+d x))^{n+1} \left((a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{2 d (n+1) (a-i b) (a+i b)}","-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (a-i b)}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}",1,"-1/2*(((a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x])^(1 + n))/((a - I*b)*(a + I*b)*d*(1 + n))","A",1
711,1,118,167,0.1408449,"\int (a+b \tan (c+d x))^n \, dx","Integrate[(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left((a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)-(a-i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)\right)}{2 d (n+1) (a+i b) (b+i a)}","\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a-\sqrt{-b^2}\right)}-\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a+\sqrt{-b^2}\right)}",1,"(((a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)])*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*(I*a + b)*d*(1 + n))","C",1
712,1,154,175,0.2487514,"\int \cot (c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(a (a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \left(a \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)-2 (a+i b) \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)\right)\right)}{2 a d (n+1) (a-i b) (a+i b)}","\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (a-i b)}+\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a d (n+1)}",1,"((a*(a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*(a*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] - 2*(a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]))*(a + b*Tan[c + d*x])^(1 + n))/(2*a*(a - I*b)*(a + I*b)*d*(1 + n))","A",1
713,1,190,245,0.7939139,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","-\frac{\tan (c+d x) (a \cot (c+d x)+b) (a+b \tan (c+d x))^n \left(a^2 (b-i a) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \left(i a^2 \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)+2 (a+i b) \left(b n \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)+a (n+1) \cot (c+d x)\right)\right)\right)}{2 a^2 d (n+1) (a-i b) (a+i b)}","-\frac{b n (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a^2 d (n+1)}-\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a-\sqrt{-b^2}\right)}+\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} d (n+1) \left(a+\sqrt{-b^2}\right)}-\frac{\cot (c+d x) (a+b \tan (c+d x))^{n+1}}{a d}",1,"-1/2*((b + a*Cot[c + d*x])*(a^2*((-I)*a + b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*(I*a^2*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] + 2*(a + I*b)*(a*(1 + n)*Cot[c + d*x] + b*n*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a])))*Tan[c + d*x]*(a + b*Tan[c + d*x])^n)/(a^2*(a - I*b)*(a + I*b)*d*(1 + n))","C",1
714,1,212,261,1.9823346,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","-\frac{\tan (c+d x) (a \cot (c+d x)+b) (a+b \tan (c+d x))^n \left(a^3 (a+i b) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)+(a-i b) \left(a^3 \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)+(a+i b) \left(\left(b^2 (n-1) n-2 a^2\right) \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)+a (n+1) \cot (c+d x) (a \cot (c+d x)+b (n-1))\right)\right)\right)}{2 a^3 d (n+1) (a-i b) (a+i b)}","\frac{b (1-n) \cot (c+d x) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}+\frac{\left(2 a^2+b^2 (1-n) n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{2 a^3 d (n+1)}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right)}{2 d (n+1) (a-i b)}-\frac{(a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right)}{2 d (n+1) (a+i b)}-\frac{\cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d}",1,"-1/2*((b + a*Cot[c + d*x])*(a^3*(a + I*b)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*(a^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)] + (a + I*b)*(a*(1 + n)*Cot[c + d*x]*(b*(-1 + n) + a*Cot[c + d*x]) + (-2*a^2 + b^2*(-1 + n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a])))*Tan[c + d*x]*(a + b*Tan[c + d*x])^n)/(a^3*(a - I*b)*(a + I*b)*d*(1 + n))","A",1
715,0,0,159,0.9496229,"\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n,x]","\int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx","\frac{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d}+\frac{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d}",1,"Integrate[Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n, x]","F",-1
716,0,0,159,1.0454025,"\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \, dx","Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n,x]","\int \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \, dx","\frac{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d}",1,"Integrate[Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n, x]","F",-1
717,0,0,153,1.02572,"\int \frac{(a+b \tan (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^n/Sqrt[Tan[c + d*x]],x]","\int \frac{(a+b \tan (c+d x))^n}{\sqrt{\tan (c+d x)}} \, dx","\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}+\frac{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}",1,"Integrate[(a + b*Tan[c + d*x])^n/Sqrt[Tan[c + d*x]], x]","F",-1
718,0,0,155,0.8014255,"\int \frac{(a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^n/Tan[c + d*x]^(3/2),x]","\int \frac{(a+b \tan (c+d x))^n}{\tan ^{\frac{3}{2}}(c+d x)} \, dx","-\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\tan (c+d x)}}-\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\tan (c+d x)}}",1,"Integrate[(a + b*Tan[c + d*x])^n/Tan[c + d*x]^(3/2), x]","F",-1
719,1,116,65,1.2049832,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x]),x]","-\frac{2 a e^{-i c} \sin (c+d x) \sqrt{\cot (c+d x)} (\cot (c+d x)+i) (\cos (d x)-i \sin (d x)) \left(\cot (c+d x)-3 i \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+3 i\right)}{3 d}","-\frac{2 a \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 i a \sqrt{\cot (c+d x)}}{d}-\frac{2 (-1)^{3/4} a \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(-2*a*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])*(Cos[d*x] - I*Sin[d*x])*Sin[c + d*x]*(3*I + Cot[c + d*x] - (3*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*E^(I*c))","A",1
720,1,67,45,0.7170654,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x]),x]","\frac{2 a \sqrt{\cot (c+d x)} \left(-1+\sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{d}","-\frac{2 a \sqrt{\cot (c+d x)}}{d}-\frac{2 \sqrt[4]{-1} a \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(2*a*Sqrt[Cot[c + d*x]]*(-1 + ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/d","C",1
721,1,111,28,0.6406178,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x]),x]","-\frac{2 i a \sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{d}","\frac{2 (-1)^{3/4} a \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"((-2*I)*a*Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/d","C",1
722,1,71,47,0.863453,"\int \frac{a+i a \tan (c+d x)}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])/Sqrt[Cot[c + d*x]],x]","\frac{a \left(2 i-\frac{2 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{i \tan (c+d x)}}\right)}{d \sqrt{\cot (c+d x)}}","\frac{2 \sqrt[4]{-1} a \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}+\frac{2 i a}{d \sqrt{\cot (c+d x)}}",1,"(a*(2*I - ((2*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/Sqrt[I*Tan[c + d*x]]))/(d*Sqrt[Cot[c + d*x]])","A",1
723,1,136,65,1.2420332,"\int \frac{a+i a \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])/Cot[c + d*x]^(3/2),x]","\frac{a e^{-i c} (\cot (c+d x)+i) \sec (c+d x) (\sin (d x)+i \cos (d x)) \left(-3 i \sin (2 (c+d x))-\cos (2 (c+d x))+6 \cos ^2(c+d x) \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+1\right)}{3 d \sqrt{\cot (c+d x)}}","\frac{2 i a}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a}{d \sqrt{\cot (c+d x)}}-\frac{2 (-1)^{3/4} a \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(a*(I + Cot[c + d*x])*Sec[c + d*x]*(I*Cos[d*x] + Sin[d*x])*(1 - Cos[2*(c + d*x)] - (3*I)*Sin[2*(c + d*x)] + 6*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Cos[c + d*x]^2*Sqrt[I*Tan[c + d*x]]))/(3*d*E^(I*c)*Sqrt[Cot[c + d*x]])","B",1
724,1,104,91,3.516003,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2,x]","-\frac{a^2 \sqrt{\cot (c+d x)} \left(60 \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+\csc ^2(c+d x) (10 i \sin (2 (c+d x))+33 \cos (2 (c+d x))-27)\right)}{15 d}","-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{4 i a^2 \cot ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 \sqrt{\cot (c+d x)}}{d}+\frac{4 \sqrt[4]{-1} a^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"-1/15*(a^2*Sqrt[Cot[c + d*x]]*(Csc[c + d*x]^2*(-27 + 33*Cos[2*(c + d*x)] + (10*I)*Sin[2*(c + d*x)]) + 60*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/d","A",1
725,1,125,71,1.6493767,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2,x]","-\frac{2 a^2 e^{-2 i c} \sqrt{\cot (c+d x)} (\cos (2 (c+d x))+i \sin (2 (c+d x))) \left(\cot (c+d x)-6 i \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+6 i\right)}{3 d (\cos (d x)+i \sin (d x))^2}","-\frac{2 a^2 \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 i a^2 \sqrt{\cot (c+d x)}}{d}-\frac{4 (-1)^{3/4} a^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(-2*a^2*Sqrt[Cot[c + d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*(6*I + Cot[c + d*x] - (6*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*E^((2*I)*c)*(Cos[d*x] + I*Sin[d*x])^2)","A",1
726,1,70,49,1.8395147,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2,x]","\frac{2 a^2 \sqrt{\cot (c+d x)} \left(-1+2 \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{d}","-\frac{2 a^2 \sqrt{\cot (c+d x)}}{d}-\frac{4 \sqrt[4]{-1} a^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(2*a^2*Sqrt[Cot[c + d*x]]*(-1 + 2*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/d","C",1
727,1,83,49,1.8387441,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2,x]","\frac{2 a^2 (i \tan (c+d x))^{3/2} \cot ^{\frac{3}{2}}(c+d x) \left(\sqrt{i \tan (c+d x)}-2 \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{d}","\frac{4 (-1)^{3/4} a^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 a^2}{d \sqrt{\cot (c+d x)}}",1,"(2*a^2*Cot[c + d*x]^(3/2)*(-2*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]] + Sqrt[I*Tan[c + d*x]])*(I*Tan[c + d*x])^(3/2))/d","C",1
728,1,90,71,2.0236726,"\int \frac{(a+i a \tan (c+d x))^2}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/Sqrt[Cot[c + d*x]],x]","-\frac{2 a^2 \left(-6 i \cot (c+d x)+6 \sqrt{i \tan (c+d x)} \cot ^2(c+d x) \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+1\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}","-\frac{2 a^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{4 i a^2}{d \sqrt{\cot (c+d x)}}+\frac{4 \sqrt[4]{-1} a^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(-2*a^2*(1 - (6*I)*Cot[c + d*x] + 6*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Cot[c + d*x]^2*Sqrt[I*Tan[c + d*x]]))/(3*d*Cot[c + d*x]^(3/2))","A",1
729,1,152,91,4.0127372,"\int \frac{(a+i a \tan (c+d x))^2}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^2/Cot[c + d*x]^(3/2),x]","\frac{a^2 e^{-2 i c} (\sin (2 (c+d x))-i \cos (2 (c+d x))) \left(2 i \sec ^2(c+d x) (10 i \sin (2 (c+d x))+33 \cos (2 (c+d x))+27)-\frac{120 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{i \tan (c+d x)}}\right)}{30 d \sqrt{\cot (c+d x)} (\cos (d x)+i \sin (d x))^2}","\frac{4 i a^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2}{d \sqrt{\cot (c+d x)}}-\frac{4 (-1)^{3/4} a^2 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(a^2*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])*((2*I)*Sec[c + d*x]^2*(27 + 33*Cos[2*(c + d*x)] + (10*I)*Sin[2*(c + d*x)]) - ((120*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/Sqrt[I*Tan[c + d*x]]))/(30*d*E^((2*I)*c)*Sqrt[Cot[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^2)","A",1
730,1,147,106,3.0154738,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3,x]","-\frac{a^3 e^{-3 i c} \sqrt{\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(40 \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+\csc ^2(c+d x) (5 i \sin (2 (c+d x))+21 \cos (2 (c+d x))-19)\right)}{5 d (\cos (d x)+i \sin (d x))^3}","-\frac{8 i a^3 \cot ^{\frac{3}{2}}(c+d x)}{5 d}-\frac{2 \cot ^{\frac{3}{2}}(c+d x) \left(a^3 \cot (c+d x)+i a^3\right)}{5 d}+\frac{8 a^3 \sqrt{\cot (c+d x)}}{d}+\frac{8 \sqrt[4]{-1} a^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"-1/5*(a^3*Sqrt[Cot[c + d*x]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*(Csc[c + d*x]^2*(-19 + 21*Cos[2*(c + d*x)] + (5*I)*Sin[2*(c + d*x)]) + 40*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(d*E^((3*I)*c)*(Cos[d*x] + I*Sin[d*x])^3)","A",1
731,1,125,88,2.7418484,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3,x]","-\frac{2 a^3 e^{-3 i c} \sqrt{\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(\cot (c+d x)-12 i \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)+9 i\right)}{3 d (\cos (d x)+i \sin (d x))^3}","-\frac{16 i a^3 \sqrt{\cot (c+d x)}}{3 d}-\frac{2 \sqrt{\cot (c+d x)} \left(a^3 \cot (c+d x)+i a^3\right)}{3 d}-\frac{8 (-1)^{3/4} a^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(-2*a^3*Sqrt[Cot[c + d*x]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*(9*I + Cot[c + d*x] - (12*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(3*d*E^((3*I)*c)*(Cos[d*x] + I*Sin[d*x])^3)","A",1
732,1,125,64,2.7197276,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3,x]","\frac{2 a^3 e^{-3 i c} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(-\cot (c+d x)+\frac{4 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)}{\sqrt{i \tan (c+d x)}}-i\right)}{d \sqrt{\cot (c+d x)} (\cos (d x)+i \sin (d x))^3}","-\frac{8 \sqrt[4]{-1} a^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}-\frac{2 \left(a^3 \cot (c+d x)+i a^3\right)}{d \sqrt{\cot (c+d x)}}",1,"(2*a^3*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*(-I - Cot[c + d*x] + ((4*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])/Sqrt[I*Tan[c + d*x]]))/(d*E^((3*I)*c)*Sqrt[Cot[c + d*x]]*(Cos[d*x] + I*Sin[d*x])^3)","A",1
733,1,147,86,2.8285093,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3,x]","\frac{i a^3 e^{-3 i c} \sqrt{\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(\sec ^2(c+d x) (9 i \sin (2 (c+d x))+\cos (2 (c+d x))-1)-24 \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{3 d (\cos (d x)+i \sin (d x))^3}","-\frac{2 \left(a^3 \cot (c+d x)+i a^3\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{16 a^3}{3 d \sqrt{\cot (c+d x)}}+\frac{8 (-1)^{3/4} a^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"((I/3)*a^3*Sqrt[Cot[c + d*x]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*(Sec[c + d*x]^2*(-1 + Cos[2*(c + d*x)] + (9*I)*Sin[2*(c + d*x)]) - 24*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(d*E^((3*I)*c)*(Cos[d*x] + I*Sin[d*x])^3)","A",1
734,1,164,106,4.0077277,"\int \frac{(a+i a \tan (c+d x))^3}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^3/Sqrt[Cot[c + d*x]],x]","\frac{a^3 e^{-3 i c} \sqrt{\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left(\sec ^3(c+d x) (17 i \sin (c+d x)+21 i \sin (3 (c+d x))-5 \cos (c+d x)+5 \cos (3 (c+d x)))-80 \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right)\right)}{10 d (\cos (d x)+i \sin (d x))^3}","-\frac{8 a^3}{5 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(a^3 \cot (c+d x)+i a^3\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{8 i a^3}{d \sqrt{\cot (c+d x)}}+\frac{8 \sqrt[4]{-1} a^3 \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{d}",1,"(a^3*Sqrt[Cot[c + d*x]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*(Sec[c + d*x]^3*(-5*Cos[c + d*x] + 5*Cos[3*(c + d*x)] + (17*I)*Sin[c + d*x] + (21*I)*Sin[3*(c + d*x)]) - 80*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*Sqrt[I*Tan[c + d*x]]))/(10*d*E^((3*I)*c)*(Cos[d*x] + I*Sin[d*x])^3)","A",1
735,1,213,220,1.0732993,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x]),x]","-\frac{\sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left(10 i \sin (2 (c+d x))+8 \cos (2 (c+d x))-(5+3 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (c+d x)+i \sin (c+d x))-(5-3 i) \sqrt{\sin (2 (c+d x))} \cos (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(3+5 i) \sin (c+d x) \sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+8\right)}{8 a d (\cot (c+d x)+i)}","\frac{\cot ^{\frac{3}{2}}(c+d x)}{2 d (a \cot (c+d x)+i a)}-\frac{5 \sqrt{\cot (c+d x)}}{2 a d}-\frac{\left(\frac{5}{8}-\frac{3 i}{8}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{5}{8}-\frac{3 i}{8}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}-\frac{\left(\frac{5}{4}+\frac{3 i}{4}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}+\frac{\left(\frac{5}{4}+\frac{3 i}{4}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"-1/8*(Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x]*(8 + 8*Cos[2*(c + d*x)] - (5 - 3*I)*Cos[c + d*x]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - (5 + 3*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*(Cos[c + d*x] + I*Sin[c + d*x])*Sqrt[Sin[2*(c + d*x)]] - (3 + 5*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[c + d*x]*Sqrt[Sin[2*(c + d*x)]] + (10*I)*Sin[2*(c + d*x)]))/(a*d*(I + Cot[c + d*x]))","A",1
736,1,174,200,0.9797724,"\int \frac{\sqrt{\cot (c+d x)}}{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x]),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\sin (2 (c+d x))} \sqrt{\cot (c+d x)} \left(\frac{2-2 i}{\sqrt{\sin (2 (c+d x))}}+(1+2 i) \sec (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(2-i) \csc (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(1-2 i) \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\csc (c+d x)+i \sec (c+d x))\right)}{a d (\cot (c+d x)+i)}","\frac{\sqrt{\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}-\frac{\left(\frac{3}{8}+\frac{i}{8}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{3}{8}+\frac{i}{8}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{3}{4}-\frac{i}{4}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}-\frac{\left(\frac{3}{4}-\frac{i}{4}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((1/8 + I/8)*Sqrt[Cot[c + d*x]]*((2 - I)*Csc[c + d*x]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]] - (1 - 2*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*(Csc[c + d*x] + I*Sec[c + d*x]) + (1 + 2*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sec[c + d*x] + (2 - 2*I)/Sqrt[Sin[2*(c + d*x)]])*Sqrt[Sin[2*(c + d*x)]])/(a*d*(I + Cot[c + d*x]))","A",1
737,1,126,68,0.889824,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])),x]","\frac{(\sin (c+d x)+i \cos (c+d x)) \left(\cos (c+d x) \sqrt{i \tan (c+d x)}-\tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (\cos (c+d x)+i \sin (c+d x))\right)}{2 a d \sqrt{i \tan (c+d x)} \sqrt{\cot (c+d x)}}","\frac{\sqrt[4]{-1} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{2 a d}+\frac{i \sqrt{\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}",1,"((I*Cos[c + d*x] + Sin[c + d*x])*(-(ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*(Cos[c + d*x] + I*Sin[c + d*x])) + Cos[c + d*x]*Sqrt[I*Tan[c + d*x]]))/(2*a*d*Sqrt[Cot[c + d*x]]*Sqrt[I*Tan[c + d*x]])","A",1
738,1,174,200,1.0666932,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\sin (2 (c+d x))} \sqrt{\cot (c+d x)} \left(-\frac{2-2 i}{\sqrt{\sin (2 (c+d x))}}-(1-2 i) \sec (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(2+i) \csc (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(1+2 i) \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\csc (c+d x)+i \sec (c+d x))\right)}{a d (\cot (c+d x)+i)}","-\frac{\sqrt{\cot (c+d x)}}{2 d (a \cot (c+d x)+i a)}-\frac{\left(\frac{1}{8}+\frac{3 i}{8}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{8}+\frac{3 i}{8}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{1}{4}-\frac{3 i}{4}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}-\frac{\left(\frac{1}{4}-\frac{3 i}{4}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"((1/8 + I/8)*Sqrt[Cot[c + d*x]]*((2 + I)*Csc[c + d*x]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]] + (1 + 2*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*(Csc[c + d*x] + I*Sec[c + d*x]) - (1 - 2*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sec[c + d*x] - (2 - 2*I)/Sqrt[Sin[2*(c + d*x)]])*Sqrt[Sin[2*(c + d*x)]])/(a*d*(I + Cot[c + d*x]))","A",1
739,1,213,222,1.0922483,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])),x]","-\frac{\sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left(10 i \sin (2 (c+d x))+8 \cos (2 (c+d x))+(3+5 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (c+d x)+i \sin (c+d x))+(3-5 i) \sqrt{\sin (2 (c+d x))} \cos (c+d x) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(5+3 i) \sin (c+d x) \sqrt{\sin (2 (c+d x))} \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-8\right)}{8 a d (\cot (c+d x)+i)}","-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)}+\frac{\left(\frac{3}{8}-\frac{5 i}{8}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}-\frac{\left(\frac{3}{8}-\frac{5 i}{8}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}+\frac{\left(\frac{3}{4}+\frac{5 i}{4}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a d}-\frac{\left(\frac{3}{4}+\frac{5 i}{4}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a d}",1,"-1/8*(Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x]*(-8 + 8*Cos[2*(c + d*x)] + (3 - 5*I)*Cos[c + d*x]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (3 + 5*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*(Cos[c + d*x] + I*Sin[c + d*x])*Sqrt[Sin[2*(c + d*x)]] + (5 + 3*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[c + d*x]*Sqrt[Sin[2*(c + d*x)]] + (10*I)*Sin[2*(c + d*x)]))/(a*d*(I + Cot[c + d*x]))","A",1
740,1,232,252,1.2102349,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^2,x]","-\frac{\cot ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sec ^2(c+d x) \left(43 i \sin (c+d x)+43 i \sin (3 (c+d x))+23 \cos (c+d x)+41 \cos (3 (c+d x))+(21-25 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+(-21-25 i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(25-21 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{32 a^2 d (\cot (c+d x)+i)^2}","\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{8 a^2 d (\cot (c+d x)+i)}-\frac{25 \sqrt{\cot (c+d x)}}{8 a^2 d}-\frac{\left(\frac{25}{32}-\frac{21 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{25}{32}-\frac{21 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{25}{16}+\frac{21 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{25}{16}+\frac{21 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\cot ^{\frac{5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}",1,"-1/32*(Cot[c + d*x]^(3/2)*Csc[c + d*x]*Sec[c + d*x]^2*(23*Cos[c + d*x] + 41*Cos[3*(c + d*x)] + (43*I)*Sin[c + d*x] - (25 - 21*I)*Cos[2*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - (21 + 25*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[2*(c + d*x)]^(3/2) + (21 - 25*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]) + (43*I)*Sin[3*(c + d*x)]))/(a^2*d*(I + Cot[c + d*x])^2)","A",1
741,1,232,232,0.9754604,"\int \frac{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^2,x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sec ^2(c+d x) \left(7 \sin (c+d x)+7 \sin (3 (c+d x))+5 i \cos (c+d x)-5 i \cos (3 (c+d x))-(5+9 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+(-5+9 i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(9+5 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{32 a^2 d (\cot (c+d x)+i)^2}","\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}-\frac{\left(\frac{9}{32}+\frac{5 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{9}{32}+\frac{5 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{9}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{9}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2}",1,"(Cot[c + d*x]^(3/2)*Csc[c + d*x]*Sec[c + d*x]^2*((5*I)*Cos[c + d*x] - (5*I)*Cos[3*(c + d*x)] + 7*Sin[c + d*x] + (9 + 5*I)*Cos[2*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - (5 - 9*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[2*(c + d*x)]^(3/2) - (5 + 9*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]) + 7*Sin[3*(c + d*x)]))/(32*a^2*d*(I + Cot[c + d*x])^2)","A",1
742,1,224,234,0.9314803,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2),x]","\frac{\csc ^3(c+d x) \left(3 i \sin (c+d x)+3 i \sin (3 (c+d x))-\cos (c+d x)+\cos (3 (c+d x))-(3+i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+(3-i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(1+3 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{32 a^2 d \sqrt{\cot (c+d x)} (\cot (c+d x)+i)^2}","\frac{3 i \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac{\left(\frac{1}{32}+\frac{3 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{32}+\frac{3 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{1}{16}-\frac{3 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}-\frac{3 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\sqrt{\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}",1,"(Csc[c + d*x]^3*(-Cos[c + d*x] + Cos[3*(c + d*x)] + (3*I)*Sin[c + d*x] - (1 + 3*I)*Cos[2*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (3 - I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[2*(c + d*x)]^(3/2) - (3 + I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]) + (3*I)*Sin[3*(c + d*x)]))/(32*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])^2)","A",1
743,1,222,234,0.8300204,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\csc ^3(c+d x) \left(\sin (c+d x)+\sin (3 (c+d x))+3 i \cos (c+d x)-3 i \cos (3 (c+d x))+(1+3 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+(-3-i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(1-3 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{32 a^2 d \sqrt{\cot (c+d x)} (\cot (c+d x)+i)^2}","\frac{\sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}+\frac{\left(\frac{1}{32}-\frac{3 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{32}-\frac{3 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{1}{16}+\frac{3 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{1}{16}+\frac{3 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{i \sqrt{\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}",1,"(Csc[c + d*x]^3*((3*I)*Cos[c + d*x] - (3*I)*Cos[3*(c + d*x)] + Sin[c + d*x] - (1 - 3*I)*Cos[2*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (1 + 3*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*Sqrt[Sin[2*(c + d*x)]] - (3 + I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[2*(c + d*x)]^(3/2) + Sin[3*(c + d*x)]))/(32*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])^2)","A",1
744,1,232,234,1.2572568,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sec ^2(c+d x) \left(5 i \sin (c+d x)+5 i \sin (3 (c+d x))-7 \cos (c+d x)+7 \cos (3 (c+d x))+(9+5 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+(5+9 i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(9-5 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{32 a^2 d (\cot (c+d x)+i)^2}","\frac{5 i \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}-\frac{\left(\frac{9}{32}-\frac{5 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{9}{32}-\frac{5 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{9}{16}+\frac{5 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{9}{16}+\frac{5 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{\sqrt{\cot (c+d x)}}{4 d (a \cot (c+d x)+i a)^2}",1,"(Cot[c + d*x]^(3/2)*Csc[c + d*x]*Sec[c + d*x]^2*(-7*Cos[c + d*x] + 7*Cos[3*(c + d*x)] + (5*I)*Sin[c + d*x] + (9 - 5*I)*Cos[2*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (9 + 5*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])*Sqrt[Sin[2*(c + d*x)]] + (5 + 9*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[2*(c + d*x)]^(3/2) + (5*I)*Sin[3*(c + d*x)]))/(32*a^2*d*(I + Cot[c + d*x])^2)","A",1
745,1,232,254,1.6081911,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sec ^2(c+d x) \left(23 \sin (c+d x)-41 \sin (3 (c+d x))-43 i \cos (c+d x)+43 i \cos (3 (c+d x))-(21+25 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+(-21+25 i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(25+21 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)\right)}{32 a^2 d (\cot (c+d x)+i)^2}","-\frac{25}{8 a^2 d \sqrt{\cot (c+d x)}}+\frac{7 i}{8 a^2 d \sqrt{\cot (c+d x)} (\cot (c+d x)+i)}-\frac{\left(\frac{25}{32}+\frac{21 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{25}{32}+\frac{21 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}+\frac{\left(\frac{25}{16}-\frac{21 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^2 d}-\frac{\left(\frac{25}{16}-\frac{21 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^2 d}-\frac{1}{4 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^2}",1,"(Cot[c + d*x]^(3/2)*Csc[c + d*x]*Sec[c + d*x]^2*((-43*I)*Cos[c + d*x] + (43*I)*Cos[3*(c + d*x)] + 23*Sin[c + d*x] + (25 + 21*I)*Cos[2*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - (21 - 25*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sin[2*(c + d*x)]^(3/2) - (21 + 25*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]) - 41*Sin[3*(c + d*x)]))/(32*a^2*d*(I + Cot[c + d*x])^2)","A",1
746,1,235,273,1.2850439,"\int \frac{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^3,x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \csc (c+d x) \sec ^3(c+d x) \left(12 \sin (2 (c+d x))+21 \sin (4 (c+d x))-19 i \cos (4 (c+d x))-(21-15 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (3 (c+d x))+i \sin (3 (c+d x)))-(15-21 i) \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+(21+15 i) \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+19 i\right)}{96 a^3 d (\cot (c+d x)+i)^3}","\frac{5 \sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}-\frac{\left(\frac{7}{32}+\frac{5 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{7}{32}+\frac{5 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{7}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{7}{16}-\frac{5 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (a \cot (c+d x)+i a)^2}",1,"(Cot[c + d*x]^(5/2)*Csc[c + d*x]*Sec[c + d*x]^3*(19*I - (19*I)*Cos[4*(c + d*x)] + (21 + 15*I)*Cos[3*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + 12*Sin[2*(c + d*x)] - (21 - 15*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) - (15 - 21*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] + 21*Sin[4*(c + d*x)]))/(96*a^3*d*(I + Cot[c + d*x])^3)","A",1
747,1,231,267,0.9782998,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3),x]","\frac{\sqrt{\cot (c+d x)} \csc ^3(c+d x) \sec (c+d x) \left(6 i \sin (2 (c+d x))+3 i \sin (4 (c+d x))+\cos (4 (c+d x))+6 i \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (3 (c+d x))+i \sin (3 (c+d x)))+6 \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-6 i \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-1\right)}{96 a^3 d (\cot (c+d x)+i)^3}","\frac{i \sqrt{\cot (c+d x)}}{4 d \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{i \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}-\frac{i \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}-\frac{i \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{8 \sqrt{2} a^3 d}+\frac{i \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a^3 d}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{\sqrt{\cot (c+d x)}}{4 a d (a \cot (c+d x)+i a)^2}",1,"(Sqrt[Cot[c + d*x]]*Csc[c + d*x]^3*Sec[c + d*x]*(-1 + Cos[4*(c + d*x)] - (6*I)*Cos[3*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (6*I)*Sin[2*(c + d*x)] + (6*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) + 6*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] + (3*I)*Sin[4*(c + d*x)]))/(96*a^3*d*(I + Cot[c + d*x])^3)","A",1
748,1,154,141,1.6293533,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{i \sqrt{\cot (c+d x)} \csc ^3(c+d x) \left(3 i \sin (c+d x)-3 i \sin (3 (c+d x))+5 \cos (c+d x)-5 \cos (3 (c+d x))+6 \sqrt{i \tan (c+d x)} \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right)}{48 a^3 d (\cot (c+d x)+i)^3}","\frac{\sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}-\frac{(-1)^{3/4} \tanh ^{-1}\left((-1)^{3/4} \sqrt{\cot (c+d x)}\right)}{8 a^3 d}+\frac{i \sqrt{\cot (c+d x)}}{6 a d (a \cot (c+d x)+i a)^2}+\frac{\sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}",1,"((I/48)*Sqrt[Cot[c + d*x]]*Csc[c + d*x]^3*(5*Cos[c + d*x] - 5*Cos[3*(c + d*x)] + (3*I)*Sin[c + d*x] - (3*I)*Sin[3*(c + d*x)] + 6*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)])*Sqrt[I*Tan[c + d*x]]))/(a^3*d*(I + Cot[c + d*x])^3)","A",1
749,1,224,222,1.917454,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{(\cos (3 (c+d x))-i \sin (3 (c+d x))) \left(6 \tan ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (\sin (3 (c+d x))-i \cos (3 (c+d x)))+\sqrt{i \tan (c+d x)} (\sin (c+d x)+\sin (3 (c+d x))+3 i \cos (c+d x)-3 i \cos (3 (c+d x)))+6 i \tanh ^{-1}\left(\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right)}{48 a^3 d \sqrt{i \tan (c+d x)} \sqrt{\cot (c+d x)}}","-\frac{\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}+\frac{\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{16 \sqrt{2} a^3 d}-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{8 \sqrt{2} a^3 d}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{8 \sqrt{2} a^3 d}+\frac{\sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2}+\frac{i \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}",1,"((Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)])*((6*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) + 6*ArcTan[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]]*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]) + ((3*I)*Cos[c + d*x] - (3*I)*Cos[3*(c + d*x)] + Sin[c + d*x] + Sin[3*(c + d*x)])*Sqrt[I*Tan[c + d*x]]))/(48*a^3*d*Sqrt[Cot[c + d*x]]*Sqrt[I*Tan[c + d*x]])","A",1
750,1,235,275,1.8665003,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3),x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \csc (c+d x) \sec ^3(c+d x) \left(-12 \sin (2 (c+d x))+21 \sin (4 (c+d x))-19 i \cos (4 (c+d x))+(21+15 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (3 (c+d x))-i \cos (3 (c+d x)))+(21-15 i) \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)-(15+21 i) \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left(\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right)+19 i\right)}{96 a^3 d (\cot (c+d x)+i)^3}","\frac{5 \sqrt{\cot (c+d x)}}{8 d \left(a^3 \cot (c+d x)+i a^3\right)}+\frac{\left(\frac{5}{32}+\frac{7 i}{32}\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{5}{32}+\frac{7 i}{32}\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}-\frac{\left(\frac{5}{16}-\frac{7 i}{16}\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} a^3 d}+\frac{\left(\frac{5}{16}-\frac{7 i}{16}\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} a^3 d}+\frac{i \sqrt{\cot (c+d x)}}{3 a d (a \cot (c+d x)+i a)^2}-\frac{\sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}",1,"(Cot[c + d*x]^(5/2)*Csc[c + d*x]*Sec[c + d*x]^3*(19*I - (19*I)*Cos[4*(c + d*x)] - (15 + 21*I)*Cos[3*(c + d*x)]*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] - 12*Sin[2*(c + d*x)] + (21 - 15*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] + (21 + 15*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]) + 21*Sin[4*(c + d*x)]))/(96*a^3*d*(I + Cot[c + d*x])^3)","A",1
751,1,149,174,1.3379935,"\int \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(30 e^{i (c+d x)}-40 e^{3 i (c+d x)}+34 e^{5 i (c+d x)}-15 \left(-1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{15 d \left(-1+e^{2 i (c+d x)}\right)^2}","-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}-\frac{2 i \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{15 d}+\frac{26 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{15 d}-\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((30*E^(I*(c + d*x)) - 40*E^((3*I)*(c + d*x)) + 34*E^((5*I)*(c + d*x)) - 15*(-1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^2)","A",1
752,1,125,140,1.0095193,"\int \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(4 e^{3 i (c+d x)}-3 \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{3 d \left(-1+e^{2 i (c+d x)}\right)}","-\frac{2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 d}-\frac{2 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{3 d}-\frac{(1-i) \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((-1/3*I)*(4*E^((3*I)*(c + d*x)) - 3*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))))","A",1
753,1,104,102,0.8562974,"\int \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-2 e^{i (c+d x)}\right) \sqrt{a+i a \tan (c+d x)}}{d}","\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"((-2*E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
754,1,118,69,0.7366237,"\int \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)} \, dx","Integrate[Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{d}","\frac{(1-i) \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((-I)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
755,0,0,144,42.6611706,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\cot (c+d x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[Cot[c + d*x]],x]","\int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\cot (c+d x)}} \, dx","-\frac{2 (-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[Cot[c + d*x]], x]","F",-1
756,1,223,175,2.7094242,"\int \frac{\sqrt{a+i a \tan (c+d x)}}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + I*a*Tan[c + d*x]]/Cot[c + d*x]^(3/2),x]","\frac{\left(8-\frac{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \left(8 \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+\sqrt{2} \left(\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)}{\sqrt{-1+e^{2 i (c+d x)}}}\right) \sqrt{a+i a \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}","-\frac{\sqrt[4]{-1} \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{\sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}-\frac{(1-i) \sqrt{a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((8 - ((1 + E^((2*I)*(c + d*x)))*(8*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(-Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/(E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]))*Sqrt[a + I*a*Tan[c + d*x]])/(8*d*Sqrt[Cot[c + d*x]])","A",1
757,1,161,218,1.8210022,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{4 a \cos (c+d x) \sqrt{\cot (c+d x)} \left(e^{i (c+d x)} \left(-10 e^{2 i (c+d x)}+9 e^{4 i (c+d x)}+5\right)-5 \left(-1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{5 d \left(-1+e^{2 i (c+d x)}\right)^2 \left(1+e^{2 i (c+d x)}\right)}","-\frac{(2+2 i) a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a+i a \tan (c+d x)}}-\frac{2 i a^2 \cot ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a+i a \tan (c+d x)}}-\frac{4 i a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{5 d}+\frac{12 a \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{5 d}",1,"(4*a*(E^(I*(c + d*x))*(5 - 10*E^((2*I)*(c + d*x)) + 9*E^((4*I)*(c + d*x))) - 5*(-1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cos[c + d*x]*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*(-1 + E^((2*I)*(c + d*x)))^2*(1 + E^((2*I)*(c + d*x))))","A",1
758,1,135,139,1.2592237,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{4 i a \cos (c+d x) \sqrt{\cot (c+d x)} \left(e^{i (c+d x)} \left(-3+5 e^{2 i (c+d x)}\right)-3 \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{3 d \left(-1+e^{4 i (c+d x)}\right)}","-\frac{(2-2 i) a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{2 i a \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"(((-4*I)/3)*a*(E^(I*(c + d*x))*(-3 + 5*E^((2*I)*(c + d*x))) - 3*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cos[c + d*x]*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*(-1 + E^((4*I)*(c + d*x))))","A",1
759,1,105,103,1.331132,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{2 a e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(e^{i (c+d x)}-\sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{d}","\frac{(2+2 i) a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"(-2*a*(E^(I*(c + d*x)) - Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x)))","A",1
760,1,255,144,1.5795391,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2} \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i a e^{-i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(8 \log \left(\sqrt{-1+e^{2 i (c+d x)}}+e^{i (c+d x)}\right)+\sqrt{2} \left(\log \left(2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)-\log \left(-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}}-3 e^{2 i (c+d x)}+1\right)\right)\right)}{2 \sqrt{2} d}","\frac{2 \sqrt[4]{-1} a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(2-2 i) a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}",1,"((-1/2*I)*a*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*(8*Log[E^(I*(c + d*x)) + Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*(-Log[1 - 3*E^((2*I)*(c + d*x)) - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]] + Log[1 - 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]])))/(Sqrt[2]*d*E^(I*(c + d*x)))","A",1
761,1,228,216,2.5027188,"\int \frac{(a+i a \tan (c+d x))^{3/2}}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(3/2)/Sqrt[Cot[c + d*x]],x]","\frac{a e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{2} e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)-2 \sqrt{2} \left(1+e^{2 i (c+d x)}\right) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3 \left(1+e^{2 i (c+d x)}\right) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{\sqrt{2} d \left(1+e^{2 i (c+d x)}\right)}","-\frac{3 (-1)^{3/4} a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{(2+2 i) a^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2}{d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}+\frac{i a^2}{d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(a*(Sqrt[2]*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))) - 2*Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 3*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x))))","A",1
762,1,167,222,2.4712017,"\int \cot ^{\frac{9}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{4 i a^2 e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(-21 e^{i (c+d x)}+70 e^{3 i (c+d x)}-77 e^{5 i (c+d x)}+40 e^{7 i (c+d x)}-21 \left(-1+e^{2 i (c+d x)}\right)^{7/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{21 d \left(-1+e^{2 i (c+d x)}\right)^3}","\frac{(4-4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}-\frac{6 i a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{7 d}+\frac{32 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{21 d}+\frac{104 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{21 d}",1,"(((4*I)/21)*a^2*(-21*E^(I*(c + d*x)) + 70*E^((3*I)*(c + d*x)) - 77*E^((5*I)*(c + d*x)) + 40*E^((7*I)*(c + d*x)) - 21*(-1 + E^((2*I)*(c + d*x)))^(7/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^3)","A",1
763,1,161,176,1.9359683,"\int \cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{16 a^2 e^{i (c+d x)} \cos ^2(c+d x) \sqrt{\cot (c+d x)} \left(e^{i (c+d x)} \left(-35 e^{2 i (c+d x)}+26 e^{4 i (c+d x)}+15\right)-15 \left(-1+e^{2 i (c+d x)}\right)^{5/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{15 d \left(-1+e^{4 i (c+d x)}\right)^2}","-\frac{(4+4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{4 a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 i a \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}{5 d}",1,"(16*a^2*E^(I*(c + d*x))*(E^(I*(c + d*x))*(15 - 35*E^((2*I)*(c + d*x)) + 26*E^((4*I)*(c + d*x))) - 15*(-1 + E^((2*I)*(c + d*x)))^(5/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cos[c + d*x]^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*(-1 + E^((4*I)*(c + d*x)))^2)","A",1
764,1,142,142,1.5685829,"\int \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{4 i a^2 e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(e^{i (c+d x)} \left(-3+4 e^{2 i (c+d x)}\right)-3 \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{3 d \left(-1+e^{2 i (c+d x)}\right)}","-\frac{(4-4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{4 i a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}",1,"(((-4*I)/3)*a^2*(E^(I*(c + d*x))*(-3 + 4*E^((2*I)*(c + d*x))) - 3*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))))","A",1
765,1,176,179,2.4654986,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{\sqrt{2} a^2 e^{-i (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{2} e^{i (c+d x)}-2 \sqrt{2} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{d}","\frac{2 (-1)^{3/4} a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(4+4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{2 a^2 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{d}",1,"-((Sqrt[2]*a^2*(Sqrt[2]*E^(I*(c + d*x)) - 2*Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*E^(I*(c + d*x))))","A",1
766,1,241,179,2.3244506,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2} \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{2 i \sqrt{2} a^2 e^{i (c+d x)} \cos ^2(c+d x) \sqrt{\cot (c+d x)} \left(\sqrt{2} e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)-4 \sqrt{2} \left(1+e^{2 i (c+d x)}\right) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+5 \left(1+e^{2 i (c+d x)}\right) \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right) \sqrt{a+i a \tan (c+d x)}}{d \left(1+e^{2 i (c+d x)}\right)^3}","\frac{5 \sqrt[4]{-1} a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}+\frac{(4-4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \sqrt{a+i a \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}",1,"((2*I)*Sqrt[2]*a^2*E^(I*(c + d*x))*(Sqrt[2]*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x))) - 4*Sqrt[2]*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 5*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Cos[c + d*x]^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*(1 + E^((2*I)*(c + d*x)))^3)","A",1
767,1,318,222,7.4215876,"\int \frac{(a+i a \tan (c+d x))^{5/2}}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Sqrt[Cot[c + d*x]],x]","-\frac{(a+i a \tan (c+d x))^{5/2} \left(\sqrt{2} \sqrt{e^{i d x}} e^{-i (3 c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\frac{i \left(1+e^{2 i (c+d x)}\right)}{-1+e^{2 i (c+d x)}}} \left(32 \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-23 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)-(\cos (2 c)-i \sin (2 c)) \sqrt{\cot (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sqrt{\cos (d x)+i \sin (d x)} (9 i \sin (2 (c+d x))+2 \cos (2 (c+d x))-2)\right)}{8 d \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2}}","-\frac{23 (-1)^{3/4} a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{4 d}-\frac{(4+4 i) a^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{d}-\frac{a^2 \sqrt{a+i a \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{9 i a^2 \sqrt{a+i a \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}",1,"-1/8*(((Sqrt[2]*Sqrt[E^(I*d*x)]*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[(I*(1 + E^((2*I)*(c + d*x))))/(-1 + E^((2*I)*(c + d*x)))]*(32*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 23*Sqrt[2]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/E^(I*(3*c + d*x)) - Sqrt[Cot[c + d*x]]*Sec[c + d*x]^(5/2)*(Cos[2*c] - I*Sin[2*c])*Sqrt[Cos[d*x] + I*Sin[d*x]]*(-2 + 2*Cos[2*(c + d*x)] + (9*I)*Sin[2*(c + d*x)]))*(a + I*a*Tan[c + d*x])^(5/2))/(d*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])^(5/2))","A",1
768,1,159,181,1.4027639,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^(5/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{i \sqrt{\cot (c+d x)} \left(-18 e^{2 i (c+d x)}+7 e^{4 i (c+d x)}+3 e^{i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3\right)}{3 \sqrt{2} d \left(-1+e^{4 i (c+d x)}\right) \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{5 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{3 a d}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{d \sqrt{a+i a \tan (c+d x)}}+\frac{7 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{3 a d}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((I/3)*(3 - 18*E^((2*I)*(c + d*x)) + 7*E^((4*I)*(c + d*x)) + 3*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*d*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((4*I)*(c + d*x))))","A",1
769,1,139,140,1.260509,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^(3/2)/Sqrt[a + I*a*Tan[c + d*x]],x]","\frac{e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-5 e^{2 i (c+d x)}+e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{\sqrt{2} a d}","-\frac{3 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{a d}+\frac{\sqrt{\cot (c+d x)}}{d \sqrt{a+i a \tan (c+d x)}}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 - 5*E^((2*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
770,1,140,105,0.941618,"\int \frac{\sqrt{\cot (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[Sqrt[Cot[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]],x]","-\frac{i e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(e^{2 i (c+d x)}+e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{\sqrt{2} a d}","\frac{1}{d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((-I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((2*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
771,1,138,108,1.1036535,"\int \frac{1}{\sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(e^{2 i (c+d x)}-e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{\sqrt{2} a d}","\frac{i}{d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((2*I)*(c + d*x)) - E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
772,1,210,180,1.8307528,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{i e^{-2 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(e^{2 i (c+d x)}+e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-2 \sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{\sqrt{2} a d}","-\frac{2 \sqrt[4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}-\frac{1}{d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 + E^((2*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 2*Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a*d*E^((2*I)*(c + d*x)))","A",1
773,1,248,217,2.2067255,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]),x]","\frac{\sqrt{\cot (c+d x)} \left(2 e^{2 i (c+d x)}-3 e^{4 i (c+d x)}+e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+\sqrt{2} e^{i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{\sqrt{2} d \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}}","-\frac{1}{d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{(-1)^{3/4} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a d \sqrt{\cot (c+d x)}}+\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{\sqrt{a} d}",1,"((1 + 2*E^((2*I)*(c + d*x)) - 3*E^((4*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + Sqrt[2]*E^(I*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*d*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))^2)","A",1
774,1,186,221,1.714306,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{i e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(18 e^{2 i (c+d x)}-87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}+3 e^{3 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{6 \sqrt{2} a^2 d \left(-1+e^{2 i (c+d x)}\right)}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{7 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{13 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{2 a^2 d}+\frac{5 \cot ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"((I/6)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + 18*E^((2*I)*(c + d*x)) - 87*E^((4*I)*(c + d*x)) + 52*E^((6*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x))))","A",1
775,1,156,182,1.4784548,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(3/2),x]","\frac{e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(13 e^{2 i (c+d x)}-38 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{6 \sqrt{2} a^2 d}","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{25 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{6 a^2 d}+\frac{11 \sqrt{\cot (c+d x)}}{6 a d \sqrt{a+i a \tan (c+d x)}}+\frac{\sqrt{\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 + 13*E^((2*I)*(c + d*x)) - 38*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(6*Sqrt[2]*a^2*d*E^((4*I)*(c + d*x)))","A",1
776,1,158,145,1.2994207,"\int \frac{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2),x]","-\frac{i e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-7 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{6 \sqrt{2} a^2 d}","\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{7}{6 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{1}{3 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}",1,"((-1/6*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 - 7*E^((2*I)*(c + d*x)) + 8*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x)))","A",1
777,1,156,147,1.4318609,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-e^{2 i (c+d x)}+2 e^{4 i (c+d x)}-3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{6 \sqrt{2} a^2 d}","-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{1}{3 d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{i}{2 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 - E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x)) - 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(6*Sqrt[2]*a^2*d*E^((4*I)*(c + d*x)))","A",1
778,1,158,147,0.8333368,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","-\frac{i e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-5 e^{2 i (c+d x)}+4 e^{4 i (c+d x)}-3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{6 \sqrt{2} a^2 d}","-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{i}{3 d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{1}{2 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"((-1/6*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 - 5*E^((2*I)*(c + d*x)) + 4*E^((4*I)*(c + d*x)) - 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^2*d*E^((4*I)*(c + d*x)))","A",1
779,1,226,221,2.5036367,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","\frac{e^{-4 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-11 e^{2 i (c+d x)}+10 e^{4 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-12 \sqrt{2} e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{6 \sqrt{2} a^2 d}","\frac{2 (-1)^{3/4} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{\left(\frac{1}{4}+\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{1}{3 d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{3 i}{2 a d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 - 11*E^((2*I)*(c + d*x)) + 10*E^((4*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] - 12*Sqrt[2]*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(6*Sqrt[2]*a^2*d*E^((4*I)*(c + d*x)))","A",1
780,1,268,258,2.8376595,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2)),x]","-\frac{i \sqrt{\cot (c+d x)} \left(16 e^{2 i (c+d x)}+13 e^{4 i (c+d x)}-28 e^{6 i (c+d x)}+3 e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+18 \sqrt{2} e^{3 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \left(1+e^{2 i (c+d x)}\right) \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{6 \sqrt{2} d \left(1+e^{2 i (c+d x)}\right)^3 \left(\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2}}","-\frac{3 \sqrt[4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{3/2} d}-\frac{7 \sqrt{a+i a \tan (c+d x)}}{2 a^2 d \sqrt{\cot (c+d x)}}+\frac{13 i}{6 a d \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}-\frac{1}{3 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}",1,"((-1/6*I)*(-1 + 16*E^((2*I)*(c + d*x)) + 13*E^((4*I)*(c + d*x)) - 28*E^((6*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 18*Sqrt[2]*E^((3*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*(1 + E^((2*I)*(c + d*x)))*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*d*((a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^3)","A",1
781,1,199,258,2.0556557,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(33 e^{2 i (c+d x)}+348 e^{4 i (c+d x)}-1527 e^{6 i (c+d x)}+983 e^{8 i (c+d x)}+15 e^{5 i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3\right)}{60 \sqrt{2} a^3 d \left(-1+e^{2 i (c+d x)}\right)}","-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"((I/60)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(3 + 33*E^((2*I)*(c + d*x)) + 348*E^((4*I)*(c + d*x)) - 1527*E^((6*I)*(c + d*x)) + 983*E^((8*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x))*(-1 + E^((2*I)*(c + d*x))))","A",1
782,1,165,219,3.0122528,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \sec (c+d x) \left((340-460 \cos (2 (c+d x))) \csc (c+d x)-i (466 \cos (2 (c+d x))+149) \sec (c+d x)+15 e^{2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \csc (2 (c+d x)) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)\right)}{60 a^2 d (\cot (c+d x)+i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{317 \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{151 \sqrt{\cot (c+d x)}}{60 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{17 \sqrt{\cot (c+d x)}}{30 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\sqrt{\cot (c+d x)}}{5 d (a+i a \tan (c+d x))^{5/2}}",1,"(Cot[c + d*x]^(3/2)*Sec[c + d*x]*((340 - 460*Cos[2*(c + d*x)])*Csc[c + d*x] + 15*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Csc[2*(c + d*x)] - I*(149 + 466*Cos[2*(c + d*x)])*Sec[c + d*x]))/(60*a^2*d*(I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
783,1,171,182,1.8679337,"\int \frac{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^(5/2),x]","-\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-16 e^{2 i (c+d x)}-64 e^{4 i (c+d x)}+83 e^{6 i (c+d x)}+15 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-3\right)}{60 \sqrt{2} a^3 d}","\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{67}{60 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{13}{30 a d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{1}{5 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}}",1,"((-1/60*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-3 - 16*E^((2*I)*(c + d*x)) - 64*E^((4*I)*(c + d*x)) + 83*E^((6*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x)))","A",1
784,1,167,188,1.6988789,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-2 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+e^{6 i (c+d x)}-5 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)-1\right)}{20 \sqrt{2} a^3 d}","-\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{i}{20 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i}{10 a d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac{i}{5 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^{5/2}}",1,"(Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(-1 - 2*E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x)) + E^((6*I)*(c + d*x)) - 5*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(20*Sqrt[2]*a^3*d*E^((6*I)*(c + d*x)))","A",1
785,1,171,184,1.2531824,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","-\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-4 e^{2 i (c+d x)}-16 e^{4 i (c+d x)}+17 e^{6 i (c+d x)}-15 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+3\right)}{60 \sqrt{2} a^3 d}","-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{1}{4 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i}{6 a d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{1}{5 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}",1,"((-1/60*I)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(3 - 4*E^((2*I)*(c + d*x)) - 16*E^((4*I)*(c + d*x)) + 17*E^((6*I)*(c + d*x)) - 15*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x)))","A",1
786,1,158,186,2.6692583,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \sec (c+d x) \left(30 e^{2 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \csc (2 (c+d x)) \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+2 \sec (c+d x) (20 \sin (2 (c+d x))-26 i \cos (2 (c+d x))+11 i)\right)}{120 a^2 d (\cot (c+d x)+i)^2 \sqrt{a+i a \tan (c+d x)}}","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}-\frac{i}{4 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{1}{6 a d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}+\frac{i}{5 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}",1,"(Cot[c + d*x]^(3/2)*Sec[c + d*x]*(30*E^((2*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]*Csc[2*(c + d*x)] + 2*Sec[c + d*x]*(11*I - (26*I)*Cos[2*(c + d*x)] + 20*Sin[2*(c + d*x)])))/(120*a^2*d*(I + Cot[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])","A",1
787,1,241,258,3.7138072,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2)),x]","\frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left(-8 e^{2 i (c+d x)}+48 e^{4 i (c+d x)}-41 e^{6 i (c+d x)}-5 e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+40 \sqrt{2} e^{5 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{\sqrt{2} e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right)+1\right)}{20 \sqrt{2} a^3 d}","\frac{2 \sqrt[4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{(-1)^{3/4} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right)}{a^{5/2} d}+\frac{7}{4 a^2 d \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{i}{2 a d \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}}-\frac{1}{5 d \cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}}",1,"((I/20)*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(1 - 8*E^((2*I)*(c + d*x)) + 48*E^((4*I)*(c + d*x)) - 41*E^((6*I)*(c + d*x)) - 5*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]] + 40*Sqrt[2]*E^((5*I)*(c + d*x))*Sqrt[-1 + E^((2*I)*(c + d*x))]*ArcTanh[(Sqrt[2]*E^(I*(c + d*x)))/Sqrt[-1 + E^((2*I)*(c + d*x))]])*Sqrt[Cot[c + d*x]])/(Sqrt[2]*a^3*d*E^((6*I)*(c + d*x)))","A",1
788,1,925,139,9.1377566,"\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^3 \, dx","Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^3,x]","-\frac{4 i \left(1+e^{2 i (e+f x)}\right)^{-n} \left(\frac{i \left(1+e^{2 i (e+f x)}\right)}{-1+e^{2 i (e+f x)}}\right)^n \cos ^3(e+f x) (d \cot (e+f x))^n \left(\left(1+e^{2 i (e+f x)}\right)^n \, _2F_1\left(1,-n;1-n;-\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)-2^n \, _2F_1\left(-n,-n;1-n;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)\right) (i \tan (e+f x) a+a)^3 \cot ^{-n}(e+f x)}{\left(e^{i e}+e^{3 i e}\right) f n (\cos (f x)+i \sin (f x))^3}-\frac{4 i e^{-3 i e} \left(-1+e^{2 i (e+f x)}\right)^{-n} \left(\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n \left(\frac{i \left(1+e^{2 i (e+f x)}\right)}{-1+e^{2 i (e+f x)}}\right)^n \cos ^3(e+f x) (d \cot (e+f x))^n \left(\frac{\left(1+e^{2 i e}\right) \left(-1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,1-n;2-n;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{n-1}}{n-1}+\frac{\, _2F_1\left(1,-n;1-n;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^n}{n}-\frac{2^n \, _2F_1\left(-n,-n;1-n;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)}{n}\right) (i \tan (e+f x) a+a)^3 \cot ^{-n}(e+f x)}{\left(1+e^{2 i e}\right) f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) (d \cot (e+f x))^n \left(\frac{(\cos (2 e)+3 i \sin (2 e)-1) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{n-1}+\frac{\sec (e+f x) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) (-\cos (e-f x)+\cos (e+f x)-3 i \sin (e-f x)+3 i \sin (e+f x)) \sec ^2(e)}{n-1}\right) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) (d \cot (e+f x))^n \left(\frac{(2 n+\cos (2 e)-3) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{(n-2) (n-1)}+\frac{(\cos (e+f x)-\cos (e-f x)) \sec (e+f x) \left(\frac{1}{2} i \cos (3 e)+\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{n-1}+\frac{\sec ^2(e+f x) (i \cos (3 e)+\sin (3 e))}{n-2}\right) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}","-\frac{4 i a^3 d^2 (d \cot (e+f x))^{n-2} \, _2F_1(1,n-2;n-1;-i \cot (e+f x))}{f (2-n)}+\frac{d^2 \left(a^3 \cot (e+f x)+i a^3\right) (d \cot (e+f x))^{n-2}}{f (1-n)}+\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{n-2}}{f (1-n) (2-n)}",1,"((-4*I)*((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^3*(d*Cot[e + f*x])^n*((1 + E^((2*I)*(e + f*x)))^n*Hypergeometric2F1[1, -n, 1 - n, -((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))] - 2^n*Hypergeometric2F1[-n, -n, 1 - n, (1 - E^((2*I)*(e + f*x)))/2])*(a + I*a*Tan[e + f*x])^3)/((E^(I*e) + E^((3*I)*e))*(1 + E^((2*I)*(e + f*x)))^n*f*n*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])^3) - ((4*I)*((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^n*((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^3*(d*Cot[e + f*x])^n*(((1 + E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(-1 + n)*Hypergeometric2F1[1, 1 - n, 2 - n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/(-1 + n) + ((1 + E^((2*I)*(e + f*x)))^n*Hypergeometric2F1[1, -n, 1 - n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/n - (2^n*Hypergeometric2F1[-n, -n, 1 - n, (1 - E^((2*I)*(e + f*x)))/2])/n)*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*(1 + E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))^n*f*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(d*Cot[e + f*x])^n*(((-3 + 2*n + Cos[2*e])*Sec[e]^2*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/((-2 + n)*(-1 + n)) + ((-Cos[e - f*x] + Cos[e + f*x])*Sec[e]^2*Sec[e + f*x]*((I/2)*Cos[3*e] + Sin[3*e]/2))/(-1 + n) + (Sec[e + f*x]^2*(I*Cos[3*e] + Sin[3*e]))/(-2 + n))*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(d*Cot[e + f*x])^n*((Sec[e]^2*(-1 + Cos[2*e] + (3*I)*Sin[2*e])*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/(-1 + n) + (Sec[e]^2*Sec[e + f*x]*((-1/2*I)*Cos[3*e] - Sin[3*e]/2)*(-Cos[e - f*x] + Cos[e + f*x] - (3*I)*Sin[e - f*x] + (3*I)*Sin[e + f*x]))/(-1 + n))*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3)","B",0
789,1,198,72,2.0440467,"\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^2 \, dx","Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^2,x]","-\frac{e^{-2 i e} \left(1+e^{2 i (e+f x)}\right)^{-n} \left(\frac{i \left(1+e^{2 i (e+f x)}\right)}{-1+e^{2 i (e+f x)}}\right)^{n-1} \cos ^2(e+f x) (a+i a \tan (e+f x))^2 \left(2^n \left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1-n,1-n;2-n;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)-\left(1+e^{2 i (e+f x)}\right)^n\right) \cot ^{-n}(e+f x) (d \cot (e+f x))^n}{f (n-1) (\cos (f x)+i \sin (f x))^2}","\frac{a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac{2 a^2 d (d \cot (e+f x))^{n-1} \, _2F_1(1,n-1;n;-i \cot (e+f x))}{f (1-n)}",1,"-((((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^(-1 + n)*Cos[e + f*x]^2*(d*Cot[e + f*x])^n*(-(1 + E^((2*I)*(e + f*x)))^n + 2^n*(1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1 - n, 1 - n, 2 - n, (1 - E^((2*I)*(e + f*x)))/2])*(a + I*a*Tan[e + f*x])^2)/(E^((2*I)*e)*(1 + E^((2*I)*(e + f*x)))^n*f*(-1 + n)*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])^2))","B",0
790,1,166,37,0.8402982,"\int (d \cot (e+f x))^n (a+i a \tan (e+f x)) \, dx","Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x]),x]","-\frac{e^{-i e} 2^{n-1} \left(1+e^{2 i (e+f x)}\right)^{1-n} \left(\frac{i \left(1+e^{2 i (e+f x)}\right)}{-1+e^{2 i (e+f x)}}\right)^{n-1} \cos (e+f x) (a+i a \tan (e+f x)) \, _2F_1\left(1-n,1-n;2-n;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right) \cot ^{-n}(e+f x) (d \cot (e+f x))^n}{f (n-1) (\cos (f x)+i \sin (f x))}","-\frac{i a (d \cot (e+f x))^n \, _2F_1(1,n;n+1;-i \cot (e+f x))}{f n}",1,"-((2^(-1 + n)*(1 + E^((2*I)*(e + f*x)))^(1 - n)*((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^(-1 + n)*Cos[e + f*x]*(d*Cot[e + f*x])^n*Hypergeometric2F1[1 - n, 1 - n, 2 - n, (1 - E^((2*I)*(e + f*x)))/2]*(a + I*a*Tan[e + f*x]))/(E^(I*e)*f*(-1 + n)*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])))","B",1
791,0,0,157,2.7040909,"\int \frac{(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx","Integrate[(d*Cot[e + f*x])^n/(a + I*a*Tan[e + f*x]),x]","\int \frac{(d \cot (e+f x))^n}{a+i a \tan (e+f x)} \, dx","\frac{(n+1) (d \cot (e+f x))^{n+3} \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right)}{2 a d^3 f (n+3)}-\frac{i n (d \cot (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\cot ^2(e+f x)\right)}{2 a d^2 f (n+2)}-\frac{(d \cot (e+f x))^{n+2}}{2 d^2 f (a \cot (e+f x)+i a)}",1,"Integrate[(d*Cot[e + f*x])^n/(a + I*a*Tan[e + f*x]), x]","F",-1
792,0,0,202,20.0143228,"\int \frac{(d \cot (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(d*Cot[e + f*x])^n/(a + I*a*Tan[e + f*x])^2,x]","\int \frac{(d \cot (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","\frac{i n (n+2) (d \cot (e+f x))^{n+4} \, _2F_1\left(1,\frac{n+4}{2};\frac{n+6}{2};-\cot ^2(e+f x)\right)}{4 a^2 d^4 f (n+4)}+\frac{(n+1)^2 (d \cot (e+f x))^{n+3} \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right)}{4 a^2 d^3 f (n+3)}-\frac{i n (d \cot (e+f x))^{n+3}}{4 a^2 d^3 f (\cot (e+f x)+i)}-\frac{(d \cot (e+f x))^{n+3}}{4 d^3 f (a \cot (e+f x)+i a)^2}",1,"Integrate[(d*Cot[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x]","F",-1
793,0,0,95,5.6802423,"\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^m \, dx","Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^m,x]","\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^m \, dx","\frac{\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m (d \cot (e+f x))^n F_1(1-n;1-m,1;2-n;-i \tan (e+f x),i \tan (e+f x))}{f (1-n)}",1,"Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^m, x]","F",-1
794,0,0,79,11.3726776,"\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n,x]","\int \cot ^{\frac{3}{2}}(c+d x) (a+i a \tan (c+d x))^n \, dx","-\frac{2 \sqrt{\cot (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(-\frac{1}{2};1-n,1;\frac{1}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d}",1,"Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n, x]","F",-1
795,0,0,79,5.6957652,"\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^n \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n,x]","\int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^n \, dx","\frac{2 (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{d \sqrt{\cot (c+d x)}}",1,"Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n, x]","F",-1
796,0,0,81,6.6514337,"\int \frac{(a+i a \tan (c+d x))^n}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/Sqrt[Cot[c + d*x]],x]","\int \frac{(a+i a \tan (c+d x))^n}{\sqrt{\cot (c+d x)}} \, dx","\frac{2 (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{3}{2};1-n,1;\frac{5}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"Integrate[(a + I*a*Tan[c + d*x])^n/Sqrt[Cot[c + d*x]], x]","F",-1
797,0,0,81,5.0894031,"\int \frac{(a+i a \tan (c+d x))^n}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + I*a*Tan[c + d*x])^n/Cot[c + d*x]^(3/2),x]","\int \frac{(a+i a \tan (c+d x))^n}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","\frac{2 (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left(\frac{5}{2};1-n,1;\frac{7}{2};-i \tan (c+d x),i \tan (c+d x)\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"Integrate[(a + I*a*Tan[c + d*x])^n/Cot[c + d*x]^(3/2), x]","F",-1
798,1,66,202,0.2024637,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x]),x]","-\frac{2 \cot ^{\frac{3}{2}}(c+d x) \left(3 a \cot (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right)+5 b \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)\right)}{15 d}","\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a \cot ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a \sqrt{\cot (c+d x)}}{d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(-2*Cot[c + d*x]^(3/2)*(3*a*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Tan[c + d*x]^2] + 5*b*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2]))/(15*d)","C",1
799,1,65,184,0.1677239,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x]),x]","-\frac{2 \sqrt{\cot (c+d x)} \left(a \cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\tan ^2(c+d x)\right)+3 b \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)\right)}{3 d}","\frac{(a-b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 b \sqrt{\cot (c+d x)}}{d}",1,"(-2*Sqrt[Cot[c + d*x]]*(a*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*b*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]))/(3*d)","C",1
800,1,153,166,0.3193878,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x]),x]","-\frac{\sqrt{\cot (c+d x)} \left(8 a \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right)+\sqrt{2} b \sqrt{\tan (c+d x)} \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{4 d}","-\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a \sqrt{\cot (c+d x)}}{d}",1,"-1/4*(Sqrt[Cot[c + d*x]]*(8*a*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2] + Sqrt[2]*b*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]]))/d","C",1
801,1,164,150,0.2560213,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x)) \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(3 \sqrt{2} a \left(-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+8 b \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)\right)}{12 d}","-\frac{(a-b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*Sqrt[2]*a*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*b*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d)","C",1
802,1,194,166,0.3672401,"\int \frac{a+b \tan (c+d x)}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(8 a \tan ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)+3 b \left(2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)\right)}{12 d}","\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b}{d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*b*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]) + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]]) + 8*a*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d)","C",1
803,1,194,184,0.6266156,"\int \frac{a+b \tan (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(3 a \left(2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-8 b \tan ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-1\right)\right)}{12 d}","\frac{(a-b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 a}{d \sqrt{\cot (c+d x)}}+\frac{2 b}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*a*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*Sqrt[Tan[c + d*x]]) - 8*b*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Tan[c + d*x]^(3/2)))/(12*d)","C",1
804,1,207,202,0.9810687,"\int \frac{a+b \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])/Cot[c + d*x]^(5/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(3 b \left(-10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)+10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+8 \tan ^{\frac{5}{2}}(c+d x)-40 \sqrt{\tan (c+d x)}-5 \sqrt{2} \log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+5 \sqrt{2} \log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-40 a \tan ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(c+d x)\right)-1\right)\right)}{60 d}","-\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 a}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 b}{5 d \cot ^{\frac{5}{2}}(c+d x)}-\frac{2 b}{d \sqrt{\cot (c+d x)}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-40*a*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Tan[c + d*x]^(3/2) + 3*b*(-10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - 40*Sqrt[Tan[c + d*x]] + 8*Tan[c + d*x]^(5/2))))/(60*d)","C",1
805,1,215,268,1.659592,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{2}{3} \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+\frac{2}{7} a^2 \cot ^{\frac{7}{2}}(c+d x)+\frac{1}{10} a b \left(8 \cot ^{\frac{5}{2}}(c+d x)-40 \sqrt{\cot (c+d x)}-5 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+5 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-10 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+10 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{d}","\frac{2 \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x)}{7 d}-\frac{4 a b \cot ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a b \sqrt{\cot (c+d x)}}{d}",1,"-(((2*a^2*Cot[c + d*x]^(7/2))/7 + (2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (a*b*(-10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 40*Sqrt[Cot[c + d*x]] + 8*Cot[c + d*x]^(5/2) - 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/10)/d)","C",1
806,1,202,249,1.5306714,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2,x]","-\frac{-\frac{1}{4} \left(a^2-b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+\frac{2}{5} a^2 \cot ^{\frac{5}{2}}(c+d x)-\frac{4}{3} a b \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)}{d}","\frac{2 \left(a^2-b^2\right) \sqrt{\cot (c+d x)}}{d}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x)}{5 d}-\frac{4 a b \cot ^{\frac{3}{2}}(c+d x)}{3 d}",1,"-(((2*a^2*Cot[c + d*x]^(5/2))/5 - (4*a*b*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 - ((a^2 - b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4)/d)","C",1
807,1,199,223,0.4540577,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2,x]","\frac{4 \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-a \left(4 a \cot ^{\frac{3}{2}}(c+d x)+24 b \sqrt{\cot (c+d x)}+3 \sqrt{2} b \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} b \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{6 d}","\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \sqrt{\cot (c+d x)}}{d}",1,"(4*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - a*(6*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*b*Sqrt[Cot[c + d*x]] + 4*a*Cot[c + d*x]^(3/2) + 3*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(6*d)","C",1
808,1,170,204,0.5652628,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{\left(a^2-b^2\right) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{2 \sqrt{2}}+2 a^2 \sqrt{\cot (c+d x)}+\frac{4}{3} a b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{d}","-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 \sqrt{\cot (c+d x)}}{d}",1,"-((2*a^2*Sqrt[Cot[c + d*x]] + (4*a*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 + ((a^2 - b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*Sqrt[2]))/d)","C",1
809,1,173,204,0.80526,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^2 \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{2 \left(a^2-b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)}{\sqrt{\cot (c+d x)}}-\frac{2 a^2}{\sqrt{\cot (c+d x)}}+\frac{a b \left(\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{\sqrt{2}}-\sqrt{2} a b \left(\tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{d}","-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2}{d \sqrt{\cot (c+d x)}}",1,"-((-(Sqrt[2]*a*b*(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])) - (2*a^2)/Sqrt[Cot[c + d*x]] + (2*(a^2 - b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])/Sqrt[Cot[c + d*x]] + (a*b*(-Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/Sqrt[2])/d)","C",1
810,1,77,223,0.2872691,"\int \frac{(a+b \tan (c+d x))^2}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Sqrt[Cot[c + d*x]],x]","\frac{2 \left(\left(b^2-a^2\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+a \left(a+6 b \cot (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}","\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{2 b^2}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(2*((-a^2 + b^2)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + a*(a + 6*b*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2])))/(3*d*Cot[c + d*x]^(3/2))","C",1
811,1,81,249,0.3140924,"\int \frac{(a+b \tan (c+d x))^2}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Cot[c + d*x]^(3/2),x]","\frac{2 a \left(3 a+10 b \cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)\right)-6 \left(a^2-b^2\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)}{15 d \cot ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(a^2-b^2\right)}{d \sqrt{\cot (c+d x)}}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{4 a b}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"(-6*(a^2 - b^2)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + 2*a*(3*a + 10*b*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2]))/(15*d*Cot[c + d*x]^(5/2))","C",1
812,1,80,268,0.5116203,"\int \frac{(a+b \tan (c+d x))^2}{\cot ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^2/Cot[c + d*x]^(5/2),x]","\frac{2 \left(a \left(5 a+14 b \cot (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)\right)-5 \left(a^2-b^2\right) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(c+d x)\right)\right)}{35 d \cot ^{\frac{7}{2}}(c+d x)}","\frac{2 \left(a^2-b^2\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{4 a b}{5 d \cot ^{\frac{5}{2}}(c+d x)}-\frac{4 a b}{d \sqrt{\cot (c+d x)}}+\frac{2 b^2}{7 d \cot ^{\frac{7}{2}}(c+d x)}",1,"(2*(-5*(a^2 - b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2] + a*(5*a + 14*b*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2])))/(35*d*Cot[c + d*x]^(7/2))","C",1
813,1,229,299,2.422519,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{2}{7} a^3 \cot ^{\frac{7}{2}}(c+d x)+\frac{2}{3} a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+\frac{1}{4} b \left(b^2-3 a^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+\frac{6}{5} a^2 b \cot ^{\frac{5}{2}}(c+d x)}{d}","\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b \left(3 a^2-b^2\right) \sqrt{\cot (c+d x)}}{d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}-\frac{32 a^2 b \cot ^{\frac{5}{2}}(c+d x)}{35 d}",1,"-(((6*a^2*b*Cot[c + d*x]^(5/2))/5 + (2*a^3*Cot[c + d*x]^(7/2))/7 + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (b*(-3*a^2 + b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4)/d)","C",1
814,1,225,270,2.3482666,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{2}{5} a^3 \cot ^{\frac{5}{2}}(c+d x)+\frac{2}{3} b \left(b^2-3 a^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-\frac{1}{4} a \left(a^2-3 b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+2 a^2 b \cot ^{\frac{3}{2}}(c+d x)}{d}","\frac{2 a \left(a^2-3 b^2\right) \sqrt{\cot (c+d x)}}{d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{8 a^2 b \cot ^{\frac{3}{2}}(c+d x)}{5 d}-\frac{2 a^2 \cot ^{\frac{3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}",1,"-((2*a^2*b*Cot[c + d*x]^(3/2) + (2*a^3*Cot[c + d*x]^(5/2))/5 + (2*b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 - (a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4)/d)","C",1
815,1,194,245,1.1626929,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3,x]","-\frac{2 \left(a^3 \cot ^{\frac{3}{2}}(c+d x)-a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-\frac{3 b \left(b^2-3 a^2\right) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 \sqrt{2}}+9 a^2 b \sqrt{\cot (c+d x)}\right)}{3 d}","\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}-\frac{2 a^2 \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)}{3 d}-\frac{16 a^2 b \sqrt{\cot (c+d x)}}{3 d}",1,"(-2*(9*a^2*b*Sqrt[Cot[c + d*x]] + a^3*Cot[c + d*x]^(3/2) - a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - (3*b*(-3*a^2 + b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*Sqrt[2])))/(3*d)","C",1
816,1,188,245,3.521133,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3,x]","-\frac{2 \left(a^3 \cot (c+d x)-b \left(b^2-3 a^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)+\frac{a \left(a^2-3 b^2\right) \sqrt{\cot (c+d x)} \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{4 \sqrt{2}}-3 a^2 b\right)}{d \sqrt{\cot (c+d x)}}","-\frac{2 a \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}{d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (a \cot (c+d x)+b)}{d \sqrt{\cot (c+d x)}}",1,"(-2*(-3*a^2*b + a^3*Cot[c + d*x] - b*(-3*a^2 + b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + (a*(a^2 - 3*b^2)*Sqrt[Cot[c + d*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*Sqrt[2])))/(d*Sqrt[Cot[c + d*x]])","C",1
817,1,100,245,0.3914338,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^3 \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3,x]","\frac{\left(2 b^3-6 a^2 b\right) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)+6 a \left(a (a \cot (c+d x)+b)-\left(a^2-3 b^2\right) \cot (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}","-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{16 a b^2}{3 d \sqrt{\cot (c+d x)}}",1,"((-6*a^2*b + 2*b^3)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + 6*a*(a*(b + a*Cot[c + d*x]) - (a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2]))/(3*d*Cot[c + d*x]^(3/2))","C",1
818,1,102,272,0.6775394,"\int \frac{(a+b \tan (c+d x))^3}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Sqrt[Cot[c + d*x]],x]","\frac{2 \left(\left(3 b^3-9 a^2 b\right) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)+a \left(a (5 a \cot (c+d x)+9 b)-5 \left(a^2-3 b^2\right) \cot (c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};-\cot ^2(c+d x)\right)\right)\right)}{15 d \cot ^{\frac{5}{2}}(c+d x)}","\frac{2 b \left(3 a^2-b^2\right)}{d \sqrt{\cot (c+d x)}}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{8 a b^2}{5 d \cot ^{\frac{3}{2}}(c+d x)}",1,"(2*((-9*a^2*b + 3*b^3)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + a*(a*(9*b + 5*a*Cot[c + d*x]) - 5*(a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*Cot[c + d*x]^(5/2))","C",1
819,1,101,299,0.6785873,"\int \frac{(a+b \tan (c+d x))^3}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^3/Cot[c + d*x]^(3/2),x]","\frac{2 \left(5 b \left(b^2-3 a^2\right) \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};-\cot ^2(c+d x)\right)+a \left(a (7 a \cot (c+d x)+15 b)-7 \left(a^2-3 b^2\right) \cot (c+d x) \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};-\cot ^2(c+d x)\right)\right)\right)}{35 d \cot ^{\frac{7}{2}}(c+d x)}","\frac{2 b \left(3 a^2-b^2\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(a^2-3 b^2\right)}{d \sqrt{\cot (c+d x)}}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d}+\frac{2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac{7}{2}}(c+d x)}+\frac{32 a b^2}{35 d \cot ^{\frac{5}{2}}(c+d x)}",1,"(2*(5*b*(-3*a^2 + b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2] + a*(a*(15*b + 7*a*Cot[c + d*x]) - 7*(a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2])))/(35*d*Cot[c + d*x]^(7/2))","C",1
820,1,303,271,0.4664234,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x]),x]","\frac{-8 a^{3/2} b^2 \cot ^{\frac{3}{2}}(c+d x)+24 a^{5/2} b \sqrt{\cot (c+d x)}+3 \sqrt{2} a^{5/2} b \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} a^{5/2} b \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} a^{5/2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} a^{5/2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+8 a^{7/2} \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-8 a^{7/2} \cot ^{\frac{3}{2}}(c+d x)-24 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)+24 \sqrt{a} b^3 \sqrt{\cot (c+d x)}}{12 a^{5/2} d \left(a^2+b^2\right)}","\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b \sqrt{\cot (c+d x)}}{a^2 d}-\frac{2 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{5/2} d \left(a^2+b^2\right)}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d}",1,"(6*Sqrt[2]*a^(5/2)*b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*a^(5/2)*b*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 24*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] + 24*a^(5/2)*b*Sqrt[Cot[c + d*x]] + 24*Sqrt[a]*b^3*Sqrt[Cot[c + d*x]] - 8*a^(7/2)*Cot[c + d*x]^(3/2) - 8*a^(3/2)*b^2*Cot[c + d*x]^(3/2) + 8*a^(7/2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 3*Sqrt[2]*a^(5/2)*b*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*a^(5/2)*b*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(12*a^(5/2)*(a^2 + b^2)*d)","C",1
821,1,264,250,0.4950806,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x]),x]","\frac{8 a^{3/2} b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \left(8 a^{5/2} \sqrt{\cot (c+d x)}+\sqrt{2} a^{5/2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} a^{5/2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} a^{5/2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} a^{5/2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-8 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)+8 \sqrt{a} b^2 \sqrt{\cot (c+d x)}\right)}{12 a^{3/2} d \left(a^2+b^2\right)}","-\frac{(a-b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2} d \left(a^2+b^2\right)}-\frac{2 \sqrt{\cot (c+d x)}}{a d}",1,"(8*a^(3/2)*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 3*(2*Sqrt[2]*a^(5/2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*a^(5/2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - 8*b^(5/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] + 8*a^(5/2)*Sqrt[Cot[c + d*x]] + 8*Sqrt[a]*b^2*Sqrt[Cot[c + d*x]] + Sqrt[2]*a^(5/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*a^(5/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(12*a^(3/2)*(a^2 + b^2)*d)","C",1
822,1,227,232,0.2591709,"\int \frac{\sqrt{\cot (c+d x)}}{a+b \tan (c+d x)} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x]),x]","\frac{-8 a^{3/2} \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 b \left(8 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)+\sqrt{2} \sqrt{a} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \sqrt{a} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \sqrt{a} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \sqrt{a} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)}{12 \sqrt{a} d \left(a^2+b^2\right)}","-\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a} d \left(a^2+b^2\right)}",1,"(-8*a^(3/2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 3*b*(2*Sqrt[2]*Sqrt[a]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*Sqrt[a]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] + Sqrt[2]*Sqrt[a]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Sqrt[a]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(12*Sqrt[a]*(a^2 + b^2)*d)","C",1
823,1,204,232,0.1744509,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])),x]","\frac{24 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)+3 \sqrt{2} a \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-3 \sqrt{2} a \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+6 \sqrt{2} a \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-6 \sqrt{2} a \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-8 b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{12 d \left(a^2+b^2\right)}","\frac{(a-b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 \sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{d \left(a^2+b^2\right)}",1,"(6*Sqrt[2]*a*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 6*Sqrt[2]*a*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] - 8*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 3*Sqrt[2]*a*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*a*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(12*(a^2 + b^2)*d)","C",1
824,1,226,232,0.1818977,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]","-\frac{-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)}-\frac{b \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{8 \left(a^2+b^2\right)}+\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} \left(a^2+b^2\right)}}{d}","\frac{(a+b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a-b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}-\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d \left(a^2+b^2\right)}",1,"-(((2*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)) - (2*a*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)) - (b*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2)))/d)","C",1
825,1,193,250,0.4345463,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])),x]","\frac{8 a^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{a \cot (c+d x)}{b}\right)+b \left(\sqrt{2} a \sqrt{\cot (c+d x)} \left(-\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+8 b \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)}{4 b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}","-\frac{(a-b) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}+\frac{(a-b) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)}-\frac{(a+b) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{(a+b) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)}+\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d \left(a^2+b^2\right)}+\frac{2}{b d \sqrt{\cot (c+d x)}}",1,"(8*a^2*Hypergeometric2F1[-1/2, 1, 1/2, -((a*Cot[c + d*x])/b)] + b*(8*b*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a*Sqrt[Cot[c + d*x]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(4*b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]])","C",1
826,1,455,398,6.2033047,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{2 a^2 \cot ^{\frac{11}{2}}(c+d x) \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};-\frac{a \cot (c+d x)}{b}\right)}{11 b^2 \left(a^2+b^2\right)}+\frac{2 \left(a^2-b^2\right) \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^2}+\frac{4 a b \cot ^{\frac{9}{2}}(c+d x)}{9 \left(a^2+b^2\right)^2}-\frac{a b \left(40 \cot ^{\frac{9}{2}}(c+d x)-72 \cot ^{\frac{5}{2}}(c+d x)+360 \sqrt{\cot (c+d x)}+45 \left(\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{90 \left(a^2+b^2\right)^2}-\frac{4 b^2 \left(15 \cot ^{\frac{7}{2}}(c+d x)-7 b \left(\frac{3 \cot ^{\frac{5}{2}}(c+d x)}{a}-\frac{5 b \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{a}-\frac{3 b \left(\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2}}\right)}{a}\right)}{a}\right)\right)}{105 \left(a^2+b^2\right)^2}}{d}","\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{a d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(2 a^2+5 b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2+b^2\right)}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{b^{7/2} \left(9 a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{7/2} d \left(a^2+b^2\right)^2}+\frac{b \left(4 a^2+5 b^2\right) \sqrt{\cot (c+d x)}}{a^3 d \left(a^2+b^2\right)}",1,"-(((4*a*b*Cot[c + d*x]^(9/2))/(9*(a^2 + b^2)^2) - (4*b^2*(15*Cot[c + d*x]^(7/2) - 7*b*((3*Cot[c + d*x]^(5/2))/a - (5*b*((-3*b*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/a^(3/2)) + Sqrt[Cot[c + d*x]]/a))/a + Cot[c + d*x]^(3/2)/a))/a)))/(105*(a^2 + b^2)^2) + (2*(a^2 - b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((a*Cot[c + d*x])/b)])/(11*b^2*(a^2 + b^2)) - (a*b*(360*Sqrt[Cot[c + d*x]] - 72*Cot[c + d*x]^(5/2) + 40*Cot[c + d*x]^(9/2) + 45*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(90*(a^2 + b^2)^2))/d)","C",1
827,1,424,357,6.1517683,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-\frac{a \cot (c+d x)}{b}\right)}{9 b^2 \left(a^2+b^2\right)}+\frac{4 a b \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^2}+\frac{4 a b \cot ^{\frac{7}{2}}(c+d x)}{7 \left(a^2+b^2\right)^2}+\frac{\left(a^2-b^2\right) \left(-8 \cot ^{\frac{5}{2}}(c+d x)+40 \sqrt{\cot (c+d x)}+\frac{5}{2} \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{20 \left(a^2+b^2\right)^2}-\frac{4 b^2 \left(3 \cot ^{\frac{5}{2}}(c+d x)-5 b \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{a}-\frac{3 b \left(\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2}}\right)}{a}\right)\right)}{15 \left(a^2+b^2\right)^2}}{d}","\frac{b^2 \cot ^{\frac{3}{2}}(c+d x)}{a d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(2 a^2+3 b^2\right) \sqrt{\cot (c+d x)}}{a^2 d \left(a^2+b^2\right)}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{b^{5/2} \left(7 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{5/2} d \left(a^2+b^2\right)^2}",1,"-(((4*a*b*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^2) - (4*b^2*(3*Cot[c + d*x]^(5/2) - 5*b*((-3*b*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/a^(3/2)) + Sqrt[Cot[c + d*x]]/a))/a + Cot[c + d*x]^(3/2)/a)))/(15*(a^2 + b^2)^2) + (4*a*b*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((a*Cot[c + d*x])/b)])/(9*b^2*(a^2 + b^2)) + ((a^2 - b^2)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2))/(20*(a^2 + b^2)^2))/d)","C",1
828,1,368,318,1.7946644,"\int \frac{\sqrt{\cot (c+d x)}}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^2,x]","\frac{7 b^2 \left(4 a^{3/2} b^2 \cot ^{\frac{3}{2}}(c+d x)-24 a^{5/2} b \sqrt{\cot (c+d x)}-3 \sqrt{2} a^{5/2} b \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+3 \sqrt{2} a^{5/2} b \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-6 \sqrt{2} a^{5/2} b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+6 \sqrt{2} a^{5/2} b \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 a^{7/2} \cot ^{\frac{3}{2}}(c+d x)+24 b^{7/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)-24 \sqrt{a} b^3 \sqrt{\cot (c+d x)}\right)-12 a^{7/2} \left(a^2+b^2\right) \cot ^{\frac{7}{2}}(c+d x) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{a \cot (c+d x)}{b}\right)-28 a^{3/2} b^2 \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{42 a^{3/2} b^2 d \left(a^2+b^2\right)^2}","\frac{b^2 \sqrt{\cot (c+d x)}}{a d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{b^{3/2} \left(5 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2} d \left(a^2+b^2\right)^2}",1,"(-28*a^(3/2)*b^2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - 12*a^(7/2)*(a^2 + b^2)*Cot[c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((a*Cot[c + d*x])/b)] + 7*b^2*(-6*Sqrt[2]*a^(5/2)*b*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 6*Sqrt[2]*a^(5/2)*b*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 24*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] - 24*a^(5/2)*b*Sqrt[Cot[c + d*x]] - 24*Sqrt[a]*b^3*Sqrt[Cot[c + d*x]] + 4*a^(7/2)*Cot[c + d*x]^(3/2) + 4*a^(3/2)*b^2*Cot[c + d*x]^(3/2) - 3*Sqrt[2]*a^(5/2)*b*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + 3*Sqrt[2]*a^(5/2)*b*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(42*a^(3/2)*b^2*(a^2 + b^2)^2*d)","C",1
829,1,301,315,5.1001994,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2),x]","-\frac{\frac{24 a^2 \left(a^2+b^2\right) \cot ^{\frac{5}{2}}(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{a \cot (c+d x)}{b}\right)}{b^2}-15 \left(a^2-b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-240 b^2 \left(\sqrt{\cot (c+d x)}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a}}\right)+80 a b \cot ^{\frac{3}{2}}(c+d x) \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-1\right)+80 a b \cot ^{\frac{3}{2}}(c+d x)}{60 d \left(a^2+b^2\right)^2}","-\frac{b \sqrt{\cot (c+d x)}}{d \left(a^2+b^2\right) (a \cot (c+d x)+b)}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\sqrt{b} \left(3 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}",1,"-1/60*(-240*b^2*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/Sqrt[a]) + Sqrt[Cot[c + d*x]]) + 80*a*b*Cot[c + d*x]^(3/2) + 80*a*b*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]) + (24*a^2*(a^2 + b^2)*Cot[c + d*x]^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((a*Cot[c + d*x])/b)])/b^2 - 15*(a^2 - b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/((a^2 + b^2)^2*d)","C",1
830,1,380,313,6.1057758,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]","-\frac{-\frac{2 \left(a^2-b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)^2}+\frac{4 a b \sqrt{\cot (c+d x)}}{\left(a^2+b^2\right)^2}-\frac{\sqrt{a} \left(\sqrt{a} \sqrt{b} \sqrt{\cot (c+d x)}-b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)-a \cot (c+d x) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)\right)}{\sqrt{b} \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{a b \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{2 \left(a^2+b^2\right)^2}-\frac{4 \sqrt{a} b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\left(a^2+b^2\right)^2}}{d}","\frac{a \sqrt{\cot (c+d x)}}{d \left(a^2+b^2\right) (a \cot (c+d x)+b)}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\sqrt{a} \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}",1,"-(((-4*Sqrt[a]*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^2 + b^2)^2 + (4*a*b*Sqrt[Cot[c + d*x]])/(a^2 + b^2)^2 - (Sqrt[a]*(-(b*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]) + Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d*x]] - a*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]*Cot[c + d*x]))/(Sqrt[b]*(a^2 + b^2)*(b + a*Cot[c + d*x])) - (2*(a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^2) - (a*b*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(2*(a^2 + b^2)^2))/d)","C",1
831,1,279,319,2.8576746,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2),x]","-\frac{96 a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)+\frac{24 a^2 \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \left(\frac{b}{a \cot (c+d x)+b}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a} \sqrt{\cot (c+d x)}}\right)}{b^2}+6 \sqrt{2} \left(a^2-b^2\right) \left(\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-32 a b \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{24 d \left(a^2+b^2\right)^2}","-\frac{a^2 \sqrt{\cot (c+d x)}}{b d \left(a^2+b^2\right) (a \cot (c+d x)+b)}-\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{a^{3/2} \left(a^2+5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d \left(a^2+b^2\right)^2}",1,"-1/24*(96*a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] + (24*a^2*(a^2 + b^2)*Sqrt[Cot[c + d*x]]*((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[a]*Sqrt[Cot[c + d*x]]) + b/(b + a*Cot[c + d*x])))/b^2 - 32*a*b*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + 6*Sqrt[2]*(a^2 - b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/((a^2 + b^2)^2*d)","C",1
832,1,239,357,0.6227636,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2),x]","\frac{8 a^2 b^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{a \cot (c+d x)}{b}\right)+4 a^2 \left(a^2+b^2\right) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};-\frac{a \cot (c+d x)}{b}\right)+b^2 \left(\sqrt{2} a b \sqrt{\cot (c+d x)} \left(-\log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+\log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)-4 \left(a^2-b^2\right) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};-\cot ^2(c+d x)\right)\right)}{2 b^2 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)}}","-\frac{a^2}{b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a \cot (c+d x)+b)}+\frac{3 a^2+2 b^2}{b^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}-\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2+2 a b-b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^2}+\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}-\frac{\left(a^2-2 a b-b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^2}+\frac{a^{5/2} \left(3 a^2+7 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{b^{5/2} d \left(a^2+b^2\right)^2}",1,"(8*a^2*b^2*Hypergeometric2F1[-1/2, 1, 1/2, -((a*Cot[c + d*x])/b)] + 4*a^2*(a^2 + b^2)*Hypergeometric2F1[-1/2, 2, 1/2, -((a*Cot[c + d*x])/b)] + b^2*(-4*(a^2 - b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a*b*Sqrt[Cot[c + d*x]]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(2*b^2*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]])","C",1
833,1,564,493,6.2804358,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{4 a^2 \cot ^{\frac{13}{2}}(c+d x) \, _2F_1\left(2,\frac{13}{2};\frac{15}{2};-\frac{a \cot (c+d x)}{b}\right)}{13 b \left(a^2+b^2\right)^2}+\frac{2 a \left(a^2-3 b^2\right) \left(-77 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)+21 \cot ^{\frac{11}{2}}(c+d x)-33 \cot ^{\frac{7}{2}}(c+d x)+77 \cot ^{\frac{3}{2}}(c+d x)\right)}{231 \left(a^2+b^2\right)^3}-\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{11}{2}}(c+d x)}{11 \left(a^2+b^2\right)^3}-\frac{b \left(3 a^2-b^2\right) \left(40 \cot ^{\frac{9}{2}}(c+d x)-72 \cot ^{\frac{5}{2}}(c+d x)+360 \sqrt{\cot (c+d x)}+45 \left(\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{180 \left(a^2+b^2\right)^3}+\frac{2 a^2 \cot ^{\frac{13}{2}}(c+d x) \, _2F_1\left(3,\frac{13}{2};\frac{15}{2};-\frac{a \cot (c+d x)}{b}\right)}{13 b^3 \left(a^2+b^2\right)}+\frac{2 b \left(a^2-3 b^2\right) \left(35 \cot ^{\frac{9}{2}}(c+d x)-3 b \left(\frac{15 \cot ^{\frac{7}{2}}(c+d x)}{a}-\frac{7 b \left(\frac{3 \cot ^{\frac{5}{2}}(c+d x)}{a}-\frac{5 b \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{a}-\frac{3 b \left(\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2}}\right)}{a}\right)}{a}\right)}{a}\right)\right)}{315 \left(a^2+b^2\right)^3}}{d}","\frac{b^2 \left(15 a^2+7 b^2\right) \cot ^{\frac{5}{2}}(c+d x)}{4 a^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{b^2 \cot ^{\frac{7}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b \left(24 a^4+67 a^2 b^2+35 b^4\right) \sqrt{\cot (c+d x)}}{4 a^4 d \left(a^2+b^2\right)^2}-\frac{b^{7/2} \left(99 a^4+102 a^2 b^2+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{9/2} d \left(a^2+b^2\right)^3}-\frac{\left(8 a^4+67 a^2 b^2+35 b^4\right) \cot ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2+b^2\right)^2}",1,"-(((-2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(11/2))/(11*(a^2 + b^2)^3) + (2*b*(a^2 - 3*b^2)*(35*Cot[c + d*x]^(9/2) - 3*b*((15*Cot[c + d*x]^(7/2))/a - (7*b*((3*Cot[c + d*x]^(5/2))/a - (5*b*((-3*b*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/a^(3/2)) + Sqrt[Cot[c + d*x]]/a))/a + Cot[c + d*x]^(3/2)/a))/a))/a)))/(315*(a^2 + b^2)^3) + (2*a*(a^2 - 3*b^2)*(77*Cot[c + d*x]^(3/2) - 33*Cot[c + d*x]^(7/2) + 21*Cot[c + d*x]^(11/2) - 77*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(231*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(13/2)*Hypergeometric2F1[2, 13/2, 15/2, -((a*Cot[c + d*x])/b)])/(13*b*(a^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(13/2)*Hypergeometric2F1[3, 13/2, 15/2, -((a*Cot[c + d*x])/b)])/(13*b^3*(a^2 + b^2)) - (b*(3*a^2 - b^2)*(360*Sqrt[Cot[c + d*x]] - 72*Cot[c + d*x]^(5/2) + 40*Cot[c + d*x]^(9/2) + 45*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(180*(a^2 + b^2)^3))/d)","C",1
834,1,530,444,6.2288843,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{4 a^2 \cot ^{\frac{11}{2}}(c+d x) \, _2F_1\left(2,\frac{11}{2};\frac{13}{2};-\frac{a \cot (c+d x)}{b}\right)}{11 b \left(a^2+b^2\right)^2}+\frac{2 b \left(3 a^2-b^2\right) \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^3}-\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{9}{2}}(c+d x)}{9 \left(a^2+b^2\right)^3}+\frac{a \left(a^2-3 b^2\right) \left(40 \cot ^{\frac{9}{2}}(c+d x)-72 \cot ^{\frac{5}{2}}(c+d x)+360 \sqrt{\cot (c+d x)}+45 \left(\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{180 \left(a^2+b^2\right)^3}+\frac{2 a^2 \cot ^{\frac{11}{2}}(c+d x) \, _2F_1\left(3,\frac{11}{2};\frac{13}{2};-\frac{a \cot (c+d x)}{b}\right)}{11 b^3 \left(a^2+b^2\right)}+\frac{2 b \left(a^2-3 b^2\right) \left(15 \cot ^{\frac{7}{2}}(c+d x)-7 b \left(\frac{3 \cot ^{\frac{5}{2}}(c+d x)}{a}-\frac{5 b \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{a}-\frac{3 b \left(\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2}}\right)}{a}\right)}{a}\right)\right)}{105 \left(a^2+b^2\right)^3}}{d}","\frac{b^2 \left(13 a^2+5 b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{b^2 \cot ^{\frac{5}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{b^{5/2} \left(63 a^4+46 a^2 b^2+15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{7/2} d \left(a^2+b^2\right)^3}-\frac{\left(8 a^4+31 a^2 b^2+15 b^4\right) \sqrt{\cot (c+d x)}}{4 a^3 d \left(a^2+b^2\right)^2}",1,"-(((-2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(9/2))/(9*(a^2 + b^2)^3) + (2*b*(a^2 - 3*b^2)*(15*Cot[c + d*x]^(7/2) - 7*b*((3*Cot[c + d*x]^(5/2))/a - (5*b*((-3*b*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/a^(3/2)) + Sqrt[Cot[c + d*x]]/a))/a + Cot[c + d*x]^(3/2)/a))/a)))/(105*(a^2 + b^2)^3) + (2*b*(3*a^2 - b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((a*Cot[c + d*x])/b)])/(11*b*(a^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(11/2)*Hypergeometric2F1[3, 11/2, 13/2, -((a*Cot[c + d*x])/b)])/(11*b^3*(a^2 + b^2)) + (a*(a^2 - 3*b^2)*(360*Sqrt[Cot[c + d*x]] - 72*Cot[c + d*x]^(5/2) + 40*Cot[c + d*x]^(9/2) + 45*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(180*(a^2 + b^2)^3))/d)","C",1
835,1,499,396,6.212961,"\int \frac{\sqrt{\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^3,x]","-\frac{\frac{4 a^2 \cot ^{\frac{9}{2}}(c+d x) \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};-\frac{a \cot (c+d x)}{b}\right)}{9 b \left(a^2+b^2\right)^2}-\frac{2 a \left(a^2-3 b^2\right) \left(-7 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-3 \cot ^{\frac{7}{2}}(c+d x)+7 \cot ^{\frac{3}{2}}(c+d x)\right)}{21 \left(a^2+b^2\right)^3}-\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{7}{2}}(c+d x)}{7 \left(a^2+b^2\right)^3}+\frac{b \left(3 a^2-b^2\right) \left(-8 \cot ^{\frac{5}{2}}(c+d x)+40 \sqrt{\cot (c+d x)}+\frac{5}{2} \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{20 \left(a^2+b^2\right)^3}+\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \, _2F_1\left(3,\frac{9}{2};\frac{11}{2};-\frac{a \cot (c+d x)}{b}\right)}{9 b^3 \left(a^2+b^2\right)}+\frac{2 b \left(a^2-3 b^2\right) \left(3 \cot ^{\frac{5}{2}}(c+d x)-5 b \left(\frac{\cot ^{\frac{3}{2}}(c+d x)}{a}-\frac{3 b \left(\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2}}\right)}{a}\right)\right)}{15 \left(a^2+b^2\right)^3}}{d}","\frac{b^2 \cot ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}+\frac{b^2 \left(11 a^2+3 b^2\right) \sqrt{\cot (c+d x)}}{4 a^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{b^{3/2} \left(35 a^4+6 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{5/2} d \left(a^2+b^2\right)^3}",1,"-(((-2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^3) + (2*b*(a^2 - 3*b^2)*(3*Cot[c + d*x]^(5/2) - 5*b*((-3*b*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/a^(3/2)) + Sqrt[Cot[c + d*x]]/a))/a + Cot[c + d*x]^(3/2)/a)))/(15*(a^2 + b^2)^3) - (2*a*(a^2 - 3*b^2)*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((a*Cot[c + d*x])/b)])/(9*b*(a^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[3, 9/2, 11/2, -((a*Cot[c + d*x])/b)])/(9*b^3*(a^2 + b^2)) + (b*(3*a^2 - b^2)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2))/(20*(a^2 + b^2)^3))/d)","C",1
836,1,462,392,6.1799643,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3),x]","-\frac{\frac{4 a^2 \cot ^{\frac{7}{2}}(c+d x) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{a \cot (c+d x)}{b}\right)}{7 b \left(a^2+b^2\right)^2}-\frac{2 b \left(3 a^2-b^2\right) \left(\cot ^{\frac{3}{2}}(c+d x)-\cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{3 \left(a^2+b^2\right)^3}-\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{5}{2}}(c+d x)}{5 \left(a^2+b^2\right)^3}-\frac{a \left(a^2-3 b^2\right) \left(-8 \cot ^{\frac{5}{2}}(c+d x)+40 \sqrt{\cot (c+d x)}+\frac{5}{2} \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)\right)}{20 \left(a^2+b^2\right)^3}+\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \, _2F_1\left(3,\frac{7}{2};\frac{9}{2};-\frac{a \cot (c+d x)}{b}\right)}{7 b^3 \left(a^2+b^2\right)}+\frac{2 b \left(a^2-3 b^2\right) \left(\cot ^{\frac{3}{2}}(c+d x)-3 b \left(\frac{\sqrt{\cot (c+d x)}}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{a^{3/2}}\right)\right)}{3 \left(a^2+b^2\right)^3}}{d}","\frac{b^2 \sqrt{\cot (c+d x)}}{2 a d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{b \left(9 a^2+b^2\right) \sqrt{\cot (c+d x)}}{4 a d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{\sqrt{b} \left(15 a^4-18 a^2 b^2-b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 a^{3/2} d \left(a^2+b^2\right)^3}",1,"-(((-2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 + b^2)^3) + (2*b*(a^2 - 3*b^2)*(-3*b*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/a^(3/2)) + Sqrt[Cot[c + d*x]]/a) + Cot[c + d*x]^(3/2)))/(3*(a^2 + b^2)^3) - (2*b*(3*a^2 - b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((a*Cot[c + d*x])/b)])/(7*b*(a^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, -((a*Cot[c + d*x])/b)])/(7*b^3*(a^2 + b^2)) - (a*(a^2 - 3*b^2)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2))/(20*(a^2 + b^2)^3))/d)","C",1
837,1,492,385,6.1914048,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]","-\frac{\frac{4 a^2 \cot ^{\frac{5}{2}}(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{a \cot (c+d x)}{b}\right)}{5 b \left(a^2+b^2\right)^2}+\frac{2 a \left(a^2-3 b^2\right) \left(\cot ^{\frac{3}{2}}(c+d x)-\cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)\right)}{3 \left(a^2+b^2\right)^3}-\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x)}{3 \left(a^2+b^2\right)^3}-\frac{\frac{2 a^2 \cot ^2(c+d x)}{(a \cot (c+d x)+b)^2}+\frac{3 a \cot (c+d x)}{a \cot (c+d x)+b}-\frac{3 \sqrt{a} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{b}}}{4 a \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}+\frac{2 b \left(a^2-3 b^2\right) \left(\sqrt{\cot (c+d x)}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a}}\right)}{\left(a^2+b^2\right)^3}-\frac{b \left(3 a^2-b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{4 \left(a^2+b^2\right)^3}}{d}","-\frac{b \sqrt{\cot (c+d x)}}{2 d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}+\frac{\left(5 a^2-3 b^2\right) \sqrt{\cot (c+d x)}}{4 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\left(3 a^4-26 a^2 b^2+3 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{a} \sqrt{b} d \left(a^2+b^2\right)^3}",1,"-(((2*b*(a^2 - 3*b^2)*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/Sqrt[a]) + Sqrt[Cot[c + d*x]]))/(a^2 + b^2)^3 - (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2))/(3*(a^2 + b^2)^3) - ((-3*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]*Sqrt[Cot[c + d*x]])/Sqrt[b] + (2*a^2*Cot[c + d*x]^2)/(b + a*Cot[c + d*x])^2 + (3*a*Cot[c + d*x])/(b + a*Cot[c + d*x]))/(4*a*(a^2 + b^2)*Sqrt[Cot[c + d*x]]) + (2*a*(a^2 - 3*b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, -((a*Cot[c + d*x])/b)])/(5*b*(a^2 + b^2)^2) - (b*(3*a^2 - b^2)*(2*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*(a^2 + b^2)^3))/d)","C",1
838,1,398,385,2.869559,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3),x]","-\frac{8 b \left(b^2-3 a^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)-24 a \left(a^2-3 b^2\right) \sqrt{\cot (c+d x)}+\frac{24 \sqrt{a} \sqrt{b} \left(a^2+b^2\right) \left((a \cot (c+d x)+b) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)-\sqrt{a} \sqrt{b} \sqrt{\cot (c+d x)}\right)}{a \cot (c+d x)+b}+24 \sqrt{a} \sqrt{b} \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)+3 a \left(a^2-3 b^2\right) \left(8 \sqrt{\cot (c+d x)}+\sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-\sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)+\frac{8 a^2 \left(a^2+b^2\right)^2 \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{2},3;\frac{5}{2};-\frac{a \cot (c+d x)}{b}\right)}{b^3}}{12 d \left(a^2+b^2\right)^3}","-\frac{a \left(a^2-7 b^2\right) \sqrt{\cot (c+d x)}}{4 b d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\frac{a \sqrt{\cot (c+d x)}}{2 d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{\sqrt{a} \left(a^4+18 a^2 b^2-15 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d \left(a^2+b^2\right)^3}",1,"-1/12*(24*Sqrt[a]*Sqrt[b]*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]] - 24*a*(a^2 - 3*b^2)*Sqrt[Cot[c + d*x]] + (24*Sqrt[a]*Sqrt[b]*(a^2 + b^2)*(-(Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d*x]]) + ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]*(b + a*Cot[c + d*x])))/(b + a*Cot[c + d*x]) + 8*b*(-3*a^2 + b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] + (8*a^2*(a^2 + b^2)^2*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, -((a*Cot[c + d*x])/b)])/b^3 + 3*a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/((a^2 + b^2)^3*d)","C",1
839,1,387,396,6.1628575,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3),x]","-\frac{\frac{2 a \left(a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\cot ^2(c+d x)\right)}{3 \left(a^2+b^2\right)^3}+\frac{2 a^2 \sqrt{\cot (c+d x)} \left(\frac{b}{a \cot (c+d x)+b}+\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{a} \sqrt{\cot (c+d x)}}\right)}{b \left(a^2+b^2\right)^2}+\frac{b \left(3 a^2-b^2\right) \left(2 \sqrt{2} \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)-2 \sqrt{2} \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)+4 \left(\sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)\right)\right)}{8 \left(a^2+b^2\right)^3}+\frac{2 a^2 \sqrt{\cot (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};-\frac{a \cot (c+d x)}{b}\right)}{b^3 \left(a^2+b^2\right)}-\frac{2 a^{3/2} \left(a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} \left(a^2+b^2\right)^3}}{d}","-\frac{a^2 \left(3 a^2+11 b^2\right) \sqrt{\cot (c+d x)}}{4 b^2 d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{a^2 \sqrt{\cot (c+d x)}}{2 b d \left(a^2+b^2\right) (a \cot (c+d x)+b)^2}-\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a-b) \left(a^2+4 a b+b^2\right) \log \left(\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{2 \sqrt{2} d \left(a^2+b^2\right)^3}+\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{\cot (c+d x)}\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{(a+b) \left(a^2-4 a b+b^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{\cot (c+d x)}+1\right)}{\sqrt{2} d \left(a^2+b^2\right)^3}-\frac{a^{3/2} \left(3 a^4+6 a^2 b^2+35 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right)}{4 b^{5/2} d \left(a^2+b^2\right)^3}",1,"-(((-2*a^(3/2)*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)^3) + (2*a^2*Sqrt[Cot[c + d*x]]*((Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[a]*Sqrt[Cot[c + d*x]]) + b/(b + a*Cot[c + d*x])))/(b*(a^2 + b^2)^2) + (2*a^2*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[1/2, 3, 3/2, -((a*Cot[c + d*x])/b)])/(b^3*(a^2 + b^2)) + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^3) + (b*(3*a^2 - b^2)*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2)^3))/d)","C",1
840,1,220,261,2.2555365,"\int \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{\cot (c+d x)} \left(-2 \sqrt{a+b \tan (c+d x)} \left(3 a^2 \cot ^2(c+d x)-15 a^2+a b \cot (c+d x)-2 b^2\right)+15 \sqrt[4]{-1} a^2 \sqrt{-a+i b} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+15 \sqrt[4]{-1} a^2 \sqrt{a+i b} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{15 a^2 d}","\frac{2 \left(15 a^2+2 b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}+\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*(15*(-1)^(1/4)*a^2*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + 15*(-1)^(1/4)*a^2*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 2*(-15*a^2 - 2*b^2 + a*b*Cot[c + d*x] + 3*a^2*Cot[c + d*x]^2)*Sqrt[a + b*Tan[c + d*x]]))/(15*a^2*d)","A",1
841,1,185,221,0.7634545,"\int \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]],x]","\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{(-1)^{3/4} \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(-1)^{3/4} \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 (a+b \tan (c+d x))^{3/2}}{3 a d \tan ^{\frac{3}{2}}(c+d x)}\right)","-\frac{2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{i \sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}+\frac{i \sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(((-1)^(3/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + ((-1)^(3/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*(a + b*Tan[c + d*x])^(3/2))/(3*a*d*Tan[c + d*x]^(3/2)))","A",1
842,1,174,179,0.181502,"\int \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]],x]","-\frac{\sqrt{\cot (c+d x)} \left(\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+2 \sqrt{a+b \tan (c+d x)}\right)}{d}","-\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"-((Sqrt[Cot[c + d*x]]*((-1)^(1/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + (-1)^(1/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] + 2*Sqrt[a + b*Tan[c + d*x]]))/d)","A",1
843,1,143,155,0.1585953,"\int \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)} \, dx","Integrate[Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]],x]","\frac{(-1)^{3/4} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","-\frac{i \sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{i \sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"((-1)^(3/4)*(Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
844,1,209,211,0.8725588,"\int \frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{\cot (c+d x)}} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+\sqrt[4]{-1} \left(\sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)\right)}{d}","\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{2 \sqrt{b} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(1/4)*(Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (2*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]]))/d","A",1
845,1,261,244,0.8119149,"\int \frac{\sqrt{a+b \tan (c+d x)}}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Tan[c + d*x]]/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{a} \sqrt{a+b \tan (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1} \left(-(-1)^{3/4} \sqrt{-a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+(-1)^{3/4} \sqrt{a+i b} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}\right)\right)}{\sqrt{b} d \sqrt{\frac{b \tan (c+d x)}{a}+1}}","\frac{i \sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{\sqrt{a+b \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{a \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}+\frac{i \sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[b]*Sqrt[1 + (b*Tan[c + d*x])/a]*(-((-1)^(3/4)*Sqrt[-a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]) + (-1)^(3/4)*Sqrt[a + I*b]*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])))/(Sqrt[b]*d*Sqrt[1 + (b*Tan[c + d*x])/a])","A",1
846,1,255,306,3.3230474,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\cot ^{\frac{7}{2}}(c+d x) \left(105 \sqrt[4]{-1} a^2 \sqrt{-a+i b} (b+i a) \tan ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-105 (-1)^{3/4} a^2 (a+i b)^{3/2} \tan ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+2 \sqrt{a+b \tan (c+d x)} \left(-15 a^3+2 b \left(70 a^2+3 b^2\right) \tan ^3(c+d x)+a \left(35 a^2-3 b^2\right) \tan ^2(c+d x)-24 a^2 b \tan (c+d x)\right)\right)}{105 a^2 d}","\frac{2 \left(35 a^2-3 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}+\frac{4 b \left(70 a^2+3 b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 a \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}-\frac{16 b \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 d}-\frac{(-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(7/2)*(105*(-1)^(1/4)*a^2*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) - 105*(-1)^(3/4)*a^2*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-15*a^3 - 24*a^2*b*Tan[c + d*x] + a*(35*a^2 - 3*b^2)*Tan[c + d*x]^2 + 2*b*(70*a^2 + 3*b^2)*Tan[c + d*x]^3)))/(105*a^2*d)","A",1
847,1,219,264,1.5745422,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \left(2 \sqrt{a+b \tan (c+d x)} \left(\left(5 a^2-b^2\right) \tan ^2(c+d x)-a^2-2 a b \tan (c+d x)\right)-5 \sqrt[4]{-1} a (-a+i b)^{3/2} \tan ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+5 \sqrt[4]{-1} a (a+i b)^{3/2} \tan ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{5 a d}","\frac{2 \left(5 a^2-b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{5 a d}-\frac{2 a \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{4 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{i (-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{i (b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(5/2)*(-5*(-1)^(1/4)*a*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 5*(-1)^(1/4)*a*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-a^2 - 2*a*b*Tan[c + d*x] + (5*a^2 - b^2)*Tan[c + d*x]^2)))/(5*a*d)","A",1
848,1,196,213,1.2940208,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \left(-3 \sqrt[4]{-1} \sqrt{-a+i b} (b+i a) \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+3 (-1)^{3/4} (a+i b)^{3/2} \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-2 \sqrt{a+b \tan (c+d x)} (a+4 b \tan (c+d x))\right)}{3 d}","-\frac{2 a \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{8 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(3/2)*(-3*(-1)^(1/4)*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + 3*(-1)^(3/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) - 2*Sqrt[a + b*Tan[c + d*x]]*(a + 4*b*Tan[c + d*x])))/(3*d)","A",1
849,1,175,185,0.5359807,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\cot (c+d x)} \left(\sqrt[4]{-1} (-a+i b)^{3/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt[4]{-1} (a+i b)^{3/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-2 a \sqrt{a+b \tan (c+d x)}\right)}{d}","\frac{i (-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{2 a \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}-\frac{i (b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*((-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - (-1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 2*a*Sqrt[a + b*Tan[c + d*x]]))/d","A",1
850,1,237,212,0.6291252,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2} \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{a} b^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt[4]{-1} \left(\sqrt{-a+i b} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\sqrt{a+i b} (b-i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right) \sqrt{a+b \tan (c+d x)}\right)}{d \sqrt{a+b \tan (c+d x)}}","\frac{2 b^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(1/4)*(Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + Sqrt[a + I*b]*((-I)*a + b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])*Sqrt[a + b*Tan[c + d*x]] + 2*Sqrt[a]*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a]))/(d*Sqrt[a + b*Tan[c + d*x]])","A",1
851,1,292,246,2.0758402,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(3 \sqrt{a} \sqrt{b} (a+b \tan (c+d x)) \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{\frac{b \tan (c+d x)}{a}+1} \left(-\sqrt[4]{-1} (-a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}+\sqrt[4]{-1} (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}+b \sqrt{\tan (c+d x)} (a+b \tan (c+d x))\right)\right)}{d \sqrt{a+b \tan (c+d x)} \sqrt{\frac{b \tan (c+d x)}{a}+1}}","-\frac{i (-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b \sqrt{a+b \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{3 a \sqrt{b} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{i (b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*(a + b*Tan[c + d*x]) + Sqrt[1 + (b*Tan[c + d*x])/a]*(-((-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]]) + (-1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + b*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))))/(d*Sqrt[a + b*Tan[c + d*x]]*Sqrt[1 + (b*Tan[c + d*x])/a])","A",1
852,1,312,286,5.8094017,"\int \frac{(a+b \tan (c+d x))^{3/2}}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(3/2)/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\sqrt{\tan (c+d x)} \left(5 a^2+7 a b \tan (c+d x)+2 b^2 \tan ^2(c+d x)\right)+\frac{\left(3 a^2-8 b^2\right) (a+b \tan (c+d x)) \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1}}+4 (-1)^{3/4} (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}-4 \sqrt[4]{-1} \sqrt{-a+i b} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}\right)}{4 d \sqrt{a+b \tan (c+d x)}}","\frac{\left(3 a^2-8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 \sqrt{b} d}+\frac{(-b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(a+b \tan (c+d x))^{3/2}}{2 d \sqrt{\cot (c+d x)}}+\frac{3 a \sqrt{a+b \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{(b+i a)^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-4*(-1)^(1/4)*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + 4*(-1)^(3/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + ((3*a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*(a + b*Tan[c + d*x]))/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*Tan[c + d*x])/a]) + Sqrt[Tan[c + d*x]]*(5*a^2 + 7*a*b*Tan[c + d*x] + 2*b^2*Tan[c + d*x]^2)))/(4*d*Sqrt[a + b*Tan[c + d*x]])","A",1
853,1,329,358,5.4508572,"\int \cot ^{\frac{11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{9}{2}}(c+d x) \left(-315 \sqrt[4]{-1} a^2 (-a+i b)^{5/2} \tan ^{\frac{9}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-315 \sqrt[4]{-1} a^2 (a+i b)^{5/2} \tan ^{\frac{9}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+\frac{1}{4} \sec ^4(c+d x) \sqrt{a+b \tan (c+d x)} \left(-987 a^4+272 a^3 b \sin (2 (c+d x))-326 a^3 b \sin (4 (c+d x))+1374 a^2 b^2+4 \left(280 a^4-483 a^2 b^2-10 b^4\right) \cos (2 (c+d x))+\left(-413 a^4+558 a^2 b^2+10 b^4\right) \cos (4 (c+d x))-10 a b^3 \sin (2 (c+d x))+5 a b^3 \sin (4 (c+d x))+30 b^4\right)\right)}{315 a^2 d}","\frac{2 \left(21 a^2-25 b^2\right) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 d}+\frac{2 b \left(231 a^2-5 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a d}-\frac{2 a^2 \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{2 \left(315 a^4-483 a^2 b^2-10 b^4\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{38 a b \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}+\frac{(-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(9/2)*(-315*(-1)^(1/4)*a^2*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(9/2) - 315*(-1)^(1/4)*a^2*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(9/2) + (Sec[c + d*x]^4*(-987*a^4 + 1374*a^2*b^2 + 30*b^4 + 4*(280*a^4 - 483*a^2*b^2 - 10*b^4)*Cos[2*(c + d*x)] + (-413*a^4 + 558*a^2*b^2 + 10*b^4)*Cos[4*(c + d*x)] + 272*a^3*b*Sin[2*(c + d*x)] - 10*a*b^3*Sin[2*(c + d*x)] - 326*a^3*b*Sin[4*(c + d*x)] + 5*a*b^3*Sin[4*(c + d*x)])*Sqrt[a + b*Tan[c + d*x]])/4))/(315*a^2*d)","A",1
854,1,274,310,3.51953,"\int \cot ^{\frac{9}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{7}{2}}(c+d x) \left(-\frac{1}{2} \sec ^3(c+d x) \sqrt{a+b \tan (c+d x)} \left(\left(10 a^3-9 a b^2\right) \cos (3 (c+d x))+a \left(2 a^2+9 b^2\right) \cos (c+d x)+2 b \sin (c+d x) \left(\left(58 a^2-3 b^2\right) \cos (2 (c+d x))-40 a^2+3 b^2\right)\right)+21 (-1)^{3/4} a (-a+i b)^{5/2} \tan ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-21 (-1)^{3/4} a (a+i b)^{5/2} \tan ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{21 a d}","\frac{2 \left(7 a^2-9 b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{21 d}+\frac{2 b \left(49 a^2-3 b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{21 a d}-\frac{2 a^2 \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}-\frac{6 a b \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}+\frac{i (-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{i (b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(7/2)*(21*(-1)^(3/4)*a*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) - 21*(-1)^(3/4)*a*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(7/2) - (Sec[c + d*x]^3*(a*(2*a^2 + 9*b^2)*Cos[c + d*x] + (10*a^3 - 9*a*b^2)*Cos[3*(c + d*x)] + 2*b*(-40*a^2 + 3*b^2 + (58*a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/2))/(21*a*d)","A",1
855,1,214,259,1.8482819,"\int \cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{5}{2}}(c+d x) \left(2 \sqrt{a+b \tan (c+d x)} \left(\left(15 a^2-23 b^2\right) \tan ^2(c+d x)-3 a^2-11 a b \tan (c+d x)\right)+15 \sqrt[4]{-1} (-a+i b)^{5/2} \tan ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+15 \sqrt[4]{-1} (a+i b)^{5/2} \tan ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{15 d}","\frac{2 \left(15 a^2-23 b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 d}-\frac{2 a^2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{22 a b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 d}-\frac{(-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(5/2)*(15*(-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 15*(-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(5/2) + 2*Sqrt[a + b*Tan[c + d*x]]*(-3*a^2 - 11*a*b*Tan[c + d*x] + (15*a^2 - 23*b^2)*Tan[c + d*x]^2)))/(15*d)","A",1
856,1,190,222,1.1048759,"\int \cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\cot ^{\frac{3}{2}}(c+d x) \left(-3 (-1)^{3/4} (-a+i b)^{5/2} \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+3 (-1)^{3/4} (a+i b)^{5/2} \tan ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-2 a \sqrt{a+b \tan (c+d x)} (a+7 b \tan (c+d x))\right)}{3 d}","-\frac{2 a^2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{i (-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{14 a b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{i (b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Cot[c + d*x]^(3/2)*(-3*(-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) + 3*(-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Tan[c + d*x]^(3/2) - 2*a*Sqrt[a + b*Tan[c + d*x]]*(a + 7*b*Tan[c + d*x])))/(3*d)","A",1
857,1,253,243,2.0888725,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\sqrt{\cot (c+d x)} \left(-2 a^2 \sqrt{a+b \tan (c+d x)}+\frac{2 \sqrt{a} b^{5/2} \sqrt{\tan (c+d x)} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}-\sqrt[4]{-1} (-a+i b)^{5/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-\sqrt[4]{-1} (a+i b)^{5/2} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","-\frac{2 a^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{(-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{(b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*(-((-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]]) - (-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 2*a^2*Sqrt[a + b*Tan[c + d*x]] + (2*Sqrt[a]*b^(5/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]]))/d","A",1
858,1,241,248,1.105345,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{5/2} \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{5 a^{3/2} b^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+b^2 \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+(-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)-(-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)\right)}{d}","\frac{5 a b^{3/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{d \sqrt{\cot (c+d x)}}+\frac{i (-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{i (b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - (-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + b^2*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + (5*a^(3/2)*b^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]]))/d","A",1
859,1,284,291,2.5491717,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Sqrt[Cot[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\sqrt{a} \sqrt{b} \left(15 a^2-8 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{a+b \tan (c+d x)}}+2 b^2 \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}+4 \sqrt[4]{-1} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+4 \sqrt[4]{-1} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+9 a b \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}\right)}{4 d}","\frac{\sqrt{b} \left(15 a^2-8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{4 d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{2 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{(-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}+\frac{9 a b \sqrt{a+b \tan (c+d x)}}{4 d \sqrt{\cot (c+d x)}}+\frac{(b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(4*(-1)^(1/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 4*(-1)^(1/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 9*a*b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 2*b^2*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + (Sqrt[a]*Sqrt[b]*(15*a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/Sqrt[a + b*Tan[c + d*x]]))/(4*d)","A",1
860,1,320,337,4.3087928,"\int \frac{(a+b \tan (c+d x))^{5/2}}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^(5/2)/Cot[c + d*x]^(3/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(3 \left(11 a^2-8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}+\frac{15 a^{3/2} \left(a^2-8 b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+8 b^2 \tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}-24 (-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+24 (-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)+26 a b \tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}\right)}{24 d}","\frac{\left(11 a^2-8 b^2\right) \sqrt{a+b \tan (c+d x)}}{8 d \sqrt{\cot (c+d x)}}+\frac{5 a \left(a^2-8 b^2\right) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{8 \sqrt{b} d}+\frac{b^2 \sqrt{a+b \tan (c+d x)}}{3 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sqrt{a+b \tan (c+d x)}}{12 d \cot ^{\frac{3}{2}}(c+d x)}-\frac{i (-b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}-\frac{i (b+i a)^{5/2} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-24*(-1)^(3/4)*(-a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 24*(-1)^(3/4)*(a + I*b)^(5/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 3*(11*a^2 - 8*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] + 26*a*b*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]] + 8*b^2*Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]] + (15*a^(3/2)*(a^2 - 8*b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(24*d)","A",1
861,1,193,220,3.5078215,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^(5/2)/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{\cot (c+d x)} \left(-\frac{2 \sqrt{a+b \tan (c+d x)} (a \cot (c+d x)-2 b)}{a^2}+\frac{3 (-1)^{3/4} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{3 (-1)^{3/4} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{3 d}","\frac{4 b \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a^2 d}-\frac{2 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 a d}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(Sqrt[Cot[c + d*x]]*((3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[-a + I*b] + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*b] - (2*(-2*b + a*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]])/a^2))/(3*d)","A",1
862,1,177,187,0.6499343,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Cot[c + d*x]^(3/2)/Sqrt[a + b*Tan[c + d*x]],x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{2 \sqrt{a+b \tan (c+d x)}}{a \sqrt{\tan (c+d x)}}\right)}{d}","-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{a d}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*Sqrt[a + b*Tan[c + d*x]])/(a*Sqrt[Tan[c + d*x]])))/d","A",1
863,1,144,149,0.1827287,"\int \frac{\sqrt{\cot (c+d x)}}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[Sqrt[Cot[c + d*x]]/Sqrt[a + b*Tan[c + d*x]],x]","\frac{(-1)^{3/4} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{d}","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(3/4)*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) - ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
864,1,143,155,0.1918381,"\int \frac{1}{\sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\sqrt[4]{-1} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)}{d}","\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"((-1)^(1/4)*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d","A",1
865,1,209,212,1.2057226,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{a} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+(-1)^{3/4} \left(\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)\right)}{d}","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{b} d}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-1)^(3/4)*(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b] + ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]) + (2*Sqrt[a]*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/d","A",1
866,1,243,248,2.4801959,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{a^{3/2} \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} b \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt[4]{-1} b \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}\right)}{b d}","-\frac{a \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{-b+i a}}+\frac{\sqrt{a+b \tan (c+d x)}}{b d \sqrt{\cot (c+d x)}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d \sqrt{b+i a}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(1/4)*b*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((-1)^(1/4)*b*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] - (a^(3/2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a])/(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(b*d)","A",1
867,1,241,281,5.7966212,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(3/2),x]","\frac{\sqrt{\cot (c+d x)} \left(\frac{2 b \left(b \left(5 a^2+8 b^2\right) \tan (c+d x)+4 a \left(a^2+b^2\right)\right)-2 a^2 \left(a^2+b^2\right) \cot (c+d x)}{a^3 \left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}-\frac{3 (-1)^{3/4} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}+\frac{3 (-1)^{3/4} \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}\right)}{3 d}","\frac{8 b \sqrt{\cot (c+d x)}}{3 a^2 d \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left(5 a^2+8 b^2\right)}{3 a^3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d \sqrt{a+b \tan (c+d x)}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*((-3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(-a + I*b)^(3/2) + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(a + I*b)^(3/2) + (-2*a^2*(a^2 + b^2)*Cot[c + d*x] + 2*b*(4*a*(a^2 + b^2) + b*(5*a^2 + 8*b^2)*Tan[c + d*x]))/(a^3*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/(3*d)","A",1
868,1,229,233,5.7091337,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{\sqrt{\cot (c+d x)} \left(\frac{\frac{2 \left(b \left(a^2+2 b^2\right) \tan (c+d x)+a \left(a^2+b^2\right)\right)}{\sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} a^2 (a-i b) \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}}{a \left(a^2+b^2\right)}+\frac{\sqrt[4]{-1} a \sqrt{\tan (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}\right)}{a d}","-\frac{2 b \left(a^2+2 b^2\right)}{a^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{2 \sqrt{\cot (c+d x)}}{a d \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"-((Sqrt[Cot[c + d*x]]*(((-1)^(1/4)*a*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(-a + I*b)^(3/2) + (((-1)^(1/4)*a^2*(a - I*b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*b] + (2*(a*(a^2 + b^2) + b*(a^2 + 2*b^2)*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2))))/(a*d))","A",1
869,1,203,199,1.6363769,"\int \frac{\sqrt{\cot (c+d x)}}{(a+b \tan (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^(3/2),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{2 b^2 \sqrt{\tan (c+d x)}}{a \sqrt{a+b \tan (c+d x)}}+\frac{(-1)^{3/4} (a+i b) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}+\frac{\sqrt[4]{-1} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{d \left(a^2+b^2\right)}","\frac{2 b^2}{a d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"-((Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(3/4)*(a + I*b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((-1)^(1/4)*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] - (2*b^2*Sqrt[Tan[c + d*x]])/(a*Sqrt[a + b*Tan[c + d*x]])))/((a^2 + b^2)*d))","A",1
870,1,183,189,2.0237241,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{2 b \sqrt{\tan (c+d x)}}{\left(a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}+\frac{\sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}\right)}{d}","-\frac{2 b}{d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) + ((-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) - (2*b*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])))/d","A",1
871,1,202,194,1.8379564,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{\frac{\sqrt[4]{-1} (b+i a) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}+\frac{2 a \sqrt{\tan (c+d x)}}{(a+i b) \sqrt{a+b \tan (c+d x)}}}{a-i b}-\frac{(-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}\right)}{d}","\frac{2 a}{d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-(((-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2)) + (((-1)^(1/4)*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) + (2*a*Sqrt[Tan[c + d*x]])/((a + I*b)*Sqrt[a + b*Tan[c + d*x]]))/(a - I*b)))/d","A",1
872,1,341,255,2.0667366,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)),x]","-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(2 \sqrt{a} \sqrt{-a+i b} \sqrt{a+i b} \left(a^2+b^2\right) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right)+\sqrt{b} \left(\sqrt{-a+i b} \left(-2 a^2 \sqrt{a+i b} \sqrt{\tan (c+d x)}-\sqrt[4]{-1} b (a-i b) \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}\right)+\sqrt[4]{-1} b (a+i b)^{3/2} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right) \sqrt{a+b \tan (c+d x)}\right)\right)}{b^{3/2} d (-a+i b)^{3/2} (a+i b)^{3/2} \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2}{b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{3/2} d}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"-((Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(2*Sqrt[a]*Sqrt[-a + I*b]*Sqrt[a + I*b]*(a^2 + b^2)*ArcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a] + Sqrt[b]*((-1)^(1/4)*(a + I*b)^(3/2)*b*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]] + Sqrt[-a + I*b]*(-2*a^2*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]] - (-1)^(1/4)*(a - I*b)*b*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[a + b*Tan[c + d*x]]))))/((-a + I*b)^(3/2)*(a + I*b)^(3/2)*b^(3/2)*d*Sqrt[a + b*Tan[c + d*x]]))","A",1
873,1,319,310,6.2414338,"\int \frac{1}{\cot ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx","Integrate[1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\frac{b \tan (c+d x)}{a}+1} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b \tan (c+d x)}{a}\right)}{5 a d \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}+\frac{(-1)^{3/4} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-a+i b)^{3/2}}-\frac{(-1)^{3/4} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (a+i b)^{3/2}}-\frac{1}{d (a-i b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{1}{d (a+i b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}","-\frac{2 a^2}{b d \left(a^2+b^2\right) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}+\frac{\left(3 a^2+b^2\right) \sqrt{a+b \tan (c+d x)}}{b^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)}}-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{b^{5/2} d}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{3/2}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{3/2}}",1,"((-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((-a + I*b)^(3/2)*d) - ((-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((a + I*b)^(3/2)*d) - 1/((a - I*b)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) - 1/((a + I*b)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*Hypergeometric2F1[3/2, 5/2, 7/2, -((b*Tan[c + d*x])/a)]*Sqrt[1 + (b*Tan[c + d*x])/a])/(5*a*d*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])","C",1
874,1,504,338,6.3144894,"\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(5/2),x]","\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(-\frac{2}{3 a d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac{2 \left(-\frac{6 b}{a d \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{3 \left(\frac{32 b^2 \sqrt{\tan (c+d x)}}{3 a^2 \sqrt{a+b \tan (c+d x)}}-\frac{\frac{b (5 a-2 i b) \sqrt{\tan (c+d x)}}{(a-i b) \sqrt{a+b \tan (c+d x)}}-\frac{3 \sqrt[4]{-1} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{3/2}}}{3 (b+i a)}+\frac{\frac{b (5 a+2 i b) \sqrt{\tan (c+d x)}}{(a+i b) \sqrt{a+b \tan (c+d x)}}-\frac{3 \sqrt[4]{-1} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{3/2}}}{3 (-b+i a)}+\frac{16 b^2 \sqrt{\tan (c+d x)}}{3 a (a+b \tan (c+d x))^{3/2}}+\frac{a b \sqrt{\tan (c+d x)}}{3 (-b+i a) (a+b \tan (c+d x))^{3/2}}-\frac{a b \sqrt{\tan (c+d x)}}{3 (b+i a) (a+b \tan (c+d x))^{3/2}}\right)}{2 a d}\right)}{3 a}\right)","\frac{4 b \sqrt{\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}+\frac{4 b^2 \left(4 a^4+15 a^2 b^2+8 b^4\right)}{3 a^4 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left(7 a^2+8 b^2\right)}{3 a^3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{2 \cot ^{\frac{3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2/(3*a*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)) - (2*((-6*b)/(a*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (3*((a*b*Sqrt[Tan[c + d*x]])/(3*(I*a - b)*(a + b*Tan[c + d*x])^(3/2)) + (16*b^2*Sqrt[Tan[c + d*x]])/(3*a*(a + b*Tan[c + d*x])^(3/2)) - (a*b*Sqrt[Tan[c + d*x]])/(3*(I*a + b)*(a + b*Tan[c + d*x])^(3/2)) + (32*b^2*Sqrt[Tan[c + d*x]])/(3*a^2*Sqrt[a + b*Tan[c + d*x]]) - ((-3*(-1)^(1/4)*a^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3/2) + ((5*a - (2*I)*b)*b*Sqrt[Tan[c + d*x]])/((a - I*b)*Sqrt[a + b*Tan[c + d*x]]))/(3*(I*a + b)) + ((-3*(-1)^(1/4)*a^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(3/2) + ((5*a + (2*I)*b)*b*Sqrt[Tan[c + d*x]])/((a + I*b)*Sqrt[a + b*Tan[c + d*x]]))/(3*(I*a - b))))/(2*a*d)))/(3*a))","A",1
875,1,296,305,4.1400515,"\int \frac{\cot ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(5/2),x]","-\frac{\sqrt{\cot (c+d x)} \left(-\frac{3 \sqrt[4]{-1} a \sqrt{\tan (c+d x)} \left(\frac{(a+i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}-\frac{(a-i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}\right)}{\left(a^2+b^2\right)^2}+\frac{2 b \left(3 a^2+4 b^2\right) \tan (c+d x)}{a \left(a^2+b^2\right) (a+b \tan (c+d x))^{3/2}}+\frac{2 b \left(3 a^4+17 a^2 b^2+8 b^4\right) \tan (c+d x)}{a^2 \left(a^2+b^2\right)^2 \sqrt{a+b \tan (c+d x)}}+\frac{6}{(a+b \tan (c+d x))^{3/2}}\right)}{3 a d}","-\frac{2 b \left(3 a^2+4 b^2\right)}{3 a^2 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{2 b \left(3 a^4+17 a^2 b^2+8 b^4\right)}{3 a^3 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{2 \sqrt{\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"-1/3*(Sqrt[Cot[c + d*x]]*((-3*(-1)^(1/4)*a*(((a + I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] - ((a - I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b])*Sqrt[Tan[c + d*x]])/(a^2 + b^2)^2 + 6/(a + b*Tan[c + d*x])^(3/2) + (2*b*(3*a^2 + 4*b^2)*Tan[c + d*x])/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(3*a^4 + 17*a^2*b^2 + 8*b^4)*Tan[c + d*x])/(a^2*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]])))/(a*d)","A",1
876,1,255,252,2.0969143,"\int \frac{\sqrt{\cot (c+d x)}}{(a+b \tan (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^(5/2),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{4 b^2 \left(4 a^2+b^2\right) \sqrt{\tan (c+d x)}}{a^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2 \left(a^2+b^2\right) \sqrt{\tan (c+d x)}}{a (a+b \tan (c+d x))^{3/2}}-3 (-1)^{3/4} \left(\frac{(a-i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{a+i b}}+\frac{(a+i b)^2 \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{\sqrt{-a+i b}}\right)\right)}{3 d \left(a^2+b^2\right)^2}","\frac{4 b^2 \left(4 a^2+b^2\right)}{3 a^2 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{2 b^2}{3 a d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-3*(-1)^(3/4)*(((a + I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[-a + I*b] + ((a - I*b)^2*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b]) + (2*b^2*(a^2 + b^2)*Sqrt[Tan[c + d*x]])/(a*(a + b*Tan[c + d*x])^(3/2)) + (4*b^2*(4*a^2 + b^2)*Sqrt[Tan[c + d*x]])/(a^2*Sqrt[a + b*Tan[c + d*x]])))/(3*(a^2 + b^2)^2*d)","A",1
877,1,214,251,5.2311908,"\int \frac{1}{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 b \sqrt{\tan (c+d x)} \left(\left(b^3-5 a^2 b\right) \tan (c+d x)-6 a^3\right)}{a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}-\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{5/2}}+\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{5/2}}\right)}{3 d}","-\frac{2 b \left(5 a^2-b^2\right)}{3 a d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{2 b}{3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((-3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(5/2) + (3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(5/2) + (2*b*Sqrt[Tan[c + d*x]]*(-6*a^3 + (-5*a^2*b + b^3)*Tan[c + d*x]))/(a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^(3/2))))/(3*d)","A",1
878,1,218,239,4.1904267,"\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 \sqrt{\tan (c+d x)} \left(2 b \left(a^2-2 b^2\right) \tan (c+d x)+3 a \left(a^2-b^2\right)\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{5/2}}+\frac{3 (-1)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{5/2}}\right)}{3 d}","\frac{2 a}{3 d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{4 \left(a^2-2 b^2\right)}{3 d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}+\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(5/2) + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(5/2) + (2*Sqrt[Tan[c + d*x]]*(3*a*(a^2 - b^2) + 2*b*(a^2 - 2*b^2)*Tan[c + d*x]))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^(3/2))))/(3*d)","A",1
879,1,209,254,4.3769576,"\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx","Integrate[1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)),x]","\frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left(\frac{2 a \sqrt{\tan (c+d x)} \left(\left(a^2+7 b^2\right) \tan (c+d x)+6 a b\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))^{3/2}}+\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{-a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(-a+i b)^{5/2}}-\frac{3 \sqrt[4]{-1} \tan ^{-1}\left(\frac{\sqrt[4]{-1} \sqrt{a+i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{(a+i b)^{5/2}}\right)}{3 d}","-\frac{2 a^2}{3 b d \left(a^2+b^2\right) \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left(a^2+7 b^2\right)}{3 b d \left(a^2+b^2\right)^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left(\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (-b+i a)^{5/2}}-\frac{i \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right)}{d (b+i a)^{5/2}}",1,"(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(5/2) - (3*(-1)^(1/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^(5/2) + (2*a*Sqrt[Tan[c + d*x]]*(6*a*b + (a^2 + 7*b^2)*Tan[c + d*x]))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^(3/2))))/(3*d)","A",1
880,1,141,206,0.9189013,"\int (d \cot (e+f x))^n (a+b \tan (e+f x))^3 \, dx","Integrate[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^3,x]","\frac{\tan ^2(e+f x) (d \cot (e+f x))^n \left(a \left((n-2) \left(a^2-3 b^2\right) \cot (e+f x) \, _2F_1\left(1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right)+a (-a (n-2) \cot (e+f x)-3 b (n-1))\right)-b (n-1) \left(b^2-3 a^2\right) \, _2F_1\left(1,\frac{n-2}{2};\frac{n}{2};-\cot ^2(e+f x)\right)\right)}{f (n-2) (n-1)}","-\frac{b d^2 \left(3 a^2-b^2\right) (d \cot (e+f x))^{n-2} \, _2F_1\left(1,\frac{n-2}{2};\frac{n}{2};-\cot ^2(e+f x)\right)}{f (2-n)}-\frac{a d \left(a^2-3 b^2\right) (d \cot (e+f x))^{n-1} \, _2F_1\left(1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right)}{f (1-n)}+\frac{a^2 d^2 (a \cot (e+f x)+b) (d \cot (e+f x))^{n-2}}{f (1-n)}+\frac{a^2 b d^2 (1-2 n) (d \cot (e+f x))^{n-2}}{f (1-n) (2-n)}",1,"((d*Cot[e + f*x])^n*(-(b*(-3*a^2 + b^2)*(-1 + n)*Hypergeometric2F1[1, (-2 + n)/2, n/2, -Cot[e + f*x]^2]) + a*(a*(-3*b*(-1 + n) - a*(-2 + n)*Cot[e + f*x]) + (a^2 - 3*b^2)*(-2 + n)*Cot[e + f*x]*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, -Cot[e + f*x]^2]))*Tan[e + f*x]^2)/(f*(-2 + n)*(-1 + n))","A",1
881,1,107,132,0.4800753,"\int (d \cot (e+f x))^n (a+b \tan (e+f x))^2 \, dx","Integrate[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^2,x]","-\frac{d (d \cot (e+f x))^{n-1} \left(a \left(a n+2 b (n-1) \cot (e+f x) \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right)\right)-n \left(a^2-b^2\right) \, _2F_1\left(1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right)\right)}{f (n-1) n}","-\frac{d \left(a^2-b^2\right) (d \cot (e+f x))^{n-1} \, _2F_1\left(1,\frac{n-1}{2};\frac{n+1}{2};-\cot ^2(e+f x)\right)}{f (1-n)}+\frac{a^2 d (d \cot (e+f x))^{n-1}}{f (1-n)}-\frac{2 a b (d \cot (e+f x))^n \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right)}{f n}",1,"-((d*(d*Cot[e + f*x])^(-1 + n)*(-((a^2 - b^2)*n*Hypergeometric2F1[1, (-1 + n)/2, (1 + n)/2, -Cot[e + f*x]^2]) + a*(a*n + 2*b*(-1 + n)*Cot[e + f*x]*Hypergeometric2F1[1, n/2, (2 + n)/2, -Cot[e + f*x]^2])))/(f*(-1 + n)*n))","A",1
882,1,88,96,0.2073997,"\int (d \cot (e+f x))^n (a+b \tan (e+f x)) \, dx","Integrate[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x]),x]","-\frac{(d \cot (e+f x))^n \left(a n \cot (e+f x) \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(e+f x)\right)+b (n+1) \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right)\right)}{f n (n+1)}","-\frac{a (d \cot (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(e+f x)\right)}{d f (n+1)}-\frac{b (d \cot (e+f x))^n \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};-\cot ^2(e+f x)\right)}{f n}",1,"-(((d*Cot[e + f*x])^n*(b*(1 + n)*Hypergeometric2F1[1, n/2, (2 + n)/2, -Cot[e + f*x]^2] + a*n*Cot[e + f*x]*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Cot[e + f*x]^2]))/(f*n*(1 + n)))","A",1
883,1,145,182,0.6159959,"\int \frac{(d \cot (e+f x))^n}{a+b \tan (e+f x)} \, dx","Integrate[(d*Cot[e + f*x])^n/(a + b*Tan[e + f*x]),x]","-\frac{\cot ^2(e+f x) (d \cot (e+f x))^n \left(a \left(a (n+3) \, _2F_1\left(1,n+2;n+3;-\frac{a \cot (e+f x)}{b}\right)-b (n+2) \cot (e+f x) \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right)\right)+b^2 (n+3) \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\cot ^2(e+f x)\right)\right)}{b f (n+2) (n+3) \left(a^2+b^2\right)}","\frac{a (d \cot (e+f x))^{n+3} \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right)}{d^3 f (n+3) \left(a^2+b^2\right)}-\frac{b (d \cot (e+f x))^{n+2} \, _2F_1\left(1,\frac{n+2}{2};\frac{n+4}{2};-\cot ^2(e+f x)\right)}{d^2 f (n+2) \left(a^2+b^2\right)}-\frac{a^2 (d \cot (e+f x))^{n+2} \, _2F_1\left(1,n+2;n+3;-\frac{a \cot (e+f x)}{b}\right)}{b d^2 f (n+2) \left(a^2+b^2\right)}",1,"-((Cot[e + f*x]^2*(d*Cot[e + f*x])^n*(b^2*(3 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2] + a*(a*(3 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, -((a*Cot[e + f*x])/b)] - b*(2 + n)*Cot[e + f*x]*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])))/(b*(a^2 + b^2)*f*(2 + n)*(3 + n)))","A",1
884,1,192,250,0.8661841,"\int \frac{(d \cot (e+f x))^n}{(a+b \tan (e+f x))^2} \, dx","Integrate[(d*Cot[e + f*x])^n/(a + b*Tan[e + f*x])^2,x]","-\frac{\cot ^3(e+f x) (d \cot (e+f x))^n \left(b^2 (n+4) \left(b^2-a^2\right) \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right)+a \left(a (n+4) \left(a^2+b^2\right) \, _2F_1\left(2,n+3;n+4;-\frac{a \cot (e+f x)}{b}\right)+2 a b^2 (n+4) \, _2F_1\left(1,n+3;n+4;-\frac{a \cot (e+f x)}{b}\right)-2 b^3 (n+3) \cot (e+f x) \, _2F_1\left(1,\frac{n+4}{2};\frac{n+6}{2};-\cot ^2(e+f x)\right)\right)\right)}{b^2 f (n+3) (n+4) \left(a^2+b^2\right)^2}","\frac{2 a b (d \cot (e+f x))^{n+4} \, _2F_1\left(1,\frac{n+4}{2};\frac{n+6}{2};-\cot ^2(e+f x)\right)}{d^4 f (n+4) \left(a^2+b^2\right)^2}+\frac{\left(a^2-b^2\right) (d \cot (e+f x))^{n+3} \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\cot ^2(e+f x)\right)}{d^3 f (n+3) \left(a^2+b^2\right)^2}+\frac{a^2 \left(a^2 (n+2)+b^2 n\right) (d \cot (e+f x))^{n+3} \, _2F_1\left(1,n+3;n+4;-\frac{a \cot (e+f x)}{b}\right)}{b^2 d^3 f (n+3) \left(a^2+b^2\right)^2}-\frac{a^2 (d \cot (e+f x))^{n+3}}{b d^3 f \left(a^2+b^2\right) (a \cot (e+f x)+b)}",1,"-((Cot[e + f*x]^3*(d*Cot[e + f*x])^n*(b^2*(-a^2 + b^2)*(4 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2] + a*(2*a*b^2*(4 + n)*Hypergeometric2F1[1, 3 + n, 4 + n, -((a*Cot[e + f*x])/b)] - 2*b^3*(3 + n)*Cot[e + f*x]*Hypergeometric2F1[1, (4 + n)/2, (6 + n)/2, -Cot[e + f*x]^2] + a*(a^2 + b^2)*(4 + n)*Hypergeometric2F1[2, 3 + n, 4 + n, -((a*Cot[e + f*x])/b)])))/(b^2*(a^2 + b^2)^2*f*(3 + n)*(4 + n)))","A",1
885,0,0,193,3.4877504,"\int (d \cot (e+f x))^n (a+b \tan (e+f x))^m \, dx","Integrate[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^m,x]","\int (d \cot (e+f x))^n (a+b \tan (e+f x))^m \, dx","\frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left(\frac{b \tan (e+f x)}{a}+1\right)^{-m} F_1\left(1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right)}{2 f (1-n)}+\frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left(\frac{b \tan (e+f x)}{a}+1\right)^{-m} F_1\left(1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right)}{2 f (1-n)}",1,"Integrate[(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^m, x]","F",-1
886,0,0,155,3.9184679,"\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n,x]","\int \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx","-\frac{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}-\frac{\sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(-\frac{1}{2};1,-n;\frac{1}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d}",1,"Integrate[Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n, x]","F",-1
887,0,0,153,4.6097622,"\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \, dx","Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n,x]","\int \sqrt{\cot (c+d x)} (a+b \tan (c+d x))^n \, dx","\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\cot (c+d x)}}+\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{1}{2};1,-n;\frac{3}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{d \sqrt{\cot (c+d x)}}",1,"Integrate[Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n, x]","F",-1
888,0,0,159,5.735941,"\int \frac{(a+b \tan (c+d x))^n}{\sqrt{\cot (c+d x)}} \, dx","Integrate[(a + b*Tan[c + d*x])^n/Sqrt[Cot[c + d*x]],x]","\int \frac{(a+b \tan (c+d x))^n}{\sqrt{\cot (c+d x)}} \, dx","\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}+\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{3}{2};1,-n;\frac{5}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{3 d \cot ^{\frac{3}{2}}(c+d x)}",1,"Integrate[(a + b*Tan[c + d*x])^n/Sqrt[Cot[c + d*x]], x]","F",-1
889,0,0,159,5.5729498,"\int \frac{(a+b \tan (c+d x))^n}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2),x]","\int \frac{(a+b \tan (c+d x))^n}{\cot ^{\frac{3}{2}}(c+d x)} \, dx","\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};-i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}+\frac{(a+b \tan (c+d x))^n \left(\frac{b \tan (c+d x)}{a}+1\right)^{-n} F_1\left(\frac{5}{2};1,-n;\frac{7}{2};i \tan (c+d x),-\frac{b \tan (c+d x)}{a}\right)}{5 d \cot ^{\frac{5}{2}}(c+d x)}",1,"Integrate[(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2), x]","F",-1
890,1,55,25,0.1954574,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x]),x]","\frac{a^3 c \left(-3 \tan ^{-1}(\tan (e+f x))-\tan ^3(e+f x)+3 i \tan ^2(e+f x)+3 \tan (e+f x)+3 f x\right)}{3 f}","-\frac{i c (a+i a \tan (e+f x))^3}{3 f}",1,"(a^3*c*(3*f*x - 3*ArcTan[Tan[e + f*x]] + 3*Tan[e + f*x] + (3*I)*Tan[e + f*x]^2 - Tan[e + f*x]^3))/(3*f)","B",1
891,1,45,25,0.1347913,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x]),x]","\frac{a^2 c \left(-2 \tan ^{-1}(\tan (e+f x))+i \tan ^2(e+f x)+2 \tan (e+f x)+2 f x\right)}{2 f}","-\frac{i c (a+i a \tan (e+f x))^2}{2 f}",1,"(a^2*c*(2*f*x - 2*ArcTan[Tan[e + f*x]] + 2*Tan[e + f*x] + I*Tan[e + f*x]^2))/(2*f)","A",1
892,1,12,12,0.0095644,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x]),x]","\frac{a c \tan (e+f x)}{f}","\frac{a c \tan (e+f x)}{f}",1,"(a*c*Tan[e + f*x])/f","A",1
893,1,32,23,0.1114652,"\int \frac{c-i c \tan (e+f x)}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])/(a + I*a*Tan[e + f*x]),x]","\frac{c (\sin (2 (e+f x))+i \cos (2 (e+f x)))}{2 a f}","\frac{i c}{f (a+i a \tan (e+f x))}",1,"(c*(I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]))/(2*a*f)","A",1
894,1,45,25,0.8775519,"\int \frac{c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^2,x]","\frac{(\tan (e+f x)-3 i) (c-i c \tan (e+f x))}{8 a^2 f (\tan (e+f x)-i)^2}","\frac{i c}{2 f (a+i a \tan (e+f x))^2}",1,"((-3*I + Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(8*a^2*f*(-I + Tan[e + f*x])^2)","A",1
895,1,56,25,0.7524516,"\int \frac{c-i c \tan (e+f x)}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^3,x]","\frac{c (2 i \sin (2 (e+f x))+4 \cos (2 (e+f x))+3) (\sin (4 (e+f x))+i \cos (4 (e+f x)))}{24 a^3 f}","\frac{i c}{3 f (a+i a \tan (e+f x))^3}",1,"(c*(3 + 4*Cos[2*(e + f*x)] + (2*I)*Sin[2*(e + f*x)])*(I*Cos[4*(e + f*x)] + Sin[4*(e + f*x)]))/(24*a^3*f)","B",1
896,1,80,58,3.3271515,"\int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^2,x]","\frac{a^4 c^2 \sec (e) \sec ^5(e+f x) (-5 \sin (2 e+f x)+5 \sin (2 e+3 f x)+\sin (4 e+5 f x)+5 i \cos (2 e+f x)+5 \sin (f x)+5 i \cos (f x))}{20 f}","\frac{i c^2 (a+i a \tan (e+f x))^5}{5 a f}-\frac{i c^2 (a+i a \tan (e+f x))^4}{2 f}",1,"(a^4*c^2*Sec[e]*Sec[e + f*x]^5*((5*I)*Cos[f*x] + (5*I)*Cos[2*e + f*x] + 5*Sin[f*x] - 5*Sin[2*e + f*x] + 5*Sin[2*e + 3*f*x] + Sin[4*e + 5*f*x]))/(20*f)","A",1
897,1,52,61,3.2621128,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2,x]","\frac{a^3 c^2 \sec (e) \sec ^4(e+f x) (4 \sin (e+2 f x)+\sin (3 e+4 f x)-3 \sin (e)+3 i \cos (e))}{12 f}","\frac{a^3 c^2 \tan ^3(e+f x)}{3 f}+\frac{a^3 c^2 \tan (e+f x)}{f}+\frac{i a^3 c^2 \sec ^4(e+f x)}{4 f}",1,"(a^3*c^2*Sec[e]*Sec[e + f*x]^4*((3*I)*Cos[e] - 3*Sin[e] + 4*Sin[e + 2*f*x] + Sin[3*e + 4*f*x]))/(12*f)","A",1
898,1,29,38,0.0520504,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^2,x]","\frac{a^2 c^2 \left(\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}","\frac{a^2 c^2 \tan ^3(e+f x)}{3 f}+\frac{a^2 c^2 \tan (e+f x)}{f}",1,"(a^2*c^2*(Tan[e + f*x] + Tan[e + f*x]^3/3))/f","A",1
899,1,45,25,0.1437294,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2,x]","\frac{a c^2 \left(-2 \tan ^{-1}(\tan (e+f x))-i \tan ^2(e+f x)+2 \tan (e+f x)+2 f x\right)}{2 f}","\frac{i a (c-i c \tan (e+f x))^2}{2 f}",1,"(a*c^2*(2*f*x - 2*ArcTan[Tan[e + f*x]] + 2*Tan[e + f*x] - I*Tan[e + f*x]^2))/(2*f)","A",1
900,1,74,55,1.3223075,"\int \frac{(c-i c \tan (e+f x))^2}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x]),x]","-\frac{c^2 \left(2 \tan ^{-1}(\tan (f x)) (\tan (e+f x)-i)+\log \left(\cos ^2(e+f x)\right)+i \tan (e+f x) \left(\log \left(\cos ^2(e+f x)\right)+2\right)-2\right)}{2 a f (\tan (e+f x)-i)}","\frac{2 i c^2}{f (a+i a \tan (e+f x))}-\frac{i c^2 \log (\cos (e+f x))}{a f}-\frac{c^2 x}{a}",1,"-1/2*(c^2*(-2 + Log[Cos[e + f*x]^2] + I*(2 + Log[Cos[e + f*x]^2])*Tan[e + f*x] + 2*ArcTan[Tan[f*x]]*(-I + Tan[e + f*x])))/(a*f*(-I + Tan[e + f*x]))","A",1
901,1,34,28,0.2658189,"\int \frac{(c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^2,x]","\frac{c^2 (\sin (4 (e+f x))+i \cos (4 (e+f x)))}{4 a^2 f}","\frac{c^2 \tan (e+f x)}{f (a+i a \tan (e+f x))^2}",1,"(c^2*(I*Cos[4*(e + f*x)] + Sin[4*(e + f*x)]))/(4*a^2*f)","A",1
902,1,53,58,1.7756385,"\int \frac{(c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^3,x]","\frac{c^2 (5 \cos (e+f x)+i \sin (e+f x)) (\sin (5 (e+f x))+i \cos (5 (e+f x)))}{24 a^3 f}","\frac{2 i c^2}{3 f (a+i a \tan (e+f x))^3}-\frac{i c^2}{2 a f (a+i a \tan (e+f x))^2}",1,"(c^2*(5*Cos[e + f*x] + I*Sin[e + f*x])*(I*Cos[5*(e + f*x)] + Sin[5*(e + f*x)]))/(24*a^3*f)","A",1
903,1,58,62,2.0157008,"\int \frac{(c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^4} \, dx","Integrate[(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^4,x]","\frac{c^2 (3 i \sin (2 (e+f x))+9 \cos (2 (e+f x))+8) (\sin (6 (e+f x))+i \cos (6 (e+f x)))}{96 a^4 f}","\frac{i c^2}{2 f (a+i a \tan (e+f x))^4}-\frac{i a^2 c^2}{3 f \left(a^2+i a^2 \tan (e+f x)\right)^3}",1,"(c^2*(8 + 9*Cos[2*(e + f*x)] + (3*I)*Sin[2*(e + f*x)])*(I*Cos[6*(e + f*x)] + Sin[6*(e + f*x)]))/(96*a^4*f)","A",1
904,1,93,88,4.8937919,"\int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^3,x]","\frac{a^5 c^3 \sec (e) \sec ^7(e+f x) (-35 \sin (2 e+f x)+42 \sin (2 e+3 f x)+14 \sin (4 e+5 f x)+2 \sin (6 e+7 f x)+35 i \cos (2 e+f x)+35 \sin (f x)+35 i \cos (f x))}{210 f}","-\frac{i c^3 (a+i a \tan (e+f x))^7}{7 a^2 f}+\frac{2 i c^3 (a+i a \tan (e+f x))^6}{3 a f}-\frac{4 i c^3 (a+i a \tan (e+f x))^5}{5 f}",1,"(a^5*c^3*Sec[e]*Sec[e + f*x]^7*((35*I)*Cos[f*x] + (35*I)*Cos[2*e + f*x] + 35*Sin[f*x] - 35*Sin[2*e + f*x] + 42*Sin[2*e + 3*f*x] + 14*Sin[4*e + 5*f*x] + 2*Sin[6*e + 7*f*x]))/(210*f)","A",1
905,1,63,82,4.1363455,"\int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^3,x]","\frac{a^4 c^3 \sec (e) \sec ^6(e+f x) (15 \sin (e+2 f x)+6 \sin (3 e+4 f x)+\sin (5 e+6 f x)-10 \sin (e)+10 i \cos (e))}{60 f}","\frac{a^4 c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^4 c^3 \tan (e+f x)}{f}+\frac{i a^4 c^3 \sec ^6(e+f x)}{6 f}",1,"(a^4*c^3*Sec[e]*Sec[e + f*x]^6*((10*I)*Cos[e] - 10*Sin[e] + 15*Sin[e + 2*f*x] + 6*Sin[3*e + 4*f*x] + Sin[5*e + 6*f*x]))/(60*f)","A",1
906,1,41,59,0.1292266,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^3,x]","\frac{a^3 c^3 \left(\frac{1}{5} \tan ^5(e+f x)+\frac{2}{3} \tan ^3(e+f x)+\tan (e+f x)\right)}{f}","\frac{a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac{2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^3 c^3 \tan (e+f x)}{f}",1,"(a^3*c^3*(Tan[e + f*x] + (2*Tan[e + f*x]^3)/3 + Tan[e + f*x]^5/5))/f","A",1
907,1,52,61,3.2995896,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^3,x]","\frac{a^2 c^3 \sec (e) \sec ^4(e+f x) (4 \sin (e+2 f x)+\sin (3 e+4 f x)-3 \sin (e)-3 i \cos (e))}{12 f}","\frac{a^2 c^3 \tan ^3(e+f x)}{3 f}+\frac{a^2 c^3 \tan (e+f x)}{f}-\frac{i a^2 c^3 \sec ^4(e+f x)}{4 f}",1,"(a^2*c^3*Sec[e]*Sec[e + f*x]^4*((-3*I)*Cos[e] - 3*Sin[e] + 4*Sin[e + 2*f*x] + Sin[3*e + 4*f*x]))/(12*f)","A",1
908,1,55,25,0.1968623,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3,x]","\frac{a c^3 \left(-3 \tan ^{-1}(\tan (e+f x))-\tan ^3(e+f x)-3 i \tan ^2(e+f x)+3 \tan (e+f x)+3 f x\right)}{3 f}","\frac{i a (c-i c \tan (e+f x))^3}{3 f}",1,"(a*c^3*(3*f*x - 3*ArcTan[Tan[e + f*x]] + 3*Tan[e + f*x] - (3*I)*Tan[e + f*x]^2 - Tan[e + f*x]^3))/(3*f)","B",1
909,1,234,71,1.9945584,"\int \frac{(c-i c \tan (e+f x))^3}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x]),x]","\frac{i c^3 \sec ^2(e+f x) \left(-2 \sin (e+2 f x)-\sin (3 e+2 f x)-i \cos (3 e+2 f x)+i \cos (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+i \cos (e+2 f x) \log \left(\cos ^2(e+f x)\right)+i \cos (e) \left(2 \log \left(\cos ^2(e+f x)\right)-3\right)-\sin (e+2 f x) \log \left(\cos ^2(e+f x)\right)-\sin (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+8 \cos (e) \tan ^{-1}(\tan (f x)) \cos (e+f x) (\cos (e+f x)+i \sin (e+f x))+\sin (e)\right)}{2 a f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) (\tan (e+f x)-i)}","\frac{c^3 \tan (e+f x)}{a f}+\frac{4 i c^3}{f (a+i a \tan (e+f x))}-\frac{4 i c^3 \log (\cos (e+f x))}{a f}-\frac{4 c^3 x}{a}",1,"((I/2)*c^3*Sec[e + f*x]^2*((-I)*Cos[3*e + 2*f*x] + I*Cos[e + 2*f*x]*Log[Cos[e + f*x]^2] + I*Cos[3*e + 2*f*x]*Log[Cos[e + f*x]^2] + I*Cos[e]*(-3 + 2*Log[Cos[e + f*x]^2]) + Sin[e] + 8*ArcTan[Tan[f*x]]*Cos[e]*Cos[e + f*x]*(Cos[e + f*x] + I*Sin[e + f*x]) - 2*Sin[e + 2*f*x] - Log[Cos[e + f*x]^2]*Sin[e + 2*f*x] - Sin[3*e + 2*f*x] - Log[Cos[e + f*x]^2]*Sin[3*e + 2*f*x]))/(a*f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(-I + Tan[e + f*x]))","B",1
910,1,115,83,2.1023284,"\int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^2,x]","-\frac{c^3 \sec ^2(e+f x) \left(\sin (2 (e+f x))+i \cos (2 (e+f x)) \left(\log \left(\cos ^2(e+f x)\right)+1\right)-\sin (2 (e+f x)) \log \left(\cos ^2(e+f x)\right)+2 \tan ^{-1}(\tan (f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))-2 i\right)}{2 a^2 f (\tan (e+f x)-i)^2}","-\frac{4 i c^3}{f \left(a^2+i a^2 \tan (e+f x)\right)}+\frac{i c^3 \log (\cos (e+f x))}{a^2 f}+\frac{c^3 x}{a^2}+\frac{2 i c^3}{f (a+i a \tan (e+f x))^2}",1,"-1/2*(c^3*Sec[e + f*x]^2*(-2*I + I*Cos[2*(e + f*x)]*(1 + Log[Cos[e + f*x]^2]) + 2*ArcTan[Tan[f*x]]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]) + Sin[2*(e + f*x)] - Log[Cos[e + f*x]^2]*Sin[2*(e + f*x)]))/(a^2*f*(-I + Tan[e + f*x])^2)","A",1
911,1,34,50,0.2957074,"\int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^3,x]","\frac{c^3 (\sin (6 (e+f x))+i \cos (6 (e+f x)))}{6 a^3 f}","\frac{i c^3 \left(a^2-i a^2 \tan (e+f x)\right)^3}{6 f \left(a^3+i a^3 \tan (e+f x)\right)^3}",1,"(c^3*(I*Cos[6*(e + f*x)] + Sin[6*(e + f*x)]))/(6*a^3*f)","A",1
912,1,64,87,2.3161539,"\int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^4} \, dx","Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^4,x]","-\frac{c^3 (\tan (e+f x)-7 i) \sec ^3(e+f x) (\cos (3 (e+f x))-i \sin (3 (e+f x)))}{48 a^4 f (\tan (e+f x)-i)^4}","\frac{i c^3}{2 f \left(a^2+i a^2 \tan (e+f x)\right)^2}-\frac{4 i c^3}{3 a f (a+i a \tan (e+f x))^3}+\frac{i c^3}{f (a+i a \tan (e+f x))^4}",1,"-1/48*(c^3*Sec[e + f*x]^3*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)])*(-7*I + Tan[e + f*x]))/(a^4*f*(-I + Tan[e + f*x])^4)","A",1
913,1,58,90,2.9306938,"\int \frac{(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^5} \, dx","Integrate[(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^5,x]","\frac{c^3 (4 i \sin (2 (e+f x))+16 \cos (2 (e+f x))+15) (\sin (8 (e+f x))+i \cos (8 (e+f x)))}{240 a^5 f}","\frac{i c^3}{3 a^2 f (a+i a \tan (e+f x))^3}-\frac{i a^3 c^3}{f \left(a^2+i a^2 \tan (e+f x)\right)^4}+\frac{4 i c^3}{5 f (a+i a \tan (e+f x))^5}",1,"(c^3*(15 + 16*Cos[2*(e + f*x)] + (4*I)*Sin[2*(e + f*x)])*(I*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(240*a^5*f)","A",1
914,1,74,100,5.41662,"\int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^4,x]","\frac{a^5 c^4 \sec (e) \sec ^8(e+f x) (56 \sin (e+2 f x)+28 \sin (3 e+4 f x)+8 \sin (5 e+6 f x)+\sin (7 e+8 f x)-35 \sin (e)+35 i \cos (e))}{280 f}","\frac{a^5 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^5 c^4 \tan ^3(e+f x)}{f}+\frac{a^5 c^4 \tan (e+f x)}{f}+\frac{i a^5 c^4 \sec ^8(e+f x)}{8 f}",1,"(a^5*c^4*Sec[e]*Sec[e + f*x]^8*((35*I)*Cos[e] - 35*Sin[e] + 56*Sin[e + 2*f*x] + 28*Sin[3*e + 4*f*x] + 8*Sin[5*e + 6*f*x] + Sin[7*e + 8*f*x]))/(280*f)","A",1
915,1,49,77,0.2398188,"\int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^4,x]","\frac{a^4 c^4 \left(\frac{1}{7} \tan ^7(e+f x)+\frac{3}{5} \tan ^5(e+f x)+\tan ^3(e+f x)+\tan (e+f x)\right)}{f}","\frac{a^4 c^4 \tan ^7(e+f x)}{7 f}+\frac{3 a^4 c^4 \tan ^5(e+f x)}{5 f}+\frac{a^4 c^4 \tan ^3(e+f x)}{f}+\frac{a^4 c^4 \tan (e+f x)}{f}",1,"(a^4*c^4*(Tan[e + f*x] + Tan[e + f*x]^3 + (3*Tan[e + f*x]^5)/5 + Tan[e + f*x]^7/7))/f","A",1
916,1,63,82,4.2398156,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4,x]","\frac{a^3 c^4 \sec (e) \sec ^6(e+f x) (15 \sin (e+2 f x)+6 \sin (3 e+4 f x)+\sin (5 e+6 f x)-10 \sin (e)-10 i \cos (e))}{60 f}","\frac{a^3 c^4 \tan ^5(e+f x)}{5 f}+\frac{2 a^3 c^4 \tan ^3(e+f x)}{3 f}+\frac{a^3 c^4 \tan (e+f x)}{f}-\frac{i a^3 c^4 \sec ^6(e+f x)}{6 f}",1,"(a^3*c^4*Sec[e]*Sec[e + f*x]^6*((-10*I)*Cos[e] - 10*Sin[e] + 15*Sin[e + 2*f*x] + 6*Sin[3*e + 4*f*x] + Sin[5*e + 6*f*x]))/(60*f)","A",1
917,1,80,58,3.4413263,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4,x]","\frac{a^2 c^4 \sec (e) \sec ^5(e+f x) (-5 \sin (2 e+f x)+5 \sin (2 e+3 f x)+\sin (4 e+5 f x)-5 i \cos (2 e+f x)+5 \sin (f x)-5 i \cos (f x))}{20 f}","\frac{i a^2 (c-i c \tan (e+f x))^4}{2 f}-\frac{i a^2 (c-i c \tan (e+f x))^5}{5 c f}",1,"(a^2*c^4*Sec[e]*Sec[e + f*x]^5*((-5*I)*Cos[f*x] - (5*I)*Cos[2*e + f*x] + 5*Sin[f*x] - 5*Sin[2*e + f*x] + 5*Sin[2*e + 3*f*x] + Sin[4*e + 5*f*x]))/(20*f)","A",1
918,1,85,25,1.5588634,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]","\frac{a c^4 \sec (e) \sec ^4(e+f x) (2 \sin (e+2 f x)-2 \sin (3 e+2 f x)+\sin (3 e+4 f x)-2 i \cos (e+2 f x)-2 i \cos (3 e+2 f x)-3 \sin (e)-3 i \cos (e))}{4 f}","\frac{i a (c-i c \tan (e+f x))^4}{4 f}",1,"(a*c^4*Sec[e]*Sec[e + f*x]^4*((-3*I)*Cos[e] - (2*I)*Cos[e + 2*f*x] - (2*I)*Cos[3*e + 2*f*x] - 3*Sin[e] + 2*Sin[e + 2*f*x] - 2*Sin[3*e + 2*f*x] + Sin[3*e + 4*f*x]))/(4*f)","B",1
919,1,194,95,3.0007867,"\int \frac{(c-i c \tan (e+f x))^4}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x]),x]","\frac{c^4 \cos (e) \sec (e+f x) (\cos (f x)+i \sin (f x)) \left(-24 i (\tan (e)-i) \tan ^{-1}(\tan (f x))-24 f x \tan ^2(e)+24 f x \sec ^2(e)-i \sec ^2(e+f x)-8 i \tan (e) \sin (2 f x)-12 i \log \left(\cos ^2(e+f x)\right)+8 (\tan (e)+i) \cos (2 f x)+\tan (e) \sec ^2(e+f x)+10 \sec (e) \sin (f x) \sec (e+f x)+12 \tan (e) \log \left(\cos ^2(e+f x)\right)+10 i \tan (e) \sec (e) \sin (f x) \sec (e+f x)-24 f x+8 \sin (2 f x)\right)}{2 f (a+i a \tan (e+f x))}","-\frac{i c^4 \tan ^2(e+f x)}{2 a f}+\frac{5 c^4 \tan (e+f x)}{a f}+\frac{8 i c^4}{f (a+i a \tan (e+f x))}-\frac{12 i c^4 \log (\cos (e+f x))}{a f}-\frac{12 c^4 x}{a}",1,"(c^4*Cos[e]*Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*(-24*f*x - (12*I)*Log[Cos[e + f*x]^2] + 24*f*x*Sec[e]^2 - I*Sec[e + f*x]^2 + 10*Sec[e]*Sec[e + f*x]*Sin[f*x] + 8*Sin[2*f*x] + 12*Log[Cos[e + f*x]^2]*Tan[e] + Sec[e + f*x]^2*Tan[e] + (10*I)*Sec[e]*Sec[e + f*x]*Sin[f*x]*Tan[e] - (8*I)*Sin[2*f*x]*Tan[e] - 24*f*x*Tan[e]^2 - (24*I)*ArcTan[Tan[f*x]]*(-I + Tan[e]) + 8*Cos[2*f*x]*(I + Tan[e])))/(2*f*(a + I*a*Tan[e + f*x]))","B",1
920,1,279,101,2.6273539,"\int \frac{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^2,x]","\frac{c^4 \sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(-24 f x \sin ^2(e)-12 i f x \sin (2 e)+2 i \sin (2 e) \sin (4 f x)+12 i f x \tan (e)-2 \sin (2 e) \cos (4 f x)+i \sec (e) \cos (2 e-f x) \sec (e+f x)-i \sec (e) \cos (2 e+f x) \sec (e+f x)-\sec (e) \sin (2 e-f x) \sec (e+f x)+\sec (e) \sin (2 e+f x) \sec (e+f x)+6 \sin (2 e) \log \left(\cos ^2(e+f x)\right)-12 (\cos (2 e)+i \sin (2 e)) \tan ^{-1}(\tan (f x))+2 i \cos (2 e) \left(6 f x \tan (e)-3 \log \left(\cos ^2(e+f x)\right)+6 i f x+i \sin (4 f x)-\cos (4 f x)\right)+12 f x+8 \sin (2 f x)+8 i \cos (2 f x)\right)}{2 a^2 f (\tan (e+f x)-i)^2}","-\frac{c^4 \tan (e+f x)}{a^2 f}-\frac{12 i c^4}{f \left(a^2+i a^2 \tan (e+f x)\right)}+\frac{6 i c^4 \log (\cos (e+f x))}{a^2 f}+\frac{6 c^4 x}{a^2}+\frac{4 i c^4}{f (a+i a \tan (e+f x))^2}",1,"(c^4*Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(12*f*x + (8*I)*Cos[2*f*x] + I*Cos[2*e - f*x]*Sec[e]*Sec[e + f*x] - I*Cos[2*e + f*x]*Sec[e]*Sec[e + f*x] - 24*f*x*Sin[e]^2 - 12*ArcTan[Tan[f*x]]*(Cos[2*e] + I*Sin[2*e]) - (12*I)*f*x*Sin[2*e] - 2*Cos[4*f*x]*Sin[2*e] + 6*Log[Cos[e + f*x]^2]*Sin[2*e] + 8*Sin[2*f*x] + (2*I)*Sin[2*e]*Sin[4*f*x] - Sec[e]*Sec[e + f*x]*Sin[2*e - f*x] + Sec[e]*Sec[e + f*x]*Sin[2*e + f*x] + (12*I)*f*x*Tan[e] + (2*I)*Cos[2*e]*((6*I)*f*x - Cos[4*f*x] - 3*Log[Cos[e + f*x]^2] + I*Sin[4*f*x] + 6*f*x*Tan[e])))/(2*a^2*f*(-I + Tan[e + f*x])^2)","B",1
921,1,121,114,1.764619,"\int \frac{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^3,x]","\frac{c^4 \sec ^3(e+f x) (-9 i \sin (e+f x)+6 f x \sin (3 (e+f x))+2 i \sin (3 (e+f x))-3 \cos (e+f x)+\cos (3 (e+f x)) (6 \log (\cos (e+f x))-6 i f x-2)+6 i \sin (3 (e+f x)) \log (\cos (e+f x)))}{6 a^3 f (\tan (e+f x)-i)^3}","\frac{6 i c^4}{f \left(a^3+i a^3 \tan (e+f x)\right)}-\frac{i c^4 \log (\cos (e+f x))}{a^3 f}-\frac{c^4 x}{a^3}-\frac{6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac{8 i c^4}{3 f (a+i a \tan (e+f x))^3}",1,"(c^4*Sec[e + f*x]^3*(-3*Cos[e + f*x] + Cos[3*(e + f*x)]*(-2 - (6*I)*f*x + 6*Log[Cos[e + f*x]]) - (9*I)*Sin[e + f*x] + (2*I)*Sin[3*(e + f*x)] + 6*f*x*Sin[3*(e + f*x)] + (6*I)*Log[Cos[e + f*x]]*Sin[3*(e + f*x)]))/(6*a^3*f*(-I + Tan[e + f*x])^3)","A",1
922,1,34,50,0.2800623,"\int \frac{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^4} \, dx","Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^4,x]","\frac{c^4 (\sin (8 (e+f x))+i \cos (8 (e+f x)))}{8 a^4 f}","\frac{i c^4 \left(a^2-i a^2 \tan (e+f x)\right)^4}{8 f \left(a^3+i a^3 \tan (e+f x)\right)^4}",1,"(c^4*(I*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(8*a^4*f)","A",1
923,1,53,87,1.9715133,"\int \frac{(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^5} \, dx","Integrate[(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^5,x]","\frac{c^4 (9 \cos (e+f x)+i \sin (e+f x)) (\sin (9 (e+f x))+i \cos (9 (e+f x)))}{80 a^5 f}","\frac{i c^4 (a-i a \tan (e+f x))^4}{80 a^5 f (a+i a \tan (e+f x))^4}+\frac{i c^4 (1-i \tan (e+f x))^4}{10 f (a+i a \tan (e+f x))^5}",1,"(c^4*(9*Cos[e + f*x] + I*Sin[e + f*x])*(I*Cos[9*(e + f*x)] + Sin[9*(e + f*x)]))/(80*a^5*f)","A",1
924,1,376,95,3.141502,"\int \frac{(a+i a \tan (e+f x))^4}{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x]),x]","-\frac{a^4 \sec (e) \sec ^2(e+f x) (\cos (e+5 f x)+i \sin (e+5 f x)) \left(-6 i f x \sin (2 e+f x)+2 \sin (2 e+f x)-6 i f x \sin (2 e+3 f x)-7 \sin (2 e+3 f x)-6 i f x \sin (4 e+3 f x)-2 \sin (4 e+3 f x)+6 f x \cos (2 e+3 f x)-3 i \cos (2 e+3 f x)+6 f x \cos (4 e+3 f x)+2 i \cos (4 e+3 f x)-3 i \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)+\cos (f x) \left(-9 i \log \left(\cos ^2(e+f x)\right)+18 f x+5 i\right)+\cos (2 e+f x) \left(-9 i \log \left(\cos ^2(e+f x)\right)+18 f x+10 i\right)-3 i \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (2 e+f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-6 i f x \sin (f x)-13 \sin (f x)\right)}{4 c f (\cos (f x)+i \sin (f x))^4}","\frac{i a^4 \tan ^2(e+f x)}{2 c f}+\frac{5 a^4 \tan (e+f x)}{c f}-\frac{8 i a^4}{f (c-i c \tan (e+f x))}+\frac{12 i a^4 \log (\cos (e+f x))}{c f}-\frac{12 a^4 x}{c}",1,"-1/4*(a^4*Sec[e]*Sec[e + f*x]^2*((-3*I)*Cos[2*e + 3*f*x] + 6*f*x*Cos[2*e + 3*f*x] + (2*I)*Cos[4*e + 3*f*x] + 6*f*x*Cos[4*e + 3*f*x] + Cos[f*x]*(5*I + 18*f*x - (9*I)*Log[Cos[e + f*x]^2]) + Cos[2*e + f*x]*(10*I + 18*f*x - (9*I)*Log[Cos[e + f*x]^2]) - (3*I)*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - (3*I)*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - 13*Sin[f*x] - (6*I)*f*x*Sin[f*x] - 3*Log[Cos[e + f*x]^2]*Sin[f*x] + 2*Sin[2*e + f*x] - (6*I)*f*x*Sin[2*e + f*x] - 3*Log[Cos[e + f*x]^2]*Sin[2*e + f*x] - 7*Sin[2*e + 3*f*x] - (6*I)*f*x*Sin[2*e + 3*f*x] - 3*Log[Cos[e + f*x]^2]*Sin[2*e + 3*f*x] - 2*Sin[4*e + 3*f*x] - (6*I)*f*x*Sin[4*e + 3*f*x] - 3*Log[Cos[e + f*x]^2]*Sin[4*e + 3*f*x])*(Cos[e + 5*f*x] + I*Sin[e + 5*f*x]))/(c*f*(Cos[f*x] + I*Sin[f*x])^4)","B",1
925,1,214,71,2.6955059,"\int \frac{(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x]),x]","\frac{a^3 \sec (e) (\tan (e+f x)-i) \left(-2 f x \sin (e+2 f x)+2 i \sin (e+2 f x)-2 f x \sin (3 e+2 f x)+i \sin (3 e+2 f x)-2 i f x \cos (3 e+2 f x)+\cos (3 e+2 f x)-\cos (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+\cos (e) \left(-2 \log \left(\cos ^2(e+f x)\right)-4 i f x+3\right)+\cos (e+2 f x) \left(-\log \left(\cos ^2(e+f x)\right)-2 i f x\right)+i \sin (e+2 f x) \log \left(\cos ^2(e+f x)\right)+i \sin (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)-i \sin (e)\right)}{2 c f}","\frac{a^3 \tan (e+f x)}{c f}-\frac{4 i a^3}{f (c-i c \tan (e+f x))}+\frac{4 i a^3 \log (\cos (e+f x))}{c f}-\frac{4 a^3 x}{c}",1,"(a^3*Sec[e]*(Cos[3*e + 2*f*x] - (2*I)*f*x*Cos[3*e + 2*f*x] + Cos[e]*(3 - (4*I)*f*x - 2*Log[Cos[e + f*x]^2]) + Cos[e + 2*f*x]*((-2*I)*f*x - Log[Cos[e + f*x]^2]) - Cos[3*e + 2*f*x]*Log[Cos[e + f*x]^2] - I*Sin[e] + (2*I)*Sin[e + 2*f*x] - 2*f*x*Sin[e + 2*f*x] + I*Log[Cos[e + f*x]^2]*Sin[e + 2*f*x] + I*Sin[3*e + 2*f*x] - 2*f*x*Sin[3*e + 2*f*x] + I*Log[Cos[e + f*x]^2]*Sin[3*e + 2*f*x])*(-I + Tan[e + f*x]))/(2*c*f)","B",1
926,1,130,55,1.7488439,"\int \frac{(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x]),x]","-\frac{a^2 (\cos (e+3 f x)+i \sin (e+3 f x)) \left(\cos (e+f x) \left(-i \log \left(\cos ^2(e+f x)\right)+4 f x+2 i\right)+\sin (e+f x) \left(-\log \left(\cos ^2(e+f x)\right)-4 i f x-2\right)-2 \tan ^{-1}(\tan (3 e+f x)) (\cos (e+f x)-i \sin (e+f x))\right)}{2 c f (\cos (f x)+i \sin (f x))^2}","-\frac{2 i a^2}{f (c-i c \tan (e+f x))}+\frac{i a^2 \log (\cos (e+f x))}{c f}-\frac{a^2 x}{c}",1,"-1/2*(a^2*(Cos[e + f*x]*(2*I + 4*f*x - I*Log[Cos[e + f*x]^2]) - 2*ArcTan[Tan[3*e + f*x]]*(Cos[e + f*x] - I*Sin[e + f*x]) + (-2 - (4*I)*f*x - Log[Cos[e + f*x]^2])*Sin[e + f*x])*(Cos[e + 3*f*x] + I*Sin[e + 3*f*x]))/(c*f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
927,1,32,23,0.1297811,"\int \frac{a+i a \tan (e+f x)}{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c - I*c*Tan[e + f*x]),x]","\frac{a (\sin (2 (e+f x))-i \cos (2 (e+f x)))}{2 c f}","-\frac{i a}{f (c-i c \tan (e+f x))}",1,"(a*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]))/(2*c*f)","A",1
928,1,29,37,0.0299485,"\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])),x]","\frac{2 (e+f x)+\sin (2 (e+f x))}{4 a c f}","\frac{\sin (e+f x) \cos (e+f x)}{2 a c f}+\frac{x}{2 a c}",1,"(2*(e + f*x) + Sin[2*(e + f*x)])/(4*a*c*f)","A",1
929,1,81,87,1.1512367,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])),x]","-\frac{2 \cos (2 (e+f x))-12 f x \tan (e+f x)+6 i \tan (e+f x)+3 i \sin (3 (e+f x)) \sec (e+f x)+12 i f x-7}{32 a^2 c f (\tan (e+f x)-i)}","\frac{i \cos ^4(e+f x)}{4 a^2 c f}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a^2 c f}+\frac{3 \sin (e+f x) \cos (e+f x)}{8 a^2 c f}+\frac{3 x}{8 a^2 c}",1,"-1/32*(-7 + (12*I)*f*x + 2*Cos[2*(e + f*x)] + (3*I)*Sec[e + f*x]*Sin[3*(e + f*x)] + (6*I)*Tan[e + f*x] - 12*f*x*Tan[e + f*x])/(a^2*c*f*(-I + Tan[e + f*x]))","A",1
930,1,101,124,1.1941909,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])),x]","-\frac{\sec ^2(e+f x) (12 i f x \sin (2 (e+f x))+3 \sin (2 (e+f x))+2 \sin (4 (e+f x))+3 (4 f x+i) \cos (2 (e+f x))-i \cos (4 (e+f x))+9 i)}{48 a^3 c f (\tan (e+f x)-i)^2}","\frac{i c^2}{12 a^3 f (c+i c \tan (e+f x))^3}+\frac{i c}{8 a^3 f (c+i c \tan (e+f x))^2}-\frac{i}{16 a^3 f (c-i c \tan (e+f x))}+\frac{3 i}{16 a^3 f (c+i c \tan (e+f x))}+\frac{x}{4 a^3 c}",1,"-1/48*(Sec[e + f*x]^2*(9*I + 3*(I + 4*f*x)*Cos[2*(e + f*x)] - I*Cos[4*(e + f*x)] + 3*Sin[2*(e + f*x)] + (12*I)*f*x*Sin[2*(e + f*x)] + 2*Sin[4*(e + f*x)]))/(a^3*c*f*(-I + Tan[e + f*x])^2)","A",1
931,1,374,101,2.9859747,"\int \frac{(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^2,x]","\frac{a^4 \sec (e) \sec (e+f x) (\cos (2 (e+3 f x))+i \sin (2 (e+3 f x))) \left(-6 i f x \sin (2 e+f x)+3 \sin (2 e+f x)-6 i f x \sin (2 e+3 f x)-\sin (2 e+3 f x)-6 i f x \sin (4 e+3 f x)+\sin (4 e+3 f x)+6 f x \cos (2 e+3 f x)-3 i \cos (2 e+3 f x)+6 f x \cos (4 e+3 f x)-i \cos (4 e+3 f x)-3 i \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)+\cos (f x) \left(-3 i \log \left(\cos ^2(e+f x)\right)+6 f x+7 i\right)+\cos (2 e+f x) \left(-3 i \log \left(\cos ^2(e+f x)\right)+6 f x+9 i\right)-3 i \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (2 e+f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 \sin (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-6 i f x \sin (f x)+\sin (f x)\right)}{4 c^2 f (\cos (f x)+i \sin (f x))^4}","-\frac{a^4 \tan (e+f x)}{c^2 f}+\frac{12 i a^4}{f \left(c^2-i c^2 \tan (e+f x)\right)}-\frac{6 i a^4 \log (\cos (e+f x))}{c^2 f}+\frac{6 a^4 x}{c^2}-\frac{4 i a^4}{f (c-i c \tan (e+f x))^2}",1,"(a^4*Sec[e]*Sec[e + f*x]*(Cos[2*(e + 3*f*x)] + I*Sin[2*(e + 3*f*x)])*((-3*I)*Cos[2*e + 3*f*x] + 6*f*x*Cos[2*e + 3*f*x] - I*Cos[4*e + 3*f*x] + 6*f*x*Cos[4*e + 3*f*x] + Cos[f*x]*(7*I + 6*f*x - (3*I)*Log[Cos[e + f*x]^2]) + Cos[2*e + f*x]*(9*I + 6*f*x - (3*I)*Log[Cos[e + f*x]^2]) - (3*I)*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - (3*I)*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] + Sin[f*x] - (6*I)*f*x*Sin[f*x] - 3*Log[Cos[e + f*x]^2]*Sin[f*x] + 3*Sin[2*e + f*x] - (6*I)*f*x*Sin[2*e + f*x] - 3*Log[Cos[e + f*x]^2]*Sin[2*e + f*x] - Sin[2*e + 3*f*x] - (6*I)*f*x*Sin[2*e + 3*f*x] - 3*Log[Cos[e + f*x]^2]*Sin[2*e + 3*f*x] + Sin[4*e + 3*f*x] - (6*I)*f*x*Sin[4*e + 3*f*x] - 3*Log[Cos[e + f*x]^2]*Sin[4*e + 3*f*x]))/(4*c^2*f*(Cos[f*x] + I*Sin[f*x])^4)","B",1
932,1,113,83,2.2942536,"\int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^2,x]","\frac{a^3 (\cos (2 e+5 f x)+i \sin (2 e+5 f x)) \left(\cos (2 (e+f x)) \left(-i \log \left(\cos ^2(e+f x)\right)+2 f x-i\right)+\sin (2 (e+f x)) \left(-\log \left(\cos ^2(e+f x)\right)-2 i f x+1\right)+2 i\right)}{2 c^2 f (\cos (f x)+i \sin (f x))^3}","\frac{4 i a^3}{f \left(c^2-i c^2 \tan (e+f x)\right)}-\frac{i a^3 \log (\cos (e+f x))}{c^2 f}+\frac{a^3 x}{c^2}-\frac{2 i a^3}{f (c-i c \tan (e+f x))^2}",1,"(a^3*(2*I + Cos[2*(e + f*x)]*(-I + 2*f*x - I*Log[Cos[e + f*x]^2]) + (1 - (2*I)*f*x - Log[Cos[e + f*x]^2])*Sin[2*(e + f*x)])*(Cos[2*e + 5*f*x] + I*Sin[2*e + 5*f*x]))/(2*c^2*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
933,1,34,28,0.2789882,"\int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^2,x]","\frac{a^2 (\sin (4 (e+f x))-i \cos (4 (e+f x)))}{4 c^2 f}","\frac{a^2 \tan (e+f x)}{f (c-i c \tan (e+f x))^2}",1,"(a^2*((-I)*Cos[4*(e + f*x)] + Sin[4*(e + f*x)]))/(4*c^2*f)","A",1
934,1,51,25,0.733222,"\int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c - I*c*Tan[e + f*x])^2,x]","\frac{a (3 \cos (e+f x)-i \sin (e+f x)) (\sin (3 (e+f x))-i \cos (3 (e+f x)))}{8 c^2 f}","-\frac{i a}{2 f (c-i c \tan (e+f x))^2}",1,"(a*(3*Cos[e + f*x] - I*Sin[e + f*x])*((-I)*Cos[3*(e + f*x)] + Sin[3*(e + f*x)]))/(8*c^2*f)","B",1
935,1,102,101,0.8286219,"\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2),x]","-\frac{(\cos (2 (e+f x))+i \sin (2 (e+f x))) (-2 \cos (2 (e+f x))+12 f x \tan (e+f x)+6 i \tan (e+f x)+3 i \sin (3 (e+f x)) \sec (e+f x)+12 i f x+7)}{32 a c^2 f (\tan (e+f x)-i)}","-\frac{i}{4 a f \left(c^2-i c^2 \tan (e+f x)\right)}+\frac{i}{8 a f \left(c^2+i c^2 \tan (e+f x)\right)}+\frac{3 x}{8 a c^2}-\frac{i}{8 a f (c-i c \tan (e+f x))^2}",1,"-1/32*((Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(7 + (12*I)*f*x - 2*Cos[2*(e + f*x)] + (3*I)*Sec[e + f*x]*Sin[3*(e + f*x)] + (6*I)*Tan[e + f*x] + 12*f*x*Tan[e + f*x]))/(a*c^2*f*(-I + Tan[e + f*x]))","A",1
936,1,39,64,0.0515146,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^2),x]","\frac{12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x))}{32 a^2 c^2 f}","\frac{\sin (e+f x) \cos ^3(e+f x)}{4 a^2 c^2 f}+\frac{3 \sin (e+f x) \cos (e+f x)}{8 a^2 c^2 f}+\frac{3 x}{8 a^2 c^2}",1,"(12*(e + f*x) + 8*Sin[2*(e + f*x)] + Sin[4*(e + f*x)])/(32*a^2*c^2*f)","A",1
937,1,135,114,1.346731,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2),x]","\frac{\sec ^3(e+f x) (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-120 f x \sin (e+f x)+60 i \sin (e+f x)+45 i \sin (3 (e+f x))+5 i \sin (5 (e+f x))+60 i (2 f x+i) \cos (e+f x)+15 \cos (3 (e+f x))+\cos (5 (e+f x)))}{384 a^3 c^2 f (\tan (e+f x)-i)^3}","\frac{i \cos ^6(e+f x)}{6 a^3 c^2 f}+\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a^3 c^2 f}+\frac{5 \sin (e+f x) \cos ^3(e+f x)}{24 a^3 c^2 f}+\frac{5 \sin (e+f x) \cos (e+f x)}{16 a^3 c^2 f}+\frac{5 x}{16 a^3 c^2}",1,"(Sec[e + f*x]^3*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*((60*I)*(I + 2*f*x)*Cos[e + f*x] + 15*Cos[3*(e + f*x)] + Cos[5*(e + f*x)] + (60*I)*Sin[e + f*x] - 120*f*x*Sin[e + f*x] + (45*I)*Sin[3*(e + f*x)] + (5*I)*Sin[5*(e + f*x)]))/(384*a^3*c^2*f*(-I + Tan[e + f*x])^3)","A",1
938,1,569,154,6.5293567,"\int \frac{(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^3,x]","-\frac{a^6 \sec (e) \sec ^2(e+f x) (\cos (3 (e+3 f x))+i \sin (3 (e+3 f x))) \left(-60 i f x \sin (2 e+f x)+70 \sin (2 e+f x)-120 i f x \sin (2 e+3 f x)+11 \sin (2 e+3 f x)-120 i f x \sin (4 e+3 f x)+65 \sin (4 e+3 f x)-60 i f x \sin (4 e+5 f x)-29 \sin (4 e+5 f x)-60 i f x \sin (6 e+5 f x)-2 \sin (6 e+5 f x)+120 f x \cos (2 e+3 f x)+i \cos (2 e+3 f x)+120 f x \cos (4 e+3 f x)+55 i \cos (4 e+3 f x)+60 f x \cos (4 e+5 f x)-25 i \cos (4 e+5 f x)+60 f x \cos (6 e+5 f x)+2 i \cos (6 e+5 f x)-60 i \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)+10 \cos (2 e+f x) \left(-3 i \log \left(\cos ^2(e+f x)\right)+6 f x+11 i\right)+\cos (f x) \left(-30 i \log \left(\cos ^2(e+f x)\right)+60 f x+83 i\right)-60 i \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-30 i \cos (4 e+5 f x) \log \left(\cos ^2(e+f x)\right)-30 i \cos (6 e+5 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (2 e+f x) \log \left(\cos ^2(e+f x)\right)-60 \sin (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-60 \sin (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (4 e+5 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (6 e+5 f x) \log \left(\cos ^2(e+f x)\right)-60 i f x \sin (f x)+43 \sin (f x)\right)}{12 c^3 f (\cos (f x)+i \sin (f x))^6}","\frac{i a^6 \tan ^2(e+f x)}{2 c^3 f}+\frac{9 a^6 \tan (e+f x)}{c^3 f}-\frac{80 i a^6}{f \left(c^3-i c^3 \tan (e+f x)\right)}+\frac{40 i a^6 \log (\cos (e+f x))}{c^3 f}-\frac{40 a^6 x}{c^3}+\frac{40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac{32 i a^6}{3 f (c-i c \tan (e+f x))^3}",1,"-1/12*(a^6*Sec[e]*Sec[e + f*x]^2*(Cos[3*(e + 3*f*x)] + I*Sin[3*(e + 3*f*x)])*(I*Cos[2*e + 3*f*x] + 120*f*x*Cos[2*e + 3*f*x] + (55*I)*Cos[4*e + 3*f*x] + 120*f*x*Cos[4*e + 3*f*x] - (25*I)*Cos[4*e + 5*f*x] + 60*f*x*Cos[4*e + 5*f*x] + (2*I)*Cos[6*e + 5*f*x] + 60*f*x*Cos[6*e + 5*f*x] + 10*Cos[2*e + f*x]*(11*I + 6*f*x - (3*I)*Log[Cos[e + f*x]^2]) + Cos[f*x]*(83*I + 60*f*x - (30*I)*Log[Cos[e + f*x]^2]) - (60*I)*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - (60*I)*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[4*e + 5*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[6*e + 5*f*x]*Log[Cos[e + f*x]^2] + 43*Sin[f*x] - (60*I)*f*x*Sin[f*x] - 30*Log[Cos[e + f*x]^2]*Sin[f*x] + 70*Sin[2*e + f*x] - (60*I)*f*x*Sin[2*e + f*x] - 30*Log[Cos[e + f*x]^2]*Sin[2*e + f*x] + 11*Sin[2*e + 3*f*x] - (120*I)*f*x*Sin[2*e + 3*f*x] - 60*Log[Cos[e + f*x]^2]*Sin[2*e + 3*f*x] + 65*Sin[4*e + 3*f*x] - (120*I)*f*x*Sin[4*e + 3*f*x] - 60*Log[Cos[e + f*x]^2]*Sin[4*e + 3*f*x] - 29*Sin[4*e + 5*f*x] - (60*I)*f*x*Sin[4*e + 5*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[4*e + 5*f*x] - 2*Sin[6*e + 5*f*x] - (60*I)*f*x*Sin[6*e + 5*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[6*e + 5*f*x]))/(c^3*f*(Cos[f*x] + I*Sin[f*x])^6)","B",1
939,1,923,134,8.775153,"\int \frac{(a+i a \tan (e+f x))^5}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^5/(c - I*c*Tan[e + f*x])^3,x]","\frac{\cos ^4(e+f x) \left(\frac{\cos (5 e)}{c^3}-\frac{i \sin (5 e)}{c^3}\right) \sin (f x) (i \tan (e+f x) a+a)^5}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))^5}+\frac{\cos ^5(e+f x) \left(\frac{6 \cos (3 e)}{c^3}-\frac{6 i \sin (3 e)}{c^3}\right) \sin (2 f x) (i \tan (e+f x) a+a)^5}{f (\cos (f x)+i \sin (f x))^5}+\frac{\cos ^5(e+f x) \left(\frac{2 i \sin (e)}{c^3}-\frac{2 \cos (e)}{c^3}\right) \sin (4 f x) (i \tan (e+f x) a+a)^5}{f (\cos (f x)+i \sin (f x))^5}+\frac{\cos ^5(e+f x) \left(\frac{2 \cos (e)}{3 c^3}+\frac{2 i \sin (e)}{3 c^3}\right) \sin (6 f x) (i \tan (e+f x) a+a)^5}{f (\cos (f x)+i \sin (f x))^5}+\frac{x \cos ^5(e+f x) \left(-\frac{4 \cos ^5(e)}{c^3}+\frac{24 i \sin (e) \cos ^4(e)}{c^3}+\frac{60 \sin ^2(e) \cos ^3(e)}{c^3}+\frac{4 \cos ^3(e)}{c^3}-\frac{80 i \sin ^3(e) \cos ^2(e)}{c^3}-\frac{16 i \sin (e) \cos ^2(e)}{c^3}-\frac{60 \sin ^4(e) \cos (e)}{c^3}-\frac{24 \sin ^2(e) \cos (e)}{c^3}+\frac{24 i \sin ^5(e)}{c^3}+\frac{16 i \sin ^3(e)}{c^3}+\frac{4 \sin ^5(e) \tan (e)}{c^3}+\frac{4 \sin ^3(e) \tan (e)}{c^3}-i \left(\frac{8 \cos (5 e)}{c^3}-\frac{8 i \sin (5 e)}{c^3}\right) \tan (e)\right) (i \tan (e+f x) a+a)^5}{(\cos (f x)+i \sin (f x))^5}-\frac{8 x \cos (5 e) \cos ^5(e+f x) (i \tan (e+f x) a+a)^5}{c^3 (\cos (f x)+i \sin (f x))^5}+\frac{4 i \cos (5 e) \cos ^5(e+f x) \log \left(\cos ^2(e+f x)\right) (i \tan (e+f x) a+a)^5}{c^3 f (\cos (f x)+i \sin (f x))^5}+\frac{\cos (6 f x) \cos ^5(e+f x) \left(\frac{2 \sin (e)}{3 c^3}-\frac{2 i \cos (e)}{3 c^3}\right) (i \tan (e+f x) a+a)^5}{f (\cos (f x)+i \sin (f x))^5}+\frac{\cos (4 f x) \cos ^5(e+f x) \left(\frac{2 i \cos (e)}{c^3}+\frac{2 \sin (e)}{c^3}\right) (i \tan (e+f x) a+a)^5}{f (\cos (f x)+i \sin (f x))^5}+\frac{\cos (2 f x) \cos ^5(e+f x) \left(-\frac{6 i \cos (3 e)}{c^3}-\frac{6 \sin (3 e)}{c^3}\right) (i \tan (e+f x) a+a)^5}{f (\cos (f x)+i \sin (f x))^5}+\frac{8 i x \cos ^5(e+f x) \sin (5 e) (i \tan (e+f x) a+a)^5}{c^3 (\cos (f x)+i \sin (f x))^5}+\frac{4 \cos ^5(e+f x) \log \left(\cos ^2(e+f x)\right) \sin (5 e) (i \tan (e+f x) a+a)^5}{c^3 f (\cos (f x)+i \sin (f x))^5}","\frac{a^5 \tan (e+f x)}{c^3 f}-\frac{24 i a^5}{f \left(c^3-i c^3 \tan (e+f x)\right)}+\frac{8 i a^5 \log (\cos (e+f x))}{c^3 f}-\frac{8 a^5 x}{c^3}+\frac{16 i a^5 c^5}{f \left(c^4-i c^4 \tan (e+f x)\right)^2}-\frac{16 i a^5}{3 f (c-i c \tan (e+f x))^3}",1,"(-8*x*Cos[5*e]*Cos[e + f*x]^5*(a + I*a*Tan[e + f*x])^5)/(c^3*(Cos[f*x] + I*Sin[f*x])^5) + ((4*I)*Cos[5*e]*Cos[e + f*x]^5*Log[Cos[e + f*x]^2]*(a + I*a*Tan[e + f*x])^5)/(c^3*f*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[6*f*x]*Cos[e + f*x]^5*((((-2*I)/3)*Cos[e])/c^3 + (2*Sin[e])/(3*c^3))*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[4*f*x]*Cos[e + f*x]^5*(((2*I)*Cos[e])/c^3 + (2*Sin[e])/c^3)*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[2*f*x]*Cos[e + f*x]^5*(((-6*I)*Cos[3*e])/c^3 - (6*Sin[3*e])/c^3)*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[f*x] + I*Sin[f*x])^5) + ((8*I)*x*Cos[e + f*x]^5*Sin[5*e]*(a + I*a*Tan[e + f*x])^5)/(c^3*(Cos[f*x] + I*Sin[f*x])^5) + (4*Cos[e + f*x]^5*Log[Cos[e + f*x]^2]*Sin[5*e]*(a + I*a*Tan[e + f*x])^5)/(c^3*f*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[e + f*x]^4*(Cos[5*e]/c^3 - (I*Sin[5*e])/c^3)*Sin[f*x]*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[e + f*x]^5*((6*Cos[3*e])/c^3 - ((6*I)*Sin[3*e])/c^3)*Sin[2*f*x]*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[e + f*x]^5*((-2*Cos[e])/c^3 + ((2*I)*Sin[e])/c^3)*Sin[4*f*x]*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[f*x] + I*Sin[f*x])^5) + (Cos[e + f*x]^5*((2*Cos[e])/(3*c^3) + (((2*I)/3)*Sin[e])/c^3)*Sin[6*f*x]*(a + I*a*Tan[e + f*x])^5)/(f*(Cos[f*x] + I*Sin[f*x])^5) + (x*Cos[e + f*x]^5*((4*Cos[e]^3)/c^3 - (4*Cos[e]^5)/c^3 - ((16*I)*Cos[e]^2*Sin[e])/c^3 + ((24*I)*Cos[e]^4*Sin[e])/c^3 - (24*Cos[e]*Sin[e]^2)/c^3 + (60*Cos[e]^3*Sin[e]^2)/c^3 + ((16*I)*Sin[e]^3)/c^3 - ((80*I)*Cos[e]^2*Sin[e]^3)/c^3 - (60*Cos[e]*Sin[e]^4)/c^3 + ((24*I)*Sin[e]^5)/c^3 + (4*Sin[e]^3*Tan[e])/c^3 + (4*Sin[e]^5*Tan[e])/c^3 - I*((8*Cos[5*e])/c^3 - ((8*I)*Sin[5*e])/c^3)*Tan[e])*(a + I*a*Tan[e + f*x])^5)/(Cos[f*x] + I*Sin[f*x])^5","B",1
940,1,143,114,2.6012631,"\int \frac{(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^3,x]","\frac{a^4 (\cos (3 e+7 f x)+i \sin (3 e+7 f x)) \left(-9 \sin (e+f x)+6 i f x \sin (3 (e+f x))+2 \sin (3 (e+f x))-3 i \cos (e+f x)+\cos (3 (e+f x)) \left(3 i \log \left(\cos ^2(e+f x)\right)-6 f x-2 i\right)+3 \sin (3 (e+f x)) \log \left(\cos ^2(e+f x)\right)\right)}{6 c^3 f (\cos (f x)+i \sin (f x))^4}","-\frac{6 i a^4}{f \left(c^3-i c^3 \tan (e+f x)\right)}+\frac{i a^4 \log (\cos (e+f x))}{c^3 f}-\frac{a^4 x}{c^3}+\frac{6 i a^4}{c f (c-i c \tan (e+f x))^2}-\frac{8 i a^4}{3 f (c-i c \tan (e+f x))^3}",1,"(a^4*((-3*I)*Cos[e + f*x] + Cos[3*(e + f*x)]*(-2*I - 6*f*x + (3*I)*Log[Cos[e + f*x]^2]) - 9*Sin[e + f*x] + 2*Sin[3*(e + f*x)] + (6*I)*f*x*Sin[3*(e + f*x)] + 3*Log[Cos[e + f*x]^2]*Sin[3*(e + f*x)])*(Cos[3*e + 7*f*x] + I*Sin[3*e + 7*f*x]))/(6*c^3*f*(Cos[f*x] + I*Sin[f*x])^4)","A",1
941,1,34,50,0.2872932,"\int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^3,x]","\frac{a^3 (\sin (6 (e+f x))-i \cos (6 (e+f x)))}{6 c^3 f}","-\frac{i a^3 \left(c^2+i c^2 \tan (e+f x)\right)^3}{6 f \left(c^3-i c^3 \tan (e+f x)\right)^3}",1,"(a^3*((-I)*Cos[6*(e + f*x)] + Sin[6*(e + f*x)]))/(6*c^3*f)","A",1
942,1,53,58,1.506245,"\int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^3,x]","\frac{a^2 (5 \cos (e+f x)-i \sin (e+f x)) (\sin (5 (e+f x))-i \cos (5 (e+f x)))}{24 c^3 f}","\frac{i a^2}{2 c f (c-i c \tan (e+f x))^2}-\frac{2 i a^2}{3 f (c-i c \tan (e+f x))^3}",1,"(a^2*(5*Cos[e + f*x] - I*Sin[e + f*x])*((-I)*Cos[5*(e + f*x)] + Sin[5*(e + f*x)]))/(24*c^3*f)","A",1
943,1,56,25,0.5020911,"\int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c - I*c*Tan[e + f*x])^3,x]","\frac{a (-2 i \sin (2 (e+f x))+4 \cos (2 (e+f x))+3) (\sin (4 (e+f x))-i \cos (4 (e+f x)))}{24 c^3 f}","-\frac{i a}{3 f (c-i c \tan (e+f x))^3}",1,"(a*(3 + 4*Cos[2*(e + f*x)] - (2*I)*Sin[2*(e + f*x)])*((-I)*Cos[4*(e + f*x)] + Sin[4*(e + f*x)]))/(24*c^3*f)","B",1
944,1,115,131,0.8332498,"\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^3} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3),x]","\frac{\sec (e+f x) (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-12 f x \sin (2 (e+f x))-3 i \sin (2 (e+f x))-2 i \sin (4 (e+f x))+(-3-12 i f x) \cos (2 (e+f x))+\cos (4 (e+f x))-9)}{48 a c^3 f (\tan (e+f x)-i)}","-\frac{3 i}{16 a f \left(c^3-i c^3 \tan (e+f x)\right)}+\frac{i}{16 a f \left(c^3+i c^3 \tan (e+f x)\right)}+\frac{x}{4 a c^3}-\frac{i}{8 a c f (c-i c \tan (e+f x))^2}-\frac{i}{12 a f (c-i c \tan (e+f x))^3}",1,"(Sec[e + f*x]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(-9 + (-3 - (12*I)*f*x)*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] - (3*I)*Sin[2*(e + f*x)] - 12*f*x*Sin[2*(e + f*x)] - (2*I)*Sin[4*(e + f*x)]))/(48*a*c^3*f*(-I + Tan[e + f*x]))","A",1
945,1,111,161,1.2421292,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^3} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^3),x]","\frac{(\cos (e+f x)+i \sin (e+f x)) (-120 i f x \sin (e+f x)+60 \sin (e+f x)+45 \sin (3 (e+f x))+5 \sin (5 (e+f x))+60 (2 f x-i) \cos (e+f x)+15 i \cos (3 (e+f x))+i \cos (5 (e+f x)))}{384 a^2 c^3 f}","-\frac{3 i}{16 a^2 f \left(c^3-i c^3 \tan (e+f x)\right)}+\frac{i}{8 a^2 f \left(c^3+i c^3 \tan (e+f x)\right)}+\frac{5 x}{16 a^2 c^3}-\frac{3 i}{32 a^2 c f (c-i c \tan (e+f x))^2}+\frac{i}{32 a^2 c f (c+i c \tan (e+f x))^2}-\frac{i}{24 a^2 f (c-i c \tan (e+f x))^3}",1,"((Cos[e + f*x] + I*Sin[e + f*x])*(60*(-I + 2*f*x)*Cos[e + f*x] + (15*I)*Cos[3*(e + f*x)] + I*Cos[5*(e + f*x)] + 60*Sin[e + f*x] - (120*I)*f*x*Sin[e + f*x] + 45*Sin[3*(e + f*x)] + 5*Sin[5*(e + f*x)]))/(384*a^2*c^3*f)","A",1
946,1,49,91,0.0634252,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^3} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^3),x]","\frac{45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x}{192 a^3 c^3 f}","\frac{\sin (e+f x) \cos ^5(e+f x)}{6 a^3 c^3 f}+\frac{5 \sin (e+f x) \cos ^3(e+f x)}{24 a^3 c^3 f}+\frac{5 \sin (e+f x) \cos (e+f x)}{16 a^3 c^3 f}+\frac{5 x}{16 a^3 c^3}",1,"(60*e + 60*f*x + 45*Sin[2*(e + f*x)] + 9*Sin[4*(e + f*x)] + Sin[6*(e + f*x)])/(192*a^3*c^3*f)","A",1
947,1,455,160,6.4496419,"\int \frac{(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^6 \sec (e) \sec (e+f x) (\cos (4 (e+f x))+i \sin (4 (e+f x))) \left(40 \sin (2 e+f x)-60 i f x \sin (2 e+3 f x)+43 \sin (2 e+3 f x)-60 i f x \sin (4 e+3 f x)+55 \sin (4 e+3 f x)-60 i f x \sin (4 e+5 f x)-9 \sin (4 e+5 f x)-60 i f x \sin (6 e+5 f x)+3 \sin (6 e+5 f x)+20 i \cos (2 e+f x)+60 f x \cos (2 e+3 f x)+53 i \cos (2 e+3 f x)+60 f x \cos (4 e+3 f x)+65 i \cos (4 e+3 f x)+60 f x \cos (4 e+5 f x)-15 i \cos (4 e+5 f x)+60 f x \cos (6 e+5 f x)-3 i \cos (6 e+5 f x)-30 i \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-30 i \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-30 i \cos (4 e+5 f x) \log \left(\cos ^2(e+f x)\right)-30 i \cos (6 e+5 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (4 e+5 f x) \log \left(\cos ^2(e+f x)\right)-30 \sin (6 e+5 f x) \log \left(\cos ^2(e+f x)\right)+40 \sin (f x)+20 i \cos (f x)\right)}{24 c^4 f}","-\frac{a^6 \tan (e+f x)}{c^4 f}+\frac{40 i a^6}{f \left(c^4-i c^4 \tan (e+f x)\right)}-\frac{10 i a^6 \log (\cos (e+f x))}{c^4 f}+\frac{10 a^6 x}{c^4}-\frac{40 i a^6}{f \left(c^2-i c^2 \tan (e+f x)\right)^2}+\frac{80 i a^6}{3 c f (c-i c \tan (e+f x))^3}-\frac{8 i a^6}{f (c-i c \tan (e+f x))^4}",1,"(a^6*Sec[e]*Sec[e + f*x]*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)])*((20*I)*Cos[f*x] + (20*I)*Cos[2*e + f*x] + (53*I)*Cos[2*e + 3*f*x] + 60*f*x*Cos[2*e + 3*f*x] + (65*I)*Cos[4*e + 3*f*x] + 60*f*x*Cos[4*e + 3*f*x] - (15*I)*Cos[4*e + 5*f*x] + 60*f*x*Cos[4*e + 5*f*x] - (3*I)*Cos[6*e + 5*f*x] + 60*f*x*Cos[6*e + 5*f*x] - (30*I)*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[4*e + 5*f*x]*Log[Cos[e + f*x]^2] - (30*I)*Cos[6*e + 5*f*x]*Log[Cos[e + f*x]^2] + 40*Sin[f*x] + 40*Sin[2*e + f*x] + 43*Sin[2*e + 3*f*x] - (60*I)*f*x*Sin[2*e + 3*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[2*e + 3*f*x] + 55*Sin[4*e + 3*f*x] - (60*I)*f*x*Sin[4*e + 3*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[4*e + 3*f*x] - 9*Sin[4*e + 5*f*x] - (60*I)*f*x*Sin[4*e + 5*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[4*e + 5*f*x] + 3*Sin[6*e + 5*f*x] - (60*I)*f*x*Sin[6*e + 5*f*x] - 30*Log[Cos[e + f*x]^2]*Sin[6*e + 5*f*x]))/(24*c^4*f)","B",1
948,1,151,146,2.8915966,"\int \frac{(a+i a \tan (e+f x))^5}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])^5/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^5 (\cos (4 e+9 f x)+i \sin (4 e+9 f x)) \left(8 \sin (2 (e+f x))-12 i f x \sin (4 (e+f x))+3 \sin (4 (e+f x))+16 i \cos (2 (e+f x))+3 \cos (4 (e+f x)) \left(-2 i \log \left(\cos ^2(e+f x)\right)+4 f x-i\right)-6 \sin (4 (e+f x)) \log \left(\cos ^2(e+f x)\right)-6 i\right)}{12 c^4 f (\cos (f x)+i \sin (f x))^5}","\frac{8 i a^5}{f \left(c^4-i c^4 \tan (e+f x)\right)}-\frac{i a^5 \log (\cos (e+f x))}{c^4 f}+\frac{a^5 x}{c^4}-\frac{12 i a^5}{f \left(c^2-i c^2 \tan (e+f x)\right)^2}+\frac{32 i a^5 c^5}{3 f \left(c^3-i c^3 \tan (e+f x)\right)^3}-\frac{4 i a^5}{f (c-i c \tan (e+f x))^4}",1,"(a^5*(-6*I + (16*I)*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)]*(-I + 4*f*x - (2*I)*Log[Cos[e + f*x]^2]) + 8*Sin[2*(e + f*x)] + 3*Sin[4*(e + f*x)] - (12*I)*f*x*Sin[4*(e + f*x)] - 6*Log[Cos[e + f*x]^2]*Sin[4*(e + f*x)])*(Cos[4*e + 9*f*x] + I*Sin[4*e + 9*f*x]))/(12*c^4*f*(Cos[f*x] + I*Sin[f*x])^5)","A",1
949,1,34,50,0.4093666,"\int \frac{(a+i a \tan (e+f x))^4}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^4 (\sin (8 (e+f x))-i \cos (8 (e+f x)))}{8 c^4 f}","-\frac{i a^4 \left(c^2+i c^2 \tan (e+f x)\right)^4}{8 f \left(c^3-i c^3 \tan (e+f x)\right)^4}",1,"(a^4*((-I)*Cos[8*(e + f*x)] + Sin[8*(e + f*x)]))/(8*c^4*f)","A",1
950,1,53,87,2.2832491,"\int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^3 (7 \cos (e+f x)-i \sin (e+f x)) (\sin (7 (e+f x))-i \cos (7 (e+f x)))}{48 c^4 f}","-\frac{i a^3}{2 f \left(c^2-i c^2 \tan (e+f x)\right)^2}+\frac{4 i a^3}{3 c f (c-i c \tan (e+f x))^3}-\frac{i a^3}{f (c-i c \tan (e+f x))^4}",1,"(a^3*(7*Cos[e + f*x] - I*Sin[e + f*x])*((-I)*Cos[7*(e + f*x)] + Sin[7*(e + f*x)]))/(48*c^4*f)","A",1
951,1,75,62,1.8680046,"\int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^4,x]","\frac{a^2 (-3 i \sin (2 (e+f x))+9 \cos (2 (e+f x))+8) (\sin (6 e+8 f x)-i \cos (6 e+8 f x))}{96 c^4 f (\cos (f x)+i \sin (f x))^2}","\frac{i a^2 c^2}{3 f \left(c^2-i c^2 \tan (e+f x)\right)^3}-\frac{i a^2}{2 f (c-i c \tan (e+f x))^4}",1,"(a^2*(8 + 9*Cos[2*(e + f*x)] - (3*I)*Sin[2*(e + f*x)])*((-I)*Cos[6*e + 8*f*x] + Sin[6*e + 8*f*x]))/(96*c^4*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
952,1,74,25,0.612993,"\int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c - I*c*Tan[e + f*x])^4,x]","\frac{a (-i (2 \sin (e+f x)+3 \sin (3 (e+f x)))+10 \cos (e+f x)+5 \cos (3 (e+f x))) (\sin (5 (e+f x))-i \cos (5 (e+f x)))}{64 c^4 f}","-\frac{i a}{4 f (c-i c \tan (e+f x))^4}",1,"(a*(10*Cos[e + f*x] + 5*Cos[3*(e + f*x)] - I*(2*Sin[e + f*x] + 3*Sin[3*(e + f*x)]))*((-I)*Cos[5*(e + f*x)] + Sin[5*(e + f*x)]))/(64*c^4*f)","B",1
953,1,134,162,1.0543135,"\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4),x]","\frac{\sec (e+f x) (\cos (4 (e+f x))+i \sin (4 (e+f x))) (60 i \sin (e+f x)-120 f x \sin (3 (e+f x))-20 i \sin (3 (e+f x))-15 i \sin (5 (e+f x))-180 \cos (e+f x)+(-20-120 i f x) \cos (3 (e+f x))+9 \cos (5 (e+f x)))}{768 a c^4 f (\tan (e+f x)-i)}","-\frac{i}{8 a f \left(c^4-i c^4 \tan (e+f x)\right)}+\frac{i}{32 a f \left(c^4+i c^4 \tan (e+f x)\right)}+\frac{5 x}{32 a c^4}-\frac{3 i}{32 a f \left(c^2-i c^2 \tan (e+f x)\right)^2}-\frac{i}{12 a c f (c-i c \tan (e+f x))^3}-\frac{i}{16 a f (c-i c \tan (e+f x))^4}",1,"(Sec[e + f*x]*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)])*(-180*Cos[e + f*x] + (-20 - (120*I)*f*x)*Cos[3*(e + f*x)] + 9*Cos[5*(e + f*x)] + (60*I)*Sin[e + f*x] - (20*I)*Sin[3*(e + f*x)] - 120*f*x*Sin[3*(e + f*x)] - (15*I)*Sin[5*(e + f*x)]))/(768*a*c^4*f*(-I + Tan[e + f*x]))","A",1
954,1,139,193,1.3127048,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^4} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4),x]","\frac{\sec ^2(e+f x) (\sin (4 (e+f x))-i \cos (4 (e+f x))) (-120 f x \sin (2 (e+f x))-30 i \sin (2 (e+f x))-32 i \sin (4 (e+f x))-3 i \sin (6 (e+f x))+(-30-120 i f x) \cos (2 (e+f x))+16 \cos (4 (e+f x))+\cos (6 (e+f x))-80)}{512 a^2 c^4 f (\tan (e+f x)-i)^2}","-\frac{5 i}{32 a^2 f \left(c^4-i c^4 \tan (e+f x)\right)}+\frac{5 i}{64 a^2 f \left(c^4+i c^4 \tan (e+f x)\right)}+\frac{15 x}{64 a^2 c^4}-\frac{3 i}{32 a^2 f \left(c^2-i c^2 \tan (e+f x)\right)^2}+\frac{i}{64 a^2 f \left(c^2+i c^2 \tan (e+f x)\right)^2}-\frac{i}{16 a^2 c f (c-i c \tan (e+f x))^3}-\frac{i}{32 a^2 f (c-i c \tan (e+f x))^4}",1,"(Sec[e + f*x]^2*((-I)*Cos[4*(e + f*x)] + Sin[4*(e + f*x)])*(-80 + (-30 - (120*I)*f*x)*Cos[2*(e + f*x)] + 16*Cos[4*(e + f*x)] + Cos[6*(e + f*x)] - (30*I)*Sin[2*(e + f*x)] - 120*f*x*Sin[2*(e + f*x)] - (32*I)*Sin[4*(e + f*x)] - (3*I)*Sin[6*(e + f*x)]))/(512*a^2*c^4*f*(-I + Tan[e + f*x])^2)","A",1
955,1,133,223,1.9676726,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4),x]","\frac{(\cos (e+f x)+i \sin (e+f x)) (-840 i f x \sin (e+f x)+420 \sin (e+f x)+378 \sin (3 (e+f x))+70 \sin (5 (e+f x))+7 \sin (7 (e+f x))+420 (2 f x-i) \cos (e+f x)+126 i \cos (3 (e+f x))+14 i \cos (5 (e+f x))+i \cos (7 (e+f x)))}{3072 a^3 c^4 f}","-\frac{5 i}{32 a^3 f \left(c^4-i c^4 \tan (e+f x)\right)}+\frac{15 i}{128 a^3 f \left(c^4+i c^4 \tan (e+f x)\right)}+\frac{35 x}{128 a^3 c^4}-\frac{5 i}{64 a^3 f \left(c^2-i c^2 \tan (e+f x)\right)^2}+\frac{5 i}{128 a^3 f \left(c^2+i c^2 \tan (e+f x)\right)^2}-\frac{i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac{i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac{i}{64 a^3 f (c-i c \tan (e+f x))^4}",1,"((Cos[e + f*x] + I*Sin[e + f*x])*(420*(-I + 2*f*x)*Cos[e + f*x] + (126*I)*Cos[3*(e + f*x)] + (14*I)*Cos[5*(e + f*x)] + I*Cos[7*(e + f*x)] + 420*Sin[e + f*x] - (840*I)*f*x*Sin[e + f*x] + 378*Sin[3*(e + f*x)] + 70*Sin[5*(e + f*x)] + 7*Sin[7*(e + f*x)]))/(3072*a^3*c^4*f)","A",1
956,1,61,92,3.3041186,"\int (a+i a \tan (e+f x))^3 \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 i a^3 \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (7 i \sin (2 (e+f x))+23 \cos (2 (e+f x))+20)}{15 f}","\frac{2 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c f}+\frac{8 i a^3 \sqrt{c-i c \tan (e+f x)}}{f}",1,"(((2*I)/15)*a^3*Sec[e + f*x]^2*(20 + 23*Cos[2*(e + f*x)] + (7*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/f","A",1
957,1,37,60,2.2054787,"\int (a+i a \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{2 a^2 (\tan (e+f x)-5 i) \sqrt{c-i c \tan (e+f x)}}{3 f}","\frac{4 i a^2 \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 c f}",1,"(-2*a^2*(-5*I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*f)","A",1
958,1,25,25,0.8800755,"\int (a+i a \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 i a \sqrt{c-i c \tan (e+f x)}}{f}","\frac{2 i a \sqrt{c-i c \tan (e+f x)}}{f}",1,"((2*I)*a*Sqrt[c - I*c*Tan[e + f*x]])/f","A",1
959,1,110,95,1.1678884,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{a+i a \tan (e+f x)} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x]),x]","\frac{(\sin (e+f x)+i \cos (e+f x)) \left(2 \cos (e+f x) \sqrt{c-i c \tan (e+f x)}+\sqrt{2} \sqrt{c} (\cos (e+f x)+i \sin (e+f x)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{4 a f}","\frac{i \sqrt{c-i c \tan (e+f x)}}{2 a f (1+i \tan (e+f x))}+\frac{i \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{2 \sqrt{2} a f}",1,"((I*Cos[e + f*x] + Sin[e + f*x])*(Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[e + f*x] + I*Sin[e + f*x]) + 2*Cos[e + f*x]*Sqrt[c - I*c*Tan[e + f*x]]))/(4*a*f)","A",1
960,1,136,138,1.7114009,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^2,x]","\frac{(\sin (2 (e+f x))+i \cos (2 (e+f x))) \left(\sqrt{c-i c \tan (e+f x)} (3 i \sin (2 (e+f x))+7 \cos (2 (e+f x))+7)+3 \sqrt{2} \sqrt{c} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{32 a^2 f}","\frac{3 i \sqrt{c-i c \tan (e+f x)}}{16 a^2 f (1+i \tan (e+f x))}+\frac{i \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{3 i \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{16 \sqrt{2} a^2 f}",1,"((I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*(3*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]) + (7 + 7*Cos[2*(e + f*x)] + (3*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]]))/(32*a^2*f)","A",1
961,1,149,181,2.109971,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3,x]","\frac{(\sin (3 (e+f x))+i \cos (3 (e+f x))) \left(\sqrt{c-i c \tan (e+f x)} \left(93 \cos (e+f x)+41 \cos (3 (e+f x))+100 i \sin (e+f x) \cos ^2(e+f x)\right)+15 \sqrt{2} \sqrt{c} (\cos (3 (e+f x))+i \sin (3 (e+f x))) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{384 a^3 f}","\frac{5 i \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac{5 i \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{i \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{5 i \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{64 \sqrt{2} a^3 f}",1,"((I*Cos[3*(e + f*x)] + Sin[3*(e + f*x)])*(15*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)]) + (93*Cos[e + f*x] + 41*Cos[3*(e + f*x)] + (100*I)*Cos[e + f*x]^2*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]))/(384*a^3*f)","A",1
962,1,94,94,3.3472899,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a^3 c \sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (e-2 f x)+i \cos (e-2 f x)) (27 i \sin (2 (e+f x))+43 \cos (2 (e+f x))+28)}{105 f (\cos (f x)+i \sin (f x))^3}","\frac{2 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 c f}+\frac{8 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 f}",1,"(2*a^3*c*Sec[e + f*x]^3*(I*Cos[e - 2*f*x] + Sin[e - 2*f*x])*(28 + 43*Cos[2*(e + f*x)] + (27*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(105*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
963,1,80,62,2.4473794,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{2 a^2 c (3 \tan (e+f x)-7 i) \sec (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (e-f x)-i \sin (e-f x))}{15 f (\cos (f x)+i \sin (f x))^2}","\frac{4 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 c f}",1,"(-2*a^2*c*Sec[e + f*x]*(Cos[e - f*x] - I*Sin[e - f*x])*(-7*I + 3*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
964,1,54,27,1.1105914,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a c (\sin (e)+i \cos (e)) \sec (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)}}{3 f}","\frac{2 i a (c-i c \tan (e+f x))^{3/2}}{3 f}",1,"(2*a*c*Sec[e + f*x]*(I*Cos[e] + Sin[e])*(Cos[f*x] - I*Sin[f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*f)","A",1
965,1,114,95,1.4916317,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x]),x]","\frac{(-c \sin (e+f x)-i c \cos (e+f x)) \left(\sqrt{2} \sqrt{c} (\cos (e+f x)+i \sin (e+f x)) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)-2 \cos (e+f x) \sqrt{c-i c \tan (e+f x)}\right)}{2 a f}","\frac{i c^2 \sqrt{c-i c \tan (e+f x)}}{a f (c+i c \tan (e+f x))}-\frac{i c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{\sqrt{2} a f}",1,"(((-I)*c*Cos[e + f*x] - c*Sin[e + f*x])*(Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[e + f*x] + I*Sin[e + f*x]) - 2*Cos[e + f*x]*Sqrt[c - I*c*Tan[e + f*x]]))/(2*a*f)","A",1
966,1,136,146,2.3139553,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{c (\cos (2 (e+f x))-i \sin (2 (e+f x))) \left(\sqrt{c-i c \tan (e+f x)} (\sin (2 (e+f x))+3 i \cos (2 (e+f x))+3 i)+\sqrt{2} \sqrt{c} (\sin (2 (e+f x))-i \cos (2 (e+f x))) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{16 a^2 f}","-\frac{i c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{8 \sqrt{2} a^2 f}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{2 a^2 f (c+i c \tan (e+f x))^2}-\frac{i c^2 \sqrt{c-i c \tan (e+f x)}}{8 a^2 f (c+i c \tan (e+f x))}",1,"(c*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*(Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)]) + (3*I + (3*I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]]))/(16*a^2*f)","A",1
967,1,152,193,2.9499221,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{c (\cos (3 (e+f x))-i \sin (3 (e+f x))) \left(\sqrt{c-i c \tan (e+f x)} \left(39 i \cos (e+f x)+11 i \cos (3 (e+f x))+20 \sin (e+f x) \cos ^2(e+f x)\right)+3 \sqrt{2} \sqrt{c} (\sin (3 (e+f x))-i \cos (3 (e+f x))) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{192 a^3 f}","-\frac{i c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{32 \sqrt{2} a^3 f}+\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{3 a^3 f (c+i c \tan (e+f x))^3}-\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{24 a^3 f (c+i c \tan (e+f x))^2}-\frac{i c^2 \sqrt{c-i c \tan (e+f x)}}{32 a^3 f (c+i c \tan (e+f x))}",1,"(c*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)])*(3*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*((-I)*Cos[3*(e + f*x)] + Sin[3*(e + f*x)]) + ((39*I)*Cos[e + f*x] + (11*I)*Cos[3*(e + f*x)] + 20*Cos[e + f*x]^2*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]))/(192*a^3*f)","A",1
968,1,100,94,4.8314312,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a^3 c^2 \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (2 e-f x)+i \cos (2 e-f x)) (55 i \sin (2 (e+f x))+71 \cos (2 (e+f x))+36)}{315 f (\cos (f x)+i \sin (f x))^3}","\frac{2 i a^3 (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{8 i a^3 (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{8 i a^3 (c-i c \tan (e+f x))^{5/2}}{5 f}",1,"(2*a^3*c^2*Sec[e + f*x]^4*(I*Cos[2*e - f*x] + Sin[2*e - f*x])*(36 + 71*Cos[2*(e + f*x)] + (55*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(315*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
969,1,78,62,3.4578079,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{2 a^2 c^2 (\cos (2 e)-i \sin (2 e)) (5 \tan (e+f x)-9 i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)}}{35 f (\cos (f x)+i \sin (f x))^2}","\frac{4 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{7/2}}{7 c f}",1,"(-2*a^2*c^2*Sec[e + f*x]^2*(Cos[2*e] - I*Sin[2*e])*(-9*I + 5*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(35*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
970,1,70,27,1.5731406,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a c^2 \sec ^2(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (2 e+f x)+i \cos (2 e+f x))}{5 f}","\frac{2 i a (c-i c \tan (e+f x))^{5/2}}{5 f}",1,"(2*a*c^2*Sec[e + f*x]^2*(Cos[f*x] - I*Sin[f*x])*(I*Cos[2*e + f*x] + Sin[2*e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(5*f)","B",1
971,-1,0,125,180.0052504,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]","\text{\$Aborted}","-\frac{3 i \sqrt{2} c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{a f}+\frac{3 i c^2 \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{i c^2 (c-i c \tan (e+f x))^{3/2}}{a f (c+i c \tan (e+f x))}",1,"$Aborted","F",-1
972,1,138,146,3.0031203,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{c^2 (\sin (2 (e+f x))+i \cos (2 (e+f x))) \left(3 \sqrt{2} \sqrt{c} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)-\sqrt{c-i c \tan (e+f x)} (5 i \sin (2 (e+f x))+\cos (2 (e+f x))+1)\right)}{8 a^2 f}","\frac{3 i c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a^2 f}-\frac{3 i c^3 \sqrt{c-i c \tan (e+f x)}}{4 a^2 f (c+i c \tan (e+f x))}+\frac{i c^3 (c-i c \tan (e+f x))^{3/2}}{2 a^2 f (c+i c \tan (e+f x))^2}",1,"(c^2*(I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*(3*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]) - (1 + Cos[2*(e + f*x)] + (5*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]]))/(8*a^2*f)","A",1
973,1,152,193,4.2227012,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{c^2 (\sin (3 (e+f x))+i \cos (3 (e+f x))) \left(\sqrt{c-i c \tan (e+f x)} \left(9 \cos (e+f x)+5 \cos (3 (e+f x))-44 i \sin (e+f x) \cos ^2(e+f x)\right)+3 \sqrt{2} \sqrt{c} (\cos (3 (e+f x))+i \sin (3 (e+f x))) \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)\right)}{96 a^3 f}","\frac{i c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{16 \sqrt{2} a^3 f}-\frac{i c^4 \sqrt{c-i c \tan (e+f x)}}{4 a^3 f (c+i c \tan (e+f x))^2}+\frac{i c^4 (c-i c \tan (e+f x))^{3/2}}{3 a^3 f (c+i c \tan (e+f x))^3}+\frac{i c^3 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (c+i c \tan (e+f x))}",1,"(c^2*(I*Cos[3*(e + f*x)] + Sin[3*(e + f*x)])*(3*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)]) + (9*Cos[e + f*x] + 5*Cos[3*(e + f*x)] - (44*I)*Cos[e + f*x]^2*Sin[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]))/(96*a^3*f)","A",1
974,1,94,90,3.0939727,"\int \frac{(a+i a \tan (e+f x))^3}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 a^3 \sec (e+f x) \sqrt{c-i c \tan (e+f x)} (-5 i \sin (2 (e+f x))+11 \cos (2 (e+f x))+12) (\sin (e+4 f x)-i \cos (e+4 f x))}{3 c f (\cos (f x)+i \sin (f x))^3}","\frac{2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}-\frac{8 i a^3 \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{8 i a^3}{f \sqrt{c-i c \tan (e+f x)}}",1,"(2*a^3*Sec[e + f*x]*(12 + 11*Cos[2*(e + f*x)] - (5*I)*Sin[2*(e + f*x)])*((-I)*Cos[e + 4*f*x] + Sin[e + 4*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
975,1,91,58,2.3584343,"\int \frac{(a+i a \tan (e+f x))^2}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 a^2 \sqrt{c-i c \tan (e+f x)} (-2 \sin (2 e)-2 i \cos (2 e)+\sin (2 f x)-i \cos (2 f x)) (\cos (e+f x)+i \sin (e+f x))^2}{c f (\cos (f x)+i \sin (f x))^2}","-\frac{2 i a^2 \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{4 i a^2}{f \sqrt{c-i c \tan (e+f x)}}",1,"(2*a^2*((-2*I)*Cos[2*e] - I*Cos[2*f*x] - 2*Sin[2*e] + Sin[2*f*x])*(Cos[e + f*x] + I*Sin[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]])/(c*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
976,1,64,25,1.0163123,"\int \frac{a+i a \tan (e+f x)}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{2 a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (e+2 f x)-i \cos (e+2 f x))}{c f}","-\frac{2 i a}{f \sqrt{c-i c \tan (e+f x)}}",1,"(2*a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*((-I)*Cos[e + 2*f*x] + Sin[e + 2*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)","B",1
977,1,117,124,1.8286451,"\int \frac{1}{(a+i a \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]),x]","-\frac{i e^{-2 i (e+f x)} \left(e^{2 i (e+f x)}+2 e^{4 i (e+f x)}-3 e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-1\right) \sqrt{c-i c \tan (e+f x)}}{8 a c f}","-\frac{3 i}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{3 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a \sqrt{c} f}",1,"((-1/8*I)*(-1 + E^((2*I)*(e + f*x)) + 2*E^((4*I)*(e + f*x)) - 3*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[c - I*c*Tan[e + f*x]])/(a*c*E^((2*I)*(e + f*x))*f)","A",1
978,1,141,167,2.4692613,"\int \frac{1}{(a+i a \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]),x]","-\frac{i e^{-4 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(-11 e^{2 i (e+f x)}-e^{4 i (e+f x)}+8 e^{6 i (e+f x)}-15 e^{4 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-2\right)}{32 \sqrt{2} a^2 c f}","-\frac{15 i}{32 a^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{5 i}{16 a^2 f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)}}+\frac{15 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{32 \sqrt{2} a^2 \sqrt{c} f}",1,"((-1/32*I)*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*(-2 - 11*E^((2*I)*(e + f*x)) - E^((4*I)*(e + f*x)) + 8*E^((6*I)*(e + f*x)) - 15*E^((4*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]]))/(Sqrt[2]*a^2*c*E^((4*I)*(e + f*x))*f)","A",1
979,1,146,210,3.0488017,"\int \frac{1}{(a+i a \tan (e+f x))^3 \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]]),x]","-\frac{i \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} \left(-7 i \sin (2 (e+f x))-56 i \sin (4 (e+f x))+85 \cos (2 (e+f x))-40 \cos (4 (e+f x))+105 e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)+125\right)}{768 a^3 c f (\tan (e+f x)-i)^2}","-\frac{35 i}{128 a^3 f \sqrt{c-i c \tan (e+f x)}}+\frac{35 i}{192 a^3 f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{7 i}{48 a^3 f (1+i \tan (e+f x))^2 \sqrt{c-i c \tan (e+f x)}}+\frac{i}{6 a^3 f (1+i \tan (e+f x))^3 \sqrt{c-i c \tan (e+f x)}}+\frac{35 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{128 \sqrt{2} a^3 \sqrt{c} f}",1,"((-1/768*I)*Sec[e + f*x]^2*(125 + 105*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]] + 85*Cos[2*(e + f*x)] - 40*Cos[4*(e + f*x)] - (7*I)*Sin[2*(e + f*x)] - (56*I)*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*c*f*(-I + Tan[e + f*x])^2)","A",1
980,1,94,90,5.3589558,"\int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a^3 \sqrt{c-i c \tan (e+f x)} (9 \sin (2 (e+f x))+7 i \cos (2 (e+f x))+4 i) (\cos (2 e+5 f x)+i \sin (2 e+5 f x))}{3 c^2 f (\cos (f x)+i \sin (f x))^3}","\frac{2 i a^3 \sqrt{c-i c \tan (e+f x)}}{c^2 f}+\frac{8 i a^3}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(2*a^3*(4*I + (7*I)*Cos[2*(e + f*x)] + 9*Sin[2*(e + f*x)])*(Cos[2*e + 5*f*x] + I*Sin[2*e + 5*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
981,1,93,60,4.1514426,"\int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a^2 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (3 \sin (e+f x)+i \cos (e+f x)) (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x)))}{3 c^2 f (\cos (f x)+i \sin (f x))^2}","\frac{2 i a^2}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(2*a^2*Cos[e + f*x]*(I*Cos[e + f*x] + 3*Sin[e + f*x])*(Cos[2*(e + 2*f*x)] + I*Sin[2*(e + 2*f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
982,1,72,27,1.7043735,"\int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a \cos ^2(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (2 e+3 f x)-i \cos (2 e+3 f x))}{3 c^2 f}","-\frac{2 i a}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(2*a*Cos[e + f*x]^2*(Cos[f*x] - I*Sin[f*x])*((-I)*Cos[2*e + 3*f*x] + Sin[2*e + 3*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f)","B",1
983,1,138,156,3.016213,"\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x)) \left(5 \sin (e+f x)+5 \sin (3 (e+f x))-27 i \cos (e+f x)+i \cos (3 (e+f x))+15 i e^{-i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)\right)}{48 a c^2 f}","\frac{5 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{8 \sqrt{2} a c^{3/2} f}-\frac{5 i}{8 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{5 i}{12 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}",1,"((Cos[e + f*x] + I*Sin[e + f*x])*(((15*I)*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])/E^(I*(e + f*x)) - (27*I)*Cos[e + f*x] + I*Cos[3*(e + f*x)] + 5*Sin[e + f*x] + 5*Sin[3*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(48*a*c^2*f)","A",1
984,1,145,199,3.3950356,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)),x]","-\frac{i e^{-4 i (e+f x)} \left(-45 e^{2 i (e+f x)}+41 e^{4 i (e+f x)}+88 e^{6 i (e+f x)}+8 e^{8 i (e+f x)}-105 e^{4 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-6\right) \sqrt{c-i c \tan (e+f x)}}{384 a^2 c^2 f}","\frac{35 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{64 \sqrt{2} a^2 c^{3/2} f}-\frac{35 i}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}",1,"((-1/384*I)*(-6 - 45*E^((2*I)*(e + f*x)) + 41*E^((4*I)*(e + f*x)) + 88*E^((6*I)*(e + f*x)) + 8*E^((8*I)*(e + f*x)) - 105*E^((4*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*c^2*E^((4*I)*(e + f*x))*f)","A",1
985,1,160,242,4.1945213,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (e+f x)-i \sin (e+f x)) \left(258 \sin (e+f x)+282 \sin (3 (e+f x))+24 \sin (5 (e+f x))+172 i \cos (e+f x)-166 i \cos (3 (e+f x))-8 i \cos (5 (e+f x))+315 i e^{i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)\right)}{1536 a^3 c^2 f}","\frac{105 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{256 \sqrt{2} a^3 c^{3/2} f}-\frac{105 i}{256 a^3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{35 i}{128 a^3 f (c-i c \tan (e+f x))^{3/2}}+\frac{21 i}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{3 i}{16 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{i}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{3/2}}",1,"((Cos[e + f*x] - I*Sin[e + f*x])*((315*I)*E^(I*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]] + (172*I)*Cos[e + f*x] - (166*I)*Cos[3*(e + f*x)] - (8*I)*Cos[5*(e + f*x)] + 258*Sin[e + f*x] + 282*Sin[3*(e + f*x)] + 24*Sin[5*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(1536*a^3*c^2*f)","A",1
986,1,98,92,8.9591805,"\int \frac{(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a^3 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (-5 i \sin (2 (e+f x))+11 \cos (2 (e+f x))-4) (\sin (3 (e+2 f x))-i \cos (3 (e+2 f x)))}{15 c^3 f (\cos (f x)+i \sin (f x))^3}","-\frac{2 i a^3}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{8 i a^3}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{8 i a^3}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(2*a^3*Cos[e + f*x]*(-4 + 11*Cos[2*(e + f*x)] - (5*I)*Sin[2*(e + f*x)])*((-I)*Cos[3*(e + 2*f*x)] + Sin[3*(e + 2*f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
987,1,95,62,6.5779365,"\int \frac{(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a^2 \cos ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (5 \sin (e+f x)-i \cos (e+f x)) (\cos (3 e+5 f x)+i \sin (3 e+5 f x))}{15 c^3 f (\cos (f x)+i \sin (f x))^2}","\frac{2 i a^2}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{4 i a^2}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(2*a^2*Cos[e + f*x]^2*((-I)*Cos[e + f*x] + 5*Sin[e + f*x])*(Cos[3*e + 5*f*x] + I*Sin[3*e + 5*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
988,1,72,27,2.5730896,"\int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a \cos ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (3 e+4 f x)-i \cos (3 e+4 f x))}{5 c^3 f}","-\frac{2 i a}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(2*a*Cos[e + f*x]^3*(Cos[f*x] - I*Sin[f*x])*((-I)*Cos[3*e + 4*f*x] + Sin[3*e + 4*f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(5*c^3*f)","B",1
989,1,149,188,3.9916192,"\int \frac{1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \left(-63 \sin (2 (e+f x))+21 \sin (4 (e+f x))-139 i \cos (2 (e+f x))+9 i \cos (4 (e+f x))+105 i e^{-2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-148 i\right)}{480 a c^3 f}","\frac{7 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{16 \sqrt{2} a c^{5/2} f}-\frac{7 i}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{7 i}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{7 i}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}",1,"((Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(-148*I + ((105*I)*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])/E^((2*I)*(e + f*x)) - (139*I)*Cos[2*(e + f*x)] + (9*I)*Cos[4*(e + f*x)] - 63*Sin[2*(e + f*x)] + 21*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(480*a*c^3*f)","A",1
990,1,182,231,6.3564318,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (3 (e+f x))-i \cos (3 (e+f x))) \left(-141 i \sin (e+f x)-159 i \sin (3 (e+f x))-18 i \sin (5 (e+f x))-547 \cos (e+f x)+31 \cos (3 (e+f x))+2 \cos (5 (e+f x))+315 e^{-i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)\right)}{1280 a^2 c^3 f (\tan (e+f x)-i)^2}","\frac{63 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{128 \sqrt{2} a^2 c^{5/2} f}-\frac{63 i}{128 a^2 c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{21 i}{64 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac{63 i}{160 a^2 f (c-i c \tan (e+f x))^{5/2}}+\frac{9 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}",1,"(Sec[e + f*x]^2*((-I)*Cos[3*(e + f*x)] + Sin[3*(e + f*x)])*((315*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])/E^(I*(e + f*x)) - 547*Cos[e + f*x] + 31*Cos[3*(e + f*x)] + 2*Cos[5*(e + f*x)] - (141*I)*Sin[e + f*x] - (159*I)*Sin[3*(e + f*x)] - (18*I)*Sin[5*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(1280*a^2*c^3*f*(-I + Tan[e + f*x])^2)","A",1
991,1,171,274,6.3585929,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2)),x]","-\frac{i e^{-6 i (e+f x)} \left(-350 e^{2 i (e+f x)}-1645 e^{4 i (e+f x)}+1433 e^{6 i (e+f x)}+3184 e^{8 i (e+f x)}+464 e^{10 i (e+f x)}+48 e^{12 i (e+f x)}-3465 e^{6 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (e+f x)}}\right)-40\right) \sqrt{c-i c \tan (e+f x)}}{15360 a^3 c^3 f}","\frac{231 i \tanh ^{-1}\left(\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right)}{512 \sqrt{2} a^3 c^{5/2} f}-\frac{231 i}{512 a^3 c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{77 i}{256 a^3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{231 i}{640 a^3 f (c-i c \tan (e+f x))^{5/2}}+\frac{33 i}{64 a^3 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac{11 i}{48 a^3 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{5/2}}+\frac{i}{6 a^3 f (1+i \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2}}",1,"((-1/15360*I)*(-40 - 350*E^((2*I)*(e + f*x)) - 1645*E^((4*I)*(e + f*x)) + 1433*E^((6*I)*(e + f*x)) + 3184*E^((8*I)*(e + f*x)) + 464*E^((10*I)*(e + f*x)) + 48*E^((12*I)*(e + f*x)) - 3465*E^((6*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*c^3*E^((6*I)*(e + f*x))*f)","A",1
992,1,87,154,3.6309038,"\int (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{a^2 c (\tan (e+f x)+i) \sqrt{a+i a \tan (e+f x)} \left(i \tan (e+f x)-6 \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)+4\right)}{2 f \sqrt{c-i c \tan (e+f x)}}","-\frac{3 i a^{5/2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}+\frac{3 i a^2 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}+\frac{i a (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{2 f}",1,"(a^2*c*(4 - 6*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x] + I*Tan[e + f*x])*(I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])/(2*f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
993,1,129,106,2.6992802,"\int (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{a c e^{-\frac{1}{2} i (4 e+f x)} \left(\sin \left(\frac{3 e}{2}\right)-i \cos \left(\frac{3 e}{2}\right)\right) \sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{1}{2} (e+f x)\right)-i \sin \left(\frac{1}{2} (e+f x)\right)\right) \left(-\sec (e+f x)+2 \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{\sqrt{2} f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}","\frac{i a \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 i a^{3/2} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}",1,"(a*c*(2*ArcTan[E^(I*(e + f*x))] - Sec[e + f*x])*((-I)*Cos[(3*e)/2] + Sin[(3*e)/2])*(Cos[(e + f*x)/2] - I*Sin[(e + f*x)/2])*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*E^((I/2)*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f)","A",1
994,1,74,63,1.5098044,"\int \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{i \sqrt{2} c e^{-i (e+f x)} \tan ^{-1}\left(e^{i (e+f x)}\right) \sqrt{a+i a \tan (e+f x)}}{f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}","-\frac{2 i \sqrt{a} \sqrt{c} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}",1,"((-I)*Sqrt[2]*c*ArcTan[E^(I*(e + f*x))]*Sqrt[a + I*a*Tan[e + f*x]])/(E^(I*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f)","A",1
995,1,41,41,1.2985407,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{i \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}","\frac{i \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}",1,"(I*Sqrt[c - I*c*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
996,1,68,90,1.6271759,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{(2+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{3 a f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{i \sqrt{c-i c \tan (e+f x)}}{3 a f \sqrt{a+i a \tan (e+f x)}}+\frac{i \sqrt{c-i c \tan (e+f x)}}{3 f (a+i a \tan (e+f x))^{3/2}}",1,"((2 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*a*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
997,1,90,136,2.2552608,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(5/2),x]","-\frac{i \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (6 i \sin (2 (e+f x))+9 \cos (2 (e+f x))+5)}{30 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{2 i \sqrt{c-i c \tan (e+f x)}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"((-1/30*I)*Sec[e + f*x]^2*(5 + 9*Cos[2*(e + f*x)] + (6*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
998,1,105,182,3.0154375,"\int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{7/2}} \, dx","Integrate[Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(7/2),x]","-\frac{\sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (7 i \sin (e+f x)+15 i \sin (3 (e+f x))+28 \cos (e+f x)+20 \cos (3 (e+f x)))}{140 a^3 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}}","\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^3 f \sqrt{a+i a \tan (e+f x)}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}}",1,"-1/140*(Sec[e + f*x]^3*(28*Cos[e + f*x] + 20*Cos[3*(e + f*x)] + (7*I)*Sin[e + f*x] + (15*I)*Sin[3*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*f*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]])","A",1
999,1,101,159,4.8260023,"\int (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{a^2 c^2 (\tan (e+f x)+i) \sec ^2(e+f x) \sqrt{a+i a \tan (e+f x)} \left(3 i \sin (2 (e+f x))+12 \cos ^3(e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)-4\right)}{12 f \sqrt{c-i c \tan (e+f x)}}","-\frac{i a^{5/2} c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}+\frac{a^2 c \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}+\frac{i a (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}",1,"-1/12*(a^2*c^2*Sec[e + f*x]^2*(-4 + 12*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]^3 + (3*I)*Sin[2*(e + f*x)])*(I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
1000,1,133,113,3.8722839,"\int (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{a c^2 e^{-2 i e} \left(\cos \left(\frac{3 e}{2}\right)+i \sin \left(\frac{3 e}{2}\right)\right) \sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{e}{2}+f x\right)-i \sin \left(\frac{e}{2}+f x\right)\right) \left(\tan (e+f x) \sec (e+f x)-2 i \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{2 \sqrt{2} f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}","\frac{a c \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}-\frac{i a^{3/2} c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}",1,"(a*c^2*(Cos[(3*e)/2] + I*Sin[(3*e)/2])*(Cos[e/2 + f*x] - I*Sin[e/2 + f*x])*Sqrt[a + I*a*Tan[e + f*x]]*((-2*I)*ArcTan[E^(I*(e + f*x))] + Sec[e + f*x]*Tan[e + f*x]))/(2*Sqrt[2]*E^((2*I)*e)*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f)","A",1
1001,1,100,106,2.4115405,"\int \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{i \sqrt{2} c e^{-i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(e^{i (e+f x)}+\left(1+e^{2 i (e+f x)}\right) \tan ^{-1}\left(e^{i (e+f x)}\right)\right) \sqrt{a+i a \tan (e+f x)}}{f}","-\frac{2 i \sqrt{a} c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}-\frac{i c \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{f}",1,"((-I)*Sqrt[2]*c*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x)) + (1 + E^((2*I)*(e + f*x)))*ArcTan[E^(I*(e + f*x))])*Sqrt[a + I*a*Tan[e + f*x]])/(E^(I*(e + f*x))*f)","A",1
1002,1,119,106,2.8681148,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{c^2 (\cos (f x)+i \sin (f x)) (\sin (f x)+i \cos (f x)) \left(-i \tan (e+f x)+\sec (e+f x) \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))+1\right)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{\frac{c}{2 i \sin (2 (e+f x))+2 \cos (2 (e+f x))+2}}}","\frac{2 i c^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{a} f}+\frac{2 i c \sqrt{c-i c \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}",1,"(c^2*(Cos[f*x] + I*Sin[f*x])*(I*Cos[f*x] + Sin[f*x])*(1 + ArcTan[Cos[e + f*x] + I*Sin[e + f*x]]*Sec[e + f*x] - I*Tan[e + f*x]))/(f*Sqrt[c/(2 + 2*Cos[2*(e + f*x)] + (2*I)*Sin[2*(e + f*x)])]*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1003,1,69,43,2.2503569,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{c (1-i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{3 a f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}",1,"(c*(1 - I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(3*a*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1004,1,79,90,3.1043121,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{c (1-i \tan (e+f x)) (\tan (e+f x)-4 i) \sqrt{c-i c \tan (e+f x)}}{15 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{i (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"(c*(1 - I*Tan[e + f*x])*(-4*I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(15*a^2*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1005,1,101,136,4.1544703,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{7/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(7/2),x]","\frac{i c (\tan (e+f x)+i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (10 i \sin (2 (e+f x))+25 \cos (2 (e+f x))+21)}{210 a^3 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}}","\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}",1,"((I/210)*c*Sec[e + f*x]^2*(21 + 25*Cos[2*(e + f*x)] + (10*I)*Sin[2*(e + f*x)])*(I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*f*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1006,1,115,182,5.5509159,"\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(9/2),x]","\frac{c (\tan (e+f x)+i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (140 \cos (2 (e+f x))+27 i \tan (e+f x)+35 i \sin (3 (e+f x)) \sec (e+f x)+92)}{1260 a^4 f (\tan (e+f x)-i)^4 \sqrt{a+i a \tan (e+f x)}}","\frac{2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}",1,"(c*Sec[e + f*x]^2*(92 + 140*Cos[2*(e + f*x)] + (35*I)*Sec[e + f*x]*Sin[3*(e + f*x)] + (27*I)*Tan[e + f*x])*(I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(1260*a^4*f*(-I + Tan[e + f*x])^4*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1007,1,110,168,6.2345622,"\int (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{a^2 c^3 (\tan (e+f x)+i) \sec ^3(e+f x) \sqrt{a+i a \tan (e+f x)} \left(11 i \sin (e+f x)+3 i \sin (3 (e+f x))+24 \cos ^4(e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)\right)}{32 f \sqrt{c-i c \tan (e+f x)}}","-\frac{3 i a^{5/2} c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{4 f}+\frac{3 a^2 c^2 \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{8 f}+\frac{a c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{4 f}",1,"-1/32*(a^2*c^3*Sec[e + f*x]^3*(24*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]^4 + (11*I)*Sin[e + f*x] + (3*I)*Sin[3*(e + f*x)])*(I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
1008,1,169,159,4.4636527,"\int (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{i e^{-2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{5/2} \left(e^{i (e+f x)} \left(8 e^{2 i (e+f x)}+3 e^{4 i (e+f x)}-3\right)+3 \left(1+e^{2 i (e+f x)}\right)^3 \tan ^{-1}\left(e^{i (e+f x)}\right)\right) (a+i a \tan (e+f x))^{3/2}}{3 f \sec ^{\frac{3}{2}}(e+f x)}","-\frac{i a^{3/2} c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}+\frac{a c^2 \tan (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}-\frac{i c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}",1,"((-1/3*I)*(c/(1 + E^((2*I)*(e + f*x))))^(5/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x))*(-3 + 8*E^((2*I)*(e + f*x)) + 3*E^((4*I)*(e + f*x))) + 3*(1 + E^((2*I)*(e + f*x)))^3*ArcTan[E^(I*(e + f*x))])*(a + I*a*Tan[e + f*x])^(3/2))/(E^((2*I)*(e + f*x))*f*Sec[e + f*x]^(3/2))","A",1
1009,1,155,154,3.0665024,"\int \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{i c e^{-i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(e^{i (e+f x)} \left(5+3 e^{2 i (e+f x)}\right)+3 \left(1+e^{2 i (e+f x)}\right)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)\right) \sqrt{a+i a \tan (e+f x)}}{f \sqrt{\sec (e+f x)}}","-\frac{3 i \sqrt{a} c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{f}-\frac{3 i c^2 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 f}-\frac{i c \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{2 f}",1,"((-I)*c*(c/(1 + E^((2*I)*(e + f*x))))^(3/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x))*(5 + 3*E^((2*I)*(e + f*x))) + 3*(1 + E^((2*I)*(e + f*x)))^2*ArcTan[E^(I*(e + f*x))])*Sqrt[a + I*a*Tan[e + f*x]])/(E^(I*(e + f*x))*f*Sqrt[Sec[e + f*x]])","A",1
1010,1,113,153,3.9569556,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{c^3 (\cos (f x)+i \sin (f x)) (\sin (f x)+i \cos (f x)) \left(\tan ^2(e+f x)-4 i \tan (e+f x)+6 \sec (e+f x) \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))+5\right)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}","\frac{6 i c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{a} f}+\frac{3 i c^2 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{2 i c (c-i c \tan (e+f x))^{3/2}}{f \sqrt{a+i a \tan (e+f x)}}",1,"(c^3*(Cos[f*x] + I*Sin[f*x])*(I*Cos[f*x] + Sin[f*x])*(5 + 6*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]]*Sec[e + f*x] - (4*I)*Tan[e + f*x] + Tan[e + f*x]^2))/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])","A",1
1011,1,109,155,4.5901708,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2),x]","-\frac{2 i \sqrt{2} c^2 e^{-2 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(3 e^{2 i (e+f x)}+3 e^{3 i (e+f x)} \tan ^{-1}\left(e^{i (e+f x)}\right)-1\right)}{3 a f \sqrt{a+i a \tan (e+f x)}}","-\frac{2 i c^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{a^{3/2} f}-\frac{2 i c^2 \sqrt{c-i c \tan (e+f x)}}{a f \sqrt{a+i a \tan (e+f x)}}+\frac{2 i c (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}",1,"(((-2*I)/3)*Sqrt[2]*c^2*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*(-1 + 3*E^((2*I)*(e + f*x)) + 3*E^((3*I)*(e + f*x))*ArcTan[E^(I*(e + f*x))]))/(a*E^((2*I)*(e + f*x))*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1012,1,90,43,4.4656538,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(5/2),x]","-\frac{i c^2 \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))-i \sin (2 (e+f x)))}{5 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{i (c-i c \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"((-1/5*I)*c^2*Sec[e + f*x]^2*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","B",1
1013,1,100,90,6.5180617,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{7/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(7/2),x]","-\frac{i c^2 (\tan (e+f x)-6 i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))-i \sin (2 (e+f x)))}{35 a^3 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}}","\frac{i (c-i c \tan (e+f x))^{5/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{7 f (a+i a \tan (e+f x))^{7/2}}",1,"((-1/35*I)*c^2*Sec[e + f*x]^2*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*(-6*I + Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*f*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1014,1,112,136,8.103706,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(9/2),x]","\frac{c^2 \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (14 i \sin (2 (e+f x))+49 \cos (2 (e+f x))+45) (\sin (2 (e+f x))+i \cos (2 (e+f x)))}{630 a^4 f (\tan (e+f x)-i)^4 \sqrt{a+i a \tan (e+f x)}}","\frac{2 i (c-i c \tan (e+f x))^{5/2}}{315 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac{2 i (c-i c \tan (e+f x))^{5/2}}{63 a f (a+i a \tan (e+f x))^{7/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{9 f (a+i a \tan (e+f x))^{9/2}}",1,"(c^2*Sec[e + f*x]^4*(45 + 49*Cos[2*(e + f*x)] + (14*I)*Sin[2*(e + f*x)])*(I*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(630*a^4*f*(-I + Tan[e + f*x])^4*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1015,1,128,182,11.1989828,"\int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx","Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(11/2),x]","\frac{c^2 \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))-i \sin (2 (e+f x))) (336 \cos (2 (e+f x))+55 i \tan (e+f x)+63 i \sin (3 (e+f x)) \sec (e+f x)+272)}{4620 a^5 f (\tan (e+f x)-i)^5 \sqrt{a+i a \tan (e+f x)}}","\frac{2 i (c-i c \tan (e+f x))^{5/2}}{1155 a^3 f (a+i a \tan (e+f x))^{5/2}}+\frac{2 i (c-i c \tan (e+f x))^{5/2}}{231 a^2 f (a+i a \tan (e+f x))^{7/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{33 a f (a+i a \tan (e+f x))^{9/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{11 f (a+i a \tan (e+f x))^{11/2}}",1,"(c^2*Sec[e + f*x]^4*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*(272 + 336*Cos[2*(e + f*x)] + (63*I)*Sec[e + f*x]*Sin[3*(e + f*x)] + (55*I)*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(4620*a^5*f*(-I + Tan[e + f*x])^5*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1016,1,340,204,10.4072014,"\int \frac{(a+i a \tan (e+f x))^{7/2}}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{\cos ^3(e+f x) (a+i a \tan (e+f x))^{7/2} \left(\cos (2 f x) \left(-\frac{4 \sin (e)}{c}-\frac{4 i \cos (e)}{c}\right)+\sin (2 f x) \left(\frac{4 \cos (e)}{c}-\frac{4 i \sin (e)}{c}\right)+\sec (e) \sin (f x) \left(\frac{\cos (3 e)}{2 c}-\frac{i \sin (3 e)}{2 c}\right) \sec (e+f x)+\sec (e) (16 \cos (e)+i \sin (e)) \left(-\frac{\sin (3 e)}{2 c}-\frac{i \cos (3 e)}{2 c}\right)\right) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^3}+\frac{15 i \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2}}{f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{7}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2}}","\frac{15 i a^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{15 i a^3 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 c f}-\frac{5 i a^2 (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{2 c f}-\frac{2 i a (a+i a \tan (e+f x))^{5/2}}{f \sqrt{c-i c \tan (e+f x)}}",1,"((15*I)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(7/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)) + (Cos[e + f*x]^3*(Cos[2*f*x]*(((-4*I)*Cos[e])/c - (4*Sin[e])/c) + Sec[e]*(16*Cos[e] + I*Sin[e])*(((-1/2*I)*Cos[3*e])/c - Sin[3*e]/(2*c)) + Sec[e]*Sec[e + f*x]*(Cos[3*e]/(2*c) - ((I/2)*Sin[3*e])/c)*Sin[f*x] + ((4*Cos[e])/c - ((4*I)*Sin[e])/c)*Sin[2*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1017,1,155,153,4.523918,"\int \frac{(a+i a \tan (e+f x))^{5/2}}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)/Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{2 i e^{-3 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(e^{i (e+f x)} \left(3+2 e^{2 i (e+f x)}\right)-3 \left(1+e^{2 i (e+f x)}\right) \tan ^{-1}\left(e^{i (e+f x)}\right)\right) (a+i a \tan (e+f x))^{5/2}}{c f \sec ^{\frac{5}{2}}(e+f x)}","\frac{6 i a^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{3 i a^2 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{2 i a (a+i a \tan (e+f x))^{3/2}}{f \sqrt{c-i c \tan (e+f x)}}",1,"((-2*I)*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x))*(3 + 2*E^((2*I)*(e + f*x))) - 3*(1 + E^((2*I)*(e + f*x)))*ArcTan[E^(I*(e + f*x))])*(a + I*a*Tan[e + f*x])^(5/2))/(c*E^((3*I)*(e + f*x))*f*Sec[e + f*x]^(5/2))","A",1
1018,1,123,106,2.9672383,"\int \frac{(a+i a \tan (e+f x))^{3/2}}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)/Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{2 i e^{-2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \left(e^{i (e+f x)}-\tan ^{-1}\left(e^{i (e+f x)}\right)\right) (a+i a \tan (e+f x))^{3/2}}{f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{3}{2}}(e+f x)}","\frac{2 i a^{3/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{\sqrt{c} f}-\frac{2 i a \sqrt{a+i a \tan (e+f x)}}{f \sqrt{c-i c \tan (e+f x)}}",1,"((-2*I)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x)) - ArcTan[E^(I*(e + f*x))])*(a + I*a*Tan[e + f*x])^(3/2))/(E^((2*I)*(e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(3/2))","A",1
1019,1,64,41,1.3607229,"\int \frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]/Sqrt[c - I*c*Tan[e + f*x]],x]","\frac{\cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\sin (e+f x)-i \cos (e+f x))}{c f}","-\frac{i \sqrt{a+i a \tan (e+f x)}}{f \sqrt{c-i c \tan (e+f x)}}",1,"(Cos[e + f*x]*((-I)*Cos[e + f*x] + Sin[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)","A",1
1020,1,64,44,1.4185757,"\int \frac{1}{\sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{\sin (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x))}{c f \sqrt{a+i a \tan (e+f x)}}","\frac{\tan (e+f x)}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}",1,"((Cos[e + f*x] + I*Sin[e + f*x])*Sin[e + f*x]*Sqrt[c - I*c*Tan[e + f*x]])/(c*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1021,1,73,94,1.8765864,"\int \frac{1}{(a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{\sqrt{c-i c \tan (e+f x)} (2 \sin (2 (e+f x))-i \cos (2 (e+f x))+3 i)}{6 a c f \sqrt{a+i a \tan (e+f x)}}","\frac{2 \tan (e+f x)}{3 a f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{3 f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}",1,"((3*I - I*Cos[2*(e + f*x)] + 2*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(6*a*c*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1022,1,99,140,2.8893711,"\int \frac{1}{(a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{\sqrt{c-i c \tan (e+f x)} (-4 \cos (2 (e+f x))+5 i \tan (e+f x)-3 i \sin (3 (e+f x)) \sec (e+f x)+12)}{20 a^2 c f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{2 \tan (e+f x)}{5 a^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}",1,"((12 - 4*Cos[2*(e + f*x)] - (3*I)*Sec[e + f*x]*Sin[3*(e + f*x)] + (5*I)*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(20*a^2*c*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1023,1,115,186,4.0425344,"\int \frac{1}{(a+i a \tan (e+f x))^{7/2} \sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]]),x]","\frac{\sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (56 \sin (2 (e+f x))-20 \sin (4 (e+f x))-84 i \cos (2 (e+f x))+15 i \cos (4 (e+f x))-35 i)}{280 a^3 c f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{8 \tan (e+f x)}{35 a^3 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 i}{35 a^2 f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{4 i}{35 a f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} \sqrt{c-i c \tan (e+f x)}}",1,"(Sec[e + f*x]^2*(-35*I - (84*I)*Cos[2*(e + f*x)] + (15*I)*Cos[4*(e + f*x)] + 56*Sin[2*(e + f*x)] - 20*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(280*a^3*c*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1024,1,386,255,13.2737971,"\int \frac{(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(9/2)/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{9/2} \left(\cos (2 f x) \left(\frac{32 \sin (2 e)}{3 c^2}+\frac{32 i \cos (2 e)}{3 c^2}\right)+\sin (2 f x) \left(-\frac{32 \cos (2 e)}{3 c^2}+\frac{32 i \sin (2 e)}{3 c^2}\right)-\sec (e) \sin (f x) \left(\frac{\cos (4 e)}{2 c^2}-\frac{i \sin (4 e)}{2 c^2}\right) \sec (e+f x)+\sec (e) (36 \cos (e)+i \sin (e)) \left(\frac{\sin (4 e)}{2 c^2}+\frac{i \cos (4 e)}{2 c^2}\right)+\frac{4 \sin (4 f x)}{3 c^2}-\frac{4 i \cos (4 f x)}{3 c^2}\right) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^4}-\frac{35 i \sqrt{e^{i f x}} e^{-i (5 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{9/2}}{c f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{9/2}}","-\frac{35 i a^{9/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{3/2} f}+\frac{35 i a^4 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 c^2 f}+\frac{35 i a^3 (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{6 c^2 f}+\frac{14 i a^2 (a+i a \tan (e+f x))^{5/2}}{3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i a (a+i a \tan (e+f x))^{7/2}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"((-35*I)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(9/2))/(c*E^(I*(5*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(9/2)) + (Cos[e + f*x]^4*((((-4*I)/3)*Cos[4*f*x])/c^2 + Cos[2*f*x]*((((32*I)/3)*Cos[2*e])/c^2 + (32*Sin[2*e])/(3*c^2)) + Sec[e]*(36*Cos[e] + I*Sin[e])*(((I/2)*Cos[4*e])/c^2 + Sin[4*e]/(2*c^2)) - Sec[e]*Sec[e + f*x]*(Cos[4*e]/(2*c^2) - ((I/2)*Sin[4*e])/c^2)*Sin[f*x] + ((-32*Cos[2*e])/(3*c^2) + (((32*I)/3)*Sin[2*e])/c^2)*Sin[2*f*x] + (4*Sin[4*f*x])/(3*c^2))*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(9/2))/(f*(Cos[f*x] + I*Sin[f*x])^4)","A",1
1025,1,348,204,12.0869678,"\int \frac{(a+i a \tan (e+f x))^{7/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\cos ^3(e+f x) (a+i a \tan (e+f x))^{7/2} \left(\cos (4 f x) \left(\frac{2 \sin (e)}{3 c^2}-\frac{2 i \cos (e)}{3 c^2}\right)+\cos (2 f x) \left(\frac{10 \sin (e)}{3 c^2}+\frac{10 i \cos (e)}{3 c^2}\right)+\sin (2 f x) \left(-\frac{10 \cos (e)}{3 c^2}+\frac{10 i \sin (e)}{3 c^2}\right)+\sin (4 f x) \left(\frac{2 \cos (e)}{3 c^2}+\frac{2 i \sin (e)}{3 c^2}\right)+\frac{5 \sin (3 e)}{c^2}+\frac{5 i \cos (3 e)}{c^2}\right) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^3}-\frac{10 i \sqrt{e^{i f x}} e^{-i (4 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{7/2}}{c f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{7}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2}}","-\frac{10 i a^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{3/2} f}+\frac{5 i a^3 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{c^2 f}+\frac{10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"((-10*I)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(7/2))/(c*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(7/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)) + (Cos[e + f*x]^3*(((5*I)*Cos[3*e])/c^2 + Cos[4*f*x]*((((-2*I)/3)*Cos[e])/c^2 + (2*Sin[e])/(3*c^2)) + Cos[2*f*x]*((((10*I)/3)*Cos[e])/c^2 + (10*Sin[e])/(3*c^2)) + (5*Sin[3*e])/c^2 + ((-10*Cos[e])/(3*c^2) + (((10*I)/3)*Sin[e])/c^2)*Sin[2*f*x] + ((2*Cos[e])/(3*c^2) + (((2*I)/3)*Sin[e])/c^2)*Sin[4*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1026,1,184,155,7.581842,"\int \frac{(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{2 a^2 \cos (e+f x) (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{1}{2} (e-2 f x)\right)-i \sin \left(\frac{1}{2} (e-2 f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+4 f x)\right)+i \cos \left(\frac{1}{2} (e+4 f x)\right)\right) \left(2 i \sin (2 (e+f x))-\cos (2 (e+f x))+3 \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right) (\cos (2 (e+f x))-i \sin (2 (e+f x)))-1\right)}{3 c f \sqrt{c-i c \tan (e+f x)}}","-\frac{2 i a^{5/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{3/2} f}+\frac{2 i a^2 \sqrt{a+i a \tan (e+f x)}}{c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i a (a+i a \tan (e+f x))^{3/2}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(2*a^2*Cos[e + f*x]*(Cos[(e - 2*f*x)/2] - I*Sin[(e - 2*f*x)/2])*(-1 - Cos[2*(e + f*x)] + 3*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]) + (2*I)*Sin[2*(e + f*x)])*(I*Cos[(e + 4*f*x)/2] + Sin[(e + 4*f*x)/2])*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
1027,1,87,43,3.131284,"\int \frac{(a+i a \tan (e+f x))^{3/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\sin (3 e+4 f x)-i \cos (3 e+4 f x))}{3 c^2 f}","-\frac{i (a+i a \tan (e+f x))^{3/2}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"(a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*((-I)*Cos[3*e + 4*f*x] + Sin[3*e + 4*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(3*c^2*f)","B",1
1028,1,89,90,2.3432348,"\int \frac{\sqrt{a+i a \tan (e+f x)}}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]/(c - I*c*Tan[e + f*x])^(3/2),x]","\frac{\sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (-i \sin (2 (e+f x))+2 \cos (2 (e+f x))+2) (\sin (2 (e+f x))-i \cos (2 (e+f x)))}{6 c^2 f}","-\frac{i \sqrt{a+i a \tan (e+f x)}}{3 c f \sqrt{c-i c \tan (e+f x)}}-\frac{i \sqrt{a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"((2 + 2*Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(6*c^2*f)","A",1
1029,1,89,137,2.8235834,"\int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{i \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-2 i \sin (2 (e+f x))+\cos (2 (e+f x))-3)}{6 c^2 f \sqrt{a+i a \tan (e+f x)}}","-\frac{2 i \sqrt{a+i a \tan (e+f x)}}{3 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}",1,"((I/6)*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(-3 + Cos[2*(e + f*x)] - (2*I)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1030,1,103,101,3.7530877,"\int \frac{1}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{(9 \sin (e+f x)+\sin (3 (e+f x))) \sec (e+f x) \sqrt{c-i c \tan (e+f x)} (\sin (2 (e+f x))-i \cos (2 (e+f x)))}{12 a c^2 f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","\frac{2 \tan (e+f x)}{3 a c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{\tan (e+f x)}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])*(9*Sin[e + f*x] + Sin[3*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(12*a*c^2*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1031,1,93,147,5.1398483,"\int \frac{1}{(a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)),x]","-\frac{i \sqrt{c-i c \tan (e+f x)} (40 i \sin (2 (e+f x))+4 i \sin (4 (e+f x))+20 \cos (2 (e+f x))+\cos (4 (e+f x))-45)}{120 a^2 c^2 f \sqrt{a+i a \tan (e+f x)}}","\frac{8 \tan (e+f x)}{15 a^2 c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{15 a f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}",1,"((-1/120*I)*(-45 + 20*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] + (40*I)*Sin[2*(e + f*x)] + (4*I)*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*c^2*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1032,1,151,193,7.1258015,"\int \frac{1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(3/2)),x]","\frac{\sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-70 i \sin (e+f x)+63 i \sin (3 (e+f x))+5 i \sin (5 (e+f x))-140 \cos (e+f x)+42 \cos (3 (e+f x))+2 \cos (5 (e+f x)))}{336 a^3 c^2 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}}","\frac{8 \tan (e+f x)}{21 a^3 c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{21 a^2 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^3*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*(-140*Cos[e + f*x] + 42*Cos[3*(e + f*x)] + 2*Cos[5*(e + f*x)] - (70*I)*Sin[e + f*x] + (63*I)*Sin[3*(e + f*x)] + (5*I)*Sin[5*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(336*a^3*c^2*f*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1033,1,459,304,17.3736408,"\int \frac{(a+i a \tan (e+f x))^{11/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(11/2)/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\cos ^5(e+f x) (a+i a \tan (e+f x))^{11/2} \left(\cos (6 f x) \left(\frac{4 \sin (e)}{5 c^3}-\frac{4 i \cos (e)}{5 c^3}\right)+\cos (4 f x) \left(\frac{16 \sin (e)}{5 c^3}+\frac{16 i \cos (e)}{5 c^3}\right)+\cos (2 f x) \left(-\frac{20 \sin (3 e)}{c^3}-\frac{20 i \cos (3 e)}{c^3}\right)+\sin (2 f x) \left(\frac{20 \cos (3 e)}{c^3}-\frac{20 i \sin (3 e)}{c^3}\right)+\sin (4 f x) \left(-\frac{16 \cos (e)}{5 c^3}+\frac{16 i \sin (e)}{5 c^3}\right)+\sin (6 f x) \left(\frac{4 \cos (e)}{5 c^3}+\frac{4 i \sin (e)}{5 c^3}\right)+\sec (e) \sin (f x) \left(\frac{\cos (5 e)}{2 c^3}-\frac{i \sin (5 e)}{2 c^3}\right) \sec (e+f x)+\sec (e) (64 \cos (e)+i \sin (e)) \left(-\frac{\sin (5 e)}{2 c^3}-\frac{i \cos (5 e)}{2 c^3}\right)\right) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^5}+\frac{63 i \sqrt{e^{i f x}} e^{-i (6 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{11/2}}{c^2 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{11}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{11/2}}","\frac{63 i a^{11/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{5/2} f}-\frac{63 i a^5 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{2 c^3 f}-\frac{21 i a^4 (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}{2 c^3 f}-\frac{42 i a^3 (a+i a \tan (e+f x))^{5/2}}{5 c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{6 i a^2 (a+i a \tan (e+f x))^{7/2}}{5 c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 i a (a+i a \tan (e+f x))^{9/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"((63*I)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(11/2))/(c^2*E^(I*(6*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(11/2)*(Cos[f*x] + I*Sin[f*x])^(11/2)) + (Cos[e + f*x]^5*(Cos[6*f*x]*((((-4*I)/5)*Cos[e])/c^3 + (4*Sin[e])/(5*c^3)) + Cos[4*f*x]*((((16*I)/5)*Cos[e])/c^3 + (16*Sin[e])/(5*c^3)) + Cos[2*f*x]*(((-20*I)*Cos[3*e])/c^3 - (20*Sin[3*e])/c^3) + Sec[e]*(64*Cos[e] + I*Sin[e])*(((-1/2*I)*Cos[5*e])/c^3 - Sin[5*e]/(2*c^3)) + Sec[e]*Sec[e + f*x]*(Cos[5*e]/(2*c^3) - ((I/2)*Sin[5*e])/c^3)*Sin[f*x] + ((20*Cos[3*e])/c^3 - ((20*I)*Sin[3*e])/c^3)*Sin[2*f*x] + ((-16*Cos[e])/(5*c^3) + (((16*I)/5)*Sin[e])/c^3)*Sin[4*f*x] + ((4*Cos[e])/(5*c^3) + (((4*I)/5)*Sin[e])/c^3)*Sin[6*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(11/2))/(f*(Cos[f*x] + I*Sin[f*x])^5)","A",0
1034,1,390,253,16.8736117,"\int \frac{(a+i a \tan (e+f x))^{9/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(9/2)/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{9/2} \left(\cos (2 f x) \left(-\frac{14 \sin (2 e)}{3 c^3}-\frac{14 i \cos (2 e)}{3 c^3}\right)+\cos (6 f x) \left(\frac{2 \sin (2 e)}{5 c^3}-\frac{2 i \cos (2 e)}{5 c^3}\right)+\sin (2 f x) \left(\frac{14 \cos (2 e)}{3 c^3}-\frac{14 i \sin (2 e)}{3 c^3}\right)+\sin (6 f x) \left(\frac{2 \cos (2 e)}{5 c^3}+\frac{2 i \sin (2 e)}{5 c^3}\right)-\frac{7 \sin (4 e)}{c^3}-\frac{7 i \cos (4 e)}{c^3}-\frac{14 \sin (4 f x)}{15 c^3}+\frac{14 i \cos (4 f x)}{15 c^3}\right) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^4}+\frac{14 i \sqrt{e^{i f x}} e^{-i (5 e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left(e^{i (e+f x)}\right) (a+i a \tan (e+f x))^{9/2}}{c^2 f \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \sec ^{\frac{9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{9/2}}","\frac{14 i a^{9/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{5/2} f}-\frac{7 i a^4 \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}{c^3 f}-\frac{14 i a^3 (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{14 i a^2 (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 i a (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"((14*I)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[e + f*x])^(9/2))/(c^2*E^(I*(5*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(9/2)) + (Cos[e + f*x]^4*(((-7*I)*Cos[4*e])/c^3 + (((14*I)/15)*Cos[4*f*x])/c^3 + Cos[2*f*x]*((((-14*I)/3)*Cos[2*e])/c^3 - (14*Sin[2*e])/(3*c^3)) + Cos[6*f*x]*((((-2*I)/5)*Cos[2*e])/c^3 + (2*Sin[2*e])/(5*c^3)) - (7*Sin[4*e])/c^3 + ((14*Cos[2*e])/(3*c^3) - (((14*I)/3)*Sin[2*e])/c^3)*Sin[2*f*x] - (14*Sin[4*f*x])/(15*c^3) + ((2*Cos[2*e])/(5*c^3) + (((2*I)/5)*Sin[2*e])/c^3)*Sin[6*f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(9/2))/(f*(Cos[f*x] + I*Sin[f*x])^4)","A",1
1035,1,205,204,14.8365758,"\int \frac{(a+i a \tan (e+f x))^{7/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(7/2)/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{2 a^3 \cos ^2(e+f x) (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)} \left(\cos \left(\frac{1}{2} (e-4 f x)\right)-i \sin \left(\frac{1}{2} (e-4 f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+6 f x)\right)+i \cos \left(\frac{1}{2} (e+6 f x)\right)\right) \left(6 \sin (e+f x)+6 \sin (3 (e+f x))+4 i \cos (e+f x)+9 i \cos (3 (e+f x))-15 i \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right) (\cos (3 (e+f x))-i \sin (3 (e+f x)))\right)}{15 c^2 f \sqrt{c-i c \tan (e+f x)}}","\frac{2 i a^{7/2} \tan ^{-1}\left(\frac{\sqrt{c} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c-i c \tan (e+f x)}}\right)}{c^{5/2} f}-\frac{2 i a^3 \sqrt{a+i a \tan (e+f x)}}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{2 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 i a (a+i a \tan (e+f x))^{5/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(2*a^3*Cos[e + f*x]^2*(Cos[(e - 4*f*x)/2] - I*Sin[(e - 4*f*x)/2])*((4*I)*Cos[e + f*x] + (9*I)*Cos[3*(e + f*x)] + 6*Sin[e + f*x] - (15*I)*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x]*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)]) + 6*Sin[3*(e + f*x)])*(I*Cos[(e + 6*f*x)/2] + Sin[(e + 6*f*x)/2])*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]])/(15*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])","A",1
1036,1,91,43,5.5973944,"\int \frac{(a+i a \tan (e+f x))^{5/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a^2 \cos (e+f x) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\sin (5 e+7 f x)-i \cos (5 e+7 f x))}{5 c^3 f (\cos (f x)+i \sin (f x))^2}","-\frac{i (a+i a \tan (e+f x))^{5/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(a^2*Cos[e + f*x]*((-I)*Cos[5*e + 7*f*x] + Sin[5*e + 7*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(5*c^3*f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1037,1,106,90,4.5074461,"\int \frac{(a+i a \tan (e+f x))^{3/2}}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (4 \cos (e+f x)-i \sin (e+f x)) (\sin (4 e+5 f x)-i \cos (4 e+5 f x))}{15 c^3 f}","-\frac{i (a+i a \tan (e+f x))^{3/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{i (a+i a \tan (e+f x))^{3/2}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"(a*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(4*Cos[e + f*x] - I*Sin[e + f*x])*((-I)*Cos[4*e + 5*f*x] + Sin[4*e + 5*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(15*c^3*f)","A",1
1038,1,102,136,3.6427525,"\int \frac{\sqrt{a+i a \tan (e+f x)}}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]/(c - I*c*Tan[e + f*x])^(5/2),x]","\frac{\sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} \left(19 \cos (e+f x)+9 \cos (3 (e+f x))-24 i \sin (e+f x) \cos ^2(e+f x)\right) (\sin (3 (e+f x))-i \cos (3 (e+f x)))}{60 c^3 f}","-\frac{2 i \sqrt{a+i a \tan (e+f x)}}{15 c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i \sqrt{a+i a \tan (e+f x)}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac{i \sqrt{a+i a \tan (e+f x)}}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"((19*Cos[e + f*x] + 9*Cos[3*(e + f*x)] - (24*I)*Cos[e + f*x]^2*Sin[e + f*x])*((-I)*Cos[3*(e + f*x)] + Sin[3*(e + f*x)])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(60*c^3*f)","A",1
1039,1,106,186,4.206015,"\int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-5 \sin (e+f x)+3 \sin (3 (e+f x))-10 i \cos (e+f x)+2 i \cos (3 (e+f x)))}{20 c^3 f \sqrt{a+i a \tan (e+f x)}}","-\frac{2 i \sqrt{a+i a \tan (e+f x)}}{5 a c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i \sqrt{a+i a \tan (e+f x)}}{5 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{3 i \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}",1,"((Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*((-10*I)*Cos[e + f*x] + (2*I)*Cos[3*(e + f*x)] - 5*Sin[e + f*x] + 3*Sin[3*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(20*c^3*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1040,1,130,234,6.0151979,"\int \frac{1}{(a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2)),x]","\frac{\sec (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) (-40 i \sin (2 (e+f x))-4 i \sin (4 (e+f x))+20 \cos (2 (e+f x))+\cos (4 (e+f x))-45)}{120 a c^3 f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}}","-\frac{8 i \sqrt{a+i a \tan (e+f x)}}{15 a^2 c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{8 i \sqrt{a+i a \tan (e+f x)}}{15 a^2 c f (c-i c \tan (e+f x))^{3/2}}-\frac{4 i \sqrt{a+i a \tan (e+f x)}}{5 a^2 f (c-i c \tan (e+f x))^{5/2}}+\frac{4 i}{3 a f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}+\frac{i}{3 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}",1,"(Sec[e + f*x]*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(-45 + 20*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] - (40*I)*Sin[2*(e + f*x)] - (4*I)*Sin[4*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(120*a*c^3*f*(-I + Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1041,1,117,154,7.8024247,"\int \frac{1}{(a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)),x]","-\frac{(150 \sin (e+f x)+25 \sin (3 (e+f x))+3 \sin (5 (e+f x))) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x)))}{240 a^2 c^3 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}}","\frac{8 \tan (e+f x)}{15 a^2 c^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{15 a c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{\tan (e+f x)}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}",1,"-1/240*(Sec[e + f*x]^2*(Cos[3*(e + f*x)] + I*Sin[3*(e + f*x)])*(150*Sin[e + f*x] + 25*Sin[3*(e + f*x)] + 3*Sin[5*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*c^3*f*(-I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1042,1,115,200,11.2928263,"\int \frac{1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(5/2)),x]","-\frac{i \sqrt{c-i c \tan (e+f x)} (350 i \sin (2 (e+f x))+56 i \sin (4 (e+f x))+6 i \sin (6 (e+f x))+175 \cos (2 (e+f x))+14 \cos (4 (e+f x))+\cos (6 (e+f x))-350)}{1120 a^3 c^3 f \sqrt{a+i a \tan (e+f x)}}","\frac{16 \tan (e+f x)}{35 a^3 c^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{8 \tan (e+f x)}{35 a^2 c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}",1,"((-1/1120*I)*(-350 + 175*Cos[2*(e + f*x)] + 14*Cos[4*(e + f*x)] + Cos[6*(e + f*x)] + (350*I)*Sin[2*(e + f*x)] + (56*I)*Sin[4*(e + f*x)] + (6*I)*Sin[6*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*c^3*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1043,1,180,134,6.6790662,"\int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^n,x]","\frac{i a^4 \sec ^2(e+f x) (\cos (4 f x)+i \sin (4 f x)) (c \sec (e+f x))^n \left(i n \tan (e+f x) \left(2 \left(n^2+6 n+11\right) \cos (2 (e+f x))+n^2+9 n+20\right)+\left(n^3+6 n^2+11 n+12\right) (2 \cos (2 (e+f x))-1)+3 \left(n^2+7 n+12\right)\right) \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f n (n+1) (n+2) (n+3) (\cos (f x)+i \sin (f x))^4}","-\frac{i a^4 (c-i c \tan (e+f x))^{n+3}}{c^3 f (n+3)}+\frac{6 i a^4 (c-i c \tan (e+f x))^{n+2}}{c^2 f (n+2)}+\frac{8 i a^4 (c-i c \tan (e+f x))^n}{f n}-\frac{12 i a^4 (c-i c \tan (e+f x))^{n+1}}{c f (n+1)}",1,"(I*a^4*E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sec[e + f*x]^2*(c*Sec[e + f*x])^n*(Cos[4*f*x] + I*Sin[4*f*x])*(3*(12 + 7*n + n^2) + (12 + 11*n + 6*n^2 + n^3)*(-1 + 2*Cos[2*(e + f*x)]) + I*n*(20 + 9*n + n^2 + 2*(11 + 6*n + n^2)*Cos[2*(e + f*x)])*Tan[e + f*x]))/(f*n*(1 + n)*(2 + n)*(3 + n)*(Cos[f*x] + I*Sin[f*x])^4)","A",1
1044,1,110,99,4.7688139,"\int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^n,x]","\frac{i a^3 \sec ^2(e+f x) (c \sec (e+f x))^n \left(\left(n^2+3 n+4\right) \cos (2 (e+f x))+i n (n+3) \sin (2 (e+f x))+2 (n+2)\right) \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f n (n+1) (n+2)}","\frac{i a^3 (c-i c \tan (e+f x))^{n+2}}{c^2 f (n+2)}+\frac{4 i a^3 (c-i c \tan (e+f x))^n}{f n}-\frac{4 i a^3 (c-i c \tan (e+f x))^{n+1}}{c f (n+1)}",1,"(I*a^3*E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*Sec[e + f*x]^2*(c*Sec[e + f*x])^n*(2*(2 + n) + (4 + 3*n + n^2)*Cos[2*(e + f*x)] + I*n*(3 + n)*Sin[2*(e + f*x)]))/(f*n*(1 + n)*(2 + n))","A",1
1045,1,72,64,2.812425,"\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^n,x]","-\frac{a^2 (n \tan (e+f x)-i (n+2)) (c \sec (e+f x))^n \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f n (n+1)}","\frac{2 i a^2 (c-i c \tan (e+f x))^n}{f n}-\frac{i a^2 (c-i c \tan (e+f x))^{n+1}}{c f (n+1)}",1,"-((a^2*E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*(c*Sec[e + f*x])^n*((-I)*(2 + n) + n*Tan[e + f*x]))/(f*n*(1 + n)))","A",1
1046,1,51,26,0.8351595,"\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n,x]","\frac{i a (c \sec (e+f x))^n \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f n}","\frac{i a (c-i c \tan (e+f x))^n}{f n}",1,"(I*a*E^(n*(-Log[c*Sec[e + f*x]] + Log[c - I*c*Tan[e + f*x]]))*(c*Sec[e + f*x])^n)/(f*n)","A",1
1047,1,79,52,58.6036916,"\int \frac{(c-i c \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx","Integrate[(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x]),x]","\frac{i 2^{n-2} \left(1+e^{2 i (e+f x)}\right)^2 \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^n \, _2F_1\left(2,2-n;3-n;1+e^{2 i (e+f x)}\right)}{a f (n-2)}","\frac{i (c-i c \tan (e+f x))^n \, _2F_1\left(2,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{4 a f n}",1,"(I*2^(-2 + n)*(c/(1 + E^((2*I)*(e + f*x))))^n*(1 + E^((2*I)*(e + f*x)))^2*Hypergeometric2F1[2, 2 - n, 3 - n, 1 + E^((2*I)*(e + f*x))])/(a*f*(-2 + n))","A",1
1048,-1,0,52,180.0032689,"\int \frac{(c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2,x]","\text{\$Aborted}","\frac{i (c-i c \tan (e+f x))^n \, _2F_1\left(3,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{8 a^2 f n}",1,"$Aborted","F",-1
1049,0,0,52,138.849487,"\int \frac{(c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3,x]","\int \frac{(c-i c \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","\frac{i (c-i c \tan (e+f x))^n \, _2F_1\left(4,n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{16 a^3 f n}",1,"Integrate[(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3, x]","F",-1
1050,1,142,66,14.4263334,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n,x]","-\frac{i c 2^{m+n-1} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{n-1} \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \, _2F_1\left(1,1-n;m+1;-e^{2 i (e+f x)}\right)}{f m}","\frac{i (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left(1,m+n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right)}{2 f n}",1,"((-I)*2^(-1 + m + n)*c*(E^(I*f*x))^m*(c/(1 + E^((2*I)*(e + f*x))))^(-1 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[1, 1 - n, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/(f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1051,0,0,134,64.8287075,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^4 \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^4,x]","\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^4 \, dx","\frac{i c^4 (a+i a \tan (e+f x))^{m+3}}{a^3 f (m+3)}-\frac{6 i c^4 (a+i a \tan (e+f x))^{m+2}}{a^2 f (m+2)}-\frac{8 i c^4 (a+i a \tan (e+f x))^m}{f m}+\frac{12 i c^4 (a+i a \tan (e+f x))^{m+1}}{a f (m+1)}",1,"Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^4, x]","F",-1
1052,1,161,99,16.6282631,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^3,x]","-\frac{i c^3 2^{m+2} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \left(2 (m+2) e^{2 i (e+f x)}+2 e^{4 i (e+f x)}+m^2+3 m+2\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m (m+1) (m+2) \left(1+e^{2 i (e+f x)}\right)^2}","-\frac{i c^3 (a+i a \tan (e+f x))^{m+2}}{a^2 f (m+2)}-\frac{4 i c^3 (a+i a \tan (e+f x))^m}{f m}+\frac{4 i c^3 (a+i a \tan (e+f x))^{m+1}}{a f (m+1)}",1,"((-I)*2^(2 + m)*c^3*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*(2 + 2*E^((4*I)*(e + f*x)) + 3*m + m^2 + 2*E^((2*I)*(e + f*x))*(2 + m))*(a + I*a*Tan[e + f*x])^m)/((1 + E^((2*I)*(e + f*x)))^2*f*m*(1 + m)*(2 + m)*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","A",1
1053,1,131,64,45.3738391,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^2,x]","-\frac{i c^2 2^{m+1} e^{-i (e+f x)} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{m+1} \left(e^{2 i (e+f x)}+m+1\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m (m+1)}","\frac{i c^2 (a+i a \tan (e+f x))^{m+1}}{a f (m+1)}-\frac{2 i c^2 (a+i a \tan (e+f x))^m}{f m}",1,"((-I)*2^(1 + m)*c^2*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1 + m)*(1 + E^((2*I)*(e + f*x)) + m)*(a + I*a*Tan[e + f*x])^m)/(E^(I*(e + f*x))*f*m*(1 + m)*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",1
1054,1,95,26,2.1290055,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x]),x]","-\frac{i c 2^m \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m}","-\frac{i c (a+i a \tan (e+f x))^m}{f m}",1,"((-I)*2^m*c*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*(a + I*a*Tan[e + f*x])^m)/(f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",1
1055,1,133,52,16.5527673,"\int \frac{(a+i a \tan (e+f x))^m}{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x]),x]","-\frac{i 2^{m-2} \left(1+e^{2 i (e+f x)}\right)^2 \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \, _2F_1\left(1,2;m+1;-e^{2 i (e+f x)}\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{c f m}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(2,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{4 c f m}",1,"((-I)*2^(-2 + m)*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*(1 + E^((2*I)*(e + f*x)))^2*Hypergeometric2F1[1, 2, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/(c*f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1056,-1,0,52,180.0086336,"\int \frac{(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^2,x]","\text{\$Aborted}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(3,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{8 c^2 f m}",1,"$Aborted","F",-1
1057,-1,0,52,180.0014583,"\int \frac{(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^3,x]","\text{\$Aborted}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(4,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{16 c^3 f m}",1,"$Aborted","F",-1
1058,-1,0,52,180.0055225,"\int \frac{(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^4,x]","\text{\$Aborted}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(5,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{32 c^4 f m}",1,"$Aborted","F",-1
1059,1,141,67,69.8556008,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2),x]","-\frac{i c 2^{m+\frac{3}{2}} \left(e^{i f x}\right)^m \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \, _2F_1\left(-\frac{3}{2},1;m+1;-e^{2 i (e+f x)}\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m}","\frac{i (c-i c \tan (e+f x))^{5/2} (a+i a \tan (e+f x))^m \, _2F_1\left(1,m+\frac{5}{2};\frac{7}{2};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 f}",1,"((-I)*2^(3/2 + m)*c*(E^(I*f*x))^m*(c/(1 + E^((2*I)*(e + f*x))))^(3/2)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[-3/2, 1, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/(f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1060,1,141,67,7.2343555,"\int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{i c 2^{m+\frac{1}{2}} \left(e^{i f x}\right)^m \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \, _2F_1\left(-\frac{1}{2},1;m+1;-e^{2 i (e+f x)}\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m}","\frac{i (c-i c \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^m \, _2F_1\left(1,m+\frac{3}{2};\frac{5}{2};\frac{1}{2} (1-i \tan (e+f x))\right)}{3 f}",1,"((-I)*2^(1/2 + m)*c*(E^(I*f*x))^m*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[-1/2, 1, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/(f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1061,1,141,65,2.1612947,"\int (a+i a \tan (e+f x))^m \sqrt{c-i c \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{i c 2^{m-\frac{1}{2}} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \, _2F_1\left(\frac{1}{2},1;m+1;-e^{2 i (e+f x)}\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m \sqrt{\frac{c}{1+e^{2 i (e+f x)}}}}","\frac{i \sqrt{c-i c \tan (e+f x)} (a+i a \tan (e+f x))^m \, _2F_1\left(1,m+\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-i \tan (e+f x))\right)}{f}",1,"((-I)*2^(-1/2 + m)*c*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[1/2, 1, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/(Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1062,1,141,65,6.8957636,"\int \frac{(a+i a \tan (e+f x))^m}{\sqrt{c-i c \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/Sqrt[c - I*c*Tan[e + f*x]],x]","-\frac{i c 2^{m-\frac{3}{2}} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \, _2F_1\left(1,\frac{3}{2};m+1;-e^{2 i (e+f x)}\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{3/2}}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(1,m-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-i \tan (e+f x))\right)}{f \sqrt{c-i c \tan (e+f x)}}",1,"((-I)*2^(-3/2 + m)*c*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[1, 3/2, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/((c/(1 + E^((2*I)*(e + f*x))))^(3/2)*f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1063,1,141,67,113.1947616,"\int \frac{(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^(3/2),x]","-\frac{i c 2^{m-\frac{5}{2}} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \, _2F_1\left(1,\frac{5}{2};m+1;-e^{2 i (e+f x)}\right) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m \left(\frac{c}{1+e^{2 i (e+f x)}}\right)^{5/2}}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(1,m-\frac{3}{2};-\frac{1}{2};\frac{1}{2} (1-i \tan (e+f x))\right)}{3 f (c-i c \tan (e+f x))^{3/2}}",1,"((-I)*2^(-5/2 + m)*c*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*Hypergeometric2F1[1, 5/2, 1 + m, -E^((2*I)*(e + f*x))]*(a + I*a*Tan[e + f*x])^m)/((c/(1 + E^((2*I)*(e + f*x))))^(5/2)*f*m*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1064,-1,0,67,180.0019501,"\int \frac{(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^(5/2),x]","\text{\$Aborted}","-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(1,m-\frac{5}{2};-\frac{3}{2};\frac{1}{2} (1-i \tan (e+f x))\right)}{5 f (c-i c \tan (e+f x))^{5/2}}",1,"$Aborted","F",-1
1065,1,331,110,4.7347857,"\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x]),x]","\frac{a^3 \sec (e) \sec ^3(e+f x) \left(3 \cos (f x) \left((-3 d-3 i c) \log \left(\cos ^2(e+f x)\right)+6 c f x-i c-6 i d f x-3 d\right)+3 \cos (2 e+f x) \left((-3 d-3 i c) \log \left(\cos ^2(e+f x)\right)+6 c f x-i c-6 i d f x-3 d\right)+9 c \sin (2 e+f x)-9 c \sin (2 e+3 f x)+6 c f x \cos (2 e+3 f x)+6 c f x \cos (4 e+3 f x)-3 i c \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 i c \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)-18 c \sin (f x)-15 i d \sin (2 e+f x)+13 i d \sin (2 e+3 f x)-6 i d f x \cos (2 e+3 f x)-6 i d f x \cos (4 e+3 f x)-3 d \cos (2 e+3 f x) \log \left(\cos ^2(e+f x)\right)-3 d \cos (4 e+3 f x) \log \left(\cos ^2(e+f x)\right)+24 i d \sin (f x)\right)}{12 f}","-\frac{2 a^3 (c-i d) \tan (e+f x)}{f}-\frac{4 a^3 (d+i c) \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)+\frac{a (d+i c) (a+i a \tan (e+f x))^2}{2 f}+\frac{d (a+i a \tan (e+f x))^3}{3 f}",1,"(a^3*Sec[e]*Sec[e + f*x]^3*(6*c*f*x*Cos[2*e + 3*f*x] - (6*I)*d*f*x*Cos[2*e + 3*f*x] + 6*c*f*x*Cos[4*e + 3*f*x] - (6*I)*d*f*x*Cos[4*e + 3*f*x] - (3*I)*c*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - 3*d*Cos[2*e + 3*f*x]*Log[Cos[e + f*x]^2] - (3*I)*c*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] - 3*d*Cos[4*e + 3*f*x]*Log[Cos[e + f*x]^2] + 3*Cos[f*x]*((-I)*c - 3*d + 6*c*f*x - (6*I)*d*f*x + ((-3*I)*c - 3*d)*Log[Cos[e + f*x]^2]) + 3*Cos[2*e + f*x]*((-I)*c - 3*d + 6*c*f*x - (6*I)*d*f*x + ((-3*I)*c - 3*d)*Log[Cos[e + f*x]^2]) - 18*c*Sin[f*x] + (24*I)*d*Sin[f*x] + 9*c*Sin[2*e + f*x] - (15*I)*d*Sin[2*e + f*x] - 9*c*Sin[2*e + 3*f*x] + (13*I)*d*Sin[2*e + 3*f*x]))/(12*f)","B",1
1066,1,263,80,2.7273705,"\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x]),x]","\frac{a^2 \sec (e) \sec ^2(e+f x) (\cos (2 f x)+i \sin (2 f x)) \left(-8 (c-i d) \cos (e) \cos ^2(e+f x) \tan ^{-1}(\tan (3 e+f x))-i \left((d+i c) \cos (e+2 f x) \left(4 f x-i \log \left(\cos ^2(e+f x)\right)\right)+2 \cos (e) \left((c-i d) \log \left(\cos ^2(e+f x)\right)+4 i c f x+4 d f x-i d\right)-2 i c \sin (e+2 f x)+4 i c f x \cos (3 e+2 f x)+c \cos (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+2 i c \sin (e)-4 d \sin (e+2 f x)+4 d f x \cos (3 e+2 f x)-i d \cos (3 e+2 f x) \log \left(\cos ^2(e+f x)\right)+4 d \sin (e)\right)\right)}{4 f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (c-i d) \tan (e+f x)}{f}-\frac{2 a^2 (d+i c) \log (\cos (e+f x))}{f}+2 a^2 x (c-i d)+\frac{d (a+i a \tan (e+f x))^2}{2 f}",1,"(a^2*Sec[e]*Sec[e + f*x]^2*(Cos[2*f*x] + I*Sin[2*f*x])*(-8*(c - I*d)*ArcTan[Tan[3*e + f*x]]*Cos[e]*Cos[e + f*x]^2 - I*((4*I)*c*f*x*Cos[3*e + 2*f*x] + 4*d*f*x*Cos[3*e + 2*f*x] + (I*c + d)*Cos[e + 2*f*x]*(4*f*x - I*Log[Cos[e + f*x]^2]) + c*Cos[3*e + 2*f*x]*Log[Cos[e + f*x]^2] - I*d*Cos[3*e + 2*f*x]*Log[Cos[e + f*x]^2] + 2*Cos[e]*((-I)*d + (4*I)*c*f*x + 4*d*f*x + (c - I*d)*Log[Cos[e + f*x]^2]) + (2*I)*c*Sin[e] + 4*d*Sin[e] - (2*I)*c*Sin[e + 2*f*x] - 4*d*Sin[e + 2*f*x])))/(4*f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1067,1,66,46,0.0385798,"\int (a+i a \tan (e+f x)) (c+d \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x]),x]","-\frac{i a c \log (\cos (e+f x))}{f}+a c x-\frac{i a d \tan ^{-1}(\tan (e+f x))}{f}+\frac{i a d \tan (e+f x)}{f}-\frac{a d \log (\cos (e+f x))}{f}","-\frac{a (d+i c) \log (\cos (e+f x))}{f}+a x (c-i d)+\frac{i a d \tan (e+f x)}{f}",1,"a*c*x - (I*a*d*ArcTan[Tan[e + f*x]])/f - (I*a*c*Log[Cos[e + f*x]])/f - (a*d*Log[Cos[e + f*x]])/f + (I*a*d*Tan[e + f*x])/f","A",1
1068,1,102,47,0.6565679,"\int \frac{c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])/(a + I*a*Tan[e + f*x]),x]","\frac{\cos (e+f x) (c+d \tan (e+f x)) ((c (2 f x-i)-2 i d f x+d) \tan (e+f x)-2 i c f x+c+d (-2 f x+i))}{4 a f (\tan (e+f x)-i) (c \cos (e+f x)+d \sin (e+f x))}","\frac{-d+i c}{2 f (a+i a \tan (e+f x))}+\frac{x (c-i d)}{2 a}",1,"(Cos[e + f*x]*(c + d*Tan[e + f*x])*(c - (2*I)*c*f*x + d*(I - 2*f*x) + (d - (2*I)*d*f*x + c*(-I + 2*f*x))*Tan[e + f*x]))/(4*a*f*(c*Cos[e + f*x] + d*Sin[e + f*x])*(-I + Tan[e + f*x]))","B",1
1069,1,94,80,0.7142775,"\int \frac{c+d \tan (e+f x)}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^2,x]","-\frac{\sec ^2(e+f x) ((4 i c f x+c+4 d f x+i d) \sin (2 (e+f x))+(c (4 f x+i)+d (-1-4 i f x)) \cos (2 (e+f x))+4 i c)}{16 a^2 f (\tan (e+f x)-i)^2}","\frac{d+i c}{4 f \left(a^2+i a^2 \tan (e+f x)\right)}+\frac{x (c-i d)}{4 a^2}+\frac{-d+i c}{4 f (a+i a \tan (e+f x))^2}",1,"-1/16*(Sec[e + f*x]^2*((4*I)*c + (d*(-1 - (4*I)*f*x) + c*(I + 4*f*x))*Cos[2*(e + f*x)] + (c + I*d + (4*I)*c*f*x + 4*d*f*x)*Sin[2*(e + f*x)]))/(a^2*f*(-I + Tan[e + f*x])^2)","A",1
1070,1,150,112,0.9766532,"\int \frac{c+d \tan (e+f x)}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) ((-27 c+3 i d) \cos (e+f x)+2 (6 i c f x-c+6 d f x-i d) \cos (3 (e+f x))-9 i c \sin (e+f x)+2 i c \sin (3 (e+f x))-12 c f x \sin (3 (e+f x))-9 d \sin (e+f x)-2 d \sin (3 (e+f x))+12 i d f x \sin (3 (e+f x)))}{96 a^3 f (\tan (e+f x)-i)^3}","\frac{d+i c}{8 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{x (c-i d)}{8 a^3}+\frac{-d+i c}{6 f (a+i a \tan (e+f x))^3}+\frac{d+i c}{8 a f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^3*((-27*c + (3*I)*d)*Cos[e + f*x] + 2*(-c - I*d + (6*I)*c*f*x + 6*d*f*x)*Cos[3*(e + f*x)] - (9*I)*c*Sin[e + f*x] - 9*d*Sin[e + f*x] + (2*I)*c*Sin[3*(e + f*x)] - 2*d*Sin[3*(e + f*x)] - 12*c*f*x*Sin[3*(e + f*x)] + (12*I)*d*f*x*Sin[3*(e + f*x)]))/(96*a^3*f*(-I + Tan[e + f*x])^3)","A",1
1071,1,948,153,9.2306647,"\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]","\frac{\left(\frac{1}{3} \cos (3 e)-\frac{1}{3} i \sin (3 e)\right) \left(-3 \sin (f x) d^2-2 i c \sin (f x) d\right) (i \tan (e+f x) a+a)^3}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^2(e+f x) \left(\frac{1}{3} \cos (3 e)-\frac{1}{3} i \sin (3 e)\right) \left(-9 \sin (f x) c^2+26 i d \sin (f x) c+15 d^2 \sin (f x)\right) (i \tan (e+f x) a+a)^3}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))^3}+\frac{x \cos ^3(e+f x) \left(2 c^2 \cos ^3(e)-2 d^2 \cos ^3(e)-4 i c d \cos ^3(e)-8 i c^2 \sin (e) \cos ^2(e)+8 i d^2 \sin (e) \cos ^2(e)-16 c d \sin (e) \cos ^2(e)-2 c^2 \cos (e)+2 d^2 \cos (e)-12 c^2 \sin ^2(e) \cos (e)+12 d^2 \sin ^2(e) \cos (e)+24 i c d \sin ^2(e) \cos (e)+4 i c d \cos (e)+8 i c^2 \sin ^3(e)-8 i d^2 \sin ^3(e)+16 c d \sin ^3(e)+4 i c^2 \sin (e)-4 i d^2 \sin (e)+8 c d \sin (e)+2 c^2 \sin ^3(e) \tan (e)-2 d^2 \sin ^3(e) \tan (e)-4 i c d \sin ^3(e) \tan (e)+2 c^2 \sin (e) \tan (e)-2 d^2 \sin (e) \tan (e)-4 i c d \sin (e) \tan (e)+i (c-i d)^2 (4 \cos (3 e)-4 i \sin (3 e)) \tan (e)\right) (i \tan (e+f x) a+a)^3}{(\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\cos \left(\frac{3 e}{2}\right) c^2-i \sin \left(\frac{3 e}{2}\right) c^2-2 i d \cos \left(\frac{3 e}{2}\right) c-2 d \sin \left(\frac{3 e}{2}\right) c-d^2 \cos \left(\frac{3 e}{2}\right)+i d^2 \sin \left(\frac{3 e}{2}\right)\right) \left(-2 i \cos \left(\frac{3 e}{2}\right) \log \left(\cos ^2(e+f x)\right)-2 \sin \left(\frac{3 e}{2}\right) \log \left(\cos ^2(e+f x)\right)\right) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{\cos (e+f x) \left(3 \cos (e) c^2-18 i d \cos (e) c+4 d \sin (e) c-15 d^2 \cos (e)-6 i d^2 \sin (e)\right) \left(-\frac{1}{6} i \cos (3 e)-\frac{1}{6} \sin (3 e)\right) (i \tan (e+f x) a+a)^3}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))^3}+\frac{\sec (e+f x) \left(-\frac{1}{4} i \cos (3 e) d^2-\frac{1}{4} \sin (3 e) d^2\right) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{(c-i d)^2 \cos ^3(e+f x) (4 f x \cos (3 e)-4 i f x \sin (3 e)) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}","-\frac{2 a^3 (c-i d)^2 \tan (e+f x)}{f}-\frac{4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^2+\frac{2 c d (a+i a \tan (e+f x))^3}{3 f}+\frac{i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}-\frac{i d^2 (a+i a \tan (e+f x))^4}{4 a f}",1,"(Cos[e + f*x]^3*(c^2*Cos[(3*e)/2] - (2*I)*c*d*Cos[(3*e)/2] - d^2*Cos[(3*e)/2] - I*c^2*Sin[(3*e)/2] - 2*c*d*Sin[(3*e)/2] + I*d^2*Sin[(3*e)/2])*((-2*I)*Cos[(3*e)/2]*Log[Cos[e + f*x]^2] - 2*Log[Cos[e + f*x]^2]*Sin[(3*e)/2])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]*(3*c^2*Cos[e] - (18*I)*c*d*Cos[e] - 15*d^2*Cos[e] + 4*c*d*Sin[e] - (6*I)*d^2*Sin[e])*((-1/6*I)*Cos[3*e] - Sin[3*e]/6)*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3) + (Sec[e + f*x]*((-1/4*I)*d^2*Cos[3*e] - (d^2*Sin[3*e])/4)*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + ((c - I*d)^2*Cos[e + f*x]^3*(4*f*x*Cos[3*e] - (4*I)*f*x*Sin[3*e])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + ((Cos[3*e]/3 - (I/3)*Sin[3*e])*((-2*I)*c*d*Sin[f*x] - 3*d^2*Sin[f*x])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^2*(Cos[3*e]/3 - (I/3)*Sin[3*e])*(-9*c^2*Sin[f*x] + (26*I)*c*d*Sin[f*x] + 15*d^2*Sin[f*x])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[f*x] + I*Sin[f*x])^3) + (x*Cos[e + f*x]^3*(-2*c^2*Cos[e] + (4*I)*c*d*Cos[e] + 2*d^2*Cos[e] + 2*c^2*Cos[e]^3 - (4*I)*c*d*Cos[e]^3 - 2*d^2*Cos[e]^3 + (4*I)*c^2*Sin[e] + 8*c*d*Sin[e] - (4*I)*d^2*Sin[e] - (8*I)*c^2*Cos[e]^2*Sin[e] - 16*c*d*Cos[e]^2*Sin[e] + (8*I)*d^2*Cos[e]^2*Sin[e] - 12*c^2*Cos[e]*Sin[e]^2 + (24*I)*c*d*Cos[e]*Sin[e]^2 + 12*d^2*Cos[e]*Sin[e]^2 + (8*I)*c^2*Sin[e]^3 + 16*c*d*Sin[e]^3 - (8*I)*d^2*Sin[e]^3 + 2*c^2*Sin[e]*Tan[e] - (4*I)*c*d*Sin[e]*Tan[e] - 2*d^2*Sin[e]*Tan[e] + 2*c^2*Sin[e]^3*Tan[e] - (4*I)*c*d*Sin[e]^3*Tan[e] - 2*d^2*Sin[e]^3*Tan[e] + I*(c - I*d)^2*(4*Cos[3*e] - (4*I)*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^3)/(Cos[f*x] + I*Sin[f*x])^3","B",1
1072,1,261,116,4.6887743,"\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2,x]","\frac{(a+i a \tan (e+f x))^2 \left(-\frac{1}{3} \left(3 c^2-12 i c d-7 d^2\right) \sec (e) (\cos (2 e)-i \sin (2 e)) \sin (f x) \cos (e+f x)+4 f x (c-i d)^2 (\cos (2 e)-i \sin (2 e)) \cos ^2(e+f x)+(c-i d)^2 (-\sin (2 e)-i \cos (2 e)) \cos ^2(e+f x) \log \left(\cos ^2(e+f x)\right)-2 (c-i d)^2 (\cos (2 e)-i \sin (2 e)) \cos ^2(e+f x) \tan ^{-1}(\tan (3 e+f x))-\frac{1}{3} d (\cos (2 e)-i \sin (2 e)) (3 c+d \tan (e)-3 i d)-\frac{1}{3} d^2 \sec (e) (\cos (2 e)-i \sin (2 e)) \sin (f x) \sec (e+f x)\right)}{f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (c-i d)^2 \tan (e+f x)}{f}-\frac{2 i a^2 (c-i d)^2 \log (\cos (e+f x))}{f}+2 a^2 x (c-i d)^2+\frac{c d (a+i a \tan (e+f x))^2}{f}-\frac{i d^2 (a+i a \tan (e+f x))^3}{3 a f}",1,"(((c - I*d)^2*Cos[e + f*x]^2*Log[Cos[e + f*x]^2]*((-I)*Cos[2*e] - Sin[2*e]) + 4*(c - I*d)^2*f*x*Cos[e + f*x]^2*(Cos[2*e] - I*Sin[2*e]) - 2*(c - I*d)^2*ArcTan[Tan[3*e + f*x]]*Cos[e + f*x]^2*(Cos[2*e] - I*Sin[2*e]) - ((3*c^2 - (12*I)*c*d - 7*d^2)*Cos[e + f*x]*Sec[e]*(Cos[2*e] - I*Sin[2*e])*Sin[f*x])/3 - (d^2*Sec[e]*Sec[e + f*x]*(Cos[2*e] - I*Sin[2*e])*Sin[f*x])/3 - (d*(Cos[2*e] - I*Sin[2*e])*(3*c - (3*I)*d + d*Tan[e]))/3)*(a + I*a*Tan[e + f*x])^2)/(f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1073,1,175,78,1.9663215,"\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^2,x]","\frac{(\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left(2 d (2 c-i d) (\tan (e)+i) \sin (f x)+4 f x (c-i d)^2 (\cos (e)-i \sin (e)) \cos (e+f x)-i (c-i d)^2 (\cos (e)-i \sin (e)) \cos (e+f x) \log \left(\cos ^2(e+f x)\right)-2 (c-i d)^2 (\cos (e)-i \sin (e)) \cos (e+f x) \tan ^{-1}(\tan (2 e+f x))+d^2 (\sin (e)+i \cos (e)) \sec (e+f x)\right)}{2 f}","\frac{i a (c+d \tan (e+f x))^2}{2 f}+\frac{a d (d+i c) \tan (e+f x)}{f}-\frac{i a (c-i d)^2 \log (\cos (e+f x))}{f}+a x (c-i d)^2",1,"((Cos[f*x] - I*Sin[f*x])*(4*(c - I*d)^2*f*x*Cos[e + f*x]*(Cos[e] - I*Sin[e]) - 2*(c - I*d)^2*ArcTan[Tan[2*e + f*x]]*Cos[e + f*x]*(Cos[e] - I*Sin[e]) - I*(c - I*d)^2*Cos[e + f*x]*Log[Cos[e + f*x]^2]*(Cos[e] - I*Sin[e]) + d^2*Sec[e + f*x]*(I*Cos[e] + Sin[e]) + 2*(2*c - I*d)*d*Sin[f*x]*(I + Tan[e]))*(a + I*a*Tan[e + f*x]))/(2*f)","B",1
1074,1,155,75,1.448818,"\int \frac{(c+d \tan (e+f x))^2}{a+i a \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x]),x]","\frac{\tan (e+f x) \left(c^2 (2 f x-i)+2 c (d-2 i d f x)+2 i d^2 \log \left(\cos ^2(e+f x)\right)+d^2 (-2 f x+i)\right)-2 i c^2 f x+c^2-4 c d f x+2 i c d+4 d^2 \tan ^{-1}(\tan (f x)) (\tan (e+f x)-i)+2 d^2 \log \left(\cos ^2(e+f x)\right)+2 i d^2 f x-d^2}{4 a f (\tan (e+f x)-i)}","\frac{x \left(c^2-2 i c d+d^2\right)}{2 a}+\frac{i (c+i d)^2}{2 f (a+i a \tan (e+f x))}+\frac{i d^2 \log (\cos (e+f x))}{a f}",1,"(c^2 + (2*I)*c*d - d^2 - (2*I)*c^2*f*x - 4*c*d*f*x + (2*I)*d^2*f*x + 2*d^2*Log[Cos[e + f*x]^2] + (d^2*(I - 2*f*x) + c^2*(-I + 2*f*x) + 2*c*(d - (2*I)*d*f*x) + (2*I)*d^2*Log[Cos[e + f*x]^2])*Tan[e + f*x] + 4*d^2*ArcTan[Tan[f*x]]*(-I + Tan[e + f*x]))/(4*a*f*(-I + Tan[e + f*x]))","B",1
1075,1,134,91,1.3130239,"\int \frac{(c+d \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^2,x]","-\frac{\sec ^2(e+f x) \left(\left(c^2 (1+4 i f x)+2 c d (4 f x+i)+d^2 (-1-4 i f x)\right) \sin (2 (e+f x))+\left(c^2 (4 f x+i)+c d (-2-8 i f x)-d^2 (4 f x+i)\right) \cos (2 (e+f x))+4 i \left(c^2+d^2\right)\right)}{16 a^2 f (\tan (e+f x)-i)^2}","\frac{(c+i d) (3 d+i c)}{4 a^2 f (1+i \tan (e+f x))}+\frac{x (c-i d)^2}{4 a^2}+\frac{i (c+i d)^2}{4 f (a+i a \tan (e+f x))^2}",1,"-1/16*(Sec[e + f*x]^2*((4*I)*(c^2 + d^2) + (c*d*(-2 - (8*I)*f*x) + c^2*(I + 4*f*x) - d^2*(I + 4*f*x))*Cos[2*(e + f*x)] + (d^2*(-1 - (4*I)*f*x) + c^2*(1 + (4*I)*f*x) + 2*c*d*(I + 4*f*x))*Sin[2*(e + f*x)]))/(a^2*f*(-I + Tan[e + f*x])^2)","A",1
1076,1,256,129,1.4097286,"\int \frac{(c+d \tan (e+f x))^2}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(12 f x (c-i d)^2 (\cos (3 e)+i \sin (3 e))+6 (3 c+i d) (c-i d) (\cos (e)+i \sin (e)) \sin (2 f x)+3 (c+i d) (d+3 i c) (\cos (e)-i \sin (e)) \cos (4 f x)+6 (3 c+i d) (d+i c) (\cos (e)+i \sin (e)) \cos (2 f x)+2 (c+i d)^2 (\sin (3 e)+i \cos (3 e)) \cos (6 f x)+3 (3 c-i d) (c+i d) (\cos (e)-i \sin (e)) \sin (4 f x)+2 (c+i d)^2 (\cos (3 e)-i \sin (3 e)) \sin (6 f x)\right)}{96 f (a+i a \tan (e+f x))^3}","\frac{i (c-i d)^2}{8 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{x (c-i d)^2}{8 a^3}+\frac{(c+i d) (3 d+i c)}{8 a f (a+i a \tan (e+f x))^2}+\frac{i (c+i d)^2}{6 f (a+i a \tan (e+f x))^3}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(3*(c + I*d)*((3*I)*c + d)*Cos[4*f*x]*(Cos[e] - I*Sin[e]) + 6*(3*c + I*d)*(I*c + d)*Cos[2*f*x]*(Cos[e] + I*Sin[e]) + 12*(c - I*d)^2*f*x*(Cos[3*e] + I*Sin[3*e]) + 2*(c + I*d)^2*Cos[6*f*x]*(I*Cos[3*e] + Sin[3*e]) + 6*(c - I*d)*(3*c + I*d)*(Cos[e] + I*Sin[e])*Sin[2*f*x] + 3*(3*c - I*d)*(c + I*d)*(Cos[e] - I*Sin[e])*Sin[4*f*x] + 2*(c + I*d)^2*(Cos[3*e] - I*Sin[3*e])*Sin[6*f*x]))/(96*f*(a + I*a*Tan[e + f*x])^3)","A",1
1077,1,1564,190,10.9474487,"\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3,x]","\frac{\sec (e) \sec ^2(e+f x) \left(\frac{1}{240} \cos (3 e)-\frac{1}{240} i \sin (3 e)\right) \left(300 f x \cos (f x) c^3-45 i \cos (f x) c^3+300 f x \cos (2 e+f x) c^3-45 i \cos (2 e+f x) c^3+150 f x \cos (2 e+3 f x) c^3-15 i \cos (2 e+3 f x) c^3+150 f x \cos (4 e+3 f x) c^3-15 i \cos (4 e+3 f x) c^3+30 f x \cos (4 e+5 f x) c^3+30 f x \cos (6 e+5 f x) c^3-270 \sin (f x) c^3+180 \sin (2 e+f x) c^3-180 \sin (2 e+3 f x) c^3+45 \sin (4 e+3 f x) c^3-45 \sin (4 e+5 f x) c^3-405 d \cos (f x) c^2-900 i d f x \cos (f x) c^2-405 d \cos (2 e+f x) c^2-900 i d f x \cos (2 e+f x) c^2-135 d \cos (2 e+3 f x) c^2-450 i d f x \cos (2 e+3 f x) c^2-135 d \cos (4 e+3 f x) c^2-450 i d f x \cos (4 e+3 f x) c^2-90 i d f x \cos (4 e+5 f x) c^2-90 i d f x \cos (6 e+5 f x) c^2+1140 i d \sin (f x) c^2-810 i d \sin (2 e+f x) c^2+750 i d \sin (2 e+3 f x) c^2-225 i d \sin (4 e+3 f x) c^2+195 i d \sin (4 e+5 f x) c^2+585 i d^2 \cos (f x) c-900 d^2 f x \cos (f x) c+585 i d^2 \cos (2 e+f x) c-900 d^2 f x \cos (2 e+f x) c+225 i d^2 \cos (2 e+3 f x) c-450 d^2 f x \cos (2 e+3 f x) c+225 i d^2 \cos (4 e+3 f x) c-450 d^2 f x \cos (4 e+3 f x) c-90 d^2 f x \cos (4 e+5 f x) c-90 d^2 f x \cos (6 e+5 f x) c+1260 d^2 \sin (f x) c-990 d^2 \sin (2 e+f x) c+810 d^2 \sin (2 e+3 f x) c-315 d^2 \sin (4 e+3 f x) c+225 d^2 \sin (4 e+5 f x) c+225 d^3 \cos (f x)+300 i d^3 f x \cos (f x)+225 d^3 \cos (2 e+f x)+300 i d^3 f x \cos (2 e+f x)+105 d^3 \cos (2 e+3 f x)+150 i d^3 f x \cos (2 e+3 f x)+105 d^3 \cos (4 e+3 f x)+150 i d^3 f x \cos (4 e+3 f x)+30 i d^3 f x \cos (4 e+5 f x)+30 i d^3 f x \cos (6 e+5 f x)-470 i d^3 \sin (f x)+360 i d^3 \sin (2 e+f x)-280 i d^3 \sin (2 e+3 f x)+135 i d^3 \sin (4 e+3 f x)-83 i d^3 \sin (4 e+5 f x)\right) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac{x \cos ^3(e+f x) \left(2 \cos ^3(e) c^3+8 i \sin ^3(e) c^3-12 \cos (e) \sin ^2(e) c^3-2 \cos (e) c^3-8 i \cos ^2(e) \sin (e) c^3+4 i \sin (e) c^3+2 \sin ^3(e) \tan (e) c^3+2 \sin (e) \tan (e) c^3-6 i d \cos ^3(e) c^2+24 d \sin ^3(e) c^2+36 i d \cos (e) \sin ^2(e) c^2+6 i d \cos (e) c^2-24 d \cos ^2(e) \sin (e) c^2+12 d \sin (e) c^2-6 i d \sin ^3(e) \tan (e) c^2-6 i d \sin (e) \tan (e) c^2-6 d^2 \cos ^3(e) c-24 i d^2 \sin ^3(e) c+36 d^2 \cos (e) \sin ^2(e) c+6 d^2 \cos (e) c-12 i d^2 \sin (e) c+24 i d^2 \cos ^2(e) \sin (e) c-6 d^2 \sin ^3(e) \tan (e) c-6 d^2 \sin (e) \tan (e) c+2 i d^3 \cos ^3(e)-8 d^3 \sin ^3(e)-12 i d^3 \cos (e) \sin ^2(e)-2 i d^3 \cos (e)-4 d^3 \sin (e)+8 d^3 \cos ^2(e) \sin (e)+2 i d^3 \sin ^3(e) \tan (e)+2 i d^3 \sin (e) \tan (e)+(-i c-d)^3 (4 \cos (3 e)-4 i \sin (3 e)) \tan (e)\right) (i \tan (e+f x) a+a)^3}{(\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(-i \cos \left(\frac{3 e}{2}\right) c^3-\sin \left(\frac{3 e}{2}\right) c^3-3 d \cos \left(\frac{3 e}{2}\right) c^2+3 i d \sin \left(\frac{3 e}{2}\right) c^2+3 i d^2 \cos \left(\frac{3 e}{2}\right) c+3 d^2 \sin \left(\frac{3 e}{2}\right) c+d^3 \cos \left(\frac{3 e}{2}\right)-i d^3 \sin \left(\frac{3 e}{2}\right)\right) \left(2 \cos \left(\frac{3 e}{2}\right) \log \left(\cos ^2(e+f x)\right)-2 i \log \left(\cos ^2(e+f x)\right) \sin \left(\frac{3 e}{2}\right)\right) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}","\frac{a^3 (-11 d+i c) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac{\left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^4}{5 d f}+\frac{4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac{2 a^3 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac{4 i a^3 d (c-i d)^2 \tan (e+f x)}{f}+\frac{4 a^3 (d+i c)^3 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^3",1,"(Cos[e + f*x]^3*((-I)*c^3*Cos[(3*e)/2] - 3*c^2*d*Cos[(3*e)/2] + (3*I)*c*d^2*Cos[(3*e)/2] + d^3*Cos[(3*e)/2] - c^3*Sin[(3*e)/2] + (3*I)*c^2*d*Sin[(3*e)/2] + 3*c*d^2*Sin[(3*e)/2] - I*d^3*Sin[(3*e)/2])*(2*Cos[(3*e)/2]*Log[Cos[e + f*x]^2] - (2*I)*Log[Cos[e + f*x]^2]*Sin[(3*e)/2])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Sec[e]*Sec[e + f*x]^2*(Cos[3*e]/240 - (I/240)*Sin[3*e])*((-45*I)*c^3*Cos[f*x] - 405*c^2*d*Cos[f*x] + (585*I)*c*d^2*Cos[f*x] + 225*d^3*Cos[f*x] + 300*c^3*f*x*Cos[f*x] - (900*I)*c^2*d*f*x*Cos[f*x] - 900*c*d^2*f*x*Cos[f*x] + (300*I)*d^3*f*x*Cos[f*x] - (45*I)*c^3*Cos[2*e + f*x] - 405*c^2*d*Cos[2*e + f*x] + (585*I)*c*d^2*Cos[2*e + f*x] + 225*d^3*Cos[2*e + f*x] + 300*c^3*f*x*Cos[2*e + f*x] - (900*I)*c^2*d*f*x*Cos[2*e + f*x] - 900*c*d^2*f*x*Cos[2*e + f*x] + (300*I)*d^3*f*x*Cos[2*e + f*x] - (15*I)*c^3*Cos[2*e + 3*f*x] - 135*c^2*d*Cos[2*e + 3*f*x] + (225*I)*c*d^2*Cos[2*e + 3*f*x] + 105*d^3*Cos[2*e + 3*f*x] + 150*c^3*f*x*Cos[2*e + 3*f*x] - (450*I)*c^2*d*f*x*Cos[2*e + 3*f*x] - 450*c*d^2*f*x*Cos[2*e + 3*f*x] + (150*I)*d^3*f*x*Cos[2*e + 3*f*x] - (15*I)*c^3*Cos[4*e + 3*f*x] - 135*c^2*d*Cos[4*e + 3*f*x] + (225*I)*c*d^2*Cos[4*e + 3*f*x] + 105*d^3*Cos[4*e + 3*f*x] + 150*c^3*f*x*Cos[4*e + 3*f*x] - (450*I)*c^2*d*f*x*Cos[4*e + 3*f*x] - 450*c*d^2*f*x*Cos[4*e + 3*f*x] + (150*I)*d^3*f*x*Cos[4*e + 3*f*x] + 30*c^3*f*x*Cos[4*e + 5*f*x] - (90*I)*c^2*d*f*x*Cos[4*e + 5*f*x] - 90*c*d^2*f*x*Cos[4*e + 5*f*x] + (30*I)*d^3*f*x*Cos[4*e + 5*f*x] + 30*c^3*f*x*Cos[6*e + 5*f*x] - (90*I)*c^2*d*f*x*Cos[6*e + 5*f*x] - 90*c*d^2*f*x*Cos[6*e + 5*f*x] + (30*I)*d^3*f*x*Cos[6*e + 5*f*x] - 270*c^3*Sin[f*x] + (1140*I)*c^2*d*Sin[f*x] + 1260*c*d^2*Sin[f*x] - (470*I)*d^3*Sin[f*x] + 180*c^3*Sin[2*e + f*x] - (810*I)*c^2*d*Sin[2*e + f*x] - 990*c*d^2*Sin[2*e + f*x] + (360*I)*d^3*Sin[2*e + f*x] - 180*c^3*Sin[2*e + 3*f*x] + (750*I)*c^2*d*Sin[2*e + 3*f*x] + 810*c*d^2*Sin[2*e + 3*f*x] - (280*I)*d^3*Sin[2*e + 3*f*x] + 45*c^3*Sin[4*e + 3*f*x] - (225*I)*c^2*d*Sin[4*e + 3*f*x] - 315*c*d^2*Sin[4*e + 3*f*x] + (135*I)*d^3*Sin[4*e + 3*f*x] - 45*c^3*Sin[4*e + 5*f*x] + (195*I)*c^2*d*Sin[4*e + 5*f*x] + 225*c*d^2*Sin[4*e + 5*f*x] - (83*I)*d^3*Sin[4*e + 5*f*x])*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (x*Cos[e + f*x]^3*(-2*c^3*Cos[e] + (6*I)*c^2*d*Cos[e] + 6*c*d^2*Cos[e] - (2*I)*d^3*Cos[e] + 2*c^3*Cos[e]^3 - (6*I)*c^2*d*Cos[e]^3 - 6*c*d^2*Cos[e]^3 + (2*I)*d^3*Cos[e]^3 + (4*I)*c^3*Sin[e] + 12*c^2*d*Sin[e] - (12*I)*c*d^2*Sin[e] - 4*d^3*Sin[e] - (8*I)*c^3*Cos[e]^2*Sin[e] - 24*c^2*d*Cos[e]^2*Sin[e] + (24*I)*c*d^2*Cos[e]^2*Sin[e] + 8*d^3*Cos[e]^2*Sin[e] - 12*c^3*Cos[e]*Sin[e]^2 + (36*I)*c^2*d*Cos[e]*Sin[e]^2 + 36*c*d^2*Cos[e]*Sin[e]^2 - (12*I)*d^3*Cos[e]*Sin[e]^2 + (8*I)*c^3*Sin[e]^3 + 24*c^2*d*Sin[e]^3 - (24*I)*c*d^2*Sin[e]^3 - 8*d^3*Sin[e]^3 + 2*c^3*Sin[e]*Tan[e] - (6*I)*c^2*d*Sin[e]*Tan[e] - 6*c*d^2*Sin[e]*Tan[e] + (2*I)*d^3*Sin[e]*Tan[e] + 2*c^3*Sin[e]^3*Tan[e] - (6*I)*c^2*d*Sin[e]^3*Tan[e] - 6*c*d^2*Sin[e]^3*Tan[e] + (2*I)*d^3*Sin[e]^3*Tan[e] + ((-I)*c - d)^3*(4*Cos[3*e] - (4*I)*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^3)/(Cos[f*x] + I*Sin[f*x])^3","B",1
1078,1,733,141,8.5995905,"\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3,x]","\frac{\sec ^2(e+f x) (a+i a \tan (e+f x))^2 \left(\frac{1}{24} \sec (e) (\cos (2 e)-i \sin (2 e)) \left(-9 c^3 \sin (e+2 f x)+3 c^3 \sin (3 e+2 f x)-3 c^3 \sin (3 e+4 f x)+12 c^3 f x \cos (3 e+2 f x)+3 c^3 f x \cos (3 e+4 f x)+3 c^3 f x \cos (5 e+4 f x)+9 c^3 \sin (e)+54 i c^2 d \sin (e+2 f x)-18 i c^2 d \sin (3 e+2 f x)+18 i c^2 d \sin (3 e+4 f x)-9 c^2 d \cos (3 e+2 f x)-36 i c^2 d f x \cos (3 e+2 f x)-9 i c^2 d f x \cos (3 e+4 f x)-9 i c^2 d f x \cos (5 e+4 f x)-54 i c^2 d \sin (e)+6 \cos (e) \left(3 c^3 f x+c^2 d (-3-9 i f x)+3 c d^2 (-3 f x+2 i)+d^3 (2+3 i f x)\right)+57 c d^2 \sin (e+2 f x)-27 c d^2 \sin (3 e+2 f x)+21 c d^2 \sin (3 e+4 f x)+18 i c d^2 \cos (3 e+2 f x)-36 c d^2 f x \cos (3 e+2 f x)-9 c d^2 f x \cos (3 e+4 f x)-9 c d^2 f x \cos (5 e+4 f x)-63 c d^2 \sin (e)+3 (c-i d)^2 (4 c f x-4 i d f x-3 d) \cos (e+2 f x)-20 i d^3 \sin (e+2 f x)+12 i d^3 \sin (3 e+2 f x)-8 i d^3 \sin (3 e+4 f x)+9 d^3 \cos (3 e+2 f x)+12 i d^3 f x \cos (3 e+2 f x)+3 i d^3 f x \cos (3 e+4 f x)+3 i d^3 f x \cos (5 e+4 f x)+24 i d^3 \sin (e)\right)+2 f x (c-i d)^3 (\cos (2 e)-i \sin (2 e)) \cos ^4(e+f x)-2 (c-i d)^3 (\cos (2 e)-i \sin (2 e)) \cos ^4(e+f x) \tan ^{-1}(\tan (3 e+f x))+(c-i d)^3 (-\sin (2 e)-i \cos (2 e)) \cos ^4(e+f x) \log \left(\cos ^2(e+f x)\right)\right)}{f (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 (c+d \tan (e+f x))^4}{4 d f}+\frac{2 i a^2 (c+d \tan (e+f x))^3}{3 f}+\frac{a^2 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac{2 i a^2 d (c-i d)^2 \tan (e+f x)}{f}+\frac{2 a^2 (d+i c)^3 \log (\cos (e+f x))}{f}+2 a^2 x (c-i d)^3",1,"(Sec[e + f*x]^2*((c - I*d)^3*Cos[e + f*x]^4*Log[Cos[e + f*x]^2]*((-I)*Cos[2*e] - Sin[2*e]) + 2*(c - I*d)^3*f*x*Cos[e + f*x]^4*(Cos[2*e] - I*Sin[2*e]) - 2*(c - I*d)^3*ArcTan[Tan[3*e + f*x]]*Cos[e + f*x]^4*(Cos[2*e] - I*Sin[2*e]) + (Sec[e]*(Cos[2*e] - I*Sin[2*e])*(6*(3*c^3*f*x + 3*c*d^2*(2*I - 3*f*x) + d^3*(2 + (3*I)*f*x) + c^2*d*(-3 - (9*I)*f*x))*Cos[e] + 3*(c - I*d)^2*(-3*d + 4*c*f*x - (4*I)*d*f*x)*Cos[e + 2*f*x] - 9*c^2*d*Cos[3*e + 2*f*x] + (18*I)*c*d^2*Cos[3*e + 2*f*x] + 9*d^3*Cos[3*e + 2*f*x] + 12*c^3*f*x*Cos[3*e + 2*f*x] - (36*I)*c^2*d*f*x*Cos[3*e + 2*f*x] - 36*c*d^2*f*x*Cos[3*e + 2*f*x] + (12*I)*d^3*f*x*Cos[3*e + 2*f*x] + 3*c^3*f*x*Cos[3*e + 4*f*x] - (9*I)*c^2*d*f*x*Cos[3*e + 4*f*x] - 9*c*d^2*f*x*Cos[3*e + 4*f*x] + (3*I)*d^3*f*x*Cos[3*e + 4*f*x] + 3*c^3*f*x*Cos[5*e + 4*f*x] - (9*I)*c^2*d*f*x*Cos[5*e + 4*f*x] - 9*c*d^2*f*x*Cos[5*e + 4*f*x] + (3*I)*d^3*f*x*Cos[5*e + 4*f*x] + 9*c^3*Sin[e] - (54*I)*c^2*d*Sin[e] - 63*c*d^2*Sin[e] + (24*I)*d^3*Sin[e] - 9*c^3*Sin[e + 2*f*x] + (54*I)*c^2*d*Sin[e + 2*f*x] + 57*c*d^2*Sin[e + 2*f*x] - (20*I)*d^3*Sin[e + 2*f*x] + 3*c^3*Sin[3*e + 2*f*x] - (18*I)*c^2*d*Sin[3*e + 2*f*x] - 27*c*d^2*Sin[3*e + 2*f*x] + (12*I)*d^3*Sin[3*e + 2*f*x] - 3*c^3*Sin[3*e + 4*f*x] + (18*I)*c^2*d*Sin[3*e + 4*f*x] + 21*c*d^2*Sin[3*e + 4*f*x] - (8*I)*d^3*Sin[3*e + 4*f*x]))/24)*(a + I*a*Tan[e + f*x])^2)/(f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1079,1,219,107,4.2832123,"\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^3,x]","\frac{(\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left(-2 d \left(-9 c^2+9 i c d+4 d^2\right) (\tan (e)+i) \sin (f x)+d^2 \cos (e) (\tan (e)+i) (9 c+2 d \tan (e)-3 i d) \sec (e+f x)+12 f x (c-i d)^3 (\cos (e)-i \sin (e)) \cos (e+f x)-3 i (c-i d)^3 (\cos (e)-i \sin (e)) \cos (e+f x) \log \left(\cos ^2(e+f x)\right)-6 (c-i d)^3 (\cos (e)-i \sin (e)) \cos (e+f x) \tan ^{-1}(\tan (2 e+f x))+2 d^3 (\tan (e)+i) \sin (f x) \sec ^2(e+f x)\right)}{6 f}","\frac{i a d (c-i d)^2 \tan (e+f x)}{f}+\frac{i a (c+d \tan (e+f x))^3}{3 f}+\frac{a (d+i c) (c+d \tan (e+f x))^2}{2 f}+\frac{a (d+i c)^3 \log (\cos (e+f x))}{f}+a x (c-i d)^3",1,"((Cos[f*x] - I*Sin[f*x])*(12*(c - I*d)^3*f*x*Cos[e + f*x]*(Cos[e] - I*Sin[e]) - 6*(c - I*d)^3*ArcTan[Tan[2*e + f*x]]*Cos[e + f*x]*(Cos[e] - I*Sin[e]) - (3*I)*(c - I*d)^3*Cos[e + f*x]*Log[Cos[e + f*x]^2]*(Cos[e] - I*Sin[e]) - 2*d*(-9*c^2 + (9*I)*c*d + 4*d^2)*Sin[f*x]*(I + Tan[e]) + 2*d^3*Sec[e + f*x]^2*Sin[f*x]*(I + Tan[e]) + d^2*Cos[e]*Sec[e + f*x]*(I + Tan[e])*(9*c - (3*I)*d + 2*d*Tan[e]))*(a + I*a*Tan[e + f*x]))/(6*f)","B",1
1080,1,236,129,2.8419149,"\int \frac{(c+d \tan (e+f x))^3}{a+i a \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x]),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(2 f x \left(c^3-3 i c^2 d+3 c d^2+3 i d^3\right) (\cos (e)+i \sin (e))-4 d^2 f x (3 c+i d) (\cos (e)+i \sin (e))+2 i d^2 (3 c+i d) (\cos (e)+i \sin (e)) \log \left(\cos ^2(e+f x)\right)+4 d^2 (3 c+i d) (\cos (e)+i \sin (e)) \tan ^{-1}(\tan (f x))+(c+i d)^3 (\sin (e)+i \cos (e)) \cos (2 f x)+(c+i d)^3 (\cos (e)-i \sin (e)) \sin (2 f x)+4 d^3 (\tan (e)-i) \sin (f x) \sec (e+f x)\right)}{4 f (a+i a \tan (e+f x))}","\frac{x \left(c^3-3 i c^2 d+3 c d^2+3 i d^3\right)}{2 a}-\frac{d^2 (c+3 i d) \tan (e+f x)}{2 a f}+\frac{d^2 (-d+3 i c) \log (\cos (e+f x))}{a f}+\frac{(-d+i c) (c+d \tan (e+f x))^2}{2 f (a+i a \tan (e+f x))}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*(-4*(3*c + I*d)*d^2*f*x*(Cos[e] + I*Sin[e]) + 2*(c^3 - (3*I)*c^2*d + 3*c*d^2 + (3*I)*d^3)*f*x*(Cos[e] + I*Sin[e]) + 4*(3*c + I*d)*d^2*ArcTan[Tan[f*x]]*(Cos[e] + I*Sin[e]) + (2*I)*(3*c + I*d)*d^2*Log[Cos[e + f*x]^2]*(Cos[e] + I*Sin[e]) + (c + I*d)^3*Cos[2*f*x]*(I*Cos[e] + Sin[e]) + (c + I*d)^3*(Cos[e] - I*Sin[e])*Sin[2*f*x] + 4*d^3*Sec[e + f*x]*Sin[f*x]*(-I + Tan[e])))/(4*f*(a + I*a*Tan[e + f*x]))","A",1
1081,1,305,136,2.4196862,"\int \frac{(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^2,x]","-\frac{\sec ^2(e+f x) \left(4 i c^3 f x \sin (2 (e+f x))+c^3 \sin (2 (e+f x))+4 i c^3+3 i c^2 d \sin (2 (e+f x))+12 c^2 d f x \sin (2 (e+f x))+\cos (2 (e+f x)) \left(c^3 (4 f x+i)+3 c^2 d (-1-4 i f x)-3 c d^2 (4 f x+i)+8 d^3 \log \left(\cos ^2(e+f x)\right)+d^3 (1+4 i f x)\right)-3 c d^2 \sin (2 (e+f x))-12 i c d^2 f x \sin (2 (e+f x))+12 i c d^2-i d^3 \sin (2 (e+f x))-4 d^3 f x \sin (2 (e+f x))+8 i d^3 \sin (2 (e+f x)) \log \left(\cos ^2(e+f x)\right)+16 d^3 \tan ^{-1}(\tan (f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))-8 d^3\right)}{16 a^2 f (\tan (e+f x)-i)^2}","\frac{x \left(c^3-3 i c^2 d-3 c d^2-3 i d^3\right)}{4 a^2}+\frac{(c+i d)^2 (3 d+i c)}{4 a^2 f (1+i \tan (e+f x))}+\frac{d^3 \log (\cos (e+f x))}{a^2 f}+\frac{(-d+i c) (c+d \tan (e+f x))^2}{4 f (a+i a \tan (e+f x))^2}",1,"-1/16*(Sec[e + f*x]^2*((4*I)*c^3 + (12*I)*c*d^2 - 8*d^3 + Cos[2*(e + f*x)]*(3*c^2*d*(-1 - (4*I)*f*x) + d^3*(1 + (4*I)*f*x) + c^3*(I + 4*f*x) - 3*c*d^2*(I + 4*f*x) + 8*d^3*Log[Cos[e + f*x]^2]) + c^3*Sin[2*(e + f*x)] + (3*I)*c^2*d*Sin[2*(e + f*x)] - 3*c*d^2*Sin[2*(e + f*x)] - I*d^3*Sin[2*(e + f*x)] + (4*I)*c^3*f*x*Sin[2*(e + f*x)] + 12*c^2*d*f*x*Sin[2*(e + f*x)] - (12*I)*c*d^2*f*x*Sin[2*(e + f*x)] - 4*d^3*f*x*Sin[2*(e + f*x)] + (8*I)*d^3*Log[Cos[e + f*x]^2]*Sin[2*(e + f*x)] + 16*d^3*ArcTan[Tan[f*x]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])))/(a^2*f*(-I + Tan[e + f*x])^2)","B",1
1082,1,260,140,2.0212538,"\int \frac{(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(12 f x (c-i d)^3 (\cos (3 e)+i \sin (3 e))+18 i (c+i d) (c-i d)^2 (\cos (e)+i \sin (e)) \cos (2 f x)+18 (c+i d) (c-i d)^2 (\cos (e)+i \sin (e)) \sin (2 f x)+9 (c+i d)^2 (c-i d) (\cos (e)-i \sin (e)) \sin (4 f x)+9 (c+i d)^2 (d+i c) (\cos (e)-i \sin (e)) \cos (4 f x)+2 (c+i d)^3 (\sin (3 e)+i \cos (3 e)) \cos (6 f x)+2 (c+i d)^3 (\cos (3 e)-i \sin (3 e)) \sin (6 f x)\right)}{96 f (a+i a \tan (e+f x))^3}","\frac{(c+i d) (c-3 i d) (d+i c)}{8 a^3 f (1+i \tan (e+f x))}+\frac{x (c-i d)^3}{8 a^3}+\frac{i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac{(c+i d)^2 (d+i c)}{8 a f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(9*(c + I*d)^2*(I*c + d)*Cos[4*f*x]*(Cos[e] - I*Sin[e]) + (18*I)*(c - I*d)^2*(c + I*d)*Cos[2*f*x]*(Cos[e] + I*Sin[e]) + 12*(c - I*d)^3*f*x*(Cos[3*e] + I*Sin[3*e]) + 2*(c + I*d)^3*Cos[6*f*x]*(I*Cos[3*e] + Sin[3*e]) + 18*(c - I*d)^2*(c + I*d)*(Cos[e] + I*Sin[e])*Sin[2*f*x] + 9*(c - I*d)*(c + I*d)^2*(Cos[e] - I*Sin[e])*Sin[4*f*x] + 2*(c + I*d)^3*(Cos[3*e] - I*Sin[3*e])*Sin[6*f*x]))/(96*f*(a + I*a*Tan[e + f*x])^3)","A",1
1083,1,229,115,6.2067942,"\int \frac{(a+i a \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x]),x]","\frac{a^3 \sec (e+f x) \left(\cos (f x) \left(-i \left(c^2+2 i c d+3 d^2\right) \log \left(\cos ^2(e+f x)\right)+i (c+i d)^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+8 d^2 f x\right)+\cos (2 e+f x) \left(-i \left(c^2+2 i c d+3 d^2\right) \log \left(\cos ^2(e+f x)\right)+i (c+i d)^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+8 d^2 f x\right)-4 i d (c-i d) \sin (f x)\right)}{4 d^2 f (c-i d) \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right)}","-\frac{a^3 (-3 d+i c) \log (\cos (e+f x))}{d^2 f}-\frac{a^3 (c+i d)^2 \log (c \cos (e+f x)+d \sin (e+f x))}{d^2 f (d+i c)}+\frac{4 a^3 x}{c-i d}-\frac{a^3+i a^3 \tan (e+f x)}{d f}",1,"(a^3*Sec[e + f*x]*(Cos[f*x]*(8*d^2*f*x - I*(c^2 + (2*I)*c*d + 3*d^2)*Log[Cos[e + f*x]^2] + I*(c + I*d)^2*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + Cos[2*e + f*x]*(8*d^2*f*x - I*(c^2 + (2*I)*c*d + 3*d^2)*Log[Cos[e + f*x]^2] + I*(c + I*d)^2*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) - (4*I)*(c - I*d)*d*Sin[f*x]))/(4*(c - I*d)*d^2*f*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2]))","A",1
1084,1,176,106,2.8507499,"\int \frac{(a+i a \tan (e+f x))^2}{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x]),x]","\frac{a^2 \left((-2 d-2 i c) \tan ^{-1}(\tan (3 e+f x))-c \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-i d \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+2 (d-i c) \tan ^{-1}\left(\frac{d \cos (3 e+f x)-c \sin (3 e+f x)}{c \cos (3 e+f x)+d \sin (3 e+f x)}\right)+c \log \left(\cos ^2(e+f x)\right)-i d \log \left(\cos ^2(e+f x)\right)+8 d f x\right)}{2 d f (c-i d)}","-\frac{a^2 c x (c+i d)}{d^2 (c-i d)}+\frac{a^2 x (c+2 i d)}{d^2}-\frac{a^2 (-d+i c) \log (c \cos (e+f x)+d \sin (e+f x))}{d f (d+i c)}+\frac{a^2 \log (\cos (e+f x))}{d f}",1,"(a^2*(8*d*f*x + 2*((-I)*c + d)*ArcTan[(d*Cos[3*e + f*x] - c*Sin[3*e + f*x])/(c*Cos[3*e + f*x] + d*Sin[3*e + f*x])] + ((-2*I)*c - 2*d)*ArcTan[Tan[3*e + f*x]] + c*Log[Cos[e + f*x]^2] - I*d*Log[Cos[e + f*x]^2] - c*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2] - I*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]))/(2*(c - I*d)*d*f)","A",1
1085,1,95,45,0.7722951,"\int \frac{a+i a \tan (e+f x)}{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x]),x]","\frac{-i a \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+2 a \tan ^{-1}\left(\frac{d \cos (2 e+f x)-c \sin (2 e+f x)}{c \cos (2 e+f x)+d \sin (2 e+f x)}\right)+4 a f x}{2 c f-2 i d f}","\frac{a \log (c \cos (e+f x)+d \sin (e+f x))}{f (d+i c)}+\frac{a x}{c-i d}",1,"(4*a*f*x + 2*a*ArcTan[(d*Cos[2*e + f*x] - c*Sin[2*e + f*x])/(c*Cos[2*e + f*x] + d*Sin[2*e + f*x])] - I*a*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2])/(2*c*f - (2*I)*d*f)","B",1
1086,1,206,128,1.2868711,"\int \frac{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])),x]","\frac{-2 i c^2 f x+c^2+2 d^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-4 d^2 (\tan (e+f x)-i) \tan ^{-1}\left(\frac{c \sin (f x)+d \cos (f x)}{d \sin (f x)-c \cos (f x)}\right)+\tan (e+f x) \left(2 i d^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+(c+i d) (2 c f x-i c+2 i d f x-d)\right)+4 c d f x+2 i d^2 f x+d^2}{4 a f (c-i d) (c+i d)^2 (\tan (e+f x)-i)}","-\frac{d^2 \log (c \cos (e+f x)+d \sin (e+f x))}{a f (-d+i c) \left(c^2+d^2\right)}-\frac{c d x}{a (-d+i c) \left(c^2+d^2\right)}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x))}+\frac{x}{2 a (c+i d)}",1,"(c^2 + d^2 - (2*I)*c^2*f*x + 4*c*d*f*x + (2*I)*d^2*f*x + 2*d^2*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2] + ((c + I*d)*((-I)*c - d + 2*c*f*x + (2*I)*d*f*x) + (2*I)*d^2*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2])*Tan[e + f*x] - 4*d^2*ArcTan[(d*Cos[f*x] + c*Sin[f*x])/(-(c*Cos[f*x]) + d*Sin[f*x])]*(-I + Tan[e + f*x]))/(4*a*(c - I*d)*(c + I*d)^2*f*(-I + Tan[e + f*x]))","A",1
1087,1,372,174,1.526097,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])),x]","-\frac{\sec ^2(e+f x) \left(4 i c^3 f x \sin (2 (e+f x))+c^3 \sin (2 (e+f x))+4 i c^3+16 d^3 (\sin (2 (e+f x))-i \cos (2 (e+f x))) \tan ^{-1}\left(\frac{\left(d^2-c^2\right) \sin (f x)-2 c d \cos (f x)}{\left(c^2-d^2\right) \cos (f x)-2 c d \sin (f x)}\right)+i c^2 d \sin (2 (e+f x))-12 c^2 d f x \sin (2 (e+f x))-8 c^2 d+\cos (2 (e+f x)) \left(-8 d^3 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+(c+i d)^2 (4 c f x+i c+4 i d f x+d)\right)-8 i d^3 \sin (2 (e+f x)) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+c d^2 \sin (2 (e+f x))-12 i c d^2 f x \sin (2 (e+f x))+4 i c d^2+i d^3 \sin (2 (e+f x))+4 d^3 f x \sin (2 (e+f x))-8 d^3\right)}{16 a^2 f (c-i d) (c+i d)^3 (\tan (e+f x)-i)^2}","\frac{x \left(c^3+3 i c^2 d-3 c d^2+3 i d^3\right)}{4 a^2 (c-i d) (c+i d)^3}-\frac{d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 f (c-i d) (c+i d)^3}+\frac{-3 d+i c}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2}",1,"-1/16*(Sec[e + f*x]^2*((4*I)*c^3 - 8*c^2*d + (4*I)*c*d^2 - 8*d^3 + Cos[2*(e + f*x)]*((c + I*d)^2*(I*c + d + 4*c*f*x + (4*I)*d*f*x) - 8*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + c^3*Sin[2*(e + f*x)] + I*c^2*d*Sin[2*(e + f*x)] + c*d^2*Sin[2*(e + f*x)] + I*d^3*Sin[2*(e + f*x)] + (4*I)*c^3*f*x*Sin[2*(e + f*x)] - 12*c^2*d*f*x*Sin[2*(e + f*x)] - (12*I)*c*d^2*f*x*Sin[2*(e + f*x)] + 4*d^3*f*x*Sin[2*(e + f*x)] - (8*I)*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[2*(e + f*x)] + 16*d^3*ArcTan[(-2*c*d*Cos[f*x] + (-c^2 + d^2)*Sin[f*x])/((c^2 - d^2)*Cos[f*x] - 2*c*d*Sin[f*x])]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])))/(a^2*(c - I*d)*(c + I*d)^3*f*(-I + Tan[e + f*x])^2)","B",1
1088,1,435,234,2.0445583,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])),x]","\frac{\sec ^3(e+f x) \left(-9 i c^4 \sin (e+f x)-12 c^4 f x \sin (3 (e+f x))+2 i c^4 \sin (3 (e+f x))+36 c^3 d \sin (e+f x)-4 c^3 d \sin (3 (e+f x))-48 i c^3 d f x \sin (3 (e+f x))+42 i c^2 d^2 \sin (e+f x)+72 c^2 d^2 f x \sin (3 (e+f x))-3 \left(9 c^4+28 i c^3 d-18 c^2 d^2+28 i c d^3-27 d^4\right) \cos (e+f x)+2 \cos (3 (e+f x)) \left(c^4 (-1+6 i f x)-2 c^3 d (12 f x+i)-36 i c^2 d^2 f x+24 d^4 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+2 c d^3 (12 f x-i)+d^4 (1-42 i f x)\right)+48 i d^4 \sin (3 (e+f x)) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+36 c d^3 \sin (e+f x)-4 c d^3 \sin (3 (e+f x))+48 i c d^3 f x \sin (3 (e+f x))+51 i d^4 \sin (e+f x)-2 i d^4 \sin (3 (e+f x))+84 d^4 f x \sin (3 (e+f x))\right)}{96 a^3 f (c-i d) (c+i d)^4 (\tan (e+f x)-i)^3}","\frac{c^2+4 i c d-7 d^2}{8 f (-d+i c)^3 \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{x \left(c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right)}{8 a^3 (c-i d) (c+i d)^4}+\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^4 (d+i c)}+\frac{-3 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3}",1,"(Sec[e + f*x]^3*(-3*(9*c^4 + (28*I)*c^3*d - 18*c^2*d^2 + (28*I)*c*d^3 - 27*d^4)*Cos[e + f*x] + 2*Cos[3*(e + f*x)]*((-36*I)*c^2*d^2*f*x + c^4*(-1 + (6*I)*f*x) + d^4*(1 - (42*I)*f*x) + 2*c*d^3*(-I + 12*f*x) - 2*c^3*d*(I + 12*f*x) + 24*d^4*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) - (9*I)*c^4*Sin[e + f*x] + 36*c^3*d*Sin[e + f*x] + (42*I)*c^2*d^2*Sin[e + f*x] + 36*c*d^3*Sin[e + f*x] + (51*I)*d^4*Sin[e + f*x] + (2*I)*c^4*Sin[3*(e + f*x)] - 4*c^3*d*Sin[3*(e + f*x)] - 4*c*d^3*Sin[3*(e + f*x)] - (2*I)*d^4*Sin[3*(e + f*x)] - 12*c^4*f*x*Sin[3*(e + f*x)] - (48*I)*c^3*d*f*x*Sin[3*(e + f*x)] + 72*c^2*d^2*f*x*Sin[3*(e + f*x)] + (48*I)*c*d^3*f*x*Sin[3*(e + f*x)] + 84*d^4*f*x*Sin[3*(e + f*x)] + (48*I)*d^4*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[3*(e + f*x)]))/(96*a^3*(c - I*d)*(c + I*d)^4*f*(-I + Tan[e + f*x])^3)","A",1
1089,1,1936,142,9.0597368,"\int \frac{(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^2,x]","\frac{x \cos ^3(e+f x) \left(\frac{3 i d \cos ^4(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}-\frac{i c^2 \cos ^4(e)}{(c-i d)^2 d (c \cos (e)+d \sin (e))}+\frac{5 c \cos ^4(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}+\frac{c^3 \cos ^4(e)}{(c-i d)^2 d^2 (c \cos (e)+d \sin (e))}+\frac{12 d \sin (e) \cos ^3(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}-\frac{4 c^2 \sin (e) \cos ^3(e)}{(c-i d)^2 d (c \cos (e)+d \sin (e))}-\frac{20 i c \sin (e) \cos ^3(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}-\frac{4 i c^3 \sin (e) \cos ^3(e)}{(c-i d)^2 d^2 (c \cos (e)+d \sin (e))}-\frac{\cos ^3(e)}{2 d^2}+\frac{2 i \sin (e) \cos ^2(e)}{d^2}-\frac{18 i d \sin ^2(e) \cos ^2(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}+\frac{6 i c^2 \sin ^2(e) \cos ^2(e)}{(c-i d)^2 d (c \cos (e)+d \sin (e))}-\frac{30 c \sin ^2(e) \cos ^2(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}-\frac{6 c^3 \sin ^2(e) \cos ^2(e)}{(c-i d)^2 d^2 (c \cos (e)+d \sin (e))}+\frac{3 \sin ^2(e) \cos (e)}{d^2}-\frac{12 d \sin ^3(e) \cos (e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}+\frac{4 c^2 \sin ^3(e) \cos (e)}{(c-i d)^2 d (c \cos (e)+d \sin (e))}+\frac{20 i c \sin ^3(e) \cos (e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}+\frac{4 i c^3 \sin ^3(e) \cos (e)}{(c-i d)^2 d^2 (c \cos (e)+d \sin (e))}+\frac{\cos (e)}{2 d^2}-\frac{2 i \sin ^3(e)}{d^2}-\frac{i \sin (e)}{d^2}+\frac{(\cos (2 e) c+i \sin (2 e) c-5 c-3 i d-3 i d \cos (2 e)+3 d \sin (2 e)) (\cos (3 e)-i \sin (3 e))}{(c-i d)^2 (\cos (2 e) c+i \sin (2 e) c+c+i d-i d \cos (2 e)+d \sin (2 e))}+\frac{(-\cos (2 e)-i \sin (2 e)+1) \left(\frac{\cos (3 e)}{d^2}-\frac{i \sin (3 e)}{d^2}\right)}{\cos (2 e)+i \sin (2 e)+1}+\frac{\left(\cos (2 e) c^3+i \sin (2 e) c^3-c^3\right) \left(\frac{\cos (3 e)}{d^2}-\frac{i \sin (3 e)}{d^2}\right)}{(c-i d)^2 (\cos (2 e) c+i \sin (2 e) c+c+i d-i d \cos (2 e)+d \sin (2 e))}+\frac{\left(3 \cos (2 e) c^2+3 i \sin (2 e) c^2-c^2\right) \left(-\frac{i \cos (3 e)}{d}-\frac{\sin (3 e)}{d}\right)}{(c-i d)^2 (\cos (2 e) c+i \sin (2 e) c+c+i d-i d \cos (2 e)+d \sin (2 e))}-\frac{\sin ^3(e) \tan (e)}{2 d^2}-\frac{\sin (e) \tan (e)}{2 d^2}+\frac{3 i d \sin ^4(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}-\frac{i c^2 \sin ^4(e)}{(c-i d)^2 d (c \cos (e)+d \sin (e))}+\frac{5 c \sin ^4(e)}{(c-i d)^2 (c \cos (e)+d \sin (e))}+\frac{c^3 \sin ^4(e)}{(c-i d)^2 d^2 (c \cos (e)+d \sin (e))}\right) (i \tan (e+f x) a+a)^3}{(\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\frac{\cos (3 e)}{d}-\frac{i \sin (3 e)}{d}\right) \left(i \sin (f x) c^2-2 d \sin (f x) c-i d^2 \sin (f x)\right) (i \tan (e+f x) a+a)^3}{(c-i d) f (c \cos (e)+d \sin (e)) (\cos (f x)+i \sin (f x))^3 (c \cos (e+f x)+d \sin (e+f x))}+\frac{i \cos (3 e) \cos ^3(e+f x) \log \left(\cos ^2(e+f x)\right) (i \tan (e+f x) a+a)^3}{2 d^2 f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\cos \left(\frac{3 e}{2}\right) c^2-i \sin \left(\frac{3 e}{2}\right) c^2-2 i d \cos \left(\frac{3 e}{2}\right) c-2 d \sin \left(\frac{3 e}{2}\right) c+3 d^2 \cos \left(\frac{3 e}{2}\right)-3 i d^2 \sin \left(\frac{3 e}{2}\right)\right) \left(\frac{\tan ^{-1}\left(\frac{-\sin (4 e+f x) c^2+2 d \cos (4 e+f x) c+d^2 \sin (4 e+f x)}{\cos (4 e+f x) c^2+2 d \sin (4 e+f x) c-d^2 \cos (4 e+f x)}\right) \cos \left(\frac{3 e}{2}\right)}{d^2}-\frac{i \tan ^{-1}\left(\frac{-\sin (4 e+f x) c^2+2 d \cos (4 e+f x) c+d^2 \sin (4 e+f x)}{\cos (4 e+f x) c^2+2 d \sin (4 e+f x) c-d^2 \cos (4 e+f x)}\right) \sin \left(\frac{3 e}{2}\right)}{d^2}\right) (i \tan (e+f x) a+a)^3}{(c-i d)^2 f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\cos \left(\frac{3 e}{2}\right) c^2-i \sin \left(\frac{3 e}{2}\right) c^2-2 i d \cos \left(\frac{3 e}{2}\right) c-2 d \sin \left(\frac{3 e}{2}\right) c+3 d^2 \cos \left(\frac{3 e}{2}\right)-3 i d^2 \sin \left(\frac{3 e}{2}\right)\right) \left(-\frac{i \cos \left(\frac{3 e}{2}\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{2 d^2}-\frac{\sin \left(\frac{3 e}{2}\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{2 d^2}\right) (i \tan (e+f x) a+a)^3}{(c-i d)^2 f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \log \left(\cos ^2(e+f x)\right) \sin (3 e) (i \tan (e+f x) a+a)^3}{2 d^2 f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) (4 f x \cos (3 e)-4 i f x \sin (3 e)) (i \tan (e+f x) a+a)^3}{(c-i d)^2 f (\cos (f x)+i \sin (f x))^3}","-\frac{a^3 (-d+i c) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{d^2 f (c-i d)^2}+\frac{(c+i d) \left(a^3+i a^3 \tan (e+f x)\right)}{d f (c-i d) (c+d \tan (e+f x))}+\frac{4 a^3 x}{(c-i d)^2}+\frac{i a^3 \log (\cos (e+f x))}{d^2 f}",1,"((I/2)*Cos[3*e]*Cos[e + f*x]^3*Log[Cos[e + f*x]^2]*(a + I*a*Tan[e + f*x])^3)/(d^2*f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(c^2*Cos[(3*e)/2] - (2*I)*c*d*Cos[(3*e)/2] + 3*d^2*Cos[(3*e)/2] - I*c^2*Sin[(3*e)/2] - 2*c*d*Sin[(3*e)/2] - (3*I)*d^2*Sin[(3*e)/2])*((ArcTan[(2*c*d*Cos[4*e + f*x] - c^2*Sin[4*e + f*x] + d^2*Sin[4*e + f*x])/(c^2*Cos[4*e + f*x] - d^2*Cos[4*e + f*x] + 2*c*d*Sin[4*e + f*x])]*Cos[(3*e)/2])/d^2 - (I*ArcTan[(2*c*d*Cos[4*e + f*x] - c^2*Sin[4*e + f*x] + d^2*Sin[4*e + f*x])/(c^2*Cos[4*e + f*x] - d^2*Cos[4*e + f*x] + 2*c*d*Sin[4*e + f*x])]*Sin[(3*e)/2])/d^2)*(a + I*a*Tan[e + f*x])^3)/((c - I*d)^2*f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(c^2*Cos[(3*e)/2] - (2*I)*c*d*Cos[(3*e)/2] + 3*d^2*Cos[(3*e)/2] - I*c^2*Sin[(3*e)/2] - 2*c*d*Sin[(3*e)/2] - (3*I)*d^2*Sin[(3*e)/2])*(((-1/2*I)*Cos[(3*e)/2]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2])/d^2 - (Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[(3*e)/2])/(2*d^2))*(a + I*a*Tan[e + f*x])^3)/((c - I*d)^2*f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*Log[Cos[e + f*x]^2]*Sin[3*e]*(a + I*a*Tan[e + f*x])^3)/(2*d^2*f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(4*f*x*Cos[3*e] - (4*I)*f*x*Sin[3*e])*(a + I*a*Tan[e + f*x])^3)/((c - I*d)^2*f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(Cos[3*e]/d - (I*Sin[3*e])/d)*(I*c^2*Sin[f*x] - 2*c*d*Sin[f*x] - I*d^2*Sin[f*x])*(a + I*a*Tan[e + f*x])^3)/((c - I*d)*f*(c*Cos[e] + d*Sin[e])*(Cos[f*x] + I*Sin[f*x])^3*(c*Cos[e + f*x] + d*Sin[e + f*x])) + (x*Cos[e + f*x]^3*(Cos[e]/(2*d^2) - Cos[e]^3/(2*d^2) - (I*Sin[e])/d^2 + ((2*I)*Cos[e]^2*Sin[e])/d^2 + (3*Cos[e]*Sin[e]^2)/d^2 - ((2*I)*Sin[e]^3)/d^2 + (5*c*Cos[e]^4)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) + (c^3*Cos[e]^4)/((c - I*d)^2*d^2*(c*Cos[e] + d*Sin[e])) - (I*c^2*Cos[e]^4)/((c - I*d)^2*d*(c*Cos[e] + d*Sin[e])) + ((3*I)*d*Cos[e]^4)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) - ((20*I)*c*Cos[e]^3*Sin[e])/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) - ((4*I)*c^3*Cos[e]^3*Sin[e])/((c - I*d)^2*d^2*(c*Cos[e] + d*Sin[e])) - (4*c^2*Cos[e]^3*Sin[e])/((c - I*d)^2*d*(c*Cos[e] + d*Sin[e])) + (12*d*Cos[e]^3*Sin[e])/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) - (30*c*Cos[e]^2*Sin[e]^2)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) - (6*c^3*Cos[e]^2*Sin[e]^2)/((c - I*d)^2*d^2*(c*Cos[e] + d*Sin[e])) + ((6*I)*c^2*Cos[e]^2*Sin[e]^2)/((c - I*d)^2*d*(c*Cos[e] + d*Sin[e])) - ((18*I)*d*Cos[e]^2*Sin[e]^2)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) + ((20*I)*c*Cos[e]*Sin[e]^3)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) + ((4*I)*c^3*Cos[e]*Sin[e]^3)/((c - I*d)^2*d^2*(c*Cos[e] + d*Sin[e])) + (4*c^2*Cos[e]*Sin[e]^3)/((c - I*d)^2*d*(c*Cos[e] + d*Sin[e])) - (12*d*Cos[e]*Sin[e]^3)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) + (5*c*Sin[e]^4)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) + (c^3*Sin[e]^4)/((c - I*d)^2*d^2*(c*Cos[e] + d*Sin[e])) - (I*c^2*Sin[e]^4)/((c - I*d)^2*d*(c*Cos[e] + d*Sin[e])) + ((3*I)*d*Sin[e]^4)/((c - I*d)^2*(c*Cos[e] + d*Sin[e])) + ((-5*c - (3*I)*d + c*Cos[2*e] - (3*I)*d*Cos[2*e] + I*c*Sin[2*e] + 3*d*Sin[2*e])*(Cos[3*e] - I*Sin[3*e]))/((c - I*d)^2*(c + I*d + c*Cos[2*e] - I*d*Cos[2*e] + I*c*Sin[2*e] + d*Sin[2*e])) + ((1 - Cos[2*e] - I*Sin[2*e])*(Cos[3*e]/d^2 - (I*Sin[3*e])/d^2))/(1 + Cos[2*e] + I*Sin[2*e]) + ((-c^3 + c^3*Cos[2*e] + I*c^3*Sin[2*e])*(Cos[3*e]/d^2 - (I*Sin[3*e])/d^2))/((c - I*d)^2*(c + I*d + c*Cos[2*e] - I*d*Cos[2*e] + I*c*Sin[2*e] + d*Sin[2*e])) + ((-c^2 + 3*c^2*Cos[2*e] + (3*I)*c^2*Sin[2*e])*(((-I)*Cos[3*e])/d - Sin[3*e]/d))/((c - I*d)^2*(c + I*d + c*Cos[2*e] - I*d*Cos[2*e] + I*c*Sin[2*e] + d*Sin[2*e])) - (Sin[e]*Tan[e])/(2*d^2) - (Sin[e]^3*Tan[e])/(2*d^2))*(a + I*a*Tan[e + f*x])^3)/(Cos[f*x] + I*Sin[f*x])^3","B",1
1090,1,253,93,2.9513546,"\int \frac{(a+i a \tan (e+f x))^2}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^2,x]","\frac{a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left(\frac{2 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{\left(d^2-c^2\right) \sin (3 e+f x)+2 c d \cos (3 e+f x)}{\left(c^2-d^2\right) \cos (3 e+f x)+2 c d \sin (3 e+f x)}\right)}{f}-\frac{(c-i d) (c+i d) (\cos (2 e)-i \sin (2 e)) \sin (f x)}{f (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{(-\sin (2 e)-i \cos (2 e)) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{f}+4 x (\cos (2 e)-i \sin (2 e))\right)}{(c-i d)^2 (\cos (f x)+i \sin (f x))^2}","\frac{a^2 (-d+i c)}{d f (d+i c) (c+d \tan (e+f x))}-\frac{2 i a^2 \log (c \cos (e+f x)+d \sin (e+f x))}{f (c-i d)^2}+\frac{2 a^2 x}{(c-i d)^2}",1,"(a^2*(Cos[e + f*x] + I*Sin[e + f*x])^2*((Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*((-I)*Cos[2*e] - Sin[2*e]))/f + 4*x*(Cos[2*e] - I*Sin[2*e]) + (2*ArcTan[(2*c*d*Cos[3*e + f*x] + (-c^2 + d^2)*Sin[3*e + f*x])/((c^2 - d^2)*Cos[3*e + f*x] + 2*c*d*Sin[3*e + f*x])]*(Cos[2*e] - I*Sin[2*e]))/f - ((c - I*d)*(c + I*d)*(Cos[2*e] - I*Sin[2*e])*Sin[f*x])/(f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/((c - I*d)^2*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1091,1,302,75,2.9332755,"\int \frac{a+i a \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x])^2,x]","\frac{(\cos (e)-i \sin (e)) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left(4 \tan ^{-1}\left(\frac{\left(d^2-c^2\right) \sin (2 e+f x)+2 c d \cos (2 e+f x)}{\left(c^2-d^2\right) \cos (2 e+f x)+2 c d \sin (2 e+f x)}\right)+\frac{\left(c^2+d^2\right) \cos (f x) \left(4 f x-i \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)\right)+\left(c^2-d^2\right) \cos (2 e+f x) \left(4 f x-i \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)\right)-2 d \left(c \sin (2 e+f x) \left(-4 f x+i \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)\right)+2 (d+i c) \sin (f x)\right)}{(c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right)}{4 f (c-i d)^2}","-\frac{a}{f (d+i c) (c+d \tan (e+f x))}-\frac{i a \log (c \cos (e+f x)+d \sin (e+f x))}{f (c-i d)^2}+\frac{a x}{(c-i d)^2}",1,"(Cos[e + f*x]*(Cos[e] - I*Sin[e])*(Cos[f*x] - I*Sin[f*x])*(4*ArcTan[(2*c*d*Cos[2*e + f*x] + (-c^2 + d^2)*Sin[2*e + f*x])/((c^2 - d^2)*Cos[2*e + f*x] + 2*c*d*Sin[2*e + f*x])] + ((c^2 + d^2)*Cos[f*x]*(4*f*x - I*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + (c^2 - d^2)*Cos[2*e + f*x]*(4*f*x - I*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) - 2*d*(2*(I*c + d)*Sin[f*x] + c*(-4*f*x + I*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2])*Sin[2*e + f*x]))/((c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])))*(a + I*a*Tan[e + f*x]))/(4*(c - I*d)^2*f)","B",1
1092,1,385,202,3.0208472,"\int \frac{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^2),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(\frac{2 x \left(c^3+3 i c^2 d+3 c d^2-3 i d^3\right) (\cos (e)+i \sin (e))}{(c-i d)^2}+\frac{4 i d^3 (c+i d) (\cos (e)+i \sin (e)) \sin (f x)}{f (c-i d) (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{2 d^2 (d+3 i c) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right)^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{f (c-i d)^2}-\frac{4 d^2 (3 c-i d) \left(\cos \left(\frac{e}{2}\right)+i \sin \left(\frac{e}{2}\right)\right)^2 \tan ^{-1}\left(\frac{c \sin (f x)+d \cos (f x)}{d \sin (f x)-c \cos (f x)}\right)}{f (c-i d)^2}-\frac{4 d^2 x (3 c-i d) (\cos (e)+i \sin (e))}{(c-i d)^2}+\frac{(c+i d) (\sin (e)+i \cos (e)) \cos (2 f x)}{f}+\frac{(c+i d) (\cos (e)-i \sin (e)) \sin (2 f x)}{f}\right)}{4 (c+i d)^3 (a+i a \tan (e+f x))}","\frac{x \left(c^3+3 i c^2 d+3 c d^2-3 i d^3\right)}{2 a (c-i d)^2 (c+i d)^3}+\frac{d^2 (3 c-i d) \log (c \cos (e+f x)+d \sin (e+f x))}{a f (-d+i c)^3 (c-i d)^2}+\frac{d (c-3 i d)}{2 a f (c-i d) (c+i d)^2 (c+d \tan (e+f x))}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-4*(3*c - I*d)*d^2*ArcTan[(d*Cos[f*x] + c*Sin[f*x])/(-(c*Cos[f*x]) + d*Sin[f*x])]*(Cos[e/2] + I*Sin[e/2])^2)/((c - I*d)^2*f) + (2*d^2*((3*I)*c + d)*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(Cos[e/2] + I*Sin[e/2])^2)/((c - I*d)^2*f) - (4*(3*c - I*d)*d^2*x*(Cos[e] + I*Sin[e]))/(c - I*d)^2 + (2*(c^3 + (3*I)*c^2*d + 3*c*d^2 - (3*I)*d^3)*x*(Cos[e] + I*Sin[e]))/(c - I*d)^2 + ((c + I*d)*Cos[2*f*x]*(I*Cos[e] + Sin[e]))/f + ((c + I*d)*(Cos[e] - I*Sin[e])*Sin[2*f*x])/f + ((4*I)*(c + I*d)*d^3*(Cos[e] + I*Sin[e])*Sin[f*x])/((c - I*d)*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(4*(c + I*d)^3*(a + I*a*Tan[e + f*x]))","A",1
1093,1,476,271,4.1181735,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2),x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(-\frac{32 i d^3 (2 c-i d) (\cos (e)+i \sin (e))^2 \tan ^{-1}\left(\frac{\left(d^2-c^2\right) \sin (f x)-2 c d \cos (f x)}{\left(c^2-d^2\right) \cos (f x)-2 c d \sin (f x)}\right)}{f (c-i d)^2}+\frac{4 x \left(c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right) (\cos (2 e)+i \sin (2 e))}{(c-i d)^2}-\frac{16 d^4 (c+i d) (\cos (2 e)+i \sin (2 e)) \sin (f x)}{f (c-i d) (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}-\frac{16 d^3 (2 c-i d) (\cos (e)+i \sin (e))^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{f (c-i d)^2}+\frac{32 d^3 x (2 c-i d) (\sin (2 e)-i \cos (2 e))}{(c-i d)^2}+\frac{(c+i d)^2 (\sin (2 e)+i \cos (2 e)) \cos (4 f x)}{f}+\frac{(c+i d)^2 (\cos (2 e)-i \sin (2 e)) \sin (4 f x)}{f}+\frac{4 (c+i d) (c+3 i d) \sin (2 f x)}{f}+\frac{4 i (c+i d) (c+3 i d) \cos (2 f x)}{f}\right)}{16 (c+i d)^4 (a+i a \tan (e+f x))^2}","\frac{d \left(c^2+4 i c d+9 d^2\right)}{4 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))}+\frac{x \left(c^4+4 i c^3 d-6 c^2 d^2+12 i c d^3+9 d^4\right)}{4 a^2 (c-i d)^2 (c+i d)^4}-\frac{2 d^3 (2 c-i d) \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 f (c-i d)^2 (c+i d)^4}+\frac{-4 d+i c}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(((4*I)*(c + I*d)*(c + (3*I)*d)*Cos[2*f*x])/f - ((32*I)*(2*c - I*d)*d^3*ArcTan[(-2*c*d*Cos[f*x] + (-c^2 + d^2)*Sin[f*x])/((c^2 - d^2)*Cos[f*x] - 2*c*d*Sin[f*x])]*(Cos[e] + I*Sin[e])^2)/((c - I*d)^2*f) - (16*(2*c - I*d)*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(Cos[e] + I*Sin[e])^2)/((c - I*d)^2*f) + (4*(c^4 + (4*I)*c^3*d - 6*c^2*d^2 + (12*I)*c*d^3 + 9*d^4)*x*(Cos[2*e] + I*Sin[2*e]))/(c - I*d)^2 + (32*(2*c - I*d)*d^3*x*((-I)*Cos[2*e] + Sin[2*e]))/(c - I*d)^2 + ((c + I*d)^2*Cos[4*f*x]*(I*Cos[2*e] + Sin[2*e]))/f + (4*(c + I*d)*(c + (3*I)*d)*Sin[2*f*x])/f + ((c + I*d)^2*(Cos[2*e] - I*Sin[2*e])*Sin[4*f*x])/f - (16*(c + I*d)*d^4*(Cos[2*e] + I*Sin[2*e])*Sin[f*x])/((c - I*d)*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(16*(c + I*d)^4*(a + I*a*Tan[e + f*x])^2)","A",1
1094,1,633,357,5.6737679,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(\frac{6 i (c+i d) \left(3 c^2+14 i c d-23 d^2\right) (\cos (e)+i \sin (e)) \cos (2 f x)}{f}+\frac{6 (c+i d) \left(3 c^2+14 i c d-23 d^2\right) (\cos (e)+i \sin (e)) \sin (2 f x)}{f}+\frac{96 d^4 (5 c-3 i d) \left(\cos \left(\frac{3 e}{2}\right)+i \sin \left(\frac{3 e}{2}\right)\right)^2 \tan ^{-1}\left(\frac{\left(d^3-3 c^2 d\right) \cos (f x)-c \left(c^2-3 d^2\right) \sin (f x)}{d \left(d^2-3 c^2\right) \sin (f x)+c \left(c^2-3 d^2\right) \cos (f x)}\right)}{f (c-i d)^2}+\frac{12 x \left(c^5+5 i c^4 d-10 c^3 d^2-10 i c^2 d^3-35 c d^4+25 i d^5\right) (\cos (3 e)+i \sin (3 e))}{(c-i d)^2}+\frac{96 d^5 (d-i c) (\cos (3 e)+i \sin (3 e)) \sin (f x)}{f (c-i d) (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}-\frac{48 i d^4 (5 c-3 i d) \left(\cos \left(\frac{3 e}{2}\right)+i \sin \left(\frac{3 e}{2}\right)\right)^2 \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{f (c-i d)^2}+\frac{96 d^4 x (5 c-3 i d) (\cos (3 e)+i \sin (3 e))}{(c-i d)^2}+\frac{3 (c+i d)^2 (3 c+7 i d) (\sin (e)+i \cos (e)) \cos (4 f x)}{f}+\frac{2 (c+i d)^3 (\sin (3 e)+i \cos (3 e)) \cos (6 f x)}{f}+\frac{3 (c+i d)^2 (3 c+7 i d) (\cos (e)-i \sin (e)) \sin (4 f x)}{f}+\frac{2 (c+i d)^3 (\cos (3 e)-i \sin (3 e)) \sin (6 f x)}{f}\right)}{96 (c+i d)^5 (a+i a \tan (e+f x))^3}","\frac{c^2+5 i c d-12 d^2}{8 f (-d+i c)^3 \left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))}+\frac{d \left(c^3+5 i c^2 d-11 c d^2+25 i d^3\right)}{8 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))}+\frac{x \left(c^5+5 i c^4 d-10 c^3 d^2-10 i c^2 d^3-35 c d^4+25 i d^5\right)}{8 a^3 (c-i d)^2 (c+i d)^5}+\frac{d^4 (5 c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (-d+i c)^5 (c-i d)^2}+\frac{-11 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(((6*I)*(c + I*d)*(3*c^2 + (14*I)*c*d - 23*d^2)*Cos[2*f*x]*(Cos[e] + I*Sin[e]))/f + (3*(c + I*d)^2*(3*c + (7*I)*d)*Cos[4*f*x]*(I*Cos[e] + Sin[e]))/f + (96*(5*c - (3*I)*d)*d^4*ArcTan[((-3*c^2*d + d^3)*Cos[f*x] - c*(c^2 - 3*d^2)*Sin[f*x])/(c*(c^2 - 3*d^2)*Cos[f*x] + d*(-3*c^2 + d^2)*Sin[f*x])]*(Cos[(3*e)/2] + I*Sin[(3*e)/2])^2)/((c - I*d)^2*f) - ((48*I)*(5*c - (3*I)*d)*d^4*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(Cos[(3*e)/2] + I*Sin[(3*e)/2])^2)/((c - I*d)^2*f) + (96*(5*c - (3*I)*d)*d^4*x*(Cos[3*e] + I*Sin[3*e]))/(c - I*d)^2 + (12*(c^5 + (5*I)*c^4*d - 10*c^3*d^2 - (10*I)*c^2*d^3 - 35*c*d^4 + (25*I)*d^5)*x*(Cos[3*e] + I*Sin[3*e]))/(c - I*d)^2 + (2*(c + I*d)^3*Cos[6*f*x]*(I*Cos[3*e] + Sin[3*e]))/f + (6*(c + I*d)*(3*c^2 + (14*I)*c*d - 23*d^2)*(Cos[e] + I*Sin[e])*Sin[2*f*x])/f + (3*(c + I*d)^2*(3*c + (7*I)*d)*(Cos[e] - I*Sin[e])*Sin[4*f*x])/f + (2*(c + I*d)^3*(Cos[3*e] - I*Sin[3*e])*Sin[6*f*x])/f + (96*d^5*((-I)*c + d)*(Cos[3*e] + I*Sin[3*e])*Sin[f*x])/((c - I*d)*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(96*(c + I*d)^5*(a + I*a*Tan[e + f*x])^3)","A",1
1095,1,595,134,3.9813519,"\int \frac{(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^3,x]","\frac{a^3 \left(-i c^3 \cos (3 e+2 f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-3 c^3 \sin (e+2 f x)+2 c^3 f x \cos (3 e+2 f x)+3 c^3 \sin (e)+\left(c^2+d^2\right) \cos (e+2 f x) \left(-i c \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+2 c f x+3 d\right)+\left(c^2+d^2\right) \cos (e) \left(-2 i c \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+4 c f x-i c-3 d\right)+4 c^2 d f x \sin (e)+2 c^2 d f x \sin (e+2 f x)+6 c^2 d f x \sin (3 e+2 f x)-2 i c^2 d \sin (e) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-i c^2 d \sin (e+2 f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-3 i c^2 d \sin (3 e+2 f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-i c^2 d \sin (e)-2 i d^3 \sin (e) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-i d^3 \sin (e+2 f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+i d^3 \sin (3 e+2 f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-3 c d^2 \sin (e+2 f x)-6 c d^2 f x \cos (3 e+2 f x)+3 i c d^2 \cos (3 e+2 f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+3 c d^2 \sin (e)+4 d^3 f x \sin (e)+2 d^3 f x \sin (e+2 f x)-2 d^3 f x \sin (3 e+2 f x)-i d^3 \sin (e)\right)}{2 f (c-i d)^3 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))^2}","\frac{2 a^3 (c+i d)}{d f (c-i d)^2 (c+d \tan (e+f x))}-\frac{4 a^3 \log (c \cos (e+f x)+d \sin (e+f x))}{f (d+i c)^3}+\frac{4 a^3 x}{(c-i d)^3}-\frac{a (a+i a \tan (e+f x))^2}{2 f (d+i c) (c+d \tan (e+f x))^2}",1,"(a^3*(2*c^3*f*x*Cos[3*e + 2*f*x] - 6*c*d^2*f*x*Cos[3*e + 2*f*x] - I*c^3*Cos[3*e + 2*f*x]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2] + (3*I)*c*d^2*Cos[3*e + 2*f*x]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2] + (c^2 + d^2)*Cos[e + 2*f*x]*(3*d + 2*c*f*x - I*c*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + (c^2 + d^2)*Cos[e]*((-I)*c - 3*d + 4*c*f*x - (2*I)*c*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + 3*c^3*Sin[e] - I*c^2*d*Sin[e] + 3*c*d^2*Sin[e] - I*d^3*Sin[e] + 4*c^2*d*f*x*Sin[e] + 4*d^3*f*x*Sin[e] - (2*I)*c^2*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e] - (2*I)*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e] - 3*c^3*Sin[e + 2*f*x] - 3*c*d^2*Sin[e + 2*f*x] + 2*c^2*d*f*x*Sin[e + 2*f*x] + 2*d^3*f*x*Sin[e + 2*f*x] - I*c^2*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e + 2*f*x] - I*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e + 2*f*x] + 6*c^2*d*f*x*Sin[3*e + 2*f*x] - 2*d^3*f*x*Sin[3*e + 2*f*x] - (3*I)*c^2*d*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[3*e + 2*f*x] + I*d^3*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[3*e + 2*f*x]))/(2*(c - I*d)^3*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])^2)","B",1
1096,1,317,125,6.2257177,"\int \frac{(a+i a \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^3,x]","\frac{a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left(-\frac{2 (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left(\frac{\left(d^3-3 c^2 d\right) \cos (3 e+f x)+c \left(c^2-3 d^2\right) \sin (3 e+f x)}{c \left(c^2-3 d^2\right) \cos (3 e+f x)-d \left(d^2-3 c^2\right) \sin (3 e+f x)}\right)}{f}-\frac{(c-i d) (c+2 i d) (\cos (2 e)-i \sin (2 e)) \sin (f x)}{f (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{d (c-i d) (\cos (2 e)-i \sin (2 e))}{2 f (c \cos (e+f x)+d \sin (e+f x))^2}+\frac{(-\sin (2 e)-i \cos (2 e)) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{f}+4 x (\cos (2 e)-i \sin (2 e))\right)}{(c-i d)^3 (\cos (f x)+i \sin (f x))^2}","\frac{2 i a^2}{f (c-i d)^2 (c+d \tan (e+f x))}+\frac{a^2 (-d+i c)}{2 d f (d+i c) (c+d \tan (e+f x))^2}-\frac{2 a^2 \log (c \cos (e+f x)+d \sin (e+f x))}{f (d+i c)^3}+\frac{2 a^2 x}{(c-i d)^3}",1,"(a^2*(Cos[e + f*x] + I*Sin[e + f*x])^2*((Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*((-I)*Cos[2*e] - Sin[2*e]))/f + 4*x*(Cos[2*e] - I*Sin[2*e]) - (2*ArcTan[((-3*c^2*d + d^3)*Cos[3*e + f*x] + c*(c^2 - 3*d^2)*Sin[3*e + f*x])/(c*(c^2 - 3*d^2)*Cos[3*e + f*x] - d*(-3*c^2 + d^2)*Sin[3*e + f*x])]*(Cos[2*e] - I*Sin[2*e]))/f + ((c - I*d)*d*(Cos[2*e] - I*Sin[2*e]))/(2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) - ((c - I*d)*(c + (2*I)*d)*(Cos[2*e] - I*Sin[2*e])*Sin[f*x])/(f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/((c - I*d)^3*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1097,1,315,104,4.2177892,"\int \frac{a+i a \tan (e+f x)}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x])^3,x]","\frac{\cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left(-\frac{(\cos (e)-i \sin (e)) \tan ^{-1}\left(\frac{\left(d^3-3 c^2 d\right) \cos (2 e+f x)+c \left(c^2-3 d^2\right) \sin (2 e+f x)}{c \left(c^2-3 d^2\right) \cos (2 e+f x)-d \left(d^2-3 c^2\right) \sin (2 e+f x)}\right)}{f}+\frac{d^2 (c-i d) (\sin (e)+i \cos (e))}{2 f (c+i d) (c \cos (e+f x)+d \sin (e+f x))^2}+\frac{d (c-i d) (d-2 i c) (\cos (e)-i \sin (e)) \sin (f x)}{f (c+i d) (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}-\frac{i (\cos (e)-i \sin (e)) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)}{2 f}+2 x (\cos (e)-i \sin (e))\right)}{(c-i d)^3}","\frac{i a}{f (c-i d)^2 (c+d \tan (e+f x))}-\frac{a}{2 f (d+i c) (c+d \tan (e+f x))^2}-\frac{a \log (c \cos (e+f x)+d \sin (e+f x))}{f (d+i c)^3}+\frac{a x}{(c-i d)^3}",1,"(Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(2*x*(Cos[e] - I*Sin[e]) - (ArcTan[((-3*c^2*d + d^3)*Cos[2*e + f*x] + c*(c^2 - 3*d^2)*Sin[2*e + f*x])/(c*(c^2 - 3*d^2)*Cos[2*e + f*x] - d*(-3*c^2 + d^2)*Sin[2*e + f*x])]*(Cos[e] - I*Sin[e]))/f - ((I/2)*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(Cos[e] - I*Sin[e]))/f + ((c - I*d)*d^2*(I*Cos[e] + Sin[e]))/(2*(c + I*d)*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + ((c - I*d)*d*((-2*I)*c + d)*(Cos[e] - I*Sin[e])*Sin[f*x])/((c + I*d)*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])))*(a + I*a*Tan[e + f*x]))/(c - I*d)^3","B",1
1098,1,1231,273,7.8246675,"\int \frac{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^3),x]","\frac{\sec (e+f x) \left(-\cos \left(\frac{e}{2}\right) d^4-i \sin \left(\frac{e}{2}\right) d^4-2 i c \cos \left(\frac{e}{2}\right) d^3+2 c \sin \left(\frac{e}{2}\right) d^3+3 c^2 \cos \left(\frac{e}{2}\right) d^2+3 i c^2 \sin \left(\frac{e}{2}\right) d^2\right) \left(-2 \tan ^{-1}\left(\frac{-d \cos (f x)-c \sin (f x)}{c \cos (f x)-d \sin (f x)}\right) \cos \left(\frac{e}{2}\right)-2 i \tan ^{-1}\left(\frac{-d \cos (f x)-c \sin (f x)}{c \cos (f x)-d \sin (f x)}\right) \sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))}{(c-i d)^3 (c+i d)^4 f (i \tan (e+f x) a+a)}+\frac{\sec (e+f x) \left(-\cos \left(\frac{e}{2}\right) d^4-i \sin \left(\frac{e}{2}\right) d^4-2 i c \cos \left(\frac{e}{2}\right) d^3+2 c \sin \left(\frac{e}{2}\right) d^3+3 c^2 \cos \left(\frac{e}{2}\right) d^2+3 i c^2 \sin \left(\frac{e}{2}\right) d^2\right) \left(i \cos \left(\frac{e}{2}\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-\log \left((c \cos (e+f x)+d \sin (e+f x))^2\right) \sin \left(\frac{e}{2}\right)\right) (\cos (f x)+i \sin (f x))}{(c-i d)^3 (c+i d)^4 f (i \tan (e+f x) a+a)}+\frac{\cos (2 f x) \sec (e+f x) \left(\frac{1}{4} i \cos (e)+\frac{\sin (e)}{4}\right) (\cos (f x)+i \sin (f x))}{(c+i d)^3 f (i \tan (e+f x) a+a)}+\frac{\left(c^4+4 i d c^3+6 d^2 c^2-12 i d^3 c-3 d^4\right) \sec (e+f x) \left(\frac{1}{2} f x \cos (e)+\frac{1}{2} i f x \sin (e)\right) (\cos (f x)+i \sin (f x))}{(c-i d)^3 (c+i d)^4 f (i \tan (e+f x) a+a)}+\frac{x \sec (e+f x) \left(\frac{2 d^4}{(c-i d)^3 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac{4 i c d^3}{(c-i d)^3 (c+i d)^3 (c \cos (e)+d \sin (e))}-\frac{6 c^2 d^2}{(c-i d)^3 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac{\left(-d^4-2 i c d^3+3 c^2 d^2\right) (2 \cos (e)+2 i \sin (e)) (-\cos (2 e) c-i \sin (2 e) c+c+i d+i d \cos (2 e)-d \sin (2 e))}{(c-i d)^3 (c+i d)^4 (\cos (2 e) c+i \sin (2 e) c+c+i d-i d \cos (2 e)+d \sin (2 e))}\right) (\cos (f x)+i \sin (f x))}{i \tan (e+f x) a+a}+\frac{\sec (e+f x) \left(\frac{\cos (e)}{4}-\frac{1}{4} i \sin (e)\right) \sin (2 f x) (\cos (f x)+i \sin (f x))}{(c+i d)^3 f (i \tan (e+f x) a+a)}+\frac{\sec (e+f x) \left(\frac{1}{2} i \cos (e-f x) d^4-\frac{1}{2} i \cos (e+f x) d^4-\frac{1}{2} \sin (e-f x) d^4+\frac{1}{2} \sin (e+f x) d^4-2 c \cos (e-f x) d^3+2 c \cos (e+f x) d^3-2 i c \sin (e-f x) d^3+2 i c \sin (e+f x) d^3\right) (\cos (f x)+i \sin (f x))}{(c-i d)^2 (c+i d)^3 f (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x)) (i \tan (e+f x) a+a)}+\frac{\sec (e+f x) \left(\frac{1}{2} d^4 \sin (e)-\frac{1}{2} i d^4 \cos (e)\right) (\cos (f x)+i \sin (f x))}{(c-i d)^2 (c+i d)^3 f (c \cos (e+f x)+d \sin (e+f x))^2 (i \tan (e+f x) a+a)}","\frac{d \left(c^2-8 i c d-3 d^2\right)}{2 a f (c-i d)^2 (c+i d)^3 (c+d \tan (e+f x))}+\frac{2 d^2 \left(3 c^2-2 i c d-d^2\right) \log (c \cos (e+f x)+d \sin (e+f x))}{a f (c+i d)^4 (d+i c)^3}+\frac{x \left(c^4+4 i c^3 d+6 c^2 d^2-12 i c d^3-3 d^4\right)}{2 a (c-i d)^3 (c+i d)^4}+\frac{d (c-2 i d)}{2 a f (c-i d) (c+i d)^2 (c+d \tan (e+f x))^2}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^2}",1,"(Sec[e + f*x]*(3*c^2*d^2*Cos[e/2] - (2*I)*c*d^3*Cos[e/2] - d^4*Cos[e/2] + (3*I)*c^2*d^2*Sin[e/2] + 2*c*d^3*Sin[e/2] - I*d^4*Sin[e/2])*(-2*ArcTan[(-(d*Cos[f*x]) - c*Sin[f*x])/(c*Cos[f*x] - d*Sin[f*x])]*Cos[e/2] - (2*I)*ArcTan[(-(d*Cos[f*x]) - c*Sin[f*x])/(c*Cos[f*x] - d*Sin[f*x])]*Sin[e/2])*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^3*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])) + (Sec[e + f*x]*(3*c^2*d^2*Cos[e/2] - (2*I)*c*d^3*Cos[e/2] - d^4*Cos[e/2] + (3*I)*c^2*d^2*Sin[e/2] + 2*c*d^3*Sin[e/2] - I*d^4*Sin[e/2])*(I*Cos[e/2]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2] - Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e/2])*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^3*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])) + (Cos[2*f*x]*Sec[e + f*x]*((I/4)*Cos[e] + Sin[e]/4)*(Cos[f*x] + I*Sin[f*x]))/((c + I*d)^3*f*(a + I*a*Tan[e + f*x])) + ((c^4 + (4*I)*c^3*d + 6*c^2*d^2 - (12*I)*c*d^3 - 3*d^4)*Sec[e + f*x]*((f*x*Cos[e])/2 + (I/2)*f*x*Sin[e])*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^3*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])) + (x*Sec[e + f*x]*((-6*c^2*d^2)/((c - I*d)^3*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + ((4*I)*c*d^3)/((c - I*d)^3*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + (2*d^4)/((c - I*d)^3*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + ((3*c^2*d^2 - (2*I)*c*d^3 - d^4)*(2*Cos[e] + (2*I)*Sin[e])*(c + I*d - c*Cos[2*e] + I*d*Cos[2*e] - I*c*Sin[2*e] - d*Sin[2*e]))/((c - I*d)^3*(c + I*d)^4*(c + I*d + c*Cos[2*e] - I*d*Cos[2*e] + I*c*Sin[2*e] + d*Sin[2*e])))*(Cos[f*x] + I*Sin[f*x]))/(a + I*a*Tan[e + f*x]) + (Sec[e + f*x]*(Cos[e]/4 - (I/4)*Sin[e])*(Cos[f*x] + I*Sin[f*x])*Sin[2*f*x])/((c + I*d)^3*f*(a + I*a*Tan[e + f*x])) + (Sec[e + f*x]*((-1/2*I)*d^4*Cos[e] + (d^4*Sin[e])/2)*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^2*(c + I*d)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + I*a*Tan[e + f*x])) + (Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*(-2*c*d^3*Cos[e - f*x] + (I/2)*d^4*Cos[e - f*x] + 2*c*d^3*Cos[e + f*x] - (I/2)*d^4*Cos[e + f*x] - (2*I)*c*d^3*Sin[e - f*x] - (d^4*Sin[e - f*x])/2 + (2*I)*c*d^3*Sin[e + f*x] + (d^4*Sin[e + f*x])/2))/((c - I*d)^2*(c + I*d)^3*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))","B",1
1099,1,4395,354,8.157745,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3),x]","\text{Result too large to show}","\frac{d (c-3 i d) \left(c^2+8 i c d+5 d^2\right)}{4 a^2 f (c-i d)^2 (c+i d)^4 (c+d \tan (e+f x))}+\frac{d \left(c^2+5 i c d+8 d^2\right)}{4 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^2}-\frac{2 d^3 \left(5 c^2-5 i c d-2 d^2\right) \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 f (-d+i c)^5 (d+i c)^3}+\frac{x \left(c^5+5 i c^4 d-10 c^3 d^2+30 i c^2 d^3+45 c d^4-15 i d^5\right)}{4 a^2 (c-i d)^3 (c+i d)^5}+\frac{-5 d+i c}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))^2}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}",1,"(Sec[e + f*x]^2*(5*c^2*d^3*Cos[e] - (5*I)*c*d^4*Cos[e] - 2*d^5*Cos[e] + (5*I)*c^2*d^3*Sin[e] + 5*c*d^4*Sin[e] - (2*I)*d^5*Sin[e])*((-2*I)*ArcTan[(-2*c*d*Cos[f*x] - c^2*Sin[f*x] + d^2*Sin[f*x])/(c^2*Cos[f*x] - d^2*Cos[f*x] - 2*c*d*Sin[f*x])]*Cos[e] + 2*ArcTan[(-2*c*d*Cos[f*x] - c^2*Sin[f*x] + d^2*Sin[f*x])/(c^2*Cos[f*x] - d^2*Cos[f*x] - 2*c*d*Sin[f*x])]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2)/((c - I*d)^3*(c + I*d)^5*f*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]^2*(5*c^2*d^3*Cos[e] - (5*I)*c*d^4*Cos[e] - 2*d^5*Cos[e] + (5*I)*c^2*d^3*Sin[e] + 5*c*d^4*Sin[e] - (2*I)*d^5*Sin[e])*(-(Cos[e]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) - I*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e])*(Cos[f*x] + I*Sin[f*x])^2)/((c - I*d)^3*(c + I*d)^5*f*(a + I*a*Tan[e + f*x])^2) + (x*Sec[e + f*x]^2*(((-10*I)*c^2*d^3*Cos[e])/((c - I*d)^3*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) - (10*c*d^4*Cos[e])/((c - I*d)^3*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + ((4*I)*d^5*Cos[e])/((c - I*d)^3*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + (10*c^2*d^3*Sin[e])/((c - I*d)^3*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) - ((10*I)*c*d^4*Sin[e])/((c - I*d)^3*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) - (4*d^5*Sin[e])/((c - I*d)^3*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + ((2*Cos[2*e] + (2*I)*Sin[2*e])*((5*I)*c^3*d^3 + (3*I)*c*d^5 + 2*d^6 - (5*I)*c^3*d^3*Cos[2*e] - 10*c^2*d^4*Cos[2*e] + (7*I)*c*d^5*Cos[2*e] + 2*d^6*Cos[2*e] + 5*c^3*d^3*Sin[2*e] - (10*I)*c^2*d^4*Sin[2*e] - 7*c*d^5*Sin[2*e] + (2*I)*d^6*Sin[2*e]))/((c - I*d)^3*(c + I*d)^5*(c + I*d + c*Cos[2*e] - I*d*Cos[2*e] + I*c*Sin[2*e] + d*Sin[2*e])))*(Cos[f*x] + I*Sin[f*x])^2)/(a + I*a*Tan[e + f*x])^2 + (Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(Cos[2*e + 4*f*x]/64 - (I/64)*Sin[2*e + 4*f*x])*((6*I)*c^8*Cos[e] - 14*c^7*d*Cos[e] + (18*I)*c^6*d^2*Cos[e] - 42*c^5*d^3*Cos[e] + (18*I)*c^4*d^4*Cos[e] - 42*c^3*d^5*Cos[e] + (6*I)*c^2*d^6*Cos[e] - 14*c*d^7*Cos[e] + (5*I)*c^8*Cos[e + 2*f*x] - 11*c^7*d*Cos[e + 2*f*x] + (31*I)*c^6*d^2*Cos[e + 2*f*x] - 33*c^5*d^3*Cos[e + 2*f*x] + (63*I)*c^4*d^4*Cos[e + 2*f*x] - 33*c^3*d^5*Cos[e + 2*f*x] + (53*I)*c^2*d^6*Cos[e + 2*f*x] - 11*c*d^7*Cos[e + 2*f*x] + (16*I)*d^8*Cos[e + 2*f*x] + 2*c^8*f*x*Cos[e + 2*f*x] + (16*I)*c^7*d*f*x*Cos[e + 2*f*x] - 56*c^6*d^2*f*x*Cos[e + 2*f*x] - (32*I)*c^5*d^3*f*x*Cos[e + 2*f*x] - 20*c^4*d^4*f*x*Cos[e + 2*f*x] + (80*I)*c^3*d^5*f*x*Cos[e + 2*f*x] - 120*c^2*d^6*f*x*Cos[e + 2*f*x] - 30*d^8*f*x*Cos[e + 2*f*x] + (5*I)*c^8*Cos[3*e + 2*f*x] - 3*c^7*d*Cos[3*e + 2*f*x] + (43*I)*c^6*d^2*Cos[3*e + 2*f*x] + 35*c^5*d^3*Cos[3*e + 2*f*x] - (25*I)*c^4*d^4*Cos[3*e + 2*f*x] + 127*c^3*d^5*Cos[3*e + 2*f*x] - (111*I)*c^2*d^6*Cos[3*e + 2*f*x] + 89*c*d^7*Cos[3*e + 2*f*x] - (48*I)*d^8*Cos[3*e + 2*f*x] + 2*c^8*f*x*Cos[3*e + 2*f*x] + (12*I)*c^7*d*f*x*Cos[3*e + 2*f*x] - 28*c^6*d^2*f*x*Cos[3*e + 2*f*x] + (52*I)*c^5*d^3*f*x*Cos[3*e + 2*f*x] + (100*I)*c^3*d^5*f*x*Cos[3*e + 2*f*x] + 60*c^2*d^6*f*x*Cos[3*e + 2*f*x] + (60*I)*c*d^7*f*x*Cos[3*e + 2*f*x] + 30*d^8*f*x*Cos[3*e + 2*f*x] + (2*I)*c^8*Cos[3*e + 4*f*x] - 2*c^7*d*Cos[3*e + 4*f*x] + (22*I)*c^6*d^2*Cos[3*e + 4*f*x] + 18*c^5*d^3*Cos[3*e + 4*f*x] + (110*I)*c^4*d^4*Cos[3*e + 4*f*x] + 10*c^3*d^5*Cos[3*e + 4*f*x] + (130*I)*c^2*d^6*Cos[3*e + 4*f*x] - 10*c*d^7*Cos[3*e + 4*f*x] + (40*I)*d^8*Cos[3*e + 4*f*x] + 4*c^8*f*x*Cos[3*e + 4*f*x] + (24*I)*c^7*d*f*x*Cos[3*e + 4*f*x] - 56*c^6*d^2*f*x*Cos[3*e + 4*f*x] + (104*I)*c^5*d^3*f*x*Cos[3*e + 4*f*x] + (200*I)*c^3*d^5*f*x*Cos[3*e + 4*f*x] + 120*c^2*d^6*f*x*Cos[3*e + 4*f*x] + (120*I)*c*d^7*f*x*Cos[3*e + 4*f*x] + 60*d^8*f*x*Cos[3*e + 4*f*x] + (2*I)*c^8*Cos[5*e + 4*f*x] + 2*c^7*d*Cos[5*e + 4*f*x] + (22*I)*c^6*d^2*Cos[5*e + 4*f*x] + 62*c^5*d^3*Cos[5*e + 4*f*x] - (130*I)*c^4*d^4*Cos[5*e + 4*f*x] - 74*c^3*d^5*Cos[5*e + 4*f*x] - (126*I)*c^2*d^6*Cos[5*e + 4*f*x] - 134*c*d^7*Cos[5*e + 4*f*x] + (24*I)*d^8*Cos[5*e + 4*f*x] + 4*c^8*f*x*Cos[5*e + 4*f*x] + (16*I)*c^7*d*f*x*Cos[5*e + 4*f*x] - 16*c^6*d^2*f*x*Cos[5*e + 4*f*x] + (176*I)*c^5*d^3*f*x*Cos[5*e + 4*f*x] + 280*c^4*d^4*f*x*Cos[5*e + 4*f*x] - (80*I)*c^3*d^5*f*x*Cos[5*e + 4*f*x] + 240*c^2*d^6*f*x*Cos[5*e + 4*f*x] - (240*I)*c*d^7*f*x*Cos[5*e + 4*f*x] - 60*d^8*f*x*Cos[5*e + 4*f*x] + (80*I)*c^4*d^4*Cos[5*e + 6*f*x] + 112*c^3*d^5*Cos[5*e + 6*f*x] + (48*I)*c^2*d^6*Cos[5*e + 6*f*x] + 112*c*d^7*Cos[5*e + 6*f*x] - (32*I)*d^8*Cos[5*e + 6*f*x] + 2*c^8*f*x*Cos[5*e + 6*f*x] + (8*I)*c^7*d*f*x*Cos[5*e + 6*f*x] - 8*c^6*d^2*f*x*Cos[5*e + 6*f*x] + (88*I)*c^5*d^3*f*x*Cos[5*e + 6*f*x] + 140*c^4*d^4*f*x*Cos[5*e + 6*f*x] - (40*I)*c^3*d^5*f*x*Cos[5*e + 6*f*x] + 120*c^2*d^6*f*x*Cos[5*e + 6*f*x] - (120*I)*c*d^7*f*x*Cos[5*e + 6*f*x] - 30*d^8*f*x*Cos[5*e + 6*f*x] + 2*c^8*f*x*Cos[7*e + 6*f*x] + (4*I)*c^7*d*f*x*Cos[7*e + 6*f*x] + 4*c^6*d^2*f*x*Cos[7*e + 6*f*x] + (92*I)*c^5*d^3*f*x*Cos[7*e + 6*f*x] + 320*c^4*d^4*f*x*Cos[7*e + 6*f*x] - (500*I)*c^3*d^5*f*x*Cos[7*e + 6*f*x] - 420*c^2*d^6*f*x*Cos[7*e + 6*f*x] + (180*I)*c*d^7*f*x*Cos[7*e + 6*f*x] + 30*d^8*f*x*Cos[7*e + 6*f*x] + (6*I)*c^7*d*Sin[e] - 14*c^6*d^2*Sin[e] + (18*I)*c^5*d^3*Sin[e] - 42*c^4*d^4*Sin[e] + (18*I)*c^3*d^5*Sin[e] - 42*c^2*d^6*Sin[e] + (6*I)*c*d^7*Sin[e] - 14*d^8*Sin[e] - 4*c^8*Sin[e + 2*f*x] - (11*I)*c^7*d*Sin[e + 2*f*x] - 27*c^6*d^2*Sin[e + 2*f*x] - (33*I)*c^5*d^3*Sin[e + 2*f*x] - 57*c^4*d^4*Sin[e + 2*f*x] - (33*I)*c^3*d^5*Sin[e + 2*f*x] - 49*c^2*d^6*Sin[e + 2*f*x] - (11*I)*c*d^7*Sin[e + 2*f*x] - 15*d^8*Sin[e + 2*f*x] + (2*I)*c^8*f*x*Sin[e + 2*f*x] - 16*c^7*d*f*x*Sin[e + 2*f*x] - (56*I)*c^6*d^2*f*x*Sin[e + 2*f*x] + 32*c^5*d^3*f*x*Sin[e + 2*f*x] - (20*I)*c^4*d^4*f*x*Sin[e + 2*f*x] - 80*c^3*d^5*f*x*Sin[e + 2*f*x] - (120*I)*c^2*d^6*f*x*Sin[e + 2*f*x] - (30*I)*d^8*f*x*Sin[e + 2*f*x] - 4*c^8*Sin[3*e + 2*f*x] - I*c^7*d*Sin[3*e + 2*f*x] - 41*c^6*d^2*Sin[3*e + 2*f*x] + (41*I)*c^5*d^3*Sin[3*e + 2*f*x] + 25*c^4*d^4*Sin[3*e + 2*f*x] + (133*I)*c^3*d^5*Sin[3*e + 2*f*x] + 109*c^2*d^6*Sin[3*e + 2*f*x] + (91*I)*c*d^7*Sin[3*e + 2*f*x] + 47*d^8*Sin[3*e + 2*f*x] + (2*I)*c^8*f*x*Sin[3*e + 2*f*x] - 12*c^7*d*f*x*Sin[3*e + 2*f*x] - (28*I)*c^6*d^2*f*x*Sin[3*e + 2*f*x] - 52*c^5*d^3*f*x*Sin[3*e + 2*f*x] - 100*c^3*d^5*f*x*Sin[3*e + 2*f*x] + (60*I)*c^2*d^6*f*x*Sin[3*e + 2*f*x] - 60*c*d^7*f*x*Sin[3*e + 2*f*x] + (30*I)*d^8*f*x*Sin[3*e + 2*f*x] - 2*c^8*Sin[3*e + 4*f*x] - (2*I)*c^7*d*Sin[3*e + 4*f*x] - 22*c^6*d^2*Sin[3*e + 4*f*x] + (18*I)*c^5*d^3*Sin[3*e + 4*f*x] - 110*c^4*d^4*Sin[3*e + 4*f*x] + (10*I)*c^3*d^5*Sin[3*e + 4*f*x] - 130*c^2*d^6*Sin[3*e + 4*f*x] - (10*I)*c*d^7*Sin[3*e + 4*f*x] - 40*d^8*Sin[3*e + 4*f*x] + (4*I)*c^8*f*x*Sin[3*e + 4*f*x] - 24*c^7*d*f*x*Sin[3*e + 4*f*x] - (56*I)*c^6*d^2*f*x*Sin[3*e + 4*f*x] - 104*c^5*d^3*f*x*Sin[3*e + 4*f*x] - 200*c^3*d^5*f*x*Sin[3*e + 4*f*x] + (120*I)*c^2*d^6*f*x*Sin[3*e + 4*f*x] - 120*c*d^7*f*x*Sin[3*e + 4*f*x] + (60*I)*d^8*f*x*Sin[3*e + 4*f*x] - 2*c^8*Sin[5*e + 4*f*x] + (2*I)*c^7*d*Sin[5*e + 4*f*x] - 22*c^6*d^2*Sin[5*e + 4*f*x] + (62*I)*c^5*d^3*Sin[5*e + 4*f*x] + 130*c^4*d^4*Sin[5*e + 4*f*x] - (74*I)*c^3*d^5*Sin[5*e + 4*f*x] + 126*c^2*d^6*Sin[5*e + 4*f*x] - (134*I)*c*d^7*Sin[5*e + 4*f*x] - 24*d^8*Sin[5*e + 4*f*x] + (4*I)*c^8*f*x*Sin[5*e + 4*f*x] - 16*c^7*d*f*x*Sin[5*e + 4*f*x] - (16*I)*c^6*d^2*f*x*Sin[5*e + 4*f*x] - 176*c^5*d^3*f*x*Sin[5*e + 4*f*x] + (280*I)*c^4*d^4*f*x*Sin[5*e + 4*f*x] + 80*c^3*d^5*f*x*Sin[5*e + 4*f*x] + (240*I)*c^2*d^6*f*x*Sin[5*e + 4*f*x] + 240*c*d^7*f*x*Sin[5*e + 4*f*x] - (60*I)*d^8*f*x*Sin[5*e + 4*f*x] - 80*c^4*d^4*Sin[5*e + 6*f*x] + (112*I)*c^3*d^5*Sin[5*e + 6*f*x] - 48*c^2*d^6*Sin[5*e + 6*f*x] + (112*I)*c*d^7*Sin[5*e + 6*f*x] + 32*d^8*Sin[5*e + 6*f*x] + (2*I)*c^8*f*x*Sin[5*e + 6*f*x] - 8*c^7*d*f*x*Sin[5*e + 6*f*x] - (8*I)*c^6*d^2*f*x*Sin[5*e + 6*f*x] - 88*c^5*d^3*f*x*Sin[5*e + 6*f*x] + (140*I)*c^4*d^4*f*x*Sin[5*e + 6*f*x] + 40*c^3*d^5*f*x*Sin[5*e + 6*f*x] + (120*I)*c^2*d^6*f*x*Sin[5*e + 6*f*x] + 120*c*d^7*f*x*Sin[5*e + 6*f*x] - (30*I)*d^8*f*x*Sin[5*e + 6*f*x] + (2*I)*c^8*f*x*Sin[7*e + 6*f*x] - 4*c^7*d*f*x*Sin[7*e + 6*f*x] + (4*I)*c^6*d^2*f*x*Sin[7*e + 6*f*x] - 92*c^5*d^3*f*x*Sin[7*e + 6*f*x] + (320*I)*c^4*d^4*f*x*Sin[7*e + 6*f*x] + 500*c^3*d^5*f*x*Sin[7*e + 6*f*x] - (420*I)*c^2*d^6*f*x*Sin[7*e + 6*f*x] - 180*c*d^7*f*x*Sin[7*e + 6*f*x] + (30*I)*d^8*f*x*Sin[7*e + 6*f*x]))/((c - I*d)^3*(c + I*d)^5*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + I*a*Tan[e + f*x])^2)","B",0
1100,1,5726,448,8.47167,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^3} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3),x]","\text{Result too large to show}","\frac{3 c^2+18 i c d-55 d^2}{24 f (-d+i c)^3 \left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^2}-\frac{d^4 \left(15 c^2-18 i c d-7 d^2\right) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^6 (d+i c)^3}+\frac{d \left(c^3+6 i c^2 d-17 c d^2+28 i d^3\right)}{8 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^2}+\frac{d \left(c^4+6 i c^3 d-16 c^2 d^2+94 i c d^3+55 d^4\right)}{8 a^3 f (c-i d)^2 (c+i d)^5 (c+d \tan (e+f x))}+\frac{x \left(c^6+6 i c^5 d-15 c^4 d^2-20 i c^3 d^3-105 c^2 d^4+150 i c d^5+55 d^6\right)}{8 a^3 (c-i d)^3 (c+i d)^6}+\frac{-13 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2}",1,"Result too large to show","B",0
1101,1,219,150,5.6234606,"\int (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]],x]","\frac{a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left(\frac{(\sin (3 e)+i \cos (3 e)) \sec ^2(e+f x) \sqrt{c+d \tan (e+f x)} \left(\left(2 c^2+15 i c d+63 d^2\right) \cos (2 (e+f x))+2 c^2-d (c-15 i d) \sin (2 (e+f x))+15 i c d+57 d^2\right)}{15 d^2}-8 i e^{-3 i e} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 (-6 d+i c) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac{8 i a^3 \sqrt{c+d \tan (e+f x)}}{f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac{8 i a^3 \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(a^3*(Cos[e + f*x] + I*Sin[e + f*x])^3*(((-8*I)*Sqrt[c - I*d]*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/E^((3*I)*e) + (Sec[e + f*x]^2*(I*Cos[3*e] + Sin[3*e])*(2*c^2 + (15*I)*c*d + 57*d^2 + (2*c^2 + (15*I)*c*d + 63*d^2)*Cos[2*(e + f*x)] - (c - (15*I)*d)*d*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(15*d^2)))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1102,1,155,100,3.6084224,"\int (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]],x]","-\frac{2 a^2 e^{-2 i e} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \left(\sqrt{c+d \tan (e+f x)} (c+d \tan (e+f x)-6 i d)+6 i d \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{3 d f (\cos (f x)+i \sin (f x))^2}","-\frac{2 a^2 (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac{4 i a^2 \sqrt{c+d \tan (e+f x)}}{f}-\frac{4 i a^2 \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(-2*a^2*(Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])*((6*I)*Sqrt[c - I*d]*d*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(c - (6*I)*d + d*Tan[e + f*x])))/(3*d*E^((2*I)*e)*f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
1103,1,88,69,1.3993469,"\int (a+i a \tan (e+f x)) \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]],x]","\frac{2 i a \left(\sqrt{c+d \tan (e+f x)}-\sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{f}","\frac{2 i a \sqrt{c+d \tan (e+f x)}}{f}-\frac{2 i a \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"((2*I)*a*(-(Sqrt[c - I*d]*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]]) + Sqrt[c + d*Tan[e + f*x]]))/f","A",1
1104,1,339,140,5.0899465,"\int \frac{\sqrt{c+d \tan (e+f x)}}{a+i a \tan (e+f x)} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x]),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(2 \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt{c+d \tan (e+f x)}-i (\cos (e)+i \sin (e)) \left(\sqrt{c-i d} \log \left(\frac{2 \left(\sqrt{c-i d} \left(1+e^{2 i (e+f x)}\right) \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+c \left(1+e^{2 i (e+f x)}\right)-i d e^{2 i (e+f x)}\right)}{\sqrt{c-i d}}\right)-\frac{c \log \left(\frac{8 i e^{-2 i f x} \left(\sqrt{c+i d} \left(1+e^{2 i (e+f x)}\right) \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+c \left(1+e^{2 i (e+f x)}\right)+i d\right)}{c \sqrt{c+i d}}\right)}{\sqrt{c+i d}}\right)\right)}{4 f (a+i a \tan (e+f x))}","\frac{i \sqrt{c+d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 a f}+\frac{i c \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 a f \sqrt{c+i d}}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-I)*(Sqrt[c - I*d]*Log[(2*((-I)*d*E^((2*I)*(e + f*x)) + c*(1 + E^((2*I)*(e + f*x))) + Sqrt[c - I*d]*(1 + E^((2*I)*(e + f*x)))*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/Sqrt[c - I*d]] - (c*Log[((8*I)*(I*d + c*(1 + E^((2*I)*(e + f*x))) + Sqrt[c + I*d]*(1 + E^((2*I)*(e + f*x)))*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(c*Sqrt[c + I*d]*E^((2*I)*f*x))])/Sqrt[c + I*d])*(Cos[e] + I*Sin[e]) + 2*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*Sqrt[c + d*Tan[e + f*x]]))/(4*f*(a + I*a*Tan[e + f*x]))","B",1
1105,1,281,211,1.8409933,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^2,x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(\frac{2 \cos (e+f x) (\sin (2 f x)+i \cos (2 f x)) \sqrt{c+d \tan (e+f x)} ((-d+2 i c) \sin (e+f x)+(4 c+3 i d) \cos (e+f x))}{c+i d}-\frac{2 (\cos (2 e)+i \sin (2 e)) \left(2 i \sqrt{-c-i d} \left(c^2+d^2\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)-i \sqrt{-c+i d} \left(2 c^2+2 i c d+d^2\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)\right)}{(-c-i d)^{3/2} \sqrt{-c+i d}}\right)}{16 f (a+i a \tan (e+f x))^2}","-\frac{\left(2 c d-i \left(2 c^2+d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{8 a^2 f (c+i d)^{3/2}}+\frac{(-d+2 i c) \sqrt{c+d \tan (e+f x)}}{8 a^2 f (c+i d) (1+i \tan (e+f x))}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{4 a^2 f}+\frac{i \sqrt{c+d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((-2*((-I)*Sqrt[-c + I*d]*(2*c^2 + (2*I)*c*d + d^2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + (2*I)*Sqrt[-c - I*d]*(c^2 + d^2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[2*e] + I*Sin[2*e]))/((-c - I*d)^(3/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[2*f*x] + Sin[2*f*x])*((4*c + (3*I)*d)*Cos[e + f*x] + ((2*I)*c - d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(c + I*d)))/(16*f*(a + I*a*Tan[e + f*x])^2)","A",1
1106,1,329,280,2.846552,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(\frac{2 \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c+d \tan (e+f x)} \left(i \left(9 c^2+14 i c d-2 d^2\right) \sin (2 (e+f x))+\left(13 c^2+22 i c d-6 d^2\right) \cos (2 (e+f x))+7 c^2+13 i c d-6 d^2\right)}{3 (c+i d)^2}+\frac{2 (\cos (3 e)+i \sin (3 e)) \left(\sqrt{-c+i d} \left(-2 i c^3+4 c^2 d+i c d^2+2 d^3\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)+2 (d+i c) (-c-i d)^{5/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{5/2} \sqrt{-c+i d}}\right)}{32 f (a+i a \tan (e+f x))^3}","\frac{\left(2 i c^3-4 c^2 d-i c d^2-2 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{16 a^3 f (c+i d)^{5/2}}+\frac{c (-3 d+2 i c) \sqrt{c+d \tan (e+f x)}}{16 f (c+i d)^2 \left(a^3+i a^3 \tan (e+f x)\right)}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{8 a^3 f}+\frac{(-2 d+3 i c) \sqrt{c+d \tan (e+f x)}}{24 a f (c+i d) (a+i a \tan (e+f x))^2}+\frac{i \sqrt{c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 4*c^2*d + I*c*d^2 + 2*d^3)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + 2*(-c - I*d)^(5/2)*(I*c + d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(5/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(7*c^2 + (13*I)*c*d - 6*d^2 + (13*c^2 + (22*I)*c*d - 6*d^2)*Cos[2*(e + f*x)] + I*(9*c^2 + (14*I)*c*d - 2*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)^2)))/(32*f*(a + I*a*Tan[e + f*x])^3)","A",1
1107,1,271,181,8.4664217,"\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2),x]","\frac{a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left(\frac{(\sin (3 e)+i \cos (3 e)) \sec ^2(e+f x) \sqrt{c+d \tan (e+f x)} \left(6 c^3+d \left(-3 c^2+126 i c d+125 d^2\right) \tan (e+f x)+63 i c^2 d+\cos (2 (e+f x)) \left(6 c^3+d \left(-3 c^2+126 i c d+155 d^2\right) \tan (e+f x)+63 i c^2 d+584 c d^2-483 i d^3\right)+536 c d^2-357 i d^3\right)}{105 d^2}-8 i e^{-3 i e} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 (-8 d+i c) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac{8 i a^3 (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{8 a^3 (d+i c) \sqrt{c+d \tan (e+f x)}}{f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac{8 i a^3 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(a^3*(Cos[e + f*x] + I*Sin[e + f*x])^3*(((-8*I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/E^((3*I)*e) + (Sec[e + f*x]^2*(I*Cos[3*e] + Sin[3*e])*Sqrt[c + d*Tan[e + f*x]]*(6*c^3 + (63*I)*c^2*d + 536*c*d^2 - (357*I)*d^3 + d*(-3*c^2 + (126*I)*c*d + 125*d^2)*Tan[e + f*x] + Cos[2*(e + f*x)]*(6*c^3 + (63*I)*c^2*d + 584*c*d^2 - (483*I)*d^3 + d*(-3*c^2 + (126*I)*c*d + 155*d^2)*Tan[e + f*x])))/(105*d^2)))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1108,1,221,131,4.474701,"\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2),x]","\frac{a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left(-\frac{(\cos (2 e)-i \sin (2 e)) \sec ^2(e+f x) \sqrt{c+d \tan (e+f x)} \left(\left(3 c^2-40 i c d-33 d^2\right) \cos (2 (e+f x))+3 c^2+2 d (3 c-5 i d) \sin (2 (e+f x))-40 i c d-27 d^2\right)}{15 d}-4 i e^{-2 i e} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{f (\cos (f x)+i \sin (f x))^2}","-\frac{2 a^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac{4 i a^2 (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{4 a^2 (d+i c) \sqrt{c+d \tan (e+f x)}}{f}-\frac{4 i a^2 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(a^2*(Cos[e + f*x] + I*Sin[e + f*x])^2*(((-4*I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/E^((2*I)*e) - (Sec[e + f*x]^2*(Cos[2*e] - I*Sin[2*e])*(3*c^2 - (40*I)*c*d - 27*d^2 + (3*c^2 - (40*I)*c*d - 33*d^2)*Cos[2*(e + f*x)] + 2*(3*c - (5*I)*d)*d*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(15*d)))/(f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
1109,1,111,98,2.1614832,"\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 a \left((4 i c+i d \tan (e+f x)+3 d) \sqrt{c+d \tan (e+f x)}-3 i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{3 f}","\frac{2 i a (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 a (d+i c) \sqrt{c+d \tan (e+f x)}}{f}-\frac{2 i a (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(2*a*((-3*I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]] + ((4*I)*c + 3*d + I*d*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(3*f)","A",1
1110,1,376,153,4.3251026,"\int \frac{(c+d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x]),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left((\cos (e)+i \sin (e)) \left(\frac{\left(i c^2+c d+2 i d^2\right) \log \left(\frac{8 i e^{-2 i f x} \left(\sqrt{c+i d} \left(1+e^{2 i (e+f x)}\right) \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+c \left(1+e^{2 i (e+f x)}\right)+i d\right)}{\sqrt{c+i d} \left(c^2-i c d+2 d^2\right)}\right)}{\sqrt{c+i d}}-i (c-i d)^{3/2} \log \left(\frac{2 \left(\sqrt{c-i d} \left(1+e^{2 i (e+f x)}\right) \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+c \left(1+e^{2 i (e+f x)}\right)-i d e^{2 i (e+f x)}\right)}{\sqrt{c-i d}}\right)\right)+2 (c+i d) \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt{c+d \tan (e+f x)}\right)}{4 f (a+i a \tan (e+f x))}","\frac{(-d+i c) \sqrt{c+d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 a f}+\frac{\sqrt{c+i d} (2 d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 a f}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*(((-I)*(c - I*d)^(3/2)*Log[(2*((-I)*d*E^((2*I)*(e + f*x)) + c*(1 + E^((2*I)*(e + f*x))) + Sqrt[c - I*d]*(1 + E^((2*I)*(e + f*x)))*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/Sqrt[c - I*d]] + ((I*c^2 + c*d + (2*I)*d^2)*Log[((8*I)*(I*d + c*(1 + E^((2*I)*(e + f*x))) + Sqrt[c + I*d]*(1 + E^((2*I)*(e + f*x)))*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[c + I*d]*(c^2 - I*c*d + 2*d^2)*E^((2*I)*f*x))])/Sqrt[c + I*d])*(Cos[e] + I*Sin[e]) + 2*(c + I*d)*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*Sqrt[c + d*Tan[e + f*x]]))/(4*f*(a + I*a*Tan[e + f*x]))","B",0
1111,1,272,209,1.8250945,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(\frac{2 (\cos (2 e)+i \sin (2 e)) \left(2 i \sqrt{-c-i d} (c-i d)^2 \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)-i \sqrt{-c+i d} \left(2 c^2-2 i c d+d^2\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)\right)}{\sqrt{-c-i d} \sqrt{-c+i d}}+2 \cos (e+f x) (\cos (2 f x)-i \sin (2 f x)) \sqrt{c+d \tan (e+f x)} ((-2 c+3 i d) \sin (e+f x)+(d+4 i c) \cos (e+f x))\right)}{16 f (a+i a \tan (e+f x))^2}","\frac{\left(2 c d+i \left(2 c^2+d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{8 a^2 f \sqrt{c+i d}}+\frac{(3 d+2 i c) \sqrt{c+d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{4 a^2 f}+\frac{(-d+i c) \sqrt{c+d \tan (e+f x)}}{4 f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*((-I)*Sqrt[-c + I*d]*(2*c^2 - (2*I)*c*d + d^2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + (2*I)*Sqrt[-c - I*d]*(c - I*d)^2*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[2*e] + I*Sin[2*e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + 2*Cos[e + f*x]*(Cos[2*f*x] - I*Sin[2*f*x])*(((4*I)*c + d)*Cos[e + f*x] + (-2*c + (3*I)*d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(16*f*(a + I*a*Tan[e + f*x])^2)","A",1
1112,1,311,274,2.7799671,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(\frac{2 i (\cos (3 e)+i \sin (3 e)) \left(c \sqrt{-c+i d} \left(2 c^2+3 d^2\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)+2 (-c-i d)^{3/2} (c-i d)^2 \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{3/2} \sqrt{-c+i d}}+\frac{2 \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c+d \tan (e+f x)} \left(\left(9 i c^2+4 c d+10 i d^2\right) \sin (2 (e+f x))+\left(13 c^2+4 i c d+6 d^2\right) \cos (2 (e+f x))+7 c (c+i d)\right)}{3 (c+i d)}\right)}{32 f (a+i a \tan (e+f x))^3}","-\frac{\left(2 c^2-i c d+2 d^2\right) \sqrt{c+d \tan (e+f x)}}{16 f (-d+i c) \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{i c \left(2 c^2+3 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{16 a^3 f (c+i d)^{3/2}}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{8 a^3 f}+\frac{(4 d+3 i c) \sqrt{c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}+\frac{(-d+i c) \sqrt{c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(((2*I)*(c*Sqrt[-c + I*d]*(2*c^2 + 3*d^2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + 2*(-c - I*d)^(3/2)*(c - I*d)^2*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(3/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(7*c*(c + I*d) + (13*c^2 + (4*I)*c*d + 6*d^2)*Cos[2*(e + f*x)] + ((9*I)*c^2 + 4*c*d + (10*I)*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d))))/(32*f*(a + I*a*Tan[e + f*x])^3)","A",1
1113,1,528,216,11.5599048,"\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2),x]","\frac{\cos ^3(e+f x) (a+i a \tan (e+f x))^3 \left(\sec (e) \left(-\frac{2}{315} \sin (3 e)-\frac{2}{315} i \cos (3 e)\right) \sec ^2(e+f x) \left(75 c^2 \cos (e)+95 c d \sin (e)-405 i c d \cos (e)-135 i d^2 \sin (e)-322 d^2 \cos (e)\right)+\sec (e) \left(\frac{2 \cos (3 e)}{315 d}-\frac{2 i \sin (3 e)}{315 d}\right) \sec (e+f x) \left(-5 i c^3 \sin (f x)-405 c^2 d \sin (f x)+1019 i c d^2 \sin (f x)+555 d^3 \sin (f x)\right)+\sec (e) \left(\frac{2 \cos (3 e)}{315 d^2}-\frac{2 i \sin (3 e)}{315 d^2}\right) \left(10 i c^4 \cos (e)-5 i c^3 d \sin (e)-135 c^3 d \cos (e)-405 c^2 d^2 \sin (e)+2007 i c^2 d^2 \cos (e)+1019 i c d^3 \sin (e)+3345 c d^3 \cos (e)+555 d^4 \sin (e)-1547 i d^4 \cos (e)\right)+\sec (e) \left(\frac{2}{63} \cos (3 e)-\frac{2}{63} i \sin (3 e)\right) \sec ^3(e+f x) \left(-27 d^2 \sin (f x)-19 i c d \sin (f x)\right)+\left(-\frac{2}{9} d^2 \sin (3 e)-\frac{2}{9} i d^2 \cos (3 e)\right) \sec ^4(e+f x)\right) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^3}-\frac{8 i e^{-3 i e} (c-i d)^{5/2} \cos ^3(e+f x) (a+i a \tan (e+f x))^3 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 (-10 d+i c) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac{8 i a^3 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{8 a^3 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{8 i a^3 (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{f}-\frac{8 i a^3 (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"((-8*I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]]*Cos[e + f*x]^3*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(Sec[e]*Sec[e + f*x]^2*(75*c^2*Cos[e] - (405*I)*c*d*Cos[e] - 322*d^2*Cos[e] + 95*c*d*Sin[e] - (135*I)*d^2*Sin[e])*(((-2*I)/315)*Cos[3*e] - (2*Sin[3*e])/315) + Sec[e]*((10*I)*c^4*Cos[e] - 135*c^3*d*Cos[e] + (2007*I)*c^2*d^2*Cos[e] + 3345*c*d^3*Cos[e] - (1547*I)*d^4*Cos[e] - (5*I)*c^3*d*Sin[e] - 405*c^2*d^2*Sin[e] + (1019*I)*c*d^3*Sin[e] + 555*d^4*Sin[e])*((2*Cos[3*e])/(315*d^2) - (((2*I)/315)*Sin[3*e])/d^2) + Sec[e + f*x]^4*(((-2*I)/9)*d^2*Cos[3*e] - (2*d^2*Sin[3*e])/9) + Sec[e]*Sec[e + f*x]^3*((2*Cos[3*e])/63 - ((2*I)/63)*Sin[3*e])*((-19*I)*c*d*Sin[f*x] - 27*d^2*Sin[f*x]) + Sec[e]*Sec[e + f*x]*((2*Cos[3*e])/(315*d) - (((2*I)/315)*Sin[3*e])/d)*((-5*I)*c^3*Sin[f*x] - 405*c^2*d*Sin[f*x] + (1019*I)*c*d^2*Sin[f*x] + 555*d^3*Sin[f*x]))*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3)","B",1
1114,1,271,166,8.2652163,"\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]","\frac{a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left(-\frac{(\cos (2 e)-i \sin (2 e)) \sec ^2(e+f x) \sqrt{c+d \tan (e+f x)} \left(15 c^3+d \left(45 c^2-154 i c d-55 d^2\right) \tan (e+f x)-322 i c^2 d+\cos (2 (e+f x)) \left(15 c^3+d \left(45 c^2-154 i c d-85 d^2\right) \tan (e+f x)-322 i c^2 d-535 c d^2+252 i d^3\right)-445 c d^2+168 i d^3\right)}{105 d}-4 i e^{-2 i e} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{f (\cos (f x)+i \sin (f x))^2}","-\frac{2 a^2 (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac{4 i a^2 (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{4 a^2 (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{4 i a^2 (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{f}-\frac{4 i a^2 (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(a^2*(Cos[e + f*x] + I*Sin[e + f*x])^2*(((-4*I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/E^((2*I)*e) - (Sec[e + f*x]^2*(Cos[2*e] - I*Sin[2*e])*Sqrt[c + d*Tan[e + f*x]]*(15*c^3 - (322*I)*c^2*d - 445*c*d^2 + (168*I)*d^3 + d*(45*c^2 - (154*I)*c*d - 55*d^2)*Tan[e + f*x] + Cos[2*(e + f*x)]*(15*c^3 - (322*I)*c^2*d - 535*c*d^2 + (252*I)*d^3 + d*(45*c^2 - (154*I)*c*d - 85*d^2)*Tan[e + f*x])))/(105*d)))/(f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
1115,1,208,131,3.5160565,"\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2),x]","\frac{\cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left(\frac{1}{15} (\sin (e)+i \cos (e)) \sec ^2(e+f x) \sqrt{c+d \tan (e+f x)} \left(\left(23 c^2-35 i c d-18 d^2\right) \cos (2 (e+f x))+23 c^2+d (11 c-5 i d) \sin (2 (e+f x))-35 i c d-12 d^2\right)-2 i e^{-i e} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)\right)}{f}","\frac{2 i a (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 i a (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 a (d+i c) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac{2 i a (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"(Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(a + I*a*Tan[e + f*x])*(((-2*I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/E^(I*e) + (Sec[e + f*x]^2*(I*Cos[e] + Sin[e])*(23*c^2 - (35*I)*c*d - 12*d^2 + (23*c^2 - (35*I)*c*d - 18*d^2)*Cos[2*(e + f*x)] + (11*c - (5*I)*d)*d*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/15))/f","A",1
1116,1,260,185,2.0376561,"\int \frac{(c+d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(2 (\sin (f x)+i \cos (f x)) \sqrt{c+d \tan (e+f x)} \left(\left(c^2+2 i c d-5 d^2\right) \cos (e+f x)-4 i d^2 \sin (e+f x)\right)+\frac{2 (\cos (e)+i \sin (e)) \left(-\sqrt{-c-i d} (d+i c)^3 \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)-i \sqrt{-c+i d} (c+i d)^2 (c-4 i d) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)\right)}{\sqrt{-c-i d} \sqrt{-c+i d}}\right)}{4 f (a+i a \tan (e+f x))}","\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}-\frac{d (c+5 i d) \sqrt{c+d \tan (e+f x)}}{2 a f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 a f}+\frac{(c+i d)^{3/2} (4 d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 a f}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((2*((-I)*Sqrt[-c + I*d]*(c + I*d)^2*(c - (4*I)*d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - Sqrt[-c - I*d]*(I*c + d)^3*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*Sin[e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + 2*(I*Cos[f*x] + Sin[f*x])*((c^2 + (2*I)*c*d - 5*d^2)*Cos[e + f*x] - (4*I)*d^2*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(4*f*(a + I*a*Tan[e + f*x]))","A",1
1117,1,291,217,2.0062528,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^2,x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(\frac{2 (\cos (2 e)+i \sin (2 e)) \left(-i \sqrt{-c+i d} \left(2 c^3-4 i c^2 d-c d^2-7 i d^3\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)-2 \sqrt{-c-i d} (d+i c)^3 \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{\sqrt{-c-i d} \sqrt{-c+i d}}+2 (c+i d) \cos (e+f x) (\cos (2 f x)-i \sin (2 f x)) \sqrt{c+d \tan (e+f x)} ((-2 c+7 i d) \sin (e+f x)+(5 d+4 i c) \cos (e+f x))\right)}{16 f (a+i a \tan (e+f x))^2}","\frac{\sqrt{c+i d} \left(2 i c^2+6 c d-7 i d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{8 a^2 f}+\frac{(c+i d) (5 d+2 i c) \sqrt{c+d \tan (e+f x)}}{8 a^2 f (1+i \tan (e+f x))}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{4 a^2 f}+\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{4 f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*((-I)*Sqrt[-c + I*d]*(2*c^3 - (4*I)*c^2*d - c*d^2 - (7*I)*d^3)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - 2*Sqrt[-c - I*d]*(I*c + d)^3*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[2*e] + I*Sin[2*e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + 2*(c + I*d)*Cos[e + f*x]*(Cos[2*f*x] - I*Sin[2*f*x])*(((4*I)*c + 5*d)*Cos[e + f*x] + (-2*c + (7*I)*d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(16*f*(a + I*a*Tan[e + f*x])^2)","A",1
1118,1,324,285,2.8010845,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3,x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(\frac{2}{3} \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c+d \tan (e+f x)} \left(\left(9 i c^2+22 c d-2 i d^2\right) \sin (2 (e+f x))+\left(13 c^2-14 i c d-6 d^2\right) \cos (2 (e+f x))+7 c^2+i c d+6 d^2\right)+\frac{2 (\cos (3 e)+i \sin (3 e)) \left(-i \sqrt{-c+i d} \left(2 c^3-4 i c^2 d-c d^2-2 i d^3\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)-2 \sqrt{-c-i d} (d+i c)^3 \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{\sqrt{-c-i d} \sqrt{-c+i d}}\right)}{32 f (a+i a \tan (e+f x))^3}","\frac{\left(2 i c^2+5 c d-4 i d^2\right) \sqrt{c+d \tan (e+f x)}}{16 f \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{\left(2 i c^3+4 c^2 d-i c d^2+2 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{16 a^3 f \sqrt{c+i d}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{8 a^3 f}+\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac{(c+i d) (2 d+i c) \sqrt{c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((2*((-I)*Sqrt[-c + I*d]*(2*c^3 - (4*I)*c^2*d - c*d^2 - (2*I)*d^3)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - 2*Sqrt[-c - I*d]*(I*c + d)^3*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(7*c^2 + I*c*d + 6*d^2 + (13*c^2 - (14*I)*c*d - 6*d^2)*Cos[2*(e + f*x)] + ((9*I)*c^2 + 22*c*d - (2*I)*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/3))/(32*f*(a + I*a*Tan[e + f*x])^3)","A",1
1119,1,168,126,5.1955185,"\int \frac{(a+i a \tan (e+f x))^3}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/Sqrt[c + d*Tan[e + f*x]],x]","\frac{a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left(\frac{2 (\sin (3 e)+i \cos (3 e)) (2 c-d \tan (e+f x)+9 i d) \sqrt{c+d \tan (e+f x)}}{3 d^2}-\frac{8 i e^{-3 i e} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}\right)}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 (-4 d+i c) \sqrt{c+d \tan (e+f x)}}{3 d^2 f}-\frac{2 \left(a^3+i a^3 \tan (e+f x)\right) \sqrt{c+d \tan (e+f x)}}{3 d f}-\frac{8 i a^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}",1,"(a^3*(Cos[e + f*x] + I*Sin[e + f*x])^3*(((-8*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*E^((3*I)*e)) + (2*(I*Cos[3*e] + Sin[3*e])*(2*c + (9*I)*d - d*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*d^2)))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1120,1,95,74,3.0781557,"\int \frac{(a+i a \tan (e+f x))^2}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/Sqrt[c + d*Tan[e + f*x]],x]","\frac{2 a^2 \left(-\frac{\sqrt{c+d \tan (e+f x)}}{d}-\frac{2 i \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}\right)}{f}","-\frac{2 a^2 \sqrt{c+d \tan (e+f x)}}{d f}-\frac{4 i a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}",1,"(2*a^2*(((-2*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/Sqrt[c - I*d] - Sqrt[c + d*Tan[e + f*x]]/d))/f","A",1
1121,1,71,46,1.3711853,"\int \frac{a+i a \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/Sqrt[c + d*Tan[e + f*x]],x]","-\frac{2 i a \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}","-\frac{2 i a \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}",1,"((-2*I)*a*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)","A",1
1122,1,222,155,1.5499505,"\int \frac{1}{(a+i a \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(\frac{2 \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt{c+d \tan (e+f x)}}{c+i d}-\frac{2 (\cos (e)+i \sin (e)) \left(\sqrt{-c+i d} (2 d-i c) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)-i (-c-i d)^{3/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{3/2} \sqrt{-c+i d}}\right)}{4 f (a+i a \tan (e+f x))}","-\frac{\sqrt{c+d \tan (e+f x)}}{2 f (-d+i c) (a+i a \tan (e+f x))}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 a f \sqrt{c-i d}}+\frac{(-2 d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 a f (c+i d)^{3/2}}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-2*(Sqrt[-c + I*d]*((-I)*c + 2*d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - I*(-c - I*d)^(3/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*Sin[e]))/((-c - I*d)^(3/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*Sqrt[c + d*Tan[e + f*x]])/(c + I*d)))/(4*f*(a + I*a*Tan[e + f*x]))","A",1
1123,1,275,221,2.1438736,"\int \frac{1}{(a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(\frac{2 (\cos (2 e)+i \sin (2 e)) \left(\sqrt{-c+i d} \left(-2 i c^2+6 c d+7 i d^2\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)+2 i (-c-i d)^{5/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{5/2} \sqrt{-c+i d}}+\frac{2 \cos (e+f x) (\sin (2 f x)+i \cos (2 f x)) \sqrt{c+d \tan (e+f x)} ((-5 d+2 i c) \sin (e+f x)+(4 c+7 i d) \cos (e+f x))}{(c+i d)^2}\right)}{16 f (a+i a \tan (e+f x))^2}","\frac{\left(2 i c^2-6 c d-7 i d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{8 a^2 f (c+i d)^{5/2}}+\frac{(-5 d+2 i c) \sqrt{c+d \tan (e+f x)}}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{4 a^2 f \sqrt{c-i d}}-\frac{\sqrt{c+d \tan (e+f x)}}{4 f (-d+i c) (a+i a \tan (e+f x))^2}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*(Sqrt[-c + I*d]*((-2*I)*c^2 + 6*c*d + (7*I)*d^2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + (2*I)*(-c - I*d)^(5/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[2*e] + I*Sin[2*e]))/((-c - I*d)^(5/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[2*f*x] + Sin[2*f*x])*((4*c + (7*I)*d)*Cos[e + f*x] + ((2*I)*c - 5*d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(c + I*d)^2))/(16*f*(a + I*a*Tan[e + f*x])^2)","A",1
1124,1,324,298,3.2841095,"\int \frac{1}{(a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(\frac{2 \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c+d \tan (e+f x)} \left(i \left(9 c^2+32 i c d-38 d^2\right) \sin (2 (e+f x))+\left(13 c^2+40 i c d-42 d^2\right) \cos (2 (e+f x))+7 c^2+19 i c d-12 d^2\right)}{3 (c+i d)^3}-\frac{2 (\cos (3 e)+i \sin (3 e)) \left(\sqrt{-c+i d} \left(-2 i c^3+8 c^2 d+13 i c d^2-12 d^3\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)-2 i (-c-i d)^{7/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{7/2} \sqrt{-c+i d}}\right)}{32 f (a+i a \tan (e+f x))^3}","\frac{\left(2 c^2+7 i c d-10 d^2\right) \sqrt{c+d \tan (e+f x)}}{16 f (-d+i c)^3 \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{\left(2 i c^3-8 c^2 d-13 i c d^2+12 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{16 a^3 f (c+i d)^{7/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{8 a^3 f \sqrt{c-i d}}+\frac{(-8 d+3 i c) \sqrt{c+d \tan (e+f x)}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{\sqrt{c+d \tan (e+f x)}}{6 f (-d+i c) (a+i a \tan (e+f x))^3}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((-2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 8*c^2*d + (13*I)*c*d^2 - 12*d^3)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - (2*I)*(-c - I*d)^(7/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(7/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(7*c^2 + (19*I)*c*d - 12*d^2 + (13*c^2 + (40*I)*c*d - 42*d^2)*Cos[2*(e + f*x)] + I*(9*c^2 + (32*I)*c*d - 38*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)^3)))/(32*f*(a + I*a*Tan[e + f*x])^3)","A",1
1125,1,219,139,5.4999184,"\int \frac{(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(3/2),x]","\frac{a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left(\frac{2 (\cos (3 e)-i \sin (3 e)) \sqrt{c+d \tan (e+f x)} \left(\left(-2 i c^2+c d+i d^2\right) \cos (e+f x)+d (-d-i c) \sin (e+f x)\right)}{d^2 (c-i d) (c \cos (e+f x)+d \sin (e+f x))}-\frac{8 i e^{-3 i e} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{(c-i d)^{3/2}}\right)}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 c \sqrt{c+d \tan (e+f x)}}{d^2 f (d+i c)}+\frac{2 (c+i d) \left(a^3+i a^3 \tan (e+f x)\right)}{d f (c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{8 i a^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}",1,"(a^3*(Cos[e + f*x] + I*Sin[e + f*x])^3*(((-8*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*E^((3*I)*e)) + (2*(Cos[3*e] - I*Sin[3*e])*(((-2*I)*c^2 + c*d + I*d^2)*Cos[e + f*x] + ((-I)*c - d)*d*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/((c - I*d)*d^2*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1126,1,189,92,4.1452745,"\int \frac{(a+i a \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2),x]","\frac{a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left(\frac{2 (c+i d) (\cos (2 e)-i \sin (2 e)) \cos (e+f x) \sqrt{c+d \tan (e+f x)}}{d (c-i d) (c \cos (e+f x)+d \sin (e+f x))}-\frac{4 i e^{-2 i e} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{(c-i d)^{3/2}}\right)}{f (\cos (f x)+i \sin (f x))^2}","\frac{2 a^2 (-d+i c)}{d f (d+i c) \sqrt{c+d \tan (e+f x)}}-\frac{4 i a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}",1,"(a^2*(Cos[e + f*x] + I*Sin[e + f*x])^2*(((-4*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*E^((2*I)*e)) + (2*(c + I*d)*Cos[e + f*x]*(Cos[2*e] - I*Sin[2*e])*Sqrt[c + d*Tan[e + f*x]])/((c - I*d)*d*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(Cos[f*x] + I*Sin[f*x])^2)","B",1
1127,1,158,76,2.7404812,"\int \frac{a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 i a e^{-i e} (\cos (e)+i \sin (e)) \left(\sqrt{c-i d} \cos (e+f x) \sqrt{c+d \tan (e+f x)}-\tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right) (c \cos (e+f x)+d \sin (e+f x))\right)}{f (c-i d)^{3/2} (c \cos (e+f x)+d \sin (e+f x))}","-\frac{2 a}{f (d+i c) \sqrt{c+d \tan (e+f x)}}-\frac{2 i a \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}",1,"((2*I)*a*(Cos[e] + I*Sin[e])*(-(ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]]*(c*Cos[e + f*x] + d*Sin[e + f*x])) + Sqrt[c - I*d]*Cos[e + f*x]*Sqrt[c + d*Tan[e + f*x]]))/((c - I*d)^(3/2)*E^(I*e)*f*(c*Cos[e + f*x] + d*Sin[e + f*x]))","B",1
1128,1,297,205,3.2391278,"\int \frac{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(\frac{2 \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt{c+d \tan (e+f x)} \left(\left(c^2-i c d-4 d^2\right) \cos (e+f x)+d (c-5 i d) \sin (e+f x)\right)}{(c-i d) (c+i d)^2 (c \cos (e+f x)+d \sin (e+f x))}-\frac{2 (\cos (e)+i \sin (e)) \left(i (-c-i d)^{5/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)-i \sqrt{-c+i d} \left(c^2+3 i c d+4 d^2\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)\right)}{(-c-i d)^{5/2} (-c+i d)^{3/2}}\right)}{4 f (a+i a \tan (e+f x))}","\frac{d (c-5 i d)}{2 a f (c-i d) (c+i d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 a f (c-i d)^{3/2}}+\frac{(-4 d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 a f (c+i d)^{5/2}}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-2*((-I)*Sqrt[-c + I*d]*(c^2 + (3*I)*c*d + 4*d^2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + I*(-c - I*d)^(5/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*Sin[e]))/((-c - I*d)^(5/2)*(-c + I*d)^(3/2)) + (2*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*((c^2 - I*c*d - 4*d^2)*Cos[e + f*x] + (c - (5*I)*d)*d*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/((c - I*d)*(c + I*d)^2*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(4*f*(a + I*a*Tan[e + f*x]))","A",1
1129,1,388,281,4.8470204,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left(\frac{(\cos (2 f x)-i \sin (2 f x)) \sqrt{c+d \tan (e+f x)} \left(\left(12 i c^3-23 c^2 d+26 i c d^2+23 d^3\right) \cos (e+f x)+\left(4 i c^3-5 c^2 d+18 i c d^2+41 d^3\right) \cos (3 (e+f x))-4 \left(2 c^3+3 i c^2 d+16 c d^2-43 i d^3\right) \sin (e+f x) \cos ^2(e+f x)\right)}{2 (c-i d) (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))}+\frac{2 (\cos (2 e)+i \sin (2 e)) \left(\sqrt{-c+i d} \left(-2 i c^3+8 c^2 d+13 i c d^2+23 d^3\right) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)-2 i (-c-i d)^{7/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{7/2} (-c+i d)^{3/2}}\right)}{16 f (a+i a \tan (e+f x))^2}","\frac{d \left(2 c^2+7 i c d+25 d^2\right)}{8 a^2 f (c-i d) (c+i d)^3 \sqrt{c+d \tan (e+f x)}}+\frac{\left(2 i c^2-10 c d-23 i d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{8 a^2 f (c+i d)^{7/2}}+\frac{-7 d+2 i c}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{4 a^2 f (c-i d)^{3/2}}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 8*c^2*d + (13*I)*c*d^2 + 23*d^3)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - (2*I)*(-c - I*d)^(7/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[2*e] + I*Sin[2*e]))/((-c - I*d)^(7/2)*(-c + I*d)^(3/2)) + ((Cos[2*f*x] - I*Sin[2*f*x])*(((12*I)*c^3 - 23*c^2*d + (26*I)*c*d^2 + 23*d^3)*Cos[e + f*x] + ((4*I)*c^3 - 5*c^2*d + (18*I)*c*d^2 + 41*d^3)*Cos[3*(e + f*x)] - 4*(2*c^3 + (3*I)*c^2*d + 16*c*d^2 - (43*I)*d^3)*Cos[e + f*x]^2*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(2*(c - I*d)*(c + I*d)^3*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(16*f*(a + I*a*Tan[e + f*x])^2)","A",1
1130,1,989,368,7.9602854,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\sec ^3(e+f x) (\cos (3 e)+i \sin (3 e)) \left(\frac{2 \left(60 i d^4-17 c d^3+9 i c^2 d^2+2 c^3 d\right) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)}{2 \sqrt{-c-i d}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right)}{2 \sqrt{i d-c}}\right) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)}-\frac{i \left(4 c^4+18 i d c^3-33 d^2 c^2-33 i d^3 c-56 d^4\right) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)}{\sqrt{-c-i d}}-\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right)}{\sqrt{i d-c}}\right) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)}\right) (\cos (f x)+i \sin (f x))^3}{32 (c-i d) (c+i d)^4 f (i \tan (e+f x) a+a)^3}+\frac{\sec ^3(e+f x) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left(\frac{\left(18 c^2+79 i d c-118 d^2\right) \cos (2 f x) \left(\frac{1}{96} i \cos (e)-\frac{\sin (e)}{96}\right)}{(c+i d)^4}+\frac{(9 c+20 i d) \cos (4 f x) \left(\frac{1}{96} i \cos (e)+\frac{\sin (e)}{96}\right)}{(c+i d)^3}+\frac{\left(11 \cos (e) c^4+43 i d \cos (e) c^3+11 d \sin (e) c^3-46 d^2 \cos (e) c^2+43 i d^2 \sin (e) c^2+100 i d^3 \cos (e) c-46 d^3 \sin (e) c+192 d^4 \cos (e)+100 i d^4 \sin (e)\right) \left(\frac{1}{96} \cos (3 e)+\frac{1}{96} i \sin (3 e)\right)}{(c-i d) (c+i d)^4 (-i c \cos (e)-i d \sin (e))}+\frac{\cos (6 f x) \left(\frac{1}{48} i \cos (3 e)+\frac{1}{48} \sin (3 e)\right)}{(c+i d)^2}+\frac{\left(18 c^2+79 i d c-118 d^2\right) \left(\frac{\cos (e)}{96}+\frac{1}{96} i \sin (e)\right) \sin (2 f x)}{(c+i d)^4}+\frac{(9 c+20 i d) \left(\frac{\cos (e)}{96}-\frac{1}{96} i \sin (e)\right) \sin (4 f x)}{(c+i d)^3}+\frac{\left(\frac{1}{48} \cos (3 e)-\frac{1}{48} i \sin (3 e)\right) \sin (6 f x)}{(c+i d)^2}+\frac{2 \left(\frac{1}{2} \cos (3 e-f x) d^5-\frac{1}{2} \cos (3 e+f x) d^5+\frac{1}{2} i \sin (3 e-f x) d^5-\frac{1}{2} i \sin (3 e+f x) d^5\right)}{(c-i d) (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right) (\cos (f x)+i \sin (f x))^3}{f (i \tan (e+f x) a+a)^3}","\frac{6 c^2+27 i c d-56 d^2}{48 f (-d+i c)^3 \left(a^3+i a^3 \tan (e+f x)\right) \sqrt{c+d \tan (e+f x)}}+\frac{d \left(2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right)}{16 a^3 f (c-i d) (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{\left(2 i c^3-12 c^2 d-33 i c d^2+58 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{16 a^3 f (c+i d)^{9/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{8 a^3 f (c-i d)^{3/2}}+\frac{-10 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((18*c^2 + (79*I)*c*d - 118*d^2)*Cos[2*f*x]*((I/96)*Cos[e] - Sin[e]/96))/(c + I*d)^4 + ((9*c + (20*I)*d)*Cos[4*f*x]*((I/96)*Cos[e] + Sin[e]/96))/(c + I*d)^3 + ((11*c^4*Cos[e] + (43*I)*c^3*d*Cos[e] - 46*c^2*d^2*Cos[e] + (100*I)*c*d^3*Cos[e] + 192*d^4*Cos[e] + 11*c^3*d*Sin[e] + (43*I)*c^2*d^2*Sin[e] - 46*c*d^3*Sin[e] + (100*I)*d^4*Sin[e])*(Cos[3*e]/96 + (I/96)*Sin[3*e]))/((c - I*d)*(c + I*d)^4*((-I)*c*Cos[e] - I*d*Sin[e])) + (Cos[6*f*x]*((I/48)*Cos[3*e] + Sin[3*e]/48))/(c + I*d)^2 + ((18*c^2 + (79*I)*c*d - 118*d^2)*(Cos[e]/96 + (I/96)*Sin[e])*Sin[2*f*x])/(c + I*d)^4 + ((9*c + (20*I)*d)*(Cos[e]/96 - (I/96)*Sin[e])*Sin[4*f*x])/(c + I*d)^3 + ((Cos[3*e]/48 - (I/48)*Sin[3*e])*Sin[6*f*x])/(c + I*d)^2 + (2*((d^5*Cos[3*e - f*x])/2 - (d^5*Cos[3*e + f*x])/2 + (I/2)*d^5*Sin[3*e - f*x] - (I/2)*d^5*Sin[3*e + f*x]))/((c - I*d)*(c + I*d)^4*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[3*e] + I*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(((-I)*(4*c^4 + (18*I)*c^3*d - 33*c^2*d^2 - (33*I)*c*d^3 - 56*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/Sqrt[-c - I*d] - ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/Sqrt[-c + I*d])*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2)) + (2*(2*c^3*d + (9*I)*c^2*d^2 - 17*c*d^3 + (60*I)*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/(2*Sqrt[-c - I*d]) + ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/(2*Sqrt[-c + I*d]))*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2))))/(32*(c - I*d)*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])^3)","B",0
1131,1,232,158,9.2369496,"\int \frac{(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]","\frac{a^3 (\cos (e+f x)+i \sin (e+f x))^3 \left(\frac{2 (c+i d) (\sin (3 e)+i \cos (3 e)) \cos (e+f x) \sqrt{c+d \tan (e+f x)} \left(\left(2 c^2-9 i c d-d^2\right) \cos (e+f x)+3 d (c-3 i d) \sin (e+f x)\right)}{3 d^2 (c-i d)^2 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac{8 i e^{-3 i e} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{(c-i d)^{5/2}}\right)}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 (-d+i c) (c-4 i d)}{3 d^2 f (c-i d)^2 \sqrt{c+d \tan (e+f x)}}+\frac{2 (c+i d) \left(a^3+i a^3 \tan (e+f x)\right)}{3 d f (c-i d) (c+d \tan (e+f x))^{3/2}}-\frac{8 i a^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}",1,"(a^3*(Cos[e + f*x] + I*Sin[e + f*x])^3*(((-8*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*E^((3*I)*e)) + (2*(c + I*d)*Cos[e + f*x]*(I*Cos[3*e] + Sin[3*e])*((2*c^2 - (9*I)*c*d - d^2)*Cos[e + f*x] + 3*(c - (3*I)*d)*d*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*(c - I*d)^2*d^2*(c*Cos[e + f*x] + d*Sin[e + f*x])^2)))/(f*(Cos[f*x] + I*Sin[f*x])^3)","A",1
1132,1,218,127,6.4669782,"\int \frac{(a+i a \tan (e+f x))^2}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(5/2),x]","\frac{a^2 (\cos (e+f x)+i \sin (e+f x))^2 \left(\frac{2 (\cos (2 e)-i \sin (2 e)) \cos (e+f x) \sqrt{c+d \tan (e+f x)} \left(\left(c^2+6 i c d+d^2\right) \cos (e+f x)+6 i d^2 \sin (e+f x)\right)}{3 d (c-i d)^2 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac{4 i e^{-2 i e} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{(c-i d)^{5/2}}\right)}{f (\cos (f x)+i \sin (f x))^2}","\frac{4 i a^2}{f (c-i d)^2 \sqrt{c+d \tan (e+f x)}}+\frac{2 a^2 (-d+i c)}{3 d f (d+i c) (c+d \tan (e+f x))^{3/2}}-\frac{4 i a^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}",1,"(a^2*(Cos[e + f*x] + I*Sin[e + f*x])^2*(((-4*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*E^((2*I)*e)) + (2*Cos[e + f*x]*(Cos[2*e] - I*Sin[2*e])*((c^2 + (6*I)*c*d + d^2)*Cos[e + f*x] + (6*I)*d^2*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*(c - I*d)^2*d*(c*Cos[e + f*x] + d*Sin[e + f*x])^2)))/(f*(Cos[f*x] + I*Sin[f*x])^2)","A",1
1133,1,198,109,4.4332837,"\int \frac{a+i a \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x])^(5/2),x]","\frac{\cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left(\frac{2 (\cos (e)-i \sin (e)) \cos (e+f x) \sqrt{c+d \tan (e+f x)} ((d+4 i c) \cos (e+f x)+3 i d \sin (e+f x))}{3 (c-i d)^2 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac{2 i e^{-i e} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{\sqrt{c-i d}}\right)}{(c-i d)^{5/2}}\right)}{f}","\frac{2 i a}{f (c-i d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 a}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}-\frac{2 i a \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}",1,"(Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(a + I*a*Tan[e + f*x])*(((-2*I)*ArcTanh[Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*E^(I*e)) + (2*Cos[e + f*x]*(Cos[e] - I*Sin[e])*(((4*I)*c + d)*Cos[e + f*x] + (3*I)*d*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*(c - I*d)^2*(c*Cos[e + f*x] + d*Sin[e + f*x])^2)))/f","A",1
1134,1,371,267,6.7083107,"\int \frac{1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(\frac{(\cos (f x)-i \sin (f x)) \sqrt{c+d \tan (e+f x)} \left((d+i c) \left(\left(3 c^3-3 i c^2 d-43 c d^2+11 i d^3\right) \cos (3 (e+f x))-8 d \left(-3 c^2+23 i c d+4 d^2\right) \sin (e+f x) \cos ^2(e+f x)\right)+3 \left(3 i c^4+6 c^3 d-42 i c^2 d^2+2 c d^3-9 i d^4\right) \cos (e+f x)\right)}{6 (c-i d)^2 (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))^2}-\frac{2 (\cos (e)+i \sin (e)) \left((-c+i d)^{5/2} (6 d-i c) \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)-i (-c-i d)^{7/2} \tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c+i d}}\right)\right)}{(-c-i d)^{7/2} (-c+i d)^{5/2}}\right)}{4 f (a+i a \tan (e+f x))}","\frac{d (7 d+3 i c)}{6 a f (-d+i c) \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}+\frac{d \left(c^2-14 i c d-5 d^2\right)}{2 a f (c-i d)^2 (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 a f (c-i d)^{5/2}}+\frac{(-6 d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 a f (c+i d)^{7/2}}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-2*((-c + I*d)^(5/2)*((-I)*c + 6*d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - I*(-c - I*d)^(7/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*Sin[e]))/((-c - I*d)^(7/2)*(-c + I*d)^(5/2)) + ((Cos[f*x] - I*Sin[f*x])*(3*((3*I)*c^4 + 6*c^3*d - (42*I)*c^2*d^2 + 2*c*d^3 - (9*I)*d^4)*Cos[e + f*x] + (I*c + d)*((3*c^3 - (3*I)*c^2*d - 43*c*d^2 + (11*I)*d^3)*Cos[3*(e + f*x)] - 8*d*(-3*c^2 + (23*I)*c*d + 4*d^2)*Cos[e + f*x]^2*Sin[e + f*x]))*Sqrt[c + d*Tan[e + f*x]])/(6*(c - I*d)^2*(c + I*d)^3*(c*Cos[e + f*x] + d*Sin[e + f*x])^2)))/(4*f*(a + I*a*Tan[e + f*x]))","A",1
1135,1,1004,351,9.9710242,"\int \frac{1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\sec ^2(e+f x) (\cos (2 e)+i \sin (2 e)) \left(\frac{2 \left(-45 i d^4+88 c d^3+9 i c^2 d^2+2 c^3 d\right) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)}{2 \sqrt{-c-i d}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right)}{2 \sqrt{i d-c}}\right) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)}-\frac{i \left(4 c^4+18 i d c^3-33 d^2 c^2+72 i d^3 c+49 d^4\right) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)}{\sqrt{-c-i d}}-\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right)}{\sqrt{i d-c}}\right) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)}\right) (\cos (f x)+i \sin (f x))^2}{16 (c-i d)^2 (c+i d)^4 f (i \tan (e+f x) a+a)^2}+\frac{\sec ^2(e+f x) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left(\frac{i (4 c+15 i d) \cos (2 f x)}{16 (c+i d)^4}+\frac{\left(9 i \cos (e) c^4-24 d \cos (e) c^3+9 i d \sin (e) c^3+75 i d^2 \cos (e) c^2-24 d^2 \sin (e) c^2+458 d^3 \cos (e) c+75 i d^3 \sin (e) c-192 i d^4 \cos (e)+10 d^4 \sin (e)\right) \left(\frac{1}{48} \cos (2 e)+\frac{1}{48} i \sin (2 e)\right)}{(c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e))}+\frac{\cos (4 f x) \left(\frac{1}{16} i \cos (2 e)+\frac{1}{16} \sin (2 e)\right)}{(c+i d)^3}+\frac{(4 c+15 i d) \sin (2 f x)}{16 (c+i d)^4}+\frac{\left(\frac{1}{16} \cos (2 e)-\frac{1}{16} i \sin (2 e)\right) \sin (4 f x)}{(c+i d)^3}-\frac{4 \left(\frac{3}{2} \cos (2 e-f x) d^5-\frac{3}{2} \cos (2 e+f x) d^5+\frac{3}{2} i \sin (2 e-f x) d^5-\frac{3}{2} i \sin (2 e+f x) d^5+\frac{7}{2} i c \cos (2 e-f x) d^4-\frac{7}{2} i c \cos (2 e+f x) d^4-\frac{7}{2} c \sin (2 e-f x) d^4+\frac{7}{2} c \sin (2 e+f x) d^4\right)}{3 (c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} \cos (2 e) d^5+\frac{2}{3} i \sin (2 e) d^5}{(c-i d)^2 (c+i d)^4 (c \cos (e+f x)+d \sin (e+f x))^2}\right) (\cos (f x)+i \sin (f x))^2}{f (i \tan (e+f x) a+a)^2}","\frac{d \left(6 c^2+27 i c d+49 d^2\right)}{24 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^{3/2}}+\frac{\left(2 i c^2-14 c d-47 i d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{8 a^2 f (c+i d)^{9/2}}+\frac{d \left(2 c^3+9 i c^2 d+88 c d^2-45 i d^3\right)}{8 a^2 f (c-i d)^2 (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{-9 d+2 i c}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{4 a^2 f (c-i d)^{5/2}}-\frac{1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((I/16)*(4*c + (15*I)*d)*Cos[2*f*x])/(c + I*d)^4 + (((9*I)*c^4*Cos[e] - 24*c^3*d*Cos[e] + (75*I)*c^2*d^2*Cos[e] + 458*c*d^3*Cos[e] - (192*I)*d^4*Cos[e] + (9*I)*c^3*d*Sin[e] - 24*c^2*d^2*Sin[e] + (75*I)*c*d^3*Sin[e] + 10*d^4*Sin[e])*(Cos[2*e]/48 + (I/48)*Sin[2*e]))/((c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + (Cos[4*f*x]*((I/16)*Cos[2*e] + Sin[2*e]/16))/(c + I*d)^3 + ((4*c + (15*I)*d)*Sin[2*f*x])/(16*(c + I*d)^4) + ((Cos[2*e]/16 - (I/16)*Sin[2*e])*Sin[4*f*x])/(c + I*d)^3 + ((2*d^5*Cos[2*e])/3 + ((2*I)/3)*d^5*Sin[2*e])/((c - I*d)^2*(c + I*d)^4*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) - (4*(((7*I)/2)*c*d^4*Cos[2*e - f*x] + (3*d^5*Cos[2*e - f*x])/2 - ((7*I)/2)*c*d^4*Cos[2*e + f*x] - (3*d^5*Cos[2*e + f*x])/2 - (7*c*d^4*Sin[2*e - f*x])/2 + ((3*I)/2)*d^5*Sin[2*e - f*x] + (7*c*d^4*Sin[2*e + f*x])/2 - ((3*I)/2)*d^5*Sin[2*e + f*x]))/(3*(c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^2) + (Sec[e + f*x]^2*(Cos[2*e] + I*Sin[2*e])*(Cos[f*x] + I*Sin[f*x])^2*(((-I)*(4*c^4 + (18*I)*c^3*d - 33*c^2*d^2 + (72*I)*c*d^3 + 49*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/Sqrt[-c - I*d] - ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/Sqrt[-c + I*d])*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2)) + (2*(2*c^3*d + (9*I)*c^2*d^2 + 88*c*d^3 - (45*I)*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/(2*Sqrt[-c - I*d]) + ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/(2*Sqrt[-c + I*d]))*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2))))/(16*(c - I*d)^2*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])^2)","B",0
1136,1,1160,446,10.8926779,"\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\sec ^3(e+f x) (\cos (3 e)+i \sin (3 e)) \left(\frac{2 \left(150 d^5+253 i c d^4-26 c^2 d^3+11 i c^3 d^2+2 c^4 d\right) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)}{2 \sqrt{-c-i d}}+\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right)}{2 \sqrt{i d-c}}\right) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)}-\frac{i \left(4 c^5+22 i d c^4-51 d^2 c^3-66 i d^3 c^2-233 d^4 c+154 i d^5\right) \left(\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{-c-i d}}\right)}{\sqrt{-c-i d}}-\frac{\tan ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{i d-c}}\right)}{\sqrt{i d-c}}\right) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left(\tan ^2(e+f x)+1\right)}\right) (\cos (f x)+i \sin (f x))^3}{32 (c-i d)^2 (c+i d)^5 f (i \tan (e+f x) a+a)^3}+\frac{\sec ^3(e+f x) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left(\frac{\left(18 c^2+103 i d c-208 d^2\right) \cos (2 f x) \left(\frac{1}{96} i \cos (e)-\frac{\sin (e)}{96}\right)}{(c+i d)^5}+\frac{(9 c+26 i d) \cos (4 f x) \left(\frac{1}{96} i \cos (e)+\frac{\sin (e)}{96}\right)}{(c+i d)^4}+\frac{\left(11 i \cos (e) c^5-50 d \cos (e) c^4+11 i d \sin (e) c^4-51 i d^2 \cos (e) c^3-50 d^2 \sin (e) c^3-296 d^3 \cos (e) c^2-51 i d^3 \sin (e) c^2+1208 i d^4 \cos (e) c-296 d^4 \sin (e) c+576 d^5 \cos (e)+120 i d^5 \sin (e)\right) \left(\frac{1}{96} \cos (3 e)+\frac{1}{96} i \sin (3 e)\right)}{(c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e))}+\frac{\cos (6 f x) \left(\frac{1}{48} i \cos (3 e)+\frac{1}{48} \sin (3 e)\right)}{(c+i d)^3}+\frac{\left(18 c^2+103 i d c-208 d^2\right) \left(\frac{\cos (e)}{96}+\frac{1}{96} i \sin (e)\right) \sin (2 f x)}{(c+i d)^5}+\frac{(9 c+26 i d) \left(\frac{\cos (e)}{96}-\frac{1}{96} i \sin (e)\right) \sin (4 f x)}{(c+i d)^4}+\frac{\left(\frac{1}{48} \cos (3 e)-\frac{1}{48} i \sin (3 e)\right) \sin (6 f x)}{(c+i d)^3}+\frac{2 \left(-\frac{9}{2} i \cos (3 e-f x) d^6+\frac{9}{2} i \cos (3 e+f x) d^6+\frac{9}{2} \sin (3 e-f x) d^6-\frac{9}{2} \sin (3 e+f x) d^6+\frac{17}{2} c \cos (3 e-f x) d^5-\frac{17}{2} c \cos (3 e+f x) d^5+\frac{17}{2} i c \sin (3 e-f x) d^5-\frac{17}{2} i c \sin (3 e+f x) d^5\right)}{3 (c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} i d^6 \cos (3 e)-\frac{2}{3} d^6 \sin (3 e)}{(c-i d)^2 (c+i d)^5 (c \cos (e+f x)+d \sin (e+f x))^2}\right) (\cos (f x)+i \sin (f x))^3}{f (i \tan (e+f x) a+a)^3}","\frac{2 c^2+11 i c d-30 d^2}{16 f (-d+i c)^3 \left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^{3/2}}+\frac{d \left(6 c^3+33 i c^2 d-83 c d^2+154 i d^3\right)}{48 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac{\left(2 i c^3-16 c^2 d-61 i c d^2+152 d^3\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{16 a^3 f (c+i d)^{11/2}}+\frac{d \left(2 c^4+11 i c^3 d-26 c^2 d^2+253 i c d^3+150 d^4\right)}{16 a^3 f (c-i d)^2 (c+i d)^5 \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{8 a^3 f (c-i d)^{5/2}}+\frac{-4 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((18*c^2 + (103*I)*c*d - 208*d^2)*Cos[2*f*x]*((I/96)*Cos[e] - Sin[e]/96))/(c + I*d)^5 + ((9*c + (26*I)*d)*Cos[4*f*x]*((I/96)*Cos[e] + Sin[e]/96))/(c + I*d)^4 + (((11*I)*c^5*Cos[e] - 50*c^4*d*Cos[e] - (51*I)*c^3*d^2*Cos[e] - 296*c^2*d^3*Cos[e] + (1208*I)*c*d^4*Cos[e] + 576*d^5*Cos[e] + (11*I)*c^4*d*Sin[e] - 50*c^3*d^2*Sin[e] - (51*I)*c^2*d^3*Sin[e] - 296*c*d^4*Sin[e] + (120*I)*d^5*Sin[e])*(Cos[3*e]/96 + (I/96)*Sin[3*e]))/((c - I*d)^2*(c + I*d)^5*(c*Cos[e] + d*Sin[e])) + (Cos[6*f*x]*((I/48)*Cos[3*e] + Sin[3*e]/48))/(c + I*d)^3 + ((18*c^2 + (103*I)*c*d - 208*d^2)*(Cos[e]/96 + (I/96)*Sin[e])*Sin[2*f*x])/(c + I*d)^5 + ((9*c + (26*I)*d)*(Cos[e]/96 - (I/96)*Sin[e])*Sin[4*f*x])/(c + I*d)^4 + ((Cos[3*e]/48 - (I/48)*Sin[3*e])*Sin[6*f*x])/(c + I*d)^3 + (((2*I)/3)*d^6*Cos[3*e] - (2*d^6*Sin[3*e])/3)/((c - I*d)^2*(c + I*d)^5*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + (2*((17*c*d^5*Cos[3*e - f*x])/2 - ((9*I)/2)*d^6*Cos[3*e - f*x] - (17*c*d^5*Cos[3*e + f*x])/2 + ((9*I)/2)*d^6*Cos[3*e + f*x] + ((17*I)/2)*c*d^5*Sin[3*e - f*x] + (9*d^6*Sin[3*e - f*x])/2 - ((17*I)/2)*c*d^5*Sin[3*e + f*x] - (9*d^6*Sin[3*e + f*x])/2))/(3*(c - I*d)^2*(c + I*d)^5*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[3*e] + I*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(((-I)*(4*c^5 + (22*I)*c^4*d - 51*c^3*d^2 - (66*I)*c^2*d^3 - 233*c*d^4 + (154*I)*d^5)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/Sqrt[-c - I*d] - ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/Sqrt[-c + I*d])*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2)) + (2*(2*c^4*d + (11*I)*c^3*d^2 - 26*c^2*d^3 + (253*I)*c*d^4 + 150*d^5)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/(2*Sqrt[-c - I*d]) + ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/(2*Sqrt[-c + I*d]))*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2))))/(32*(c - I*d)^2*(c + I*d)^5*f*(a + I*a*Tan[e + f*x])^3)","B",0
1137,1,589,263,7.9002432,"\int (a+i a \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]],x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \cos ^2(e+f x) (a+i a \tan (e+f x))^{5/2} \left(\frac{(1+i) (\sin (2 e)+i \cos (2 e)) \sqrt{c+d \tan (e+f x)} (c+2 d \tan (e+f x)-9 i d)}{d}-\frac{(\cos (2 e)-i \sin (2 e)) \cos (e+f x) \left(\left(c^2+10 i c d+23 d^2\right) \left(\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{\sqrt{d} \left(c^2+10 i c d+23 d^2\right) \left(e^{i (e+f x)}+i\right)}\right)-\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} \left(c^2+10 i c d+23 d^2\right) \left(e^{i (e+f x)}-i\right)}\right)\right)+(32+32 i) d^{3/2} \sqrt{c-i d} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{d^{3/2} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f (\cos (f x)+i \sin (f x))^2}","-\frac{\sqrt[4]{-1} a^{5/2} \left(c^2+10 i c d+23 d^2\right) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{4 d^{3/2} f}-\frac{4 i \sqrt{2} a^{5/2} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}-\frac{a^2 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac{a^2 (c+9 i d) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d f}",1,"((1/8 + I/8)*Cos[e + f*x]^2*(a + I*a*Tan[e + f*x])^(5/2)*(-((Cos[e + f*x]*((c^2 + (10*I)*c*d + 23*d^2)*(Log[((2 + 2*I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(c^2 + (10*I)*c*d + 23*d^2)*(I + E^(I*(e + f*x))))] - Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(c^2 + (10*I)*c*d + 23*d^2)*(-I + E^(I*(e + f*x))))]) + (32 + 32*I)*Sqrt[c - I*d]*d^(3/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] - I*Sin[2*e]))/(d^(3/2)*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]])) + ((1 + I)*(I*Cos[2*e] + Sin[2*e])*Sqrt[c + d*Tan[e + f*x]]*(c - (9*I)*d + 2*d*Tan[e + f*x]))/d))/(f*(Cos[f*x] + I*Sin[f*x])^2)","B",0
1138,1,559,250,6.0949268,"\int (a+i a \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]],x]","\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \left((1-i) \sin (e) \sqrt{c+d \tan (e+f x)}+(1+i) \cos (e) \sqrt{c+d \tan (e+f x)}+\frac{(\cos (e)-i \sin (e)) \cos (e+f x) \left((-3 d-i c) \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{\sqrt{d} (3 d+i c) \left(e^{i (e+f x)}+i\right)}\right)+(3 d+i c) \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} (3 d+i c) \left(e^{i (e+f x)}-i\right)}\right)-(4+4 i) \sqrt{d} \sqrt{c-i d} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{d} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f}","-\frac{\sqrt[4]{-1} a^{3/2} (3 d+i c) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{d} f}-\frac{2 i \sqrt{2} a^{3/2} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}-\frac{a^2 (c+d \tan (e+f x))^{3/2}}{d f \sqrt{a+i a \tan (e+f x)}}+\frac{a^2 (c+i d) \sqrt{c+d \tan (e+f x)}}{d f \sqrt{a+i a \tan (e+f x)}}",1,"((1/2 + I/2)*Cos[e + f*x]*(Cos[f*x] - I*Sin[f*x])*(a + I*a*Tan[e + f*x])^(3/2)*((Cos[e + f*x]*(((-I)*c - 3*d)*Log[((2 + 2*I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I*c + 3*d)*(I + E^(I*(e + f*x))))] + (I*c + 3*d)*Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I*c + 3*d)*(-I + E^(I*(e + f*x))))] - (4 + 4*I)*Sqrt[c - I*d]*Sqrt[d]*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[e] - I*Sin[e]))/(Sqrt[d]*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (1 + I)*Cos[e]*Sqrt[c + d*Tan[e + f*x]] + (1 - I)*Sin[e]*Sqrt[c + d*Tan[e + f*x]]))/f","B",0
1139,1,442,151,3.5744201,"\int \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]],x]","-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{-i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \sqrt{a+i a \tan (e+f x)} \left(\sqrt{d} \left(\log \left(\frac{(1+i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{d^{3/2} \left(e^{i (e+f x)}+i\right)}\right)-\log \left(\frac{(1+i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{d^{3/2} \left(e^{i (e+f x)}-i\right)}\right)\right)+(1+i) \sqrt{c-i d} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)\right)}{f}","-\frac{2 \sqrt[4]{-1} \sqrt{a} \sqrt{d} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{f}-\frac{i \sqrt{2} \sqrt{a} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}",1,"((-1/2 - I/2)*Sqrt[1 + E^((2*I)*(e + f*x))]*((1 + I)*Sqrt[c - I*d]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] + Sqrt[d]*(Log[((1 + I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(3/2)*(I + E^(I*(e + f*x))))] - Log[((1 + I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(3/2)*(-I + E^(I*(e + f*x))))]))*Sqrt[a + I*a*Tan[e + f*x]])/(E^(I*(e + f*x))*f)","B",0
1140,1,182,121,3.4913385,"\int \frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{i \left(\sqrt{1+e^{2 i (e+f x)}} \sqrt{c+d \tan (e+f x)}-\sqrt{c-i d} e^{i (e+f x)} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)\right)}{f \sqrt{1+e^{2 i (e+f x)}} \sqrt{a+i a \tan (e+f x)}}","\frac{i \sqrt{c+d \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{\sqrt{2} \sqrt{a} f}",1,"(I*(-(Sqrt[c - I*d]*E^(I*(e + f*x))*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c + d*Tan[e + f*x]]))/(Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[a + I*a*Tan[e + f*x]])","A",1
1141,1,249,177,4.4240128,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{\sec ^{\frac{3}{2}}(e+f x) \left(-i \sqrt{2} \sqrt{c-i d} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(1+e^{2 i (e+f x)}\right)^{3/2} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)-\frac{2 ((3 c+i d) \tan (e+f x)-5 i c+3 d) \sqrt{c+d \tan (e+f x)}}{3 (c+i d) \sec ^{\frac{3}{2}}(e+f x)}\right)}{4 f (a+i a \tan (e+f x))^{3/2}}","-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}-\frac{(c+d \tan (e+f x))^{3/2}}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2}}+\frac{i \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}",1,"(Sec[e + f*x]^(3/2)*((-I)*Sqrt[2]*Sqrt[c - I*d]*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2)*(1 + E^((2*I)*(e + f*x)))^(3/2)*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] - (2*((-5*I)*c + 3*d + (3*c + I*d)*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)*Sec[e + f*x]^(3/2))))/(4*f*(a + I*a*Tan[e + f*x])^(3/2))","A",1
1142,1,302,254,5.8579898,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(e+f x) \left(\frac{2 i \sqrt{c+d \tan (e+f x)} \left(\left(26 c^2+40 i c d-6 d^2\right) \cos (2 (e+f x))+11 c^2+4 i c (5 c+7 i d) \sin (2 (e+f x))+20 i c d-9 d^2\right)}{15 (c+i d)^2 \sqrt{\sec (e+f x)}}-i \sqrt{2} \sqrt{c-i d} e^{2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)\right)}{8 f (a+i a \tan (e+f x))^{5/2}}","-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{4 \sqrt{2} a^{5/2} f}-\frac{\left(20 c d-i \left(15 c^2+3 d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{60 a^2 f (c+i d)^2 \sqrt{a+i a \tan (e+f x)}}+\frac{(-3 d+5 i c) \sqrt{c+d \tan (e+f x)}}{30 a f (c+i d) (a+i a \tan (e+f x))^{3/2}}+\frac{i \sqrt{c+d \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}",1,"(Sec[e + f*x]^(5/2)*((-I)*Sqrt[2]*Sqrt[c - I*d]*E^((2*I)*(e + f*x))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] + (((2*I)/15)*(11*c^2 + (20*I)*c*d - 9*d^2 + (26*c^2 + (40*I)*c*d - 6*d^2)*Cos[2*(e + f*x)] + (4*I)*c*(5*c + (7*I)*d)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/((c + I*d)^2*Sqrt[Sec[e + f*x]])))/(8*f*(a + I*a*Tan[e + f*x])^(5/2))","A",0
1143,1,686,329,10.0070055,"\int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2),x]","\frac{\left(\frac{1}{16}+\frac{i}{16}\right) \cos ^2(e+f x) (a+i a \tan (e+f x))^{5/2} \left(\frac{\left(\frac{1}{6}+\frac{i}{6}\right) (\sin (2 e)+i \cos (2 e)) \sec ^2(e+f x) \sqrt{c+d \tan (e+f x)} \left(\left(3 c^2-68 i c d-65 d^2\right) \cos (2 (e+f x))+3 c^2+2 d (7 c-13 i d) \sin (2 (e+f x))-68 i c d-49 d^2\right)}{d}-\frac{i (\cos (2 e)-i \sin (2 e)) \cos (e+f x) \left(\left(-i c^3+15 c^2 d-69 i c d^2-45 d^3\right) \left(\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{\sqrt{d} \left(i c^3-15 c^2 d+69 i c d^2+45 d^3\right) \left(e^{i (e+f x)}+i\right)}\right)-\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} \left(i c^3-15 c^2 d+69 i c d^2+45 d^3\right) \left(e^{i (e+f x)}-i\right)}\right)\right)+(64-64 i) d^{3/2} (c-i d)^{3/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{d^{3/2} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f (\cos (f x)+i \sin (f x))^2}","-\frac{\sqrt[4]{-1} a^{5/2} (c-3 i d) \left(c^2+18 i c d+15 d^2\right) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{8 d^{3/2} f}-\frac{4 i \sqrt{2} a^{5/2} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}+\frac{a^2 \left(c^2+14 i c d+19 d^2\right) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 d f}-\frac{a^2 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac{a^2 (c+13 i d) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 d f}",1,"((1/16 + I/16)*Cos[e + f*x]^2*(a + I*a*Tan[e + f*x])^(5/2)*(((-I)*Cos[e + f*x]*(((-I)*c^3 + 15*c^2*d - (69*I)*c*d^2 - 45*d^3)*(Log[((2 + 2*I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I*c^3 - 15*c^2*d + (69*I)*c*d^2 + 45*d^3)*(I + E^(I*(e + f*x))))] - Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I*c^3 - 15*c^2*d + (69*I)*c*d^2 + 45*d^3)*(-I + E^(I*(e + f*x))))]) + (64 - 64*I)*(c - I*d)^(3/2)*d^(3/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] - I*Sin[2*e]))/(d^(3/2)*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + ((1/6 + I/6)*Sec[e + f*x]^2*(I*Cos[2*e] + Sin[2*e])*(3*c^2 - (68*I)*c*d - 49*d^2 + (3*c^2 - (68*I)*c*d - 65*d^2)*Cos[2*(e + f*x)] + 2*(7*c - (13*I)*d)*d*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/d))/(f*(Cos[f*x] + I*Sin[f*x])^2)","B",0
1144,1,574,315,6.9588051,"\int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) (\cos (e)-i \sin (e)) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \left((1+i) \sqrt{c+d \tan (e+f x)} (5 c+2 d \tan (e+f x)-5 i d)-\frac{\cos (e+f x) \left(\left(3 i c^2+18 c d-11 i d^2\right) \left(\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1-i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}-c \left(e^{i (e+f x)}+i\right)+i d e^{i (e+f x)}+d\right)}{\sqrt{d} \left(-3 c^2+18 i c d+11 d^2\right) \left(e^{i (e+f x)}+i\right)}\right)-\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} \left(3 i c^2+18 c d-11 i d^2\right) \left(e^{i (e+f x)}-i\right)}\right)\right)+(16+16 i) \sqrt{d} (c-i d)^{3/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{d} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f}","-\frac{\sqrt[4]{-1} a^{3/2} \left(3 i c^2+18 c d-11 i d^2\right) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{4 \sqrt{d} f}-\frac{2 i \sqrt{2} a^{3/2} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}-\frac{a^2 (c+d \tan (e+f x))^{5/2}}{2 d f \sqrt{a+i a \tan (e+f x)}}+\frac{a^2 (c+i d) (c+d \tan (e+f x))^{3/2}}{2 d f \sqrt{a+i a \tan (e+f x)}}+\frac{a (5 d+3 i c) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 f}",1,"((1/8 + I/8)*Cos[e + f*x]*(Cos[e] - I*Sin[e])*(Cos[f*x] - I*Sin[f*x])*(a + I*a*Tan[e + f*x])^(3/2)*(-((Cos[e + f*x]*(((3*I)*c^2 + 18*c*d - (11*I)*d^2)*(Log[((2 + 2*I)*E^((I/2)*e)*(d + I*d*E^(I*(e + f*x)) - c*(I + E^(I*(e + f*x))) + (1 - I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(-3*c^2 + (18*I)*c*d + 11*d^2)*(I + E^(I*(e + f*x))))] - Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*((3*I)*c^2 + 18*c*d - (11*I)*d^2)*(-I + E^(I*(e + f*x))))]) + (16 + 16*I)*(c - I*d)^(3/2)*Sqrt[d]*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])]))/(Sqrt[d]*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]])) + (1 + I)*Sqrt[c + d*Tan[e + f*x]]*(5*c - (5*I)*d + 2*d*Tan[e + f*x])))/f","A",0
1145,1,507,196,7.2767016,"\int \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2),x]","\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{a+i a \tan (e+f x)} \left((1-i) d \sqrt{c+d \tan (e+f x)}-\frac{\cos (e+f x) \left(\sqrt{d} (3 c-i d) \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1-i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}-c \left(e^{i (e+f x)}+i\right)+i d e^{i (e+f x)}+d\right)}{d^{3/2} (d+3 i c) \left(e^{i (e+f x)}+i\right)}\right)+i \sqrt{d} (d+3 i c) \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{d^{3/2} (3 c-i d) \left(e^{i (e+f x)}-i\right)}\right)+(2+2 i) (c-i d)^{3/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f}","\frac{d \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}-\frac{i \sqrt{2} \sqrt{a} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}-\frac{\sqrt[4]{-1} \sqrt{a} \sqrt{d} (3 c-i d) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{f}",1,"((1/2 + I/2)*Sqrt[a + I*a*Tan[e + f*x]]*(-((Cos[e + f*x]*((3*c - I*d)*Sqrt[d]*Log[((2 + 2*I)*E^((I/2)*e)*(d + I*d*E^(I*(e + f*x)) - c*(I + E^(I*(e + f*x))) + (1 - I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(3/2)*((3*I)*c + d)*(I + E^(I*(e + f*x))))] + I*Sqrt[d]*((3*I)*c + d)*Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/((3*c - I*d)*d^(3/2)*(-I + E^(I*(e + f*x))))] + (2 + 2*I)*(c - I*d)^(3/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])]))/Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (1 - I)*d*Sqrt[c + d*Tan[e + f*x]]))/f","B",0
1146,1,518,195,7.5444945,"\int \frac{(c+d \tan (e+f x))^{3/2}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{\sqrt{\sec (e+f x)} \left(\sqrt{2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \left(-(1-i) d^{3/2} \left(\log \left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{\frac{i e}{2}} \left((1-i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}-c \left(e^{i (e+f x)}+i\right)+i d e^{i (e+f x)}+d\right)}{d^{5/2} \left(e^{i (e+f x)}+i\right)}\right)-\log \left(-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{d^{5/2} \left(e^{i (e+f x)}-i\right)}\right)\right)-i (c-i d)^{3/2} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)\right)+\frac{2 i (c+i d) \sqrt{c+d \tan (e+f x)}}{\sqrt{\sec (e+f x)}}\right)}{2 f \sqrt{a+i a \tan (e+f x)}}","\frac{2 (-1)^{3/4} d^{3/2} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a} f}+\frac{(-d+i c) \sqrt{c+d \tan (e+f x)}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{\sqrt{2} \sqrt{a} f}",1,"(Sqrt[Sec[e + f*x]]*(Sqrt[2]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*((-I)*(c - I*d)^(3/2)*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] - (1 - I)*d^(3/2)*(Log[((1/2 + I/2)*E^((I/2)*e)*(d + I*d*E^(I*(e + f*x)) - c*(I + E^(I*(e + f*x))) + (1 - I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(5/2)*(I + E^(I*(e + f*x))))] - Log[((-1/2 + I/2)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(5/2)*(-I + E^(I*(e + f*x))))])) + ((2*I)*(c + I*d)*Sqrt[c + d*Tan[e + f*x]])/Sqrt[Sec[e + f*x]]))/(2*f*Sqrt[a + I*a*Tan[e + f*x]])","B",0
1147,1,240,173,4.2096059,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{\sec ^{\frac{3}{2}}(e+f x) \left(-i \sqrt{2} (c-i d)^{3/2} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(1+e^{2 i (e+f x)}\right)^{3/2} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)-\frac{2 ((3 c-5 i d) \tan (e+f x)-5 i c-3 d) \sqrt{c+d \tan (e+f x)}}{3 \sec ^{\frac{3}{2}}(e+f x)}\right)}{4 f (a+i a \tan (e+f x))^{3/2}}","-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{i (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{(d+i c) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}",1,"(Sec[e + f*x]^(3/2)*((-I)*Sqrt[2]*(c - I*d)^(3/2)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2)*(1 + E^((2*I)*(e + f*x)))^(3/2)*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] - (2*((-5*I)*c - 3*d + (3*c - (5*I)*d)*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*Sec[e + f*x]^(3/2))))/(4*f*(a + I*a*Tan[e + f*x])^(3/2))","A",0
1148,1,302,225,5.8271846,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(e+f x) \left(\frac{2 i \sqrt{c+d \tan (e+f x)} \left(4 \left(5 i c^2+3 c d+5 i d^2\right) \sin (2 (e+f x))+2 \left(13 c^2+7 d^2\right) \cos (2 (e+f x))+11 c^2+10 i c d+d^2\right)}{15 (c+i d) \sqrt{\sec (e+f x)}}-i \sqrt{2} (c-i d)^{3/2} e^{2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)\right)}{8 f (a+i a \tan (e+f x))^{5/2}}","-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{4 \sqrt{2} a^{5/2} f}+\frac{(d+i c) \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}-\frac{(c+d \tan (e+f x))^{5/2}}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2}}+\frac{i (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^(5/2)*((-I)*Sqrt[2]*(c - I*d)^(3/2)*E^((2*I)*(e + f*x))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] + (((2*I)/15)*(11*c^2 + (10*I)*c*d + d^2 + 2*(13*c^2 + 7*d^2)*Cos[2*(e + f*x)] + 4*((5*I)*c^2 + 3*c*d + (5*I)*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/((c + I*d)*Sqrt[Sec[e + f*x]])))/(8*f*(a + I*a*Tan[e + f*x])^(5/2))","A",0
1149,1,849,415,12.8349833,"\int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2),x]","\frac{\cos ^2(e+f x) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left(\left(-\frac{1}{4} i \cos (3 e+f x) d^2-\frac{1}{4} \sin (3 e+f x) d^2\right) \sec ^3(e+f x)+(17 i c+23 d) \left(\frac{1}{24} i d \cos (2 e)+\frac{1}{24} d \sin (2 e)\right) \sec ^2(e+f x)+\left(59 c^2-226 i d c-131 d^2\right) \left(-\frac{1}{96} i \cos (3 e+f x)-\frac{1}{96} \sin (3 e+f x)\right) \sec (e+f x)+\left(-15 c^3+719 i d c^2+1621 d^2 c-845 i d^3\right) \left(\frac{\cos (2 e)}{192 d}-\frac{i \sin (2 e)}{192 d}\right)\right) (i \tan (e+f x) a+a)^{5/2}}{f (\cos (f x)+i \sin (f x))^2}-\frac{\left(\frac{1}{128}+\frac{i}{128}\right) \cos ^3(e+f x) \left((512+512 i) d^{3/2} \log \left(2 \left(\sqrt{c-i d} \cos (e+f x)+i \sqrt{c-i d} \sin (e+f x)+\sqrt{\cos (2 e+2 f x)+i \sin (2 e+2 f x)+1} \sqrt{c+d \tan (e+f x)}\right)\right) (c-i d)^{5/2}+\left(5 c^4+100 i d c^3+690 d^2 c^2-900 i d^3 c-363 d^4\right) \left(\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left(-i e^{i (e+f x)} c+c-d e^{i (e+f x)}+i d+(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)}{\sqrt{d} \left(-5 c^4-100 i d c^3-690 d^2 c^2+900 i d^3 c+363 d^4\right) \left(i+e^{i (e+f x)}\right)}\right)-\log \left(-\frac{(2+2 i) e^{\frac{i e}{2}} \left(i e^{i (e+f x)} c+c+d e^{i (e+f x)}+i d+(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)}{\sqrt{d} \left(-5 c^4-100 i d c^3-690 d^2 c^2+900 i d^3 c+363 d^4\right) \left(-i+e^{i (e+f x)}\right)}\right)\right)\right) (\cos (2 e)-i \sin (2 e)) (i \tan (e+f x) a+a)^{5/2}}{d^{3/2} f (\cos (f x)+i \sin (f x))^2 \sqrt{\cos (2 (e+f x))+i \sin (2 (e+f x))+1}}","-\frac{\sqrt[4]{-1} a^{5/2} \left(5 c^4+100 i c^3 d+690 c^2 d^2-900 i c d^3-363 d^4\right) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{64 d^{3/2} f}-\frac{4 i \sqrt{2} a^{5/2} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}+\frac{a^2 \left(5 c^2+90 i c d+107 d^2\right) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 d f}+\frac{a^2 \left(5 c^3+95 i c^2 d+273 c d^2-149 i d^3\right) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 d f}-\frac{a^2 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{a^2 (c+17 i d) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d f}",1,"((-1/128 - I/128)*Cos[e + f*x]^3*((5*c^4 + (100*I)*c^3*d + 690*c^2*d^2 - (900*I)*c*d^3 - 363*d^4)*(Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d - I*c*E^(I*(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(-5*c^4 - (100*I)*c^3*d - 690*c^2*d^2 + (900*I)*c*d^3 + 363*d^4)*(I + E^(I*(e + f*x))))] - Log[((-2 - 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(-5*c^4 - (100*I)*c^3*d - 690*c^2*d^2 + (900*I)*c*d^3 + 363*d^4)*(-I + E^(I*(e + f*x))))]) + (512 + 512*I)*(c - I*d)^(5/2)*d^(3/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*e + 2*f*x] + I*Sin[2*e + 2*f*x]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] - I*Sin[2*e])*(a + I*a*Tan[e + f*x])^(5/2))/(d^(3/2)*f*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (Cos[e + f*x]^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*((-15*c^3 + (719*I)*c^2*d + 1621*c*d^2 - (845*I)*d^3)*(Cos[2*e]/(192*d) - ((I/192)*Sin[2*e])/d) + ((17*I)*c + 23*d)*Sec[e + f*x]^2*((I/24)*d*Cos[2*e] + (d*Sin[2*e])/24) + (59*c^2 - (226*I)*c*d - 131*d^2)*Sec[e + f*x]*((-1/96*I)*Cos[3*e + f*x] - Sin[3*e + f*x]/96) + Sec[e + f*x]^3*((-1/4*I)*d^2*Cos[3*e + f*x] - (d^2*Sin[3*e + f*x])/4))*(a + I*a*Tan[e + f*x])^(5/2))/(f*(Cos[f*x] + I*Sin[f*x])^2)","B",0
1150,1,645,378,8.5673062,"\int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2),x]","\frac{i (\cos (e)-i \sin (e)) \sec (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \left(\sqrt{c+d \tan (e+f x)} \left(\left(33 c^2-68 i c d-35 d^2\right) \cos (2 (e+f x))+33 c^2+2 d (13 c-7 i d) \sin (2 (e+f x))-68 i c d-19 d^2\right)-\frac{(3-3 i) \cos ^3(e+f x) \left(\left(5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right) \left(\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}-i c e^{i (e+f x)}+c-d e^{i (e+f x)}+i d\right)}{\sqrt{d} \left(-5 i c^3-45 c^2 d+55 i c d^2+23 d^3\right) \left(e^{i (e+f x)}+i\right)}\right)-\log \left(-\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} \left(-5 i c^3-45 c^2 d+55 i c d^2+23 d^3\right) \left(e^{i (e+f x)}-i\right)}\right)\right)+(32+32 i) \sqrt{d} (c-i d)^{5/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{d} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{48 f}","-\frac{\sqrt[4]{-1} a^{3/2} \left(5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{8 \sqrt{d} f}-\frac{2 i \sqrt{2} a^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}-\frac{a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt{a+i a \tan (e+f x)}}+\frac{a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt{a+i a \tan (e+f x)}}+\frac{a (7 d+5 i c) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac{a (c-3 i d) (3 d+5 i c) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 f}",1,"((I/48)*Sec[e + f*x]*(Cos[e] - I*Sin[e])*(Cos[f*x] - I*Sin[f*x])*(a + I*a*Tan[e + f*x])^(3/2)*(((-3 + 3*I)*Cos[e + f*x]^3*(((5*I)*c^3 + 45*c^2*d - (55*I)*c*d^2 - 23*d^3)*(Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d - I*c*E^(I*(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*((-5*I)*c^3 - 45*c^2*d + (55*I)*c*d^2 + 23*d^3)*(I + E^(I*(e + f*x))))] - Log[((-2 - 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*((-5*I)*c^3 - 45*c^2*d + (55*I)*c*d^2 + 23*d^3)*(-I + E^(I*(e + f*x))))]) + (32 + 32*I)*(c - I*d)^(5/2)*Sqrt[d]*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])]))/(Sqrt[d]*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (33*c^2 - (68*I)*c*d - 19*d^2 + (33*c^2 - (68*I)*c*d - 35*d^2)*Cos[2*(e + f*x)] + 2*(13*c - (7*I)*d)*d*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]]))/f","A",0
1151,1,539,257,5.9352738,"\int \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2),x]","\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \sqrt{a+i a \tan (e+f x)} \left((1-i) d \sqrt{c+d \tan (e+f x)} (9 c+2 d \tan (e+f x)-i d)-\frac{\cos (e+f x) \left(\sqrt{d} \left(15 c^2-10 i c d-7 d^2\right) \left(\log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}-i c e^{i (e+f x)}+c-d e^{i (e+f x)}+i d\right)}{d^{3/2} \left(-15 c^2+10 i c d+7 d^2\right) \left(e^{i (e+f x)}+i\right)}\right)-\log \left(-\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{d^{3/2} \left(-15 c^2+10 i c d+7 d^2\right) \left(e^{i (e+f x)}-i\right)}\right)\right)+(8+8 i) (c-i d)^{5/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f}","-\frac{\sqrt[4]{-1} \sqrt{a} \sqrt{d} \left(15 c^2-10 i c d-7 d^2\right) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{4 f}+\frac{d \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac{d (7 c-i d) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 f}-\frac{i \sqrt{2} \sqrt{a} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f}",1,"((1/8 + I/8)*Sqrt[a + I*a*Tan[e + f*x]]*(-((Cos[e + f*x]*(Sqrt[d]*(15*c^2 - (10*I)*c*d - 7*d^2)*(Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d - I*c*E^(I*(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(3/2)*(-15*c^2 + (10*I)*c*d + 7*d^2)*(I + E^(I*(e + f*x))))] - Log[((-2 - 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(3/2)*(-15*c^2 + (10*I)*c*d + 7*d^2)*(-I + E^(I*(e + f*x))))]) + (8 + 8*I)*(c - I*d)^(5/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])]))/Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (1 - I)*d*Sqrt[c + d*Tan[e + f*x]]*(9*c - I*d + 2*d*Tan[e + f*x])))/f","B",0
1152,1,549,250,7.6609232,"\int \frac{(c+d \tan (e+f x))^{5/2}}{\sqrt{a+i a \tan (e+f x)}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/Sqrt[a + I*a*Tan[e + f*x]],x]","\frac{\left(\frac{1}{2}+\frac{i}{2}\right) (\cos (f x)+i \sin (f x)) \left((1+i) (\cos (f x)-i \sin (f x)) \sqrt{c+d \tan (e+f x)} \left(c^2+2 i c d-i d^2 \tan (e+f x)-2 d^2\right)-\frac{(\cos (e)+i \sin (e)) \left(d^{3/2} (d-5 i c) \left(\log \left(\frac{(1+i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{d^{5/2} (d-5 i c) \left(e^{i (e+f x)}+i\right)}\right)-\log \left(\frac{(1+i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{d^{5/2} (d-5 i c) \left(e^{i (e+f x)}-i\right)}\right)\right)+(1+i) (c-i d)^{5/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}\right)}{f \sqrt{a+i a \tan (e+f x)}}","\frac{\sqrt[4]{-1} d^{3/2} (-d+5 i c) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a} f}+\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{d (c+2 i d) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{a f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{\sqrt{2} \sqrt{a} f}",1,"((1/2 + I/2)*(Cos[f*x] + I*Sin[f*x])*(-(((d^(3/2)*((-5*I)*c + d)*(Log[((1 + I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(5/2)*((-5*I)*c + d)*(I + E^(I*(e + f*x))))] - Log[((1 + I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(5/2)*((-5*I)*c + d)*(-I + E^(I*(e + f*x))))]) + (1 + I)*(c - I*d)^(5/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[e] + I*Sin[e]))/Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) + (1 + I)*(Cos[f*x] - I*Sin[f*x])*Sqrt[c + d*Tan[e + f*x]]*(c^2 + (2*I)*c*d - 2*d^2 - I*d^2*Tan[e + f*x])))/(f*Sqrt[a + I*a*Tan[e + f*x]])","B",0
1153,1,560,257,8.0121233,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x))^2 \left(\frac{(\cos (2 e)+i \sin (2 e)) \left((2+2 i) d^{5/2} \log \left(\frac{\left(\frac{1}{4}+\frac{i}{4}\right) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}-i c e^{i (e+f x)}+c-d e^{i (e+f x)}+i d\right)}{d^{7/2} \left(e^{i (e+f x)}+i\right)}\right)-(2+2 i) d^{5/2} \log \left(-\frac{\left(\frac{1}{4}+\frac{i}{4}\right) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{d^{7/2} \left(e^{i (e+f x)}-i\right)}\right)-i (c-i d)^{5/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}+\frac{1}{3} (c+i d) (\sin (2 f x)+i \cos (2 f x)) \sqrt{c+d \tan (e+f x)} ((11 d+3 i c) \sin (e+f x)+(5 c-9 i d) \cos (e+f x))\right)}{2 f (a+i a \tan (e+f x))^{3/2}}","\frac{2 \sqrt[4]{-1} d^{5/2} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{a^{3/2} f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f}+\frac{(-d+i c) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac{(c+i d) (3 d+i c) \sqrt{c+d \tan (e+f x)}}{2 a f \sqrt{a+i a \tan (e+f x)}}",1,"(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*((((2 + 2*I)*d^(5/2)*Log[((1/4 + I/4)*E^((I/2)*e)*(c + I*d - I*c*E^(I*(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(7/2)*(I + E^(I*(e + f*x))))] - (2 + 2*I)*d^(5/2)*Log[((-1/4 - I/4)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(7/2)*(-I + E^(I*(e + f*x))))] - I*(c - I*d)^(5/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] + I*Sin[2*e]))/Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + ((c + I*d)*(I*Cos[2*f*x] + Sin[2*f*x])*((5*c - (9*I)*d)*Cos[e + f*x] + ((3*I)*c + 11*d)*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/3))/(2*f*(a + I*a*Tan[e + f*x])^(3/2))","B",0
1154,1,292,225,6.3385811,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(e+f x) \left(\frac{2 \sqrt{c+d \tan (e+f x)} \left(\left(-20 c^2+52 i c d+20 d^2\right) \sin (2 (e+f x))+\left(26 i c^2+40 c d-26 i d^2\right) \cos (2 (e+f x))+11 i \left(c^2+d^2\right)\right)}{15 \sqrt{\sec (e+f x)}}-i \sqrt{2} (c-i d)^{5/2} e^{2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)\right)}{8 f (a+i a \tan (e+f x))^{5/2}}","-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{4 \sqrt{2} a^{5/2} f}+\frac{i (c-i d)^2 \sqrt{c+d \tan (e+f x)}}{4 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{i (c+d \tan (e+f x))^{5/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(d+i c) (c+d \tan (e+f x))^{3/2}}{6 a f (a+i a \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^(5/2)*((-I)*Sqrt[2]*(c - I*d)^(5/2)*E^((2*I)*(e + f*x))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])] + (2*((11*I)*(c^2 + d^2) + ((26*I)*c^2 + 40*c*d - (26*I)*d^2)*Cos[2*(e + f*x)] + (-20*c^2 + (52*I)*c*d + 20*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(15*Sqrt[Sec[e + f*x]])))/(8*f*(a + I*a*Tan[e + f*x])^(5/2))","A",0
1155,1,602,200,7.2221045,"\int \frac{(a+i a \tan (e+f x))^{5/2}}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \cos ^2(e+f x) (a+i a \tan (e+f x))^{5/2} \left(\frac{(-\cos (2 e)+i \sin (2 e)) \cos (e+f x) \left((8+8 i) d^{3/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)+\sqrt{c-i d} (c+5 i d) \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{\sqrt{d} (5 d-i c) \left(e^{i (e+f x)}+i\right)}\right)-\sqrt{c-i d} (c+5 i d) \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} (5 d-i c) \left(e^{i (e+f x)}-i\right)}\right)\right)}{\sqrt{c-i d} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1}}+(1+i) \sqrt{d} \sin (2 e) \sqrt{c+d \tan (e+f x)}+(-1+i) \sqrt{d} \cos (2 e) \sqrt{c+d \tan (e+f x)}\right)}{d^{3/2} f (\cos (f x)+i \sin (f x))^2}","-\frac{\sqrt[4]{-1} a^{5/2} (c+5 i d) \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{d^{3/2} f}-\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f \sqrt{c-i d}}-\frac{a^2 \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{d f}",1,"((1/2 + I/2)*Cos[e + f*x]^2*(a + I*a*Tan[e + f*x])^(5/2)*((Cos[e + f*x]*(Sqrt[c - I*d]*(c + (5*I)*d)*Log[((2 + 2*I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*((-I)*c + 5*d)*(I + E^(I*(e + f*x))))] - Sqrt[c - I*d]*(c + (5*I)*d)*Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*((-I)*c + 5*d)*(-I + E^(I*(e + f*x))))] + (8 + 8*I)*d^(3/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(-Cos[2*e] + I*Sin[2*e]))/(Sqrt[c - I*d]*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]) - (1 - I)*Sqrt[d]*Cos[2*e]*Sqrt[c + d*Tan[e + f*x]] + (1 + I)*Sqrt[d]*Sin[2*e]*Sqrt[c + d*Tan[e + f*x]]))/(d^(3/2)*f*(Cos[f*x] + I*Sin[f*x])^2)","B",0
1156,1,505,151,5.3897748,"\int \frac{(a+i a \tan (e+f x))^{3/2}}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \cos (e+f x) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} (\cos (2 e+f x)-i \sin (2 e+f x)) \left(\sqrt{c-i d} \log \left(\frac{(2-2 i) e^{\frac{3 i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{\sqrt{d} \left(e^{i (e+f x)}+i\right)}\right)-\sqrt{c-i d} \log \left(\frac{(2+2 i) e^{\frac{3 i e}{2}} \left((1-i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+c \left(e^{i (e+f x)}-i\right)-i d e^{i (e+f x)}+d\right)}{\sqrt{d} \left(e^{i (e+f x)}-i\right)}\right)+(2-2 i) \sqrt{d} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))+1} \sqrt{c+d \tan (e+f x)}\right)\right)\right)}{\sqrt{d} f \sqrt{c-i d}}","-\frac{2 (-1)^{3/4} a^{3/2} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{d} f}-\frac{2 i \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f \sqrt{c-i d}}",1,"((1/2 - I/2)*Cos[e + f*x]*(Sqrt[c - I*d]*Log[((2 - 2*I)*E^(((3*I)/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I + E^(I*(e + f*x))))] - Sqrt[c - I*d]*Log[((2 + 2*I)*E^(((3*I)/2)*e)*(d - I*d*E^(I*(e + f*x)) + c*(-I + E^(I*(e + f*x))) + (1 - I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(-I + E^(I*(e + f*x))))] + (2 - 2*I)*Sqrt[d]*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[f*x] - I*Sin[f*x])*Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]]*(Cos[2*e + f*x] - I*Sin[2*e + f*x])*(a + I*a*Tan[e + f*x])^(3/2))/(Sqrt[c - I*d]*Sqrt[d]*f)","B",0
1157,1,147,82,2.6970822,"\int \frac{\sqrt{a+i a \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]],x]","-\frac{i e^{-i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \sqrt{a+i a \tan (e+f x)} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{f \sqrt{c-i d}}","-\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f \sqrt{c-i d}}",1,"((-I)*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[c - I*d]*E^(I*(e + f*x))*f)","A",1
1158,1,195,174,3.2347657,"\int \frac{1}{\sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\frac{2 (d-i c) e^{i (e+f x)} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{\sqrt{1+e^{2 i (e+f x)}}}+2 i \sqrt{c-i d} \sqrt{c+d \tan (e+f x)}}{2 f \sqrt{c-i d} (c+i d) \sqrt{a+i a \tan (e+f x)}}","\frac{2 d \sqrt{c+d \tan (e+f x)}}{f \left(c^2+d^2\right) \sqrt{a+i a \tan (e+f x)}}-\frac{\sqrt{c+d \tan (e+f x)}}{f (d+i c) \sqrt{a+i a \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{\sqrt{2} \sqrt{a} f \sqrt{c-i d}}",1,"((2*((-I)*c + d)*E^(I*(e + f*x))*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/Sqrt[1 + E^((2*I)*(e + f*x))] + (2*I)*Sqrt[c - I*d]*Sqrt[c + d*Tan[e + f*x]])/(2*Sqrt[c - I*d]*(c + I*d)*f*Sqrt[a + I*a*Tan[e + f*x]])","A",0
1159,1,249,193,3.8593277,"\int \frac{1}{(a+i a \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\sec ^{\frac{3}{2}}(e+f x) \left(-\frac{i \sqrt{2} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(1+e^{2 i (e+f x)}\right)^{3/2} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{\sqrt{c-i d}}-\frac{2 ((3 c+7 i d) \tan (e+f x)-5 i c+9 d) \sqrt{c+d \tan (e+f x)}}{3 (c+i d)^2 \sec ^{\frac{3}{2}}(e+f x)}\right)}{4 f (a+i a \tan (e+f x))^{3/2}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f \sqrt{c-i d}}+\frac{(-7 d+3 i c) \sqrt{c+d \tan (e+f x)}}{6 a f (c+i d)^2 \sqrt{a+i a \tan (e+f x)}}-\frac{\sqrt{c+d \tan (e+f x)}}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2}}",1,"(Sec[e + f*x]^(3/2)*(((-I)*Sqrt[2]*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2)*(1 + E^((2*I)*(e + f*x)))^(3/2)*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/Sqrt[c - I*d] - (2*((-5*I)*c + 9*d + (3*c + (7*I)*d)*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)^2*Sec[e + f*x]^(3/2))))/(4*f*(a + I*a*Tan[e + f*x])^(3/2))","A",1
1160,1,309,262,5.3143697,"\int \frac{1}{(a+i a \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\sec ^{\frac{5}{2}}(e+f x) \left(\frac{2 i \sqrt{c+d \tan (e+f x)} \left(4 i \left(5 c^2+17 i c d-20 d^2\right) \sin (2 (e+f x))+\left(26 c^2+80 i c d-86 d^2\right) \cos (2 (e+f x))+11 c^2+30 i c d-19 d^2\right)}{15 (c+i d)^3 \sqrt{\sec (e+f x)}}-\frac{i \sqrt{2} e^{2 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{\sqrt{c-i d}}\right)}{8 f (a+i a \tan (e+f x))^{5/2}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{4 \sqrt{2} a^{5/2} f \sqrt{c-i d}}+\frac{\left(15 c^2+50 i c d-67 d^2\right) \sqrt{c+d \tan (e+f x)}}{60 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)}}+\frac{(-13 d+5 i c) \sqrt{c+d \tan (e+f x)}}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2}}-\frac{\sqrt{c+d \tan (e+f x)}}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2}}",1,"(Sec[e + f*x]^(5/2)*(((-I)*Sqrt[2]*E^((2*I)*(e + f*x))*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/Sqrt[c - I*d] + (((2*I)/15)*(11*c^2 + (30*I)*c*d - 19*d^2 + (26*c^2 + (80*I)*c*d - 86*d^2)*Cos[2*(e + f*x)] + (4*I)*(5*c^2 + (17*I)*c*d - 20*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/((c + I*d)^3*Sqrt[Sec[e + f*x]])))/(8*f*(a + I*a*Tan[e + f*x])^(5/2))","A",0
1161,1,718,209,11.2350452,"\int \frac{(a+i a \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(3/2),x]","\frac{(1+i) (\cos (2 e)-i \sin (2 e)) \cos ^3(e+f x) (a+i a \tan (e+f x))^{5/2} \left(-(4+4 i) d^{3/2} \log \left(2 \left(i \sqrt{c-i d} \sin (e+f x)+\sqrt{c-i d} \cos (e+f x)+\sqrt{i \sin (2 e+2 f x)+\cos (2 e+2 f x)+1} \sqrt{c+d \tan (e+f x)}\right)\right)+(c-i d)^{3/2} \log \left(\frac{(2-2 i) e^{\frac{i e}{2}} \left(-(1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c \left(e^{i (e+f x)}+i\right)+d e^{i (e+f x)}-i d\right)}{\sqrt{d} (d+i c) \left(e^{i (e+f x)}+i\right)}\right)-(c-i d)^{3/2} \log \left(\frac{(2+2 i) e^{\frac{i e}{2}} \left((1+i) \sqrt{d} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}+i c e^{i (e+f x)}+c+d e^{i (e+f x)}+i d\right)}{\sqrt{d} (d+i c) \left(1+i e^{i (e+f x)}\right)}\right)\right)}{d^{3/2} f (c-i d)^{3/2} (\cos (f x)+i \sin (f x))^2 \sqrt{i \sin (2 e+2 f x)+\cos (2 e+2 f x)+1}}+\frac{\cos ^2(e+f x) (a+i a \tan (e+f x))^{5/2} \left(\frac{(-2 \cos (2 e)+2 i \sin (2 e)) (c \sin (f x)+i d \sin (f x))}{(c-i d) (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{(c+i d) \cos (e) \left(\frac{2 \cos (2 e)}{d}-\frac{2 i \sin (2 e)}{d}\right)}{(c-i d) (c \cos (e)+d \sin (e))}\right) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))}}{f (\cos (f x)+i \sin (f x))^2}","\frac{2 \sqrt[4]{-1} a^{5/2} \tanh ^{-1}\left(\frac{(-1)^{3/4} \sqrt{d} \sqrt{a+i a \tan (e+f x)}}{\sqrt{a} \sqrt{c+d \tan (e+f x)}}\right)}{d^{3/2} f}-\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}+\frac{2 a^2 (c+i d) \sqrt{a+i a \tan (e+f x)}}{d f (c-i d) \sqrt{c+d \tan (e+f x)}}",1,"(Cos[e + f*x]^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((c + I*d)*Cos[e]*((2*Cos[2*e])/d - ((2*I)*Sin[2*e])/d))/((c - I*d)*(c*Cos[e] + d*Sin[e])) + ((-2*Cos[2*e] + (2*I)*Sin[2*e])*(c*Sin[f*x] + I*d*Sin[f*x]))/((c - I*d)*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])))*(a + I*a*Tan[e + f*x])^(5/2))/(f*(Cos[f*x] + I*Sin[f*x])^2) + ((1 + I)*Cos[e + f*x]^3*((c - I*d)^(3/2)*Log[((2 - 2*I)*E^((I/2)*e)*((-I)*d + d*E^(I*(e + f*x)) + I*c*(I + E^(I*(e + f*x))) - (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I*c + d)*(I + E^(I*(e + f*x))))] - (c - I*d)^(3/2)*Log[((2 + 2*I)*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(Sqrt[d]*(I*c + d)*(1 + I*E^(I*(e + f*x))))] - (4 + 4*I)*d^(3/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*e + 2*f*x] + I*Sin[2*e + 2*f*x]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] - I*Sin[2*e])*(a + I*a*Tan[e + f*x])^(5/2))/((c - I*d)^(3/2)*d^(3/2)*f*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[1 + Cos[2*e + 2*f*x] + I*Sin[2*e + 2*f*x]])","B",0
1162,1,209,129,4.8623008,"\int \frac{(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 i a e^{-i (e+f x)} \sqrt{a+i a \tan (e+f x)} \left(\sqrt{c-i d} e^{i (e+f x)}-\sqrt{1+e^{2 i (e+f x)}} \log \left(2 e^{-i e} \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right) \sqrt{c+d \tan (e+f x)}\right)}{f (c-i d)^{3/2} \sqrt{c+d \tan (e+f x)}}","-\frac{2 i \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}-\frac{2 a \sqrt{a+i a \tan (e+f x)}}{f (d+i c) \sqrt{c+d \tan (e+f x)}}",1,"((2*I)*a*Sqrt[a + I*a*Tan[e + f*x]]*(Sqrt[c - I*d]*E^(I*(e + f*x)) - Sqrt[1 + E^((2*I)*(e + f*x))]*Log[(2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/E^(I*e)]*Sqrt[c + d*Tan[e + f*x]]))/((c - I*d)^(3/2)*E^(I*(e + f*x))*f*Sqrt[c + d*Tan[e + f*x]])","A",1
1163,1,337,129,4.5560743,"\int \frac{\sqrt{a+i a \tan (e+f x)}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(3/2),x]","\frac{\sqrt{2} \sqrt{e^{i f x}} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sqrt{a+i a \tan (e+f x)} \left(-\frac{2 d \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}}{(c-i d) (c+i d) \left(c \left(1+e^{2 i (e+f x)}\right)-i d \left(-1+e^{2 i (e+f x)}\right)\right)}-\frac{i e^{-i (e+f x)} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{(c-i d)^{3/2}}\right)}{f \sqrt{\sec (e+f x)} \sqrt{\cos (f x)+i \sin (f x)}}","-\frac{2 d \sqrt{a+i a \tan (e+f x)}}{f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}",1,"(Sqrt[2]*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*((-2*d*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])/((c - I*d)*(c + I*d)*((-I)*d*(-1 + E^((2*I)*(e + f*x))) + c*(1 + E^((2*I)*(e + f*x))))) - (I*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/((c - I*d)^(3/2)*E^(I*(e + f*x))))*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[Sec[e + f*x]]*Sqrt[Cos[f*x] + I*Sin[f*x]])","B",1
1164,1,267,194,4.6647129,"\int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\sqrt{\sec (e+f x)} \left(\frac{2 i c^2+2 d (3 d+i c) \tan (e+f x)+2 c d-4 i d^2}{(c-i d) (c+i d)^2 \sqrt{\sec (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \sqrt{2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{(c-i d)^{3/2}}\right)}{2 f \sqrt{a+i a \tan (e+f x)}}","\frac{d (c-3 i d) \sqrt{a+i a \tan (e+f x)}}{a f (c-i d) (c+i d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{1}{f (-d+i c) \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{\sqrt{2} \sqrt{a} f (c-i d)^{3/2}}",1,"(Sqrt[Sec[e + f*x]]*(((-I)*Sqrt[2]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/(c - I*d)^(3/2) + ((2*I)*c^2 + 2*c*d - (4*I)*d^2 + 2*d*(I*c + 3*d)*Tan[e + f*x])/((c - I*d)*(c + I*d)^2*Sqrt[Sec[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])))/(2*f*Sqrt[a + I*a*Tan[e + f*x]])","A",0
1165,1,333,269,7.0698746,"\int \frac{1}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\sec ^{\frac{3}{2}}(e+f x) \left(\frac{\sqrt{\sec (e+f x)} \left(5 i c^3-13 c^2 d-\left(3 c^3+5 i c^2 d+23 c d^2-39 i d^3\right) \sin (2 (e+f x))+\left(5 i c^3-7 c^2 d+25 i c d^2+37 d^3\right) \cos (2 (e+f x))+5 i c d^2-13 d^3\right)}{3 (c-i d) (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{i \sqrt{2} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(1+e^{2 i (e+f x)}\right)^{3/2} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{(c-i d)^{3/2}}\right)}{4 f (a+i a \tan (e+f x))^{3/2}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-i d)^{3/2}}+\frac{d (3 c-5 i d) (c+5 i d) \sqrt{a+i a \tan (e+f x)}}{6 a^2 f (c-i d) (c+i d)^3 \sqrt{c+d \tan (e+f x)}}+\frac{-11 d+3 i c}{6 a f (c+i d)^2 \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{1}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}",1,"(Sec[e + f*x]^(3/2)*(((-I)*Sqrt[2]*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2)*(1 + E^((2*I)*(e + f*x)))^(3/2)*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/(c - I*d)^(3/2) + (Sqrt[Sec[e + f*x]]*((5*I)*c^3 - 13*c^2*d + (5*I)*c*d^2 - 13*d^3 + ((5*I)*c^3 - 7*c^2*d + (25*I)*c*d^2 + 37*d^3)*Cos[2*(e + f*x)] - (3*c^3 + (5*I)*c^2*d + 23*c*d^2 - (39*I)*d^3)*Sin[2*(e + f*x)]))/(3*(c - I*d)*(c + I*d)^3*Sqrt[c + d*Tan[e + f*x]])))/(4*f*(a + I*a*Tan[e + f*x])^(3/2))","A",0
1166,1,788,349,9.2360386,"\int \frac{1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left(\frac{\left(17 c^2+77 i c d-126 d^2\right) \left(-\frac{\sin (e)}{60}+\frac{1}{60} i \cos (e)\right) \cos (2 f x)}{(c+i d)^4}+\frac{\left(17 c^2+77 i c d-126 d^2\right) \left(\frac{\cos (e)}{60}+\frac{1}{60} i \sin (e)\right) \sin (2 f x)}{(c+i d)^4}+\frac{\left(\frac{1}{120} \cos (3 e)+\frac{1}{120} i \sin (3 e)\right) \left(23 c^4 \cos (e)+23 c^3 d \sin (e)+91 i c^3 d \cos (e)+91 i c^2 d^2 \sin (e)-109 c^2 d^2 \cos (e)-109 c d^3 \sin (e)+223 i c d^3 \cos (e)+223 i d^4 \sin (e)+240 d^4 \cos (e)\right)}{(c-i d) (c+i d)^4 (-i c \cos (e)-i d \sin (e))}+\frac{2 \left(\frac{1}{2} i d^5 \sin (3 e-f x)-\frac{1}{2} i d^5 \sin (3 e+f x)+\frac{1}{2} d^5 \cos (3 e-f x)-\frac{1}{2} d^5 \cos (3 e+f x)\right)}{(c-i d) (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{(7 c+16 i d) \left(\frac{\sin (e)}{60}+\frac{1}{60} i \cos (e)\right) \cos (4 f x)}{(c+i d)^3}+\frac{\left(\frac{1}{40} \sin (3 e)+\frac{1}{40} i \cos (3 e)\right) \cos (6 f x)}{(c+i d)^2}+\frac{(7 c+16 i d) \left(\frac{\cos (e)}{60}-\frac{1}{60} i \sin (e)\right) \sin (4 f x)}{(c+i d)^3}+\frac{\left(\frac{1}{40} \cos (3 e)-\frac{1}{40} i \sin (3 e)\right) \sin (6 f x)}{(c+i d)^2}\right) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))}}{f (a+i a \tan (e+f x))^{5/2}}-\frac{i e^{3 i e} \sqrt{e^{i f x}} \sec ^{\frac{5}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{4 \sqrt{2} f (c-i d)^{3/2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} (a+i a \tan (e+f x))^{5/2}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{4 \sqrt{2} a^{5/2} f (c-i d)^{3/2}}+\frac{d \left(15 c^3+65 i c^2 d-117 c d^2+317 i d^3\right) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{15 c^2+70 i c d-151 d^2}{60 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{-17 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}",1,"((-1/4*I)*E^((3*I)*e)*Sqrt[E^(I*f*x)]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]*Sec[e + f*x]^(5/2)*(Cos[f*x] + I*Sin[f*x])^(5/2))/(Sqrt[2]*(c - I*d)^(3/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*(a + I*a*Tan[e + f*x])^(5/2)) + (Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((17*c^2 + (77*I)*c*d - 126*d^2)*Cos[2*f*x]*((I/60)*Cos[e] - Sin[e]/60))/(c + I*d)^4 + ((7*c + (16*I)*d)*Cos[4*f*x]*((I/60)*Cos[e] + Sin[e]/60))/(c + I*d)^3 + ((23*c^4*Cos[e] + (91*I)*c^3*d*Cos[e] - 109*c^2*d^2*Cos[e] + (223*I)*c*d^3*Cos[e] + 240*d^4*Cos[e] + 23*c^3*d*Sin[e] + (91*I)*c^2*d^2*Sin[e] - 109*c*d^3*Sin[e] + (223*I)*d^4*Sin[e])*(Cos[3*e]/120 + (I/120)*Sin[3*e]))/((c - I*d)*(c + I*d)^4*((-I)*c*Cos[e] - I*d*Sin[e])) + (Cos[6*f*x]*((I/40)*Cos[3*e] + Sin[3*e]/40))/(c + I*d)^2 + ((17*c^2 + (77*I)*c*d - 126*d^2)*(Cos[e]/60 + (I/60)*Sin[e])*Sin[2*f*x])/(c + I*d)^4 + ((7*c + (16*I)*d)*(Cos[e]/60 - (I/60)*Sin[e])*Sin[4*f*x])/(c + I*d)^3 + ((Cos[3*e]/40 - (I/40)*Sin[3*e])*Sin[6*f*x])/(c + I*d)^2 + (2*((d^5*Cos[3*e - f*x])/2 - (d^5*Cos[3*e + f*x])/2 + (I/2)*d^5*Sin[3*e - f*x] - (I/2)*d^5*Sin[3*e + f*x]))/((c - I*d)*(c + I*d)^4*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^(5/2))","B",0
1167,1,283,181,7.1438908,"\int \frac{(a+i a \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(5/2),x]","\frac{(a+i a \tan (e+f x))^{5/2} \left(-\frac{4 i \sqrt{2} e^{-3 i (e+f x)} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \log \left(2 e^{-i e} \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{(c-i d)^{5/2}}-\frac{2 \sqrt{\sec (e+f x)} (\cos (2 (e+f x))-i \sin (2 (e+f x))) ((c-7 i d) \tan (e+f x)-7 i c-d)}{3 (c-i d)^2 (c+d \tan (e+f x))^{3/2}}\right)}{f \sec ^{\frac{5}{2}}(e+f x)}","-\frac{4 i \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}+\frac{4 i a^2 \sqrt{a+i a \tan (e+f x)}}{f (c-i d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 a (a+i a \tan (e+f x))^{3/2}}{3 f (d+i c) (c+d \tan (e+f x))^{3/2}}",1,"((a + I*a*Tan[e + f*x])^(5/2)*(((-4*I)*Sqrt[2]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*Log[(2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/E^(I*e)])/((c - I*d)^(5/2)*E^((3*I)*(e + f*x))) - (2*Sqrt[Sec[e + f*x]]*(Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)])*((-7*I)*c - d + (c - (7*I)*d)*Tan[e + f*x]))/(3*(c - I*d)^2*(c + d*Tan[e + f*x])^(3/2))))/(f*Sec[e + f*x]^(5/2))","A",0
1168,1,278,179,6.6942824,"\int \frac{(a+i a \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(5/2),x]","\frac{(a+i a \tan (e+f x))^{3/2} \left(\frac{2 (\tan (e+f x)+i) \left(3 c^2+2 d (c+2 i d) \tan (e+f x)+4 i c d+d^2\right)}{3 (c-i d)^2 (c+i d) \sqrt{\sec (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 i \sqrt{2} \log \left(2 e^{-i e} \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{(c-i d)^{5/2} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{3/2} \left(1+e^{2 i (e+f x)}\right)^{3/2}}\right)}{f \sec ^{\frac{3}{2}}(e+f x)}","-\frac{2 i \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}-\frac{2 d (a+i a \tan (e+f x))^{3/2}}{3 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}+\frac{2 i a \sqrt{a+i a \tan (e+f x)}}{f (c-i d)^2 \sqrt{c+d \tan (e+f x)}}",1,"((a + I*a*Tan[e + f*x])^(3/2)*(((-2*I)*Sqrt[2]*Log[(2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/E^(I*e)])/((c - I*d)^(5/2)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2)*(1 + E^((2*I)*(e + f*x)))^(3/2)) + (2*(I + Tan[e + f*x])*(3*c^2 + (4*I)*c*d + d^2 + 2*(c + (2*I)*d)*d*Tan[e + f*x]))/(3*(c - I*d)^2*(c + I*d)*Sqrt[Sec[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))))/(f*Sec[e + f*x]^(3/2))","A",0
1169,1,394,188,5.2968642,"\int \frac{\sqrt{a+i a \tan (e+f x)}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + I*a*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(5/2),x]","\frac{\sqrt{2} \sqrt{e^{i f x}} \sqrt{a+i a \tan (e+f x)} \left(-\frac{4 d e^{i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}} \left(3 c^2 \left(1+e^{2 i (e+f x)}\right)-i c d \left(-3+2 e^{2 i (e+f x)}\right)+d^2 e^{2 i (e+f x)}\right)}{3 (c-i d)^2 (c+i d)^2 \left(c \left(1+e^{2 i (e+f x)}\right)-i d \left(-1+e^{2 i (e+f x)}\right)\right)^2}-\frac{i \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{(c-i d)^{5/2}}\right)}{f \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sqrt{\sec (e+f x)} \sqrt{\cos (f x)+i \sin (f x)}}","-\frac{2 d (5 c+i d) \sqrt{a+i a \tan (e+f x)}}{3 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 d \sqrt{a+i a \tan (e+f x)}}{3 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}",1,"(Sqrt[2]*Sqrt[E^(I*f*x)]*((-4*d*E^(I*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]*(d^2*E^((2*I)*(e + f*x)) + 3*c^2*(1 + E^((2*I)*(e + f*x))) - I*c*d*(-3 + 2*E^((2*I)*(e + f*x)))))/(3*(c - I*d)^2*(c + I*d)^2*((-I)*d*(-1 + E^((2*I)*(e + f*x))) + c*(1 + E^((2*I)*(e + f*x))))^2) - (I*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])])/(c - I*d)^(5/2))*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[Sec[e + f*x]]*Sqrt[Cos[f*x] + I*Sin[f*x]])","B",1
1170,1,687,277,9.0823334,"\int \frac{1}{\sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\sec (e+f x) (\cos (f x)+i \sin (f x)) \left(\frac{\left(\frac{\cos (e)}{6}+\frac{1}{6} i \sin (e)\right) \left(3 i c^3 \cos (e)+3 i c^2 d \sin (e)+6 c^2 d \cos (e)+6 c d^2 \sin (e)-39 i c d^2 \cos (e)+i d^3 \sin (e)-8 d^3 \cos (e)\right)}{(c-i d)^2 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac{\frac{2}{3} d^4 \sin (e)-\frac{2}{3} i d^4 \cos (e)}{(c-i d)^2 (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))^2}+\frac{4 \left(-\frac{5}{2} i c d^3 \sin (e-f x)+\frac{5}{2} i c d^3 \sin (e+f x)-\frac{5}{2} c d^3 \cos (e-f x)+\frac{5}{2} c d^3 \cos (e+f x)-\frac{1}{2} d^4 \sin (e-f x)+\frac{1}{2} d^4 \sin (e+f x)+\frac{1}{2} i d^4 \cos (e-f x)-\frac{1}{2} i d^4 \cos (e+f x)\right)}{3 (c-i d)^2 (c+i d)^3 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\left(\frac{\sin (e)}{2}+\frac{1}{2} i \cos (e)\right) \cos (2 f x)}{(c+i d)^3}+\frac{\left(\frac{\cos (e)}{2}-\frac{1}{2} i \sin (e)\right) \sin (2 f x)}{(c+i d)^3}\right) \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))}}{f \sqrt{a+i a \tan (e+f x)}}-\frac{i e^{i e} \sqrt{e^{i f x}} \sqrt{\sec (e+f x)} \sqrt{\cos (f x)+i \sin (f x)} \log \left(2 \left(\sqrt{c-i d} e^{i (e+f x)}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right)}{\sqrt{2} f (c-i d)^{5/2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sqrt{a+i a \tan (e+f x)}}","\frac{d (5 d+3 i c) \sqrt{a+i a \tan (e+f x)}}{3 a f (-d+i c) \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}+\frac{d (3 c-i d) (c-7 i d) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i d)^2 (c+i d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{1}{f (-d+i c) \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{\sqrt{2} \sqrt{a} f (c-i d)^{5/2}}",1,"((-I)*E^(I*e)*Sqrt[E^(I*f*x)]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]*Sqrt[Sec[e + f*x]]*Sqrt[Cos[f*x] + I*Sin[f*x]])/(Sqrt[2]*(c - I*d)^(5/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[a + I*a*Tan[e + f*x]]) + (Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*((Cos[2*f*x]*((I/2)*Cos[e] + Sin[e]/2))/(c + I*d)^3 + ((Cos[e]/6 + (I/6)*Sin[e])*((3*I)*c^3*Cos[e] + 6*c^2*d*Cos[e] - (39*I)*c*d^2*Cos[e] - 8*d^3*Cos[e] + (3*I)*c^2*d*Sin[e] + 6*c*d^2*Sin[e] + I*d^3*Sin[e]))/((c - I*d)^2*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + ((Cos[e]/2 - (I/2)*Sin[e])*Sin[2*f*x])/(c + I*d)^3 + (((-2*I)/3)*d^4*Cos[e] + (2*d^4*Sin[e])/3)/((c - I*d)^2*(c + I*d)^3*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + (4*((-5*c*d^3*Cos[e - f*x])/2 + (I/2)*d^4*Cos[e - f*x] + (5*c*d^3*Cos[e + f*x])/2 - (I/2)*d^4*Cos[e + f*x] - ((5*I)/2)*c*d^3*Sin[e - f*x] - (d^4*Sin[e - f*x])/2 + ((5*I)/2)*c*d^3*Sin[e + f*x] + (d^4*Sin[e + f*x])/2))/(3*(c - I*d)^2*(c + I*d)^3*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*Sqrt[a + I*a*Tan[e + f*x]])","B",0
1171,1,803,354,9.627565,"\int \frac{1}{(a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left(\frac{i (5 c+21 i d) \cos (2 f x)}{12 (c+i d)^4}+\frac{\left(i \cos (e) c^4-3 d \cos (e) c^3+i d \sin (e) c^3+9 i d^2 \cos (e) c^2-3 d^2 \sin (e) c^2+29 d^3 \cos (e) c+9 i d^3 \sin (e) c-10 i d^4 \cos (e)+3 d^4 \sin (e)\right) \left(\frac{1}{3} \cos (2 e)+\frac{1}{3} i \sin (2 e)\right)}{(c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e))}+\frac{\cos (4 f x) \left(\frac{1}{12} i \cos (2 e)+\frac{1}{12} \sin (2 e)\right)}{(c+i d)^3}+\frac{(5 c+21 i d) \sin (2 f x)}{12 (c+i d)^4}+\frac{\left(\frac{1}{12} \cos (2 e)-\frac{1}{12} i \sin (2 e)\right) \sin (4 f x)}{(c+i d)^3}-\frac{2 \left(\frac{5}{2} \cos (2 e-f x) d^5-\frac{5}{2} \cos (2 e+f x) d^5+\frac{5}{2} i \sin (2 e-f x) d^5-\frac{5}{2} i \sin (2 e+f x) d^5+\frac{13}{2} i c \cos (2 e-f x) d^4-\frac{13}{2} i c \cos (2 e+f x) d^4-\frac{13}{2} c \sin (2 e-f x) d^4+\frac{13}{2} c \sin (2 e+f x) d^4\right)}{3 (c-i d)^2 (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} \cos (2 e) d^5+\frac{2}{3} i \sin (2 e) d^5}{(c-i d)^2 (c+i d)^4 (c \cos (e+f x)+d \sin (e+f x))^2}\right)}{f (i \tan (e+f x) a+a)^{3/2}}-\frac{i e^{2 i e} \sqrt{e^{i f x}} \log \left(2 \left(e^{i (e+f x)} \sqrt{c-i d}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right) \sec ^{\frac{3}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{3/2}}{2 \sqrt{2} (c-i d)^{5/2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} f (i \tan (e+f x) a+a)^{3/2}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{2 \sqrt{2} a^{3/2} f (c-i d)^{5/2}}+\frac{d (c-3 i d) \left(3 c^2+22 i c d+13 d^2\right) \sqrt{a+i a \tan (e+f x)}}{6 a^2 f (c-i d)^2 (c+i d)^4 \sqrt{c+d \tan (e+f x)}}+\frac{d \left(3 c^2+14 i c d+21 d^2\right) \sqrt{a+i a \tan (e+f x)}}{6 a^2 f (c-i d) (c+i d)^3 (c+d \tan (e+f x))^{3/2}}+\frac{-5 d+i c}{2 a f (c+i d)^2 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{1}{3 f (-d+i c) (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}",1,"((-1/2*I)*E^((2*I)*e)*Sqrt[E^(I*f*x)]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]*Sec[e + f*x]^(3/2)*(Cos[f*x] + I*Sin[f*x])^(3/2))/(Sqrt[2]*(c - I*d)^(5/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*(a + I*a*Tan[e + f*x])^(3/2)) + (Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((I/12)*(5*c + (21*I)*d)*Cos[2*f*x])/(c + I*d)^4 + ((I*c^4*Cos[e] - 3*c^3*d*Cos[e] + (9*I)*c^2*d^2*Cos[e] + 29*c*d^3*Cos[e] - (10*I)*d^4*Cos[e] + I*c^3*d*Sin[e] - 3*c^2*d^2*Sin[e] + (9*I)*c*d^3*Sin[e] + 3*d^4*Sin[e])*(Cos[2*e]/3 + (I/3)*Sin[2*e]))/((c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])) + (Cos[4*f*x]*((I/12)*Cos[2*e] + Sin[2*e]/12))/(c + I*d)^3 + ((5*c + (21*I)*d)*Sin[2*f*x])/(12*(c + I*d)^4) + ((Cos[2*e]/12 - (I/12)*Sin[2*e])*Sin[4*f*x])/(c + I*d)^3 + ((2*d^5*Cos[2*e])/3 + ((2*I)/3)*d^5*Sin[2*e])/((c - I*d)^2*(c + I*d)^4*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) - (2*(((13*I)/2)*c*d^4*Cos[2*e - f*x] + (5*d^5*Cos[2*e - f*x])/2 - ((13*I)/2)*c*d^4*Cos[2*e + f*x] - (5*d^5*Cos[2*e + f*x])/2 - (13*c*d^4*Sin[2*e - f*x])/2 + ((5*I)/2)*d^5*Sin[2*e - f*x] + (13*c*d^4*Sin[2*e + f*x])/2 - ((5*I)/2)*d^5*Sin[2*e + f*x]))/(3*(c - I*d)^2*(c + I*d)^4*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^(3/2))","B",0
1172,1,928,444,10.519847,"\int \frac{1}{(a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \sqrt{\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left(\frac{\left(17 c^2+102 i d c-231 d^2\right) \cos (2 f x) \left(\frac{1}{60} i \cos (e)-\frac{\sin (e)}{60}\right)}{(c+i d)^5}+\frac{(c+3 i d) \cos (4 f x) \left(\frac{7}{60} i \cos (e)+\frac{7 \sin (e)}{60}\right)}{(c+i d)^4}+\frac{\left(23 i \cos (e) c^5-108 d \cos (e) c^4+23 i d \sin (e) c^4-138 i d^2 \cos (e) c^3-108 d^2 \sin (e) c^3-692 d^3 \cos (e) c^2-138 i d^3 \sin (e) c^2+1623 i d^4 \cos (e) c-692 d^4 \sin (e) c+640 d^5 \cos (e)+343 i d^5 \sin (e)\right) \left(\frac{1}{120} \cos (3 e)+\frac{1}{120} i \sin (3 e)\right)}{(c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e))}+\frac{\cos (6 f x) \left(\frac{1}{40} i \cos (3 e)+\frac{1}{40} \sin (3 e)\right)}{(c+i d)^3}+\frac{\left(17 c^2+102 i d c-231 d^2\right) \left(\frac{\cos (e)}{60}+\frac{1}{60} i \sin (e)\right) \sin (2 f x)}{(c+i d)^5}+\frac{(c+3 i d) \left(\frac{7 \cos (e)}{60}-\frac{7}{60} i \sin (e)\right) \sin (4 f x)}{(c+i d)^4}+\frac{\left(\frac{1}{40} \cos (3 e)-\frac{1}{40} i \sin (3 e)\right) \sin (6 f x)}{(c+i d)^3}+\frac{16 \left(-\frac{1}{2} i \cos (3 e-f x) d^6+\frac{1}{2} i \cos (3 e+f x) d^6+\frac{1}{2} \sin (3 e-f x) d^6-\frac{1}{2} \sin (3 e+f x) d^6+c \cos (3 e-f x) d^5-c \cos (3 e+f x) d^5+i c \sin (3 e-f x) d^5-i c \sin (3 e+f x) d^5\right)}{3 (c-i d)^2 (c+i d)^5 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}+\frac{\frac{2}{3} i d^6 \cos (3 e)-\frac{2}{3} d^6 \sin (3 e)}{(c-i d)^2 (c+i d)^5 (c \cos (e+f x)+d \sin (e+f x))^2}\right)}{f (i \tan (e+f x) a+a)^{5/2}}-\frac{i e^{3 i e} \sqrt{e^{i f x}} \log \left(2 \left(e^{i (e+f x)} \sqrt{c-i d}+\sqrt{1+e^{2 i (e+f x)}} \sqrt{c-\frac{i d \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}}\right)\right) \sec ^{\frac{5}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2}}{4 \sqrt{2} (c-i d)^{5/2} \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} f (i \tan (e+f x) a+a)^{5/2}}","-\frac{i \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d} \sqrt{a+i a \tan (e+f x)}}\right)}{4 \sqrt{2} a^{5/2} f (c-i d)^{5/2}}+\frac{d \left(15 c^3+85 i c^2 d-221 c d^2+361 i d^3\right) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^{3/2}}+\frac{d \left(15 c^4+80 i c^3 d-182 c^2 d^2+1224 i c d^3+707 d^4\right) \sqrt{a+i a \tan (e+f x)}}{60 a^3 f (c-i d)^2 (c+i d)^5 \sqrt{c+d \tan (e+f x)}}+\frac{5 c^2+30 i c d-89 d^2}{20 a^2 f (-d+i c)^3 \sqrt{a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{-21 d+5 i c}{30 a f (c+i d)^2 (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{1}{5 f (-d+i c) (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}",1,"((-1/4*I)*E^((3*I)*e)*Sqrt[E^(I*f*x)]*Log[2*(Sqrt[c - I*d]*E^(I*(e + f*x)) + Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))])]*Sec[e + f*x]^(5/2)*(Cos[f*x] + I*Sin[f*x])^(5/2))/(Sqrt[2]*(c - I*d)^(5/2)*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*(a + I*a*Tan[e + f*x])^(5/2)) + (Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((17*c^2 + (102*I)*c*d - 231*d^2)*Cos[2*f*x]*((I/60)*Cos[e] - Sin[e]/60))/(c + I*d)^5 + ((c + (3*I)*d)*Cos[4*f*x]*(((7*I)/60)*Cos[e] + (7*Sin[e])/60))/(c + I*d)^4 + (((23*I)*c^5*Cos[e] - 108*c^4*d*Cos[e] - (138*I)*c^3*d^2*Cos[e] - 692*c^2*d^3*Cos[e] + (1623*I)*c*d^4*Cos[e] + 640*d^5*Cos[e] + (23*I)*c^4*d*Sin[e] - 108*c^3*d^2*Sin[e] - (138*I)*c^2*d^3*Sin[e] - 692*c*d^4*Sin[e] + (343*I)*d^5*Sin[e])*(Cos[3*e]/120 + (I/120)*Sin[3*e]))/((c - I*d)^2*(c + I*d)^5*(c*Cos[e] + d*Sin[e])) + (Cos[6*f*x]*((I/40)*Cos[3*e] + Sin[3*e]/40))/(c + I*d)^3 + ((17*c^2 + (102*I)*c*d - 231*d^2)*(Cos[e]/60 + (I/60)*Sin[e])*Sin[2*f*x])/(c + I*d)^5 + ((c + (3*I)*d)*((7*Cos[e])/60 - ((7*I)/60)*Sin[e])*Sin[4*f*x])/(c + I*d)^4 + ((Cos[3*e]/40 - (I/40)*Sin[3*e])*Sin[6*f*x])/(c + I*d)^3 + (((2*I)/3)*d^6*Cos[3*e] - (2*d^6*Sin[3*e])/3)/((c - I*d)^2*(c + I*d)^5*(c*Cos[e + f*x] + d*Sin[e + f*x])^2) + (16*(c*d^5*Cos[3*e - f*x] - (I/2)*d^6*Cos[3*e - f*x] - c*d^5*Cos[3*e + f*x] + (I/2)*d^6*Cos[3*e + f*x] + I*c*d^5*Sin[3*e - f*x] + (d^6*Sin[3*e - f*x])/2 - I*c*d^5*Sin[3*e + f*x] - (d^6*Sin[3*e + f*x])/2))/(3*(c - I*d)^2*(c + I*d)^5*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(f*(a + I*a*Tan[e + f*x])^(5/2))","B",0
1173,0,0,114,7.8715684,"\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n,x]","\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx","-\frac{i (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left(\frac{c+d \tan (e+f x)}{c+i d}\right)^{-n} F_1\left(m;-n,1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m}",1,"Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x]","F",-1
1174,0,0,157,21.4573128,"\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^n,x]","\int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx","\frac{a^3 (-d (2 n+5)+i c) (c+d \tan (e+f x))^{n+1}}{d^2 f (n+1) (n+2)}+\frac{4 a^3 (c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right)}{f (n+1) (d+i c)}-\frac{\left(a^3+i a^3 \tan (e+f x)\right) (c+d \tan (e+f x))^{n+1}}{d f (n+2)}",1,"Integrate[(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^n, x]","F",-1
1175,0,0,95,4.9193124,"\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^n,x]","\int (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^n \, dx","-\frac{a^2 (c+d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac{2 a^2 (c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right)}{f (n+1) (d+i c)}",1,"Integrate[(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^n, x]","F",-1
1176,0,0,61,1.9579341,"\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n \, dx","Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^n,x]","\int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n \, dx","\frac{a (c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right)}{f (n+1) (d+i c)}",1,"Integrate[(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^n, x]","F",-1
1177,0,0,193,30.7042936,"\int \frac{(c+d \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x]),x]","\int \frac{(c+d \tan (e+f x))^n}{a+i a \tan (e+f x)} \, dx","\frac{(c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right)}{4 a f (n+1) (d+i c)}+\frac{(i c+2 d n-d) (c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c+i d}\right)}{4 a f (n+1) (c+i d)^2}-\frac{(c+d \tan (e+f x))^{n+1}}{2 f (-d+i c) (a+i a \tan (e+f x))}",1,"Integrate[(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x]), x]","F",-1
1178,0,0,273,8.6272548,"\int \frac{(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2,x]","\int \frac{(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^2} \, dx","\frac{\left(c^2+2 i c d (1-n)-d^2 \left(2 n^2-4 n+1\right)\right) (c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c+i d}\right)}{8 a^2 f (n+1) (-d+i c)^3}+\frac{(c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right)}{8 a^2 f (n+1) (d+i c)}+\frac{(-d (2-n)+i c) (c+d \tan (e+f x))^{n+1}}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac{(c+d \tan (e+f x))^{n+1}}{4 f (-d+i c) (a+i a \tan (e+f x))^2}",1,"Integrate[(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x]","F",-1
1179,0,0,381,35.4726878,"\int \frac{(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3,x]","\int \frac{(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx","\frac{\left(3 i c^2-3 c d (3-n)-i d^2 \left(2 n^2-9 n+10\right)\right) (c+d \tan (e+f x))^{n+1}}{24 f (c+i d)^3 \left(a^3+i a^3 \tan (e+f x)\right)}+\frac{\left(3 i c^3-c^2 d (9-6 n)-3 i c d^2 \left(2 n^2-6 n+3\right)+d^3 \left(-4 n^3+18 n^2-20 n+3\right)\right) (c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c+i d}\right)}{48 a^3 f (n+1) (c+i d)^4}+\frac{(c+d \tan (e+f x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right)}{16 a^3 f (n+1) (d+i c)}+\frac{(-d (7-2 n)+3 i c) (c+d \tan (e+f x))^{n+1}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{(c+d \tan (e+f x))^{n+1}}{6 f (-d+i c) (a+i a \tan (e+f x))^3}",1,"Integrate[(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3, x]","F",-1
1180,0,0,192,52.4691837,"\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3,x]","\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx","-\frac{2 d \left(-\left(c^2 (m+3)\right)+i c d m+d^2\right) (a+i a \tan (e+f x))^m}{f m (m+2)}-\frac{d^2 (d m+i c (m+4)) (a+i a \tan (e+f x))^{m+1}}{a f (m+1) (m+2)}+\frac{(d+i c)^3 (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m}+\frac{d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (m+2)}",1,"Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3, x]","F",-1
1181,1,246,119,17.6726514,"\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2,x]","-\frac{i 2^{m-1} \left(e^{i f x}\right)^m e^{-i (e m+e+2 f m x+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{m+1} \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left(\frac{2 \left(c^2+d^2\right) e^{i (e (m+2)+2 f (m+1) x)}}{m+1}+\frac{(c-i d)^2 e^{i (e (m+4)+2 f (m+2) x)} \, _2F_1\left(1,1;m+3;-e^{2 i (e+f x)}\right)}{m+2}+\frac{(c+i d)^2 e^{i m (e+2 f x)} \left(e^{2 i (e+f x)}+m+1\right)}{m (m+1)}\right)}{f}","-\frac{i (c-i d)^2 (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m}+\frac{2 c d (a+i a \tan (e+f x))^m}{f m}-\frac{i d^2 (a+i a \tan (e+f x))^{m+1}}{a f (m+1)}",1,"((-I)*2^(-1 + m)*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1 + m)*((2*(c^2 + d^2)*E^(I*(e*(2 + m) + 2*f*(1 + m)*x)))/(1 + m) + ((c + I*d)^2*E^(I*m*(e + 2*f*x))*(1 + E^((2*I)*(e + f*x)) + m))/(m*(1 + m)) + ((c - I*d)^2*E^(I*(e*(4 + m) + 2*f*(2 + m)*x))*Hypergeometric2F1[1, 1, 3 + m, -E^((2*I)*(e + f*x))])/(2 + m))*(a + I*a*Tan[e + f*x])^m)/(E^(I*(e + e*m + f*x + 2*f*m*x))*f*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","B",0
1182,1,152,78,6.0870177,"\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x)) \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x]),x]","\frac{2^{m-1} \left(e^{i f x}\right)^m \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^m \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left((m+1) (d-i c)-i m (c-i d) e^{2 i (e+f x)} \, _2F_1\left(1,1;m+2;-e^{2 i (e+f x)}\right)\right)}{f m (m+1)}","\frac{d (a+i a \tan (e+f x))^m}{f m}-\frac{(d+i c) (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m}",1,"(2^(-1 + m)*(E^(I*f*x))^m*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^m*(((-I)*c + d)*(1 + m) - I*(c - I*d)*E^((2*I)*(e + f*x))*m*Hypergeometric2F1[1, 1, 2 + m, -E^((2*I)*(e + f*x))])*(a + I*a*Tan[e + f*x])^m)/(f*m*(1 + m)*Sec[e + f*x]^m*(Cos[f*x] + I*Sin[f*x])^m)","A",0
1183,0,0,122,14.3181136,"\int \frac{(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x]),x]","\int \frac{(a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx","\frac{(a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m (d+i c)}-\frac{d (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d}\right)}{f m \left(c^2+d^2\right)}",1,"Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x]), x]","F",-1
1184,0,0,180,28.5455325,"\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^2,x]","\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx","-\frac{d (c (2-m)+i d m) (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d}\right)}{f m \left(c^2+d^2\right)^2}-\frac{d (a+i a \tan (e+f x))^m}{f \left(c^2+d^2\right) (c+d \tan (e+f x))}-\frac{i (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m (c-i d)^2}",1,"Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^2, x]","F",-1
1185,0,0,264,47.2425749,"\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^3,x]","\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx","-\frac{d \left(c^2 \left(m^2-5 m+6\right)+2 i c d (3-m) m-d^2 \left(m^2-m+2\right)\right) (a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d}\right)}{2 f m \left(c^2+d^2\right)^3}-\frac{d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}-\frac{d (a+i a \tan (e+f x))^m}{2 f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}-\frac{(a+i a \tan (e+f x))^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m (d+i c)^3}",1,"Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^3, x]","F",-1
1186,0,0,123,27.779502,"\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2),x]","\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx","-\frac{(-d+i c) (a+i a \tan (e+f x))^m \sqrt{c+d \tan (e+f x)} F_1\left(m;-\frac{3}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}}}",1,"Integrate[(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2), x]","F",-1
1187,0,0,116,2.481394,"\int (a+i a \tan (e+f x))^m \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]],x]","\int (a+i a \tan (e+f x))^m \sqrt{c+d \tan (e+f x)} \, dx","-\frac{i (a+i a \tan (e+f x))^m \sqrt{c+d \tan (e+f x)} F_1\left(m;-\frac{1}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}}}",1,"Integrate[(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]], x]","F",-1
1188,0,0,116,21.010194,"\int \frac{(a+i a \tan (e+f x))^m}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/Sqrt[c + d*Tan[e + f*x]],x]","\int \frac{(a+i a \tan (e+f x))^m}{\sqrt{c+d \tan (e+f x)}} \, dx","-\frac{i (a+i a \tan (e+f x))^m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}} F_1\left(m;\frac{1}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m \sqrt{c+d \tan (e+f x)}}",1,"Integrate[(a + I*a*Tan[e + f*x])^m/Sqrt[c + d*Tan[e + f*x]], x]","F",-1
1189,0,0,125,11.3942067,"\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2),x]","\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx","\frac{(a+i a \tan (e+f x))^m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}} F_1\left(m;\frac{3}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m (-d+i c) \sqrt{c+d \tan (e+f x)}}",1,"Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2), x]","F",-1
1190,0,0,125,17.9855477,"\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2),x]","\int \frac{(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx","-\frac{i (a+i a \tan (e+f x))^m \sqrt{\frac{c+d \tan (e+f x)}{c+i d}} F_1\left(m;\frac{5}{2},1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right)}{2 f m (c+i d)^2 \sqrt{c+d \tan (e+f x)}}",1,"Integrate[(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2), x]","F",-1
1191,1,130,140,1.081482,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x)) \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x]),x]","\frac{6 b \left(3 a^2 d+3 a b c-b^2 d\right) \tan (e+f x)+3 b^2 (3 a d+b c) \tan ^2(e+f x)+3 (a-i b)^3 (d+i c) \log (\tan (e+f x)+i)+3 (a+i b)^3 (d-i c) \log (-\tan (e+f x)+i)+2 b^3 d \tan ^3(e+f x)}{6 f}","\frac{b \left(a^2 d+2 a b c-b^2 d\right) \tan (e+f x)}{f}-\frac{\left(a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right) \log (\cos (e+f x))}{f}+x \left(a^3 c-3 a^2 b d-3 a b^2 c+b^3 d\right)+\frac{(a d+b c) (a+b \tan (e+f x))^2}{2 f}+\frac{d (a+b \tan (e+f x))^3}{3 f}",1,"(3*(a + I*b)^3*((-I)*c + d)*Log[I - Tan[e + f*x]] + 3*(a - I*b)^3*(I*c + d)*Log[I + Tan[e + f*x]] + 6*b*(3*a*b*c + 3*a^2*d - b^2*d)*Tan[e + f*x] + 3*b^2*(b*c + 3*a*d)*Tan[e + f*x]^2 + 2*b^3*d*Tan[e + f*x]^3)/(6*f)","C",1
1192,1,96,87,0.4487276,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x)) \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x]),x]","\frac{2 b (2 a d+b c) \tan (e+f x)+(a-i b)^2 (d+i c) \log (\tan (e+f x)+i)+(a+i b)^2 (d-i c) \log (-\tan (e+f x)+i)+b^2 d \tan ^2(e+f x)}{2 f}","-\frac{\left(a^2 d+2 a b c-b^2 d\right) \log (\cos (e+f x))}{f}+x \left(a^2 c-2 a b d-b^2 c\right)+\frac{b (a d+b c) \tan (e+f x)}{f}+\frac{d (a+b \tan (e+f x))^2}{2 f}",1,"((a + I*b)^2*((-I)*c + d)*Log[I - Tan[e + f*x]] + (a - I*b)^2*(I*c + d)*Log[I + Tan[e + f*x]] + 2*b*(b*c + 2*a*d)*Tan[e + f*x] + b^2*d*Tan[e + f*x]^2)/(2*f)","C",1
1193,1,59,42,0.0376271,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x)) \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]),x]","a c x-\frac{a d \log (\cos (e+f x))}{f}-\frac{b c \log (\cos (e+f x))}{f}-\frac{b d \tan ^{-1}(\tan (e+f x))}{f}+\frac{b d \tan (e+f x)}{f}","-\frac{(a d+b c) \log (\cos (e+f x))}{f}+x (a c-b d)+\frac{b d \tan (e+f x)}{f}",1,"a*c*x - (b*d*ArcTan[Tan[e + f*x]])/f - (b*c*Log[Cos[e + f*x]])/f - (a*d*Log[Cos[e + f*x]])/f + (b*d*Tan[e + f*x])/f","A",1
1194,1,66,58,0.1295008,"\int \frac{c+d \tan (e+f x)}{a+b \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])/(a + b*Tan[e + f*x]),x]","\frac{2 (a c+b d) \tan ^{-1}(\tan (e+f x))-(b c-a d) \left(\log \left(\sec ^2(e+f x)\right)-2 \log (a+b \tan (e+f x))\right)}{2 f \left(a^2+b^2\right)}","\frac{(b c-a d) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)}+\frac{x (a c+b d)}{a^2+b^2}",1,"(2*(a*c + b*d)*ArcTan[Tan[e + f*x]] - (b*c - a*d)*(Log[Sec[e + f*x]^2] - 2*Log[a + b*Tan[e + f*x]]))/(2*(a^2 + b^2)*f)","A",1
1195,1,190,111,2.221906,"\int \frac{c+d \tan (e+f x)}{(a+b \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])/(a + b*Tan[e + f*x])^2,x]","\frac{\frac{d ((-b-i a) \log (-\tan (e+f x)+i)+i (a+i b) \log (\tan (e+f x)+i)+2 b \log (a+b \tan (e+f x)))}{a^2+b^2}-(b c-a d) \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (e+f x)}-2 a \log (a+b \tan (e+f x))\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (e+f x)+i)}{(a+i b)^2}-\frac{i \log (\tan (e+f x)+i)}{(a-i b)^2}\right)}{2 b f}","-\frac{b c-a d}{f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{\left(a^2 (-d)+2 a b c+b^2 d\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^2}+\frac{x \left(a^2 c+2 a b d-b^2 c\right)}{\left(a^2+b^2\right)^2}",1,"((d*(((-I)*a - b)*Log[I - Tan[e + f*x]] + I*(a + I*b)*Log[I + Tan[e + f*x]] + 2*b*Log[a + b*Tan[e + f*x]]))/(a^2 + b^2) - (b*c - a*d)*((I*Log[I - Tan[e + f*x]])/(a + I*b)^2 - (I*Log[I + Tan[e + f*x]])/(a - I*b)^2 + (2*b*(-2*a*Log[a + b*Tan[e + f*x]] + (a^2 + b^2)/(a + b*Tan[e + f*x])))/(a^2 + b^2)^2))/(2*b*f)","C",1
1196,1,243,175,4.4850054,"\int \frac{c+d \tan (e+f x)}{(a+b \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])/(a + b*Tan[e + f*x])^3,x]","-\frac{(b c-a d) \left(\frac{b \left(\frac{\left(a^2+b^2\right) \left(5 a^2+4 a b \tan (e+f x)+b^2\right)}{(a+b \tan (e+f x))^2}+\left(2 b^2-6 a^2\right) \log (a+b \tan (e+f x))\right)}{\left(a^2+b^2\right)^3}+\frac{i \log (-\tan (e+f x)+i)}{(a+i b)^3}-\frac{\log (\tan (e+f x)+i)}{(b+i a)^3}\right)+d \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (e+f x)}-2 a \log (a+b \tan (e+f x))\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (e+f x)+i)}{(a+i b)^2}-\frac{i \log (\tan (e+f x)+i)}{(a-i b)^2}\right)}{2 b f}","-\frac{b c-a d}{2 f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{a^2 (-d)+2 a b c+b^2 d}{f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}+\frac{\left(a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^3}+\frac{x \left(a^3 c+3 a^2 b d-3 a b^2 c-b^3 d\right)}{\left(a^2+b^2\right)^3}",1,"-1/2*(d*((I*Log[I - Tan[e + f*x]])/(a + I*b)^2 - (I*Log[I + Tan[e + f*x]])/(a - I*b)^2 + (2*b*(-2*a*Log[a + b*Tan[e + f*x]] + (a^2 + b^2)/(a + b*Tan[e + f*x])))/(a^2 + b^2)^2) + (b*c - a*d)*((I*Log[I - Tan[e + f*x]])/(a + I*b)^3 - Log[I + Tan[e + f*x]]/(I*a + b)^3 + (b*((-6*a^2 + 2*b^2)*Log[a + b*Tan[e + f*x]] + ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[e + f*x]))/(a + b*Tan[e + f*x])^2))/(a^2 + b^2)^3))/(b*f)","C",1
1197,1,221,215,2.5536736,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]","\frac{-4 c d \left(6 b^2 \left(b^2-6 a^2\right) \tan (e+f x)-12 a b^3 \tan ^2(e+f x)-3 i (a-i b)^4 \log (\tan (e+f x)+i)+3 i (a+i b)^4 \log (-\tan (e+f x)+i)-2 b^4 \tan ^3(e+f x)\right)-6 \left(2 a c d+b \left(d^2-c^2\right)\right) \left(6 a b^2 \tan (e+f x)+(-b+i a)^3 \log (-\tan (e+f x)+i)-(b+i a)^3 \log (\tan (e+f x)+i)+b^3 \tan ^2(e+f x)\right)+3 d^2 (a+b \tan (e+f x))^4}{12 b f}","-\frac{\left(2 a^3 c d+3 a^2 b \left(c^2-d^2\right)-6 a b^2 c d-b^3 \left(c^2-d^2\right)\right) \log (\cos (e+f x))}{f}-x \left(-\left(a^3 \left(c^2-d^2\right)\right)+6 a^2 b c d+3 a b^2 \left(c^2-d^2\right)-2 b^3 c d\right)+\frac{\left(2 a c d+b \left(c^2-d^2\right)\right) (a+b \tan (e+f x))^2}{2 f}+\frac{2 c d (a+b \tan (e+f x))^3}{3 f}+\frac{2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac{d^2 (a+b \tan (e+f x))^4}{4 b f}",1,"(3*d^2*(a + b*Tan[e + f*x])^4 - 6*(2*a*c*d + b*(-c^2 + d^2))*((I*a - b)^3*Log[I - Tan[e + f*x]] - (I*a + b)^3*Log[I + Tan[e + f*x]] + 6*a*b^2*Tan[e + f*x] + b^3*Tan[e + f*x]^2) - 4*c*d*((3*I)*(a + I*b)^4*Log[I - Tan[e + f*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[e + f*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[e + f*x] - 12*a*b^3*Tan[e + f*x]^2 - 2*b^4*Tan[e + f*x]^3))/(12*b*f)","C",1
1198,1,185,131,1.1098909,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2,x]","\frac{3 \left(2 a c d+b \left(d^2-c^2\right)\right) \left(-2 b^2 \tan (e+f x)+i \left((a+i b)^2 \log (-\tan (e+f x)+i)-(a-i b)^2 \log (\tan (e+f x)+i)\right)\right)+6 c d \left(6 a b^2 \tan (e+f x)+(-b+i a)^3 \log (-\tan (e+f x)+i)-(b+i a)^3 \log (\tan (e+f x)+i)+b^3 \tan ^2(e+f x)\right)+2 d^2 (a+b \tan (e+f x))^3}{6 b f}","\frac{b \left(2 a c d+b \left(c^2-d^2\right)\right) \tan (e+f x)}{f}+\frac{c d (a+b \tan (e+f x))^2}{f}-\frac{2 (a d+b c) (a c-b d) \log (\cos (e+f x))}{f}+x (a c-a d-b c-b d) (a c+a d+b c-b d)+\frac{d^2 (a+b \tan (e+f x))^3}{3 b f}",1,"(2*d^2*(a + b*Tan[e + f*x])^3 + 3*(2*a*c*d + b*(-c^2 + d^2))*(I*((a + I*b)^2*Log[I - Tan[e + f*x]] - (a - I*b)^2*Log[I + Tan[e + f*x]]) - 2*b^2*Tan[e + f*x]) + 6*c*d*((I*a - b)^3*Log[I - Tan[e + f*x]] - (I*a + b)^3*Log[I + Tan[e + f*x]] + 6*a*b^2*Tan[e + f*x] + b^3*Tan[e + f*x]^2))/(6*b*f)","C",1
1199,1,96,89,0.4683324,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^2 \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2,x]","\frac{2 d (a d+2 b c) \tan (e+f x)+(b+i a) (c-i d)^2 \log (\tan (e+f x)+i)+(b-i a) (c+i d)^2 \log (-\tan (e+f x)+i)+b d^2 \tan ^2(e+f x)}{2 f}","-\frac{\left(2 a c d+b \left(c^2-d^2\right)\right) \log (\cos (e+f x))}{f}-x \left(2 b c d-a \left(c^2-d^2\right)\right)+\frac{d (a d+b c) \tan (e+f x)}{f}+\frac{b (c+d \tan (e+f x))^2}{2 f}",1,"(((-I)*a + b)*(c + I*d)^2*Log[I - Tan[e + f*x]] + (I*a + b)*(c - I*d)^2*Log[I + Tan[e + f*x]] + 2*d*(2*b*c + a*d)*Tan[e + f*x] + b*d^2*Tan[e + f*x]^2)/(2*f)","C",1
1200,1,108,103,0.2179192,"\int \frac{(c+d \tan (e+f x))^2}{a+b \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^2/(a + b*Tan[e + f*x]),x]","\frac{\frac{2 (b c-a d)^2 \log (a+b \tan (e+f x))}{b \left(a^2+b^2\right)}-\frac{(c-i d)^2 \log (\tan (e+f x)+i)}{b+i a}+\frac{(c+i d)^2 \log (-\tan (e+f x)+i)}{-b+i a}}{2 f}","\frac{(b c-a d)^2 \log (a \cos (e+f x)+b \sin (e+f x))}{b f \left(a^2+b^2\right)}+\frac{a x (b c-a d)^2}{b^2 \left(a^2+b^2\right)}+\frac{d x (2 b c-a d)}{b^2}-\frac{d^2 \log (\cos (e+f x))}{b f}",1,"(((c + I*d)^2*Log[I - Tan[e + f*x]])/(I*a - b) - ((c - I*d)^2*Log[I + Tan[e + f*x]])/(I*a + b) + (2*(b*c - a*d)^2*Log[a + b*Tan[e + f*x]])/(b*(a^2 + b^2)))/(2*f)","C",1
1201,1,321,126,2.1991865,"\int \frac{(c+d \tan (e+f x))^2}{(a+b \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^2/(a + b*Tan[e + f*x])^2,x]","\frac{(c+d \tan (e+f x))^2 (a \cos (e+f x)+b \sin (e+f x)) \left(-2 i (e+f x) \left(a^2 c d+a b \left(d^2-c^2\right)-b^2 c d\right) (a \cos (e+f x)+b \sin (e+f x))-\left(a^2 c d+a b \left(d^2-c^2\right)-b^2 c d\right) (a \cos (e+f x)+b \sin (e+f x)) \log \left((a \cos (e+f x)+b \sin (e+f x))^2\right)+2 i \left(a^2 c d+a b \left(d^2-c^2\right)-b^2 c d\right) \tan ^{-1}(\tan (e+f x)) (a \cos (e+f x)+b \sin (e+f x))+\frac{\left(a^2+b^2\right) (b c-a d)^2 \sin (e+f x)}{a}+(e+f x) (a (c+d)+b (d-c)) (a (c-d)+b (c+d)) (a \cos (e+f x)+b \sin (e+f x))\right)}{f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))^2 (c \cos (e+f x)+d \sin (e+f x))^2}","-\frac{(b c-a d)^2}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{2 (a c+b d) (b c-a d) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^2}-\frac{x (b (c-d)-a (c+d)) (a (c-d)+b (c+d))}{\left(a^2+b^2\right)^2}",1,"((a*Cos[e + f*x] + b*Sin[e + f*x])*(((a^2 + b^2)*(b*c - a*d)^2*Sin[e + f*x])/a + (b*(-c + d) + a*(c + d))*(a*(c - d) + b*(c + d))*(e + f*x)*(a*Cos[e + f*x] + b*Sin[e + f*x]) - (2*I)*(a^2*c*d - b^2*c*d + a*b*(-c^2 + d^2))*(e + f*x)*(a*Cos[e + f*x] + b*Sin[e + f*x]) + (2*I)*(a^2*c*d - b^2*c*d + a*b*(-c^2 + d^2))*ArcTan[Tan[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x]) - (a^2*c*d - b^2*c*d + a*b*(-c^2 + d^2))*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]*(a*Cos[e + f*x] + b*Sin[e + f*x]))*(c + d*Tan[e + f*x])^2)/((a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2)","C",1
1202,1,291,214,3.7088946,"\int \frac{(c+d \tan (e+f x))^2}{(a+b \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^2/(a + b*Tan[e + f*x])^3,x]","\frac{(b c-a d) \left(-\frac{2 (b c-a d) \left(a^2 (-d)+2 a b c+b^2 d\right)}{b \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{(b+i a)^3 (c+i d)^2 \log (-\tan (e+f x)+i)}{\left(a^2+b^2\right)^2}-\frac{2 \left(2 a^3 c d+3 a^2 b \left(d^2-c^2\right)-6 a b^2 c d+b^3 \left(c^2-d^2\right)\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right)^2}+\frac{i (a+i b) (c-i d)^2 \log (\tan (e+f x)+i)}{(a-i b)^2}\right)-\frac{b^2 (c+d \tan (e+f x))^3}{(a+b \tan (e+f x))^2}+\frac{b d (c+d \tan (e+f x))^2}{a+b \tan (e+f x)}}{2 f \left(a^2+b^2\right) (b c-a d)}","-\frac{(b c-a d)^2}{2 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{2 (a c+b d) (b c-a d)}{f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{\left(2 a^3 c d-3 a^2 b \left(c^2-d^2\right)-6 a b^2 c d+b^3 \left(c^2-d^2\right)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^3}+\frac{x \left(a^3 \left(c^2-d^2\right)+6 a^2 b c d-3 a b^2 \left(c^2-d^2\right)-2 b^3 c d\right)}{\left(a^2+b^2\right)^3}",1,"((b*d*(c + d*Tan[e + f*x])^2)/(a + b*Tan[e + f*x]) - (b^2*(c + d*Tan[e + f*x])^3)/(a + b*Tan[e + f*x])^2 + (b*c - a*d)*(((I*a + b)^3*(c + I*d)^2*Log[I - Tan[e + f*x]])/(a^2 + b^2)^2 + (I*(a + I*b)*(c - I*d)^2*Log[I + Tan[e + f*x]])/(a - I*b)^2 - (2*(2*a^3*c*d - 6*a*b^2*c*d + b^3*(c^2 - d^2) + 3*a^2*b*(-c^2 + d^2))*Log[a + b*Tan[e + f*x]])/(a^2 + b^2)^2 - (2*(b*c - a*d)*(2*a*b*c - a^2*d + b^2*d))/(b*(a^2 + b^2)*(a + b*Tan[e + f*x]))))/(2*(a^2 + b^2)*(b*c - a*d)*f)","C",1
1203,1,299,302,6.447667,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3 \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3,x]","\frac{b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^4}{5 d f}+\frac{-\frac{b^2 (b c-11 a d) (c+d \tan (e+f x))^4}{4 d f}-\frac{5 \left(b \left(3 a^2-b^2\right) \left(-6 d^2 \left(6 c^2-d^2\right) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-3 i (c-i d)^4 \log (\tan (e+f x)+i)+3 i (c+i d)^4 \log (-\tan (e+f x)+i)-2 d^4 \tan ^3(e+f x)\right)+3 \left(a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right) \left(6 c d^2 \tan (e+f x)+(-d+i c)^3 \log (-\tan (e+f x)+i)-(d+i c)^3 \log (\tan (e+f x)+i)+d^3 \tan ^2(e+f x)\right)\right)}{6 f}}{5 d}","\frac{(a d+b c) \left(-\left(a^2 \left(3 c^2-d^2\right)\right)+8 a b c d+b^2 \left(c^2-3 d^2\right)\right) \log (\cos (e+f x))}{f}-x (a c-b d) \left(-\left(a^2 \left(c^2-3 d^2\right)\right)+8 a b c d+b^2 \left(3 c^2-d^2\right)\right)+\frac{b \left(3 a^2-b^2\right) (c+d \tan (e+f x))^3}{3 f}+\frac{d \left(2 a^3 c d+3 a^2 b \left(c^2-d^2\right)-6 a b^2 c d-b^3 \left(c^2-d^2\right)\right) \tan (e+f x)}{f}+\frac{\left(a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right) (c+d \tan (e+f x))^2}{2 f}-\frac{b^2 (b c-11 a d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac{b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^4}{5 d f}",1,"(b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^4)/(5*d*f) + (-1/4*(b^2*(b*c - 11*a*d)*(c + d*Tan[e + f*x])^4)/(d*f) - (5*(3*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*Log[I + Tan[e + f*x]] + 6*c*d^2*Tan[e + f*x] + d^3*Tan[e + f*x]^2) + b*(3*a^2 - b^2)*((3*I)*(c + I*d)^4*Log[I - Tan[e + f*x]] - (3*I)*(c - I*d)^4*Log[I + Tan[e + f*x]] - 6*d^2*(6*c^2 - d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e + f*x]^2 - 2*d^4*Tan[e + f*x]^3)))/(6*f))/(5*d)","C",1
1204,1,221,219,2.3172082,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3,x]","\frac{-6 \left(a^2 (-d)+2 a b c+b^2 d\right) \left(6 c d^2 \tan (e+f x)+(-d+i c)^3 \log (-\tan (e+f x)+i)-(d+i c)^3 \log (\tan (e+f x)+i)+d^3 \tan ^2(e+f x)\right)-4 a b \left(6 d^2 \left(d^2-6 c^2\right) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-3 i (c-i d)^4 \log (\tan (e+f x)+i)+3 i (c+i d)^4 \log (-\tan (e+f x)+i)-2 d^4 \tan ^3(e+f x)\right)+3 b^2 (c+d \tan (e+f x))^4}{12 d f}","-\frac{\left(a^2 \left(3 c^2 d-d^3\right)+2 a b c \left(c^2-3 d^2\right)-b^2 d \left(3 c^2-d^2\right)\right) \log (\cos (e+f x))}{f}-x \left(-\left(a^2 \left(c^3-3 c d^2\right)\right)+2 a b d \left(3 c^2-d^2\right)+b^2 c \left(c^2-3 d^2\right)\right)+\frac{\left(a^2 d+2 a b c-b^2 d\right) (c+d \tan (e+f x))^2}{2 f}+\frac{2 a b (c+d \tan (e+f x))^3}{3 f}+\frac{2 d (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac{b^2 (c+d \tan (e+f x))^4}{4 d f}",1,"(3*b^2*(c + d*Tan[e + f*x])^4 - 6*(2*a*b*c - a^2*d + b^2*d)*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*Log[I + Tan[e + f*x]] + 6*c*d^2*Tan[e + f*x] + d^3*Tan[e + f*x]^2) - 4*a*b*((3*I)*(c + I*d)^4*Log[I - Tan[e + f*x]] - (3*I)*(c - I*d)^4*Log[I + Tan[e + f*x]] + 6*d^2*(-6*c^2 + d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e + f*x]^2 - 2*d^4*Tan[e + f*x]^3))/(12*d*f)","C",1
1205,1,130,144,1.0095094,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3,x]","\frac{6 d \left(3 a c d+3 b c^2-b d^2\right) \tan (e+f x)+3 d^2 (a d+3 b c) \tan ^2(e+f x)+3 (b+i a) (c-i d)^3 \log (\tan (e+f x)+i)+3 (b-i a) (c+i d)^3 \log (-\tan (e+f x)+i)+2 b d^3 \tan ^3(e+f x)}{6 f}","\frac{d \left(2 a c d+b \left(c^2-d^2\right)\right) \tan (e+f x)}{f}-x \left(b d \left(3 c^2-d^2\right)-a \left(c^3-3 c d^2\right)\right)-\frac{\left(3 a c^2 d-a d^3+b c^3-3 b c d^2\right) \log (\cos (e+f x))}{f}+\frac{(a d+b c) (c+d \tan (e+f x))^2}{2 f}+\frac{b (c+d \tan (e+f x))^3}{3 f}",1,"(3*((-I)*a + b)*(c + I*d)^3*Log[I - Tan[e + f*x]] + 3*(I*a + b)*(c - I*d)^3*Log[I + Tan[e + f*x]] + 6*d*(3*b*c^2 + 3*a*c*d - b*d^2)*Tan[e + f*x] + 3*d^2*(3*b*c + a*d)*Tan[e + f*x]^2 + 2*b*d^3*Tan[e + f*x]^3)/(6*f)","C",1
1206,1,126,140,0.7166662,"\int \frac{(c+d \tan (e+f x))^3}{a+b \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x]),x]","\frac{\frac{2 (b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 \left(a^2+b^2\right)}-\frac{(c-i d)^3 \log (\tan (e+f x)+i)}{b+i a}+\frac{(c+i d)^3 \log (-\tan (e+f x)+i)}{-b+i a}+\frac{2 d^2 (c+d \tan (e+f x))}{b}}{2 f}","\frac{\left(-3 a c^2 d+a d^3+b c^3-3 b c d^2\right) \log (\cos (e+f x))}{f \left(a^2+b^2\right)}+\frac{x \left(a c^3-3 a c d^2+3 b c^2 d-b d^3\right)}{a^2+b^2}+\frac{(b c-a d)^3 \log (a+b \tan (e+f x))}{b^2 f \left(a^2+b^2\right)}+\frac{d^2 (c+d \tan (e+f x))}{b f}",1,"(((c + I*d)^3*Log[I - Tan[e + f*x]])/(I*a - b) - ((c - I*d)^3*Log[I + Tan[e + f*x]])/(I*a + b) + (2*(b*c - a*d)^3*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)) + (2*d^2*(c + d*Tan[e + f*x]))/b)/(2*f)","C",1
1207,1,535,230,4.6436334,"\int \frac{(c+d \tan (e+f x))^3}{(a+b \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x])^2,x]","\frac{\cos (e+f x) (c+d \tan (e+f x))^3 (a \cos (e+f x)+b \sin (e+f x)) \left(a^2 \cos (e+f x) \left(2 (a+i b)^2 (e+f x) \left(i a^2 d^3+2 a b d^3+b^2 c \left(c^2-3 i c d-3 d^2\right)\right)+(b c-a d)^2 \left(a^2 d+2 a b c+3 b^2 d\right) \log \left((a \cos (e+f x)+b \sin (e+f x))^2\right)-2 d^3 \left(a^2+b^2\right)^2 \log (\cos (e+f x))\right)-2 i a (b c-a d)^2 \left(a^2 d+2 a b c+3 b^2 d\right) \tan ^{-1}(\tan (e+f x)) (a \cos (e+f x)+b \sin (e+f x))+b \sin (e+f x) \left(a (b c-a d)^2 \left(a^2 d+2 a b c+3 b^2 d\right) \log \left((a \cos (e+f x)+b \sin (e+f x))^2\right)-2 a d^3 \left(a^2+b^2\right)^2 \log (\cos (e+f x))+2 (a+i b) \left(i a^4 d^3 (e+f x+i)+a^3 b d^2 (3 c+d (e+f x+i))+a^2 b^2 \left(c^3 (e+f x)-3 i c^2 d (e+f x-i)-3 c d^2 (e+f x+i)+2 i d^3 (e+f x)\right)+a b^3 c \left(c^2 (i e+i f x+1)+3 c d (e+f x+i)-3 i d^2 (e+f x)\right)-i b^4 c^3\right)\right)\right)}{2 a b^2 f (a-i b)^2 (a+i b)^2 (a+b \tan (e+f x))^2 (c \cos (e+f x)+d \sin (e+f x))^3}","\frac{\left(-\left(a^2 \left(3 c^2 d-d^3\right)\right)+2 a b c \left(c^2-3 d^2\right)+b^2 d \left(3 c^2-d^2\right)\right) \log (\cos (e+f x))}{f \left(a^2+b^2\right)^2}-\frac{x \left(-\left(a^2 \left(c^3-3 c d^2\right)\right)-2 a b d \left(3 c^2-d^2\right)+b^2 c \left(c^2-3 d^2\right)\right)}{\left(a^2+b^2\right)^2}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{\left(a^2 d+2 a b c+3 b^2 d\right) (b c-a d)^2 \log (a+b \tan (e+f x))}{b^2 f \left(a^2+b^2\right)^2}",1,"(Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(a^2*Cos[e + f*x]*(2*(a + I*b)^2*(I*a^2*d^3 + 2*a*b*d^3 + b^2*c*(c^2 - (3*I)*c*d - 3*d^2))*(e + f*x) - 2*(a^2 + b^2)^2*d^3*Log[Cos[e + f*x]] + (b*c - a*d)^2*(2*a*b*c + a^2*d + 3*b^2*d)*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]) + b*(2*(a + I*b)*((-I)*b^4*c^3 + I*a^4*d^3*(I + e + f*x) + a^3*b*d^2*(3*c + d*(I + e + f*x)) + a*b^3*c*(c^2*(1 + I*e + I*f*x) - (3*I)*d^2*(e + f*x) + 3*c*d*(I + e + f*x)) + a^2*b^2*(c^3*(e + f*x) + (2*I)*d^3*(e + f*x) - (3*I)*c^2*d*(-I + e + f*x) - 3*c*d^2*(I + e + f*x))) - 2*a*(a^2 + b^2)^2*d^3*Log[Cos[e + f*x]] + a*(b*c - a*d)^2*(2*a*b*c + a^2*d + 3*b^2*d)*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2])*Sin[e + f*x] - (2*I)*a*(b*c - a*d)^2*(2*a*b*c + a^2*d + 3*b^2*d)*ArcTan[Tan[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x]))*(c + d*Tan[e + f*x])^3)/(2*a*(a - I*b)^2*(a + I*b)^2*b^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)","C",1
1208,1,327,239,5.606157,"\int \frac{(c+d \tan (e+f x))^3}{(a+b \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x])^3,x]","\frac{2 b d \left(3 c^2-d^2\right) \left(\frac{b \left(2 a \log (a+b \tan (e+f x))-\frac{a^2+b^2}{a+b \tan (e+f x)}\right)}{\left(a^2+b^2\right)^2}-\frac{i \log (-\tan (e+f x)+i)}{2 (a+i b)^2}+\frac{i \log (\tan (e+f x)+i)}{2 (a-i b)^2}\right)+b \left(a d \left(d^2-3 c^2\right)+b \left(c^3-3 c d^2\right)\right) \left(\frac{b \left(\left(6 a^2-2 b^2\right) \log (a+b \tan (e+f x))-\frac{\left(a^2+b^2\right) \left(5 a^2+4 a b \tan (e+f x)+b^2\right)}{(a+b \tan (e+f x))^2}\right)}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (e+f x)+i)}{(b-i a)^3}+\frac{\log (\tan (e+f x)+i)}{(b+i a)^3}\right)-\frac{2 b d^2 (c+d \tan (e+f x))}{(a+b \tan (e+f x))^2}-\frac{d^2 (a d+b c)}{(a+b \tan (e+f x))^2}}{2 b^2 f}","\frac{\left(a^2 \left(3 c^2-d^2\right)+8 a b c d-b^2 \left(c^2-3 d^2\right)\right) (b c-a d) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^3}+\frac{x (a c+b d) \left(a^2 \left(c^2-3 d^2\right)+8 a b c d-b^2 \left(3 c^2-d^2\right)\right)}{\left(a^2+b^2\right)^3}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{\left(a^2 d+4 a b c+5 b^2 d\right) (b c-a d)^2}{2 b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}",1,"(-((d^2*(b*c + a*d))/(a + b*Tan[e + f*x])^2) - (2*b*d^2*(c + d*Tan[e + f*x]))/(a + b*Tan[e + f*x])^2 + 2*b*d*(3*c^2 - d^2)*(((-1/2*I)*Log[I - Tan[e + f*x]])/(a + I*b)^2 + ((I/2)*Log[I + Tan[e + f*x]])/(a - I*b)^2 + (b*(2*a*Log[a + b*Tan[e + f*x]] - (a^2 + b^2)/(a + b*Tan[e + f*x])))/(a^2 + b^2)^2) + b*(a*d*(-3*c^2 + d^2) + b*(c^3 - 3*c*d^2))*(Log[I - Tan[e + f*x]]/((-I)*a + b)^3 + Log[I + Tan[e + f*x]]/(I*a + b)^3 + (b*((6*a^2 - 2*b^2)*Log[a + b*Tan[e + f*x]] - ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[e + f*x]))/(a + b*Tan[e + f*x])^2))/(a^2 + b^2)^3))/(2*b^2*f)","C",1
1209,1,160,190,1.288825,"\int \frac{(a+b \tan (e+f x))^4}{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x]),x]","\frac{-\frac{2 b^3 (b c-3 a d) \tan (e+f x)}{d}+b^2 (a+b \tan (e+f x))^2+\frac{\frac{2 (b c-a d)^4 \log (c+d \tan (e+f x))}{d \left(c^2+d^2\right)}-\frac{d^2 (a-i b)^4 \log (\tan (e+f x)+i)}{d+i c}+\frac{d^2 (a+i b)^4 \log (-\tan (e+f x)+i)}{-d+i c}}{d}}{2 d f}","-\frac{\left(a^4 (-d)+4 a^3 b c+6 a^2 b^2 d-4 a b^3 c-b^4 d\right) \log (\cos (e+f x))}{f \left(c^2+d^2\right)}+\frac{x \left(a^4 c+4 a^3 b d-6 a^2 b^2 c-4 a b^3 d+b^4 c\right)}{c^2+d^2}-\frac{b^3 (b c-3 a d) \tan (e+f x)}{d^2 f}+\frac{b^2 (a+b \tan (e+f x))^2}{2 d f}+\frac{(b c-a d)^4 \log (c+d \tan (e+f x))}{d^3 f \left(c^2+d^2\right)}",1,"((((a + I*b)^4*d^2*Log[I - Tan[e + f*x]])/(I*c - d) - ((a - I*b)^4*d^2*Log[I + Tan[e + f*x]])/(I*c + d) + (2*(b*c - a*d)^4*Log[c + d*Tan[e + f*x]])/(d*(c^2 + d^2)))/d - (2*b^3*(b*c - 3*a*d)*Tan[e + f*x])/d + b^2*(a + b*Tan[e + f*x])^2)/(2*d*f)","C",1
1210,1,126,144,0.7668238,"\int \frac{(a+b \tan (e+f x))^3}{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x]),x]","\frac{\frac{2 b^2 (a+b \tan (e+f x))}{d}+\frac{2 (a d-b c)^3 \log (c+d \tan (e+f x))}{d^2 \left(c^2+d^2\right)}+\frac{(a+i b)^3 \log (-\tan (e+f x)+i)}{-d+i c}-\frac{(b+i a)^3 \log (\tan (e+f x)+i)}{c-i d}}{2 f}","-\frac{\left(a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right) \log (\cos (e+f x))}{f \left(c^2+d^2\right)}+\frac{x \left(a^3 c+3 a^2 b d-3 a b^2 c-b^3 d\right)}{c^2+d^2}+\frac{b^2 (a+b \tan (e+f x))}{d f}-\frac{(b c-a d)^3 \log (c+d \tan (e+f x))}{d^2 f \left(c^2+d^2\right)}",1,"(((a + I*b)^3*Log[I - Tan[e + f*x]])/(I*c - d) - ((I*a + b)^3*Log[I + Tan[e + f*x]])/(c - I*d) + (2*(-(b*c) + a*d)^3*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)) + (2*b^2*(a + b*Tan[e + f*x]))/d)/(2*f)","C",1
1211,1,108,103,0.1637025,"\int \frac{(a+b \tan (e+f x))^2}{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x]),x]","\frac{\frac{2 (b c-a d)^2 \log (c+d \tan (e+f x))}{d \left(c^2+d^2\right)}-\frac{(a-i b)^2 \log (\tan (e+f x)+i)}{d+i c}+\frac{(a+i b)^2 \log (-\tan (e+f x)+i)}{-d+i c}}{2 f}","\frac{(b c-a d)^2 \log (c \cos (e+f x)+d \sin (e+f x))}{d f \left(c^2+d^2\right)}+\frac{c x (b c-a d)^2}{d^2 \left(c^2+d^2\right)}-\frac{b x (b c-2 a d)}{d^2}-\frac{b^2 \log (\cos (e+f x))}{d f}",1,"(((a + I*b)^2*Log[I - Tan[e + f*x]])/(I*c - d) - ((a - I*b)^2*Log[I + Tan[e + f*x]])/(I*c + d) + (2*(b*c - a*d)^2*Log[c + d*Tan[e + f*x]])/(d*(c^2 + d^2)))/(2*f)","C",1
1212,1,65,59,0.1209623,"\int \frac{a+b \tan (e+f x)}{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x]),x]","\frac{2 (a c+b d) \tan ^{-1}(\tan (e+f x))+(b c-a d) \left(\log \left(\sec ^2(e+f x)\right)-2 \log (c+d \tan (e+f x))\right)}{2 f \left(c^2+d^2\right)}","\frac{x (a c+b d)}{c^2+d^2}-\frac{(b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)}",1,"(2*(a*c + b*d)*ArcTan[Tan[e + f*x]] + (b*c - a*d)*(Log[Sec[e + f*x]^2] - 2*Log[c + d*Tan[e + f*x]]))/(2*(c^2 + d^2)*f)","A",1
1213,1,143,118,0.3574958,"\int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx","Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])),x]","\frac{\frac{2 b^2 \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (b c-a d)}+\frac{2 d^2 \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (a d-b c)}+\frac{\log (-\tan (e+f x)+i)}{(a+i b) (-d+i c)}-\frac{\log (\tan (e+f x)+i)}{(b+i a) (c-i d)}}{2 f}","\frac{x (a c-b d)}{\left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{b^2 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right) (b c-a d)}-\frac{d^2 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right) (b c-a d)}",1,"(Log[I - Tan[e + f*x]]/((a + I*b)*(I*c - d)) - Log[I + Tan[e + f*x]]/((I*a + b)*(c - I*d)) + (2*b^2*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + (2*d^2*Log[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)))/(2*f)","C",1
1214,1,302,183,3.3303155,"\int \frac{1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx","Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])),x]","-\frac{\frac{\left(\frac{\sqrt{-b^2} \left(a^2 c-2 a b d-b^2 c\right)}{b}+a^2 d+2 a b c-b^2 d\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right)}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)}+\frac{\left(\frac{\sqrt{-b^2} \left(a^2 (-c)+2 a b d+b^2 c\right)}{b}+a^2 d+2 a b c-b^2 d\right) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right)}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)}+\frac{2 b^2}{\left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))}+\frac{2 b^2 \left(3 a^2 d-2 a b c+b^2 d\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right)^2 (b c-a d)^2}-\frac{2 d^3 \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)^2}}{2 f}","\frac{x \left(a^2 c-2 a b d-b^2 c\right)}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)}-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))}+\frac{b^2 \left(-3 a^2 d+2 a b c-b^2 d\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^2 (b c-a d)^2}+\frac{d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right) (b c-a d)^2}",1,"-1/2*(((2*a*b*c + a^2*d - b^2*d + (Sqrt[-b^2]*(a^2*c - b^2*c - 2*a*b*d))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)^2*(c^2 + d^2)) + (2*b^2*(-2*a*b*c + 3*a^2*d + b^2*d)*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^2) + ((2*a*b*c + a^2*d - b^2*d + (Sqrt[-b^2]*(-(a^2*c) + b^2*c + 2*a*b*d))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)^2*(c^2 + d^2)) - (2*d^3*Log[c + d*Tan[e + f*x]])/((b*c - a*d)^2*(c^2 + d^2)) + (2*b^2)/((a^2 + b^2)*(b*c - a*d)*(a + b*Tan[e + f*x])))/f","A",1
1215,1,529,279,6.8922272,"\int \frac{1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx","Integrate[1/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])),x]","-\frac{b^2}{2 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^2}-\frac{-\frac{-2 b^2 \left(a^2 (-d)+a b c-b^2 d\right)-a \left(2 b^2 (b c-a d)-2 a b^2 d\right)}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))}-\frac{-\frac{2 b d^4 \left(a^2+b^2\right)^2 \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)}-\frac{b (b c-a d)^2 \left(a^3 d+3 a^2 b c+\frac{\sqrt{-b^2} \left(a^3 c-3 a^2 b d-3 a b^2 c+b^3 d\right)}{b}-3 a b^2 d-b^3 c\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{b (b c-a d)^2 \left(a^3 d+3 a^2 b c+\frac{b \left(a^3 c-3 a^2 b d-3 a b^2 c+b^3 d\right)}{\sqrt{-b^2}}-3 a b^2 d-b^3 c\right) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{2 b^3 \left(-6 a^4 d^2+8 a^3 b c d-3 a^2 b^2 \left(c^2+d^2\right)+b^4 \left(c^2-d^2\right)\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (b c-a d)}}{b f \left(a^2+b^2\right) (b c-a d)}}{2 \left(a^2+b^2\right) (b c-a d)}","-\frac{b^2 \left(-3 a^2 d+2 a b c-b^2 d\right)}{f \left(a^2+b^2\right)^2 (b c-a d)^2 (a+b \tan (e+f x))}-\frac{b^2}{2 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^2}+\frac{x \left(a^3 c-3 a^2 b d-3 a b^2 c+b^3 d\right)}{\left(a^2+b^2\right)^3 \left(c^2+d^2\right)}-\frac{b^2 \left(-6 a^4 d^2+8 a^3 b c d-3 a^2 b^2 \left(c^2+d^2\right)+b^4 \left(c^2-d^2\right)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^3 (b c-a d)^3}-\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right) (b c-a d)^3}",1,"-1/2*b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((-((b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d + (Sqrt[-b^2]*(a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2))) - (2*b^3*(8*a^3*b*c*d - 6*a^4*d^2 + b^4*(c^2 - d^2) - 3*a^2*b^2*(c^2 + d^2))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d + (b*(a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d))/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)^2*d^4*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/(b*(a^2 + b^2)*(b*c - a*d)*f)) - (-2*b^2*(a*b*c - a^2*d - b^2*d) - a*(-2*a*b^2*d + 2*b^2*(b*c - a*d)))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])))/(2*(a^2 + b^2)*(b*c - a*d))","A",1
1216,1,1789,285,7.074717,"\int \frac{(a+b \tan (e+f x))^4}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^2,x]","\frac{2 \left(-i b^4 d^2 c^{10}-b^4 d^3 c^9+2 i a b^3 d^3 c^9-3 i b^4 d^4 c^8+2 a b^3 d^4 c^8-3 b^4 d^5 c^7+8 i a b^3 d^5 c^7-2 i a^3 b d^5 c^7+i a^4 d^6 c^6-2 i b^4 d^6 c^6+8 a b^3 d^6 c^6-6 i a^2 b^2 d^6 c^6-2 a^3 b d^6 c^6+a^4 d^7 c^5-2 b^4 d^7 c^5+6 i a b^3 d^7 c^5-6 a^2 b^2 d^7 c^5+i a^4 d^8 c^4+6 a b^3 d^8 c^4-6 i a^2 b^2 d^8 c^4+a^4 d^9 c^3-6 a^2 b^2 d^9 c^3+2 i a^3 b d^9 c^3+2 a^3 b d^{10} c^2\right) (e+f x) \cos ^2(e+f x) (c \cos (e+f x)+d \sin (e+f x))^2 (a+b \tan (e+f x))^4}{c^2 (c-i d)^4 (c+i d)^3 d^5 f (a \cos (e+f x)+b \sin (e+f x))^4 (c+d \tan (e+f x))^2}-\frac{2 i \left(-b^4 c^5+2 a b^3 d c^4-2 b^4 d^2 c^3+6 a b^3 d^3 c^2-2 a^3 b d^3 c^2+a^4 d^4 c-6 a^2 b^2 d^4 c+2 a^3 b d^5\right) \tan ^{-1}(\tan (e+f x)) \cos ^2(e+f x) (c \cos (e+f x)+d \sin (e+f x))^2 (a+b \tan (e+f x))^4}{d^3 \left(c^2+d^2\right)^2 f (a \cos (e+f x)+b \sin (e+f x))^4 (c+d \tan (e+f x))^2}-\frac{2 \left(2 a b^3 d-b^4 c\right) \cos ^2(e+f x) \log (\cos (e+f x)) (c \cos (e+f x)+d \sin (e+f x))^2 (a+b \tan (e+f x))^4}{d^3 f (a \cos (e+f x)+b \sin (e+f x))^4 (c+d \tan (e+f x))^2}+\frac{\left(-b^4 c^5+2 a b^3 d c^4-2 b^4 d^2 c^3+6 a b^3 d^3 c^2-2 a^3 b d^3 c^2+a^4 d^4 c-6 a^2 b^2 d^4 c+2 a^3 b d^5\right) \cos ^2(e+f x) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right) (c \cos (e+f x)+d \sin (e+f x))^2 (a+b \tan (e+f x))^4}{d^3 \left(c^2+d^2\right)^2 f (a \cos (e+f x)+b \sin (e+f x))^4 (c+d \tan (e+f x))^2}+\frac{\cos (e+f x) (c \cos (e+f x)+d \sin (e+f x)) \left(2 b^4 \sin (2 (e+f x)) c^6+b^4 d c^5-b^4 d \cos (2 (e+f x)) c^5-4 a b^3 d \sin (2 (e+f x)) c^5+a^4 d^2 (e+f x) c^4+b^4 d^2 (e+f x) c^4-6 a^2 b^2 d^2 (e+f x) c^4+a^4 d^2 (e+f x) \cos (2 (e+f x)) c^4+b^4 d^2 (e+f x) \cos (2 (e+f x)) c^4-6 a^2 b^2 d^2 (e+f x) \cos (2 (e+f x)) c^4+3 b^4 d^2 \sin (2 (e+f x)) c^4+6 a^2 b^2 d^2 \sin (2 (e+f x)) c^4+2 b^4 d^3 c^3-8 a b^3 d^3 (e+f x) c^3+8 a^3 b d^3 (e+f x) c^3-2 b^4 d^3 \cos (2 (e+f x)) c^3-8 a b^3 d^3 (e+f x) \cos (2 (e+f x)) c^3+8 a^3 b d^3 (e+f x) \cos (2 (e+f x)) c^3-4 a b^3 d^3 \sin (2 (e+f x)) c^3-4 a^3 b d^3 \sin (2 (e+f x)) c^3+a^4 d^3 (e+f x) \sin (2 (e+f x)) c^3+b^4 d^3 (e+f x) \sin (2 (e+f x)) c^3-6 a^2 b^2 d^3 (e+f x) \sin (2 (e+f x)) c^3-a^4 d^4 (e+f x) c^2-b^4 d^4 (e+f x) c^2+6 a^2 b^2 d^4 (e+f x) c^2-a^4 d^4 (e+f x) \cos (2 (e+f x)) c^2-b^4 d^4 (e+f x) \cos (2 (e+f x)) c^2+6 a^2 b^2 d^4 (e+f x) \cos (2 (e+f x)) c^2+a^4 d^4 \sin (2 (e+f x)) c^2+b^4 d^4 \sin (2 (e+f x)) c^2+6 a^2 b^2 d^4 \sin (2 (e+f x)) c^2-8 a b^3 d^4 (e+f x) \sin (2 (e+f x)) c^2+8 a^3 b d^4 (e+f x) \sin (2 (e+f x)) c^2+b^4 d^5 c-b^4 d^5 \cos (2 (e+f x)) c-4 a^3 b d^5 \sin (2 (e+f x)) c-a^4 d^5 (e+f x) \sin (2 (e+f x)) c-b^4 d^5 (e+f x) \sin (2 (e+f x)) c+6 a^2 b^2 d^5 (e+f x) \sin (2 (e+f x)) c+a^4 d^6 \sin (2 (e+f x))\right) (a+b \tan (e+f x))^4}{2 c (c-i d)^2 (c+i d)^2 d^2 f (a \cos (e+f x)+b \sin (e+f x))^4 (c+d \tan (e+f x))^2}","-\frac{2 \left(a^2 c+2 a b d-b^2 c\right) \left(a^2 (-d)+2 a b c+b^2 d\right) \log (\cos (e+f x))}{f \left(c^2+d^2\right)^2}+\frac{x \left(a^4 \left(c^2-d^2\right)+8 a^3 b c d-6 a^2 b^2 \left(c^2-d^2\right)-8 a b^3 c d+b^4 \left(c^2-d^2\right)\right)}{\left(c^2+d^2\right)^2}-\frac{b^2 \left(a d (2 b c-a d)-b^2 \left(2 c^2+d^2\right)\right) \tan (e+f x)}{d^2 f \left(c^2+d^2\right)}-\frac{(b c-a d)^2 (a+b \tan (e+f x))^2}{d f \left(c^2+d^2\right) (c+d \tan (e+f x))}-\frac{2 \left(a c d+b \left(c^2+2 d^2\right)\right) (b c-a d)^3 \log (c+d \tan (e+f x))}{d^3 f \left(c^2+d^2\right)^2}",1,"(2*((-I)*b^4*c^10*d^2 + (2*I)*a*b^3*c^9*d^3 - b^4*c^9*d^3 + 2*a*b^3*c^8*d^4 - (3*I)*b^4*c^8*d^4 - (2*I)*a^3*b*c^7*d^5 + (8*I)*a*b^3*c^7*d^5 - 3*b^4*c^7*d^5 + I*a^4*c^6*d^6 - 2*a^3*b*c^6*d^6 - (6*I)*a^2*b^2*c^6*d^6 + 8*a*b^3*c^6*d^6 - (2*I)*b^4*c^6*d^6 + a^4*c^5*d^7 - 6*a^2*b^2*c^5*d^7 + (6*I)*a*b^3*c^5*d^7 - 2*b^4*c^5*d^7 + I*a^4*c^4*d^8 - (6*I)*a^2*b^2*c^4*d^8 + 6*a*b^3*c^4*d^8 + a^4*c^3*d^9 + (2*I)*a^3*b*c^3*d^9 - 6*a^2*b^2*c^3*d^9 + 2*a^3*b*c^2*d^10)*(e + f*x)*Cos[e + f*x]^2*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^4)/(c^2*(c - I*d)^4*(c + I*d)^3*d^5*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^2) - ((2*I)*(-(b^4*c^5) + 2*a*b^3*c^4*d - 2*b^4*c^3*d^2 - 2*a^3*b*c^2*d^3 + 6*a*b^3*c^2*d^3 + a^4*c*d^4 - 6*a^2*b^2*c*d^4 + 2*a^3*b*d^5)*ArcTan[Tan[e + f*x]]*Cos[e + f*x]^2*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^4)/(d^3*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^2) - (2*(-(b^4*c) + 2*a*b^3*d)*Cos[e + f*x]^2*Log[Cos[e + f*x]]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^4)/(d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^2) + ((-(b^4*c^5) + 2*a*b^3*c^4*d - 2*b^4*c^3*d^2 - 2*a^3*b*c^2*d^3 + 6*a*b^3*c^2*d^3 + a^4*c*d^4 - 6*a^2*b^2*c*d^4 + 2*a^3*b*d^5)*Cos[e + f*x]^2*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^4)/(d^3*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^2) + (Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])*(b^4*c^5*d + 2*b^4*c^3*d^3 + b^4*c*d^5 + a^4*c^4*d^2*(e + f*x) - 6*a^2*b^2*c^4*d^2*(e + f*x) + b^4*c^4*d^2*(e + f*x) + 8*a^3*b*c^3*d^3*(e + f*x) - 8*a*b^3*c^3*d^3*(e + f*x) - a^4*c^2*d^4*(e + f*x) + 6*a^2*b^2*c^2*d^4*(e + f*x) - b^4*c^2*d^4*(e + f*x) - b^4*c^5*d*Cos[2*(e + f*x)] - 2*b^4*c^3*d^3*Cos[2*(e + f*x)] - b^4*c*d^5*Cos[2*(e + f*x)] + a^4*c^4*d^2*(e + f*x)*Cos[2*(e + f*x)] - 6*a^2*b^2*c^4*d^2*(e + f*x)*Cos[2*(e + f*x)] + b^4*c^4*d^2*(e + f*x)*Cos[2*(e + f*x)] + 8*a^3*b*c^3*d^3*(e + f*x)*Cos[2*(e + f*x)] - 8*a*b^3*c^3*d^3*(e + f*x)*Cos[2*(e + f*x)] - a^4*c^2*d^4*(e + f*x)*Cos[2*(e + f*x)] + 6*a^2*b^2*c^2*d^4*(e + f*x)*Cos[2*(e + f*x)] - b^4*c^2*d^4*(e + f*x)*Cos[2*(e + f*x)] + 2*b^4*c^6*Sin[2*(e + f*x)] - 4*a*b^3*c^5*d*Sin[2*(e + f*x)] + 6*a^2*b^2*c^4*d^2*Sin[2*(e + f*x)] + 3*b^4*c^4*d^2*Sin[2*(e + f*x)] - 4*a^3*b*c^3*d^3*Sin[2*(e + f*x)] - 4*a*b^3*c^3*d^3*Sin[2*(e + f*x)] + a^4*c^2*d^4*Sin[2*(e + f*x)] + 6*a^2*b^2*c^2*d^4*Sin[2*(e + f*x)] + b^4*c^2*d^4*Sin[2*(e + f*x)] - 4*a^3*b*c*d^5*Sin[2*(e + f*x)] + a^4*d^6*Sin[2*(e + f*x)] + a^4*c^3*d^3*(e + f*x)*Sin[2*(e + f*x)] - 6*a^2*b^2*c^3*d^3*(e + f*x)*Sin[2*(e + f*x)] + b^4*c^3*d^3*(e + f*x)*Sin[2*(e + f*x)] + 8*a^3*b*c^2*d^4*(e + f*x)*Sin[2*(e + f*x)] - 8*a*b^3*c^2*d^4*(e + f*x)*Sin[2*(e + f*x)] - a^4*c*d^5*(e + f*x)*Sin[2*(e + f*x)] + 6*a^2*b^2*c*d^5*(e + f*x)*Sin[2*(e + f*x)] - b^4*c*d^5*(e + f*x)*Sin[2*(e + f*x)])*(a + b*Tan[e + f*x])^4)/(2*c*(c - I*d)^2*(c + I*d)^2*d^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^2)","C",1
1217,1,538,223,4.8023336,"\int \frac{(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^2,x]","\frac{\cos (e+f x) (a+b \tan (e+f x))^3 (c \cos (e+f x)+d \sin (e+f x)) \left(c^2 \cos (e+f x) \left(2 (c+i d)^2 (e+f x) \left(a^3 d^2-3 i a^2 b d^2-3 a b^2 d^2+b^3 c (2 d+i c)\right)+(b c-a d)^2 \left(2 a c d+b \left(c^2+3 d^2\right)\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-2 b^3 \left(c^2+d^2\right)^2 \log (\cos (e+f x))\right)+d \sin (e+f x) \left(2 (c+i d) \left(a^3 d^2 \left(c^2 (e+f x)+c d (i e+i f x+1)-i d^2\right)+3 a^2 b c d^2 (d (e+f x+i)-i c (e+f x-i))+3 a b^2 c d \left(c^2-c d (e+f x+i)-i d^2 (e+f x)\right)+b^3 c^2 \left(i c^2 (e+f x+i)+c d (e+f x+i)+2 i d^2 (e+f x)\right)\right)+c (b c-a d)^2 \left(2 a c d+b \left(c^2+3 d^2\right)\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-2 b^3 c \left(c^2+d^2\right)^2 \log (\cos (e+f x))\right)-2 i c (b c-a d)^2 \left(2 a c d+b \left(c^2+3 d^2\right)\right) \tan ^{-1}(\tan (e+f x)) (c \cos (e+f x)+d \sin (e+f x))\right)}{2 c d^2 f (c-i d)^2 (c+i d)^2 (c+d \tan (e+f x))^2 (a \cos (e+f x)+b \sin (e+f x))^3}","\frac{\left(2 a^3 c d-3 a^2 b \left(c^2-d^2\right)-6 a b^2 c d+b^3 \left(c^2-d^2\right)\right) \log (\cos (e+f x))}{f \left(c^2+d^2\right)^2}+\frac{x \left(a^3 \left(c^2-d^2\right)+6 a^2 b c d-3 a b^2 \left(c^2-d^2\right)-2 b^3 c d\right)}{\left(c^2+d^2\right)^2}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d f \left(c^2+d^2\right) (c+d \tan (e+f x))}+\frac{\left(2 a c d+b \left(c^2+3 d^2\right)\right) (b c-a d)^2 \log (c+d \tan (e+f x))}{d^2 f \left(c^2+d^2\right)^2}",1,"(Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])*(c^2*Cos[e + f*x]*(2*(c + I*d)^2*(a^3*d^2 - (3*I)*a^2*b*d^2 - 3*a*b^2*d^2 + b^3*c*(I*c + 2*d))*(e + f*x) - 2*b^3*(c^2 + d^2)^2*Log[Cos[e + f*x]] + (b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + d*(2*(c + I*d)*(a^3*d^2*((-I)*d^2 + c*d*(1 + I*e + I*f*x) + c^2*(e + f*x)) + 3*a^2*b*c*d^2*((-I)*c*(-I + e + f*x) + d*(I + e + f*x)) + 3*a*b^2*c*d*(c^2 - I*d^2*(e + f*x) - c*d*(I + e + f*x)) + b^3*c^2*((2*I)*d^2*(e + f*x) + I*c^2*(I + e + f*x) + c*d*(I + e + f*x))) - 2*b^3*c*(c^2 + d^2)^2*Log[Cos[e + f*x]] + c*(b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2])*Sin[e + f*x] - (2*I)*c*(b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*ArcTan[Tan[e + f*x]]*(c*Cos[e + f*x] + d*Sin[e + f*x]))*(a + b*Tan[e + f*x])^3)/(2*c*(c - I*d)^2*(c + I*d)^2*d^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)","C",1
1218,1,320,126,2.1388237,"\int \frac{(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^2,x]","\frac{(a+b \tan (e+f x))^2 (c \cos (e+f x)+d \sin (e+f x)) \left(2 i (e+f x) \left(a^2 c d+a b \left(d^2-c^2\right)-b^2 c d\right) (c \cos (e+f x)+d \sin (e+f x))+\left(a^2 c d+a b \left(d^2-c^2\right)-b^2 c d\right) (c \cos (e+f x)+d \sin (e+f x)) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+2 i \left(a^2 (-c) d+a b \left(c^2-d^2\right)+b^2 c d\right) \tan ^{-1}(\tan (e+f x)) (c \cos (e+f x)+d \sin (e+f x))+\frac{\left(c^2+d^2\right) (b c-a d)^2 \sin (e+f x)}{c}+(e+f x) (a (c+d)+b (d-c)) (a (c-d)+b (c+d)) (c \cos (e+f x)+d \sin (e+f x))\right)}{f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))^2 (a \cos (e+f x)+b \sin (e+f x))^2}","-\frac{(b c-a d)^2}{d f \left(c^2+d^2\right) (c+d \tan (e+f x))}-\frac{2 (a c+b d) (b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^2}-\frac{x (b (c-d)-a (c+d)) (a (c-d)+b (c+d))}{\left(c^2+d^2\right)^2}",1,"((c*Cos[e + f*x] + d*Sin[e + f*x])*(((b*c - a*d)^2*(c^2 + d^2)*Sin[e + f*x])/c + (b*(-c + d) + a*(c + d))*(a*(c - d) + b*(c + d))*(e + f*x)*(c*Cos[e + f*x] + d*Sin[e + f*x]) + (2*I)*(a^2*c*d - b^2*c*d + a*b*(-c^2 + d^2))*(e + f*x)*(c*Cos[e + f*x] + d*Sin[e + f*x]) + (2*I)*(-(a^2*c*d) + b^2*c*d + a*b*(c^2 - d^2))*ArcTan[Tan[e + f*x]]*(c*Cos[e + f*x] + d*Sin[e + f*x]) + (a^2*c*d - b^2*c*d + a*b*(-c^2 + d^2))*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(c*Cos[e + f*x] + d*Sin[e + f*x]))*(a + b*Tan[e + f*x])^2)/((c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)","C",1
1219,1,189,111,2.1521425,"\int \frac{a+b \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^2,x]","\frac{(b c-a d) \left(\frac{2 d \left(\frac{c^2+d^2}{c+d \tan (e+f x)}-2 c \log (c+d \tan (e+f x))\right)}{\left(c^2+d^2\right)^2}+\frac{i \log (-\tan (e+f x)+i)}{(c+i d)^2}-\frac{i \log (\tan (e+f x)+i)}{(c-i d)^2}\right)+\frac{b ((-d-i c) \log (-\tan (e+f x)+i)+i (c+i d) \log (\tan (e+f x)+i)+2 d \log (c+d \tan (e+f x)))}{c^2+d^2}}{2 d f}","\frac{b c-a d}{f \left(c^2+d^2\right) (c+d \tan (e+f x))}+\frac{\left(2 a c d-b \left(c^2-d^2\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^2}+\frac{x \left(a \left(c^2-d^2\right)+2 b c d\right)}{\left(c^2+d^2\right)^2}",1,"((b*(((-I)*c - d)*Log[I - Tan[e + f*x]] + I*(c + I*d)*Log[I + Tan[e + f*x]] + 2*d*Log[c + d*Tan[e + f*x]]))/(c^2 + d^2) + (b*c - a*d)*((I*Log[I - Tan[e + f*x]])/(c + I*d)^2 - (I*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (2*d*(-2*c*Log[c + d*Tan[e + f*x]] + (c^2 + d^2)/(c + d*Tan[e + f*x])))/(c^2 + d^2)^2))/(2*d*f)","C",1
1220,1,306,184,2.2520499,"\int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx","Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2),x]","\frac{\frac{2 b d^2 \left(a^2+b^2\right) \left(b \left(3 c^2+d^2\right)-2 a c d\right) \log (c+d \tan (e+f x))-2 b^4 \left(c^2+d^2\right)^2 \log (a+b \tan (e+f x))+(b c-a d)^2 \left(a \sqrt{-b^2} \left(c^2-d^2\right)+2 b c d \left(a-\sqrt{-b^2}\right)+b^2 \left(c^2-d^2\right)\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right)+(b c-a d)^2 \left(a \sqrt{-b^2} \left(d^2-c^2\right)+2 b c d \left(a+\sqrt{-b^2}\right)+b^2 \left(c^2-d^2\right)\right) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right)}{2 b \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)}-\frac{d^2}{c+d \tan (e+f x)}}{f \left(c^2+d^2\right) (a d-b c)}","-\frac{x \left(2 b c d-a \left(c^2-d^2\right)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)^2}+\frac{b^3 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right) (b c-a d)^2}+\frac{d^2}{f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))}+\frac{d^2 \left(2 a c d-b \left(3 c^2+d^2\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^2 (b c-a d)^2}",1,"(((b*c - a*d)^2*(2*b*(a - Sqrt[-b^2])*c*d + b^2*(c^2 - d^2) + a*Sqrt[-b^2]*(c^2 - d^2))*Log[Sqrt[-b^2] - b*Tan[e + f*x]] - 2*b^4*(c^2 + d^2)^2*Log[a + b*Tan[e + f*x]] + (b*c - a*d)^2*(2*b*(a + Sqrt[-b^2])*c*d + b^2*(c^2 - d^2) + a*Sqrt[-b^2]*(-c^2 + d^2))*Log[Sqrt[-b^2] + b*Tan[e + f*x]] + 2*b*(a^2 + b^2)*d^2*(-2*a*c*d + b*(3*c^2 + d^2))*Log[c + d*Tan[e + f*x]])/(2*b*(a^2 + b^2)*(b*c - a*d)*(c^2 + d^2)) - d^2/(c + d*Tan[e + f*x]))/((-(b*c) + a*d)*(c^2 + d^2)*f)","A",1
1221,1,556,290,6.9417797,"\int \frac{1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx","Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2),x]","-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{-\frac{d^2 \left(a^2 d-a b c+2 b^2 d\right)-c \left(b d (b c-a d)-2 b^2 c d\right)}{f \left(c^2+d^2\right) (a d-b c) (c+d \tan (e+f x))}-\frac{\frac{b (b c-a d)^2 \left(\frac{b \left(-\left(a^2 \left(c^2-d^2\right)\right)+4 a b c d+b^2 \left(c^2-d^2\right)\right)}{\sqrt{-b^2}}+2 a^2 c d+2 a b c^2-2 a b d^2-2 b^2 c d\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right)}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{b (b c-a d)^2 \left(\frac{\sqrt{-b^2} \left(-\left(a^2 \left(c^2-d^2\right)\right)+4 a b c d+b^2 \left(c^2-d^2\right)\right)}{b}+2 a^2 c d+2 a b c^2-2 a b d^2-2 b^2 c d\right) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right)}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{2 b d^3 \left(a^2+b^2\right) \left(a c d-b \left(2 c^2+d^2\right)\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)}-\frac{2 b^4 \left(c^2+d^2\right) \left(-2 a^2 d+a b c-b^2 d\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (b c-a d)}}{b f \left(c^2+d^2\right) (a d-b c)}}{\left(a^2+b^2\right) (b c-a d)}","-\frac{d \left(a^2 d^2+b^2 \left(c^2+2 d^2\right)\right)}{f \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)^2 (c+d \tan (e+f x))}+\frac{x (a (c+d)+b (c-d)) (a (c-d)-b (c+d))}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)^2}-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac{2 b^3 \left(-2 a^2 d+a b c-b^2 d\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^2 (b c-a d)^3}-\frac{2 d^3 \left(a c d-b \left(2 c^2+d^2\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^2 (b c-a d)^3}",1,"-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))) - (-(((b*(b*c - a*d)^2*(2*a*b*c^2 + 2*a^2*c*d - 2*b^2*c*d - 2*a*b*d^2 + (b*(4*a*b*c*d - a^2*(c^2 - d^2) + b^2*(c^2 - d^2)))/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (2*b^4*(a*b*c - 2*a^2*d - b^2*d)*(c^2 + d^2)*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + (b*(b*c - a*d)^2*(2*a*b*c^2 + 2*a^2*c*d - 2*b^2*c*d - 2*a*b*d^2 + (Sqrt[-b^2]*(4*a*b*c*d - a^2*(c^2 - d^2) + b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) + (2*b*(a^2 + b^2)*d^3*(a*c*d - b*(2*c^2 + d^2))*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (d^2*(-(a*b*c) + a^2*d + 2*b^2*d) - c*(-2*b^2*c*d + b*d*(b*c - a*d)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d))","A",1
1222,1,840,457,7.3438836,"\int \frac{1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx","Integrate[1/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]","-\frac{b^2}{2 \left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac{-\frac{b^2 \left(2 d a^2-2 b c a+3 b^2 d\right)-a \left(2 b^2 (b c-a d)-3 a b^2 d\right)}{\left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{-\frac{-2 \left(-d^2 a^4+2 b c d a^3-b^2 \left(c^2+6 d^2\right) a^2+2 b^3 c d a+b^4 \left(c^2-3 d^2\right)\right) d^2-c \left(2 b^2 c d \left(-7 d a^2+4 b c a-3 b^2 d\right)-4 a b d (b c-a d)^2\right)}{(a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))}-\frac{-\frac{2 \left(c^2+d^2\right) \left(-10 d^2 a^4+10 b c d a^3-3 b^2 \left(c^2+3 d^2\right) a^2+2 b^3 c d a+b^4 \left(c^2-3 d^2\right)\right) \log (a+b \tan (e+f x)) b^4}{\left(a^2+b^2\right) (b c-a d)}-\frac{(b c-a d)^3 \left(2 c d a^3+3 b c^2 a^2-3 b d^2 a^2-6 b^2 c d a-b^3 c^2+b^3 d^2-\frac{\sqrt{-b^2} \left(-\left(\left(c^2-d^2\right) a^3\right)+6 b c d a^2+3 b^2 \left(c^2-d^2\right) a-2 b^3 c d\right)}{b}\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right) b}{\left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{(b c-a d)^3 \left(2 c d a^3+3 b c^2 a^2-3 b d^2 a^2-6 b^2 c d a-b^3 c^2+b^3 d^2+\frac{\sqrt{-b^2} \left(-\left(\left(c^2-d^2\right) a^3\right)+6 b c d a^2+3 b^2 \left(c^2-d^2\right) a-2 b^3 c d\right)}{b}\right) \log \left(b \tan (e+f x)+\sqrt{-b^2}\right) b}{\left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{2 \left(a^2+b^2\right)^2 d^4 \left(5 b c^2-2 a d c+3 b d^2\right) \log (c+d \tan (e+f x)) b}{(b c-a d) \left(c^2+d^2\right)}}{b (a d-b c) \left(c^2+d^2\right) f}}{\left(a^2+b^2\right) (b c-a d)}}{2 \left(a^2+b^2\right) (b c-a d)}","-\frac{b^2 \left(-7 a^2 d+4 a b c-3 b^2 d\right)}{2 f \left(a^2+b^2\right)^2 (b c-a d)^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{b^2}{2 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac{d \left(a^4 d^3+2 a^2 b^2 d \left(2 c^2+3 d^2\right)-2 a b^3 c \left(c^2+d^2\right)+b^4 d \left(2 c^2+3 d^2\right)\right)}{f \left(a^2+b^2\right)^2 \left(c^2+d^2\right) (b c-a d)^3 (c+d \tan (e+f x))}-\frac{x \left(-\left(a^3 \left(c^2-d^2\right)\right)+6 a^2 b c d+3 a b^2 \left(c^2-d^2\right)-2 b^3 c d\right)}{\left(a^2+b^2\right)^3 \left(c^2+d^2\right)^2}-\frac{b^3 \left(-10 a^4 d^2+10 a^3 b c d-3 a^2 b^2 \left(c^2+3 d^2\right)+2 a b^3 c d+b^4 \left(c^2-3 d^2\right)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^3 (b c-a d)^4}-\frac{d^4 \left(-2 a c d+5 b c^2+3 b d^2\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^2 (b c-a d)^4}",1,"-1/2*b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (-((b^2*(-2*a*b*c + 2*a^2*d + 3*b^2*d) - a*(-3*a*b^2*d + 2*b^2*(b*c - a*d)))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))) - (-((-((b*(b*c - a*d)^3*(3*a^2*b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*d^2 + b^3*d^2 - (Sqrt[-b^2]*(6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2))) - (2*b^4*(c^2 + d^2)*(10*a^3*b*c*d + 2*a*b^3*c*d - 10*a^4*d^2 + b^4*(c^2 - 3*d^2) - 3*a^2*b^2*(c^2 + 3*d^2))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b*c - a*d)^3*(3*a^2*b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*d^2 + b^3*d^2 + (Sqrt[-b^2]*(6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)^2*d^4*(5*b*c^2 - 2*a*c*d + 3*b*d^2)*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (-(c*(-4*a*b*d*(b*c - a*d)^2 + 2*b^2*c*d*(4*a*b*c - 7*a^2*d - 3*b^2*d))) - 2*d^2*(2*a^3*b*c*d + 2*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 3*d^2) - a^2*b^2*(c^2 + 6*d^2)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d)))/(2*(a^2 + b^2)*(b*c - a*d))","A",1
1223,1,2775,406,7.0390052,"\int \frac{(a+b \tan (e+f x))^4}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^3,x]","\text{Result too large to show}","\frac{\left(a^2 d^2 \left(3 c^2-d^2\right)+2 a b c d \left(c^2+5 d^2\right)+b^2 \left(c^4+3 c^2 d^2+6 d^4\right)\right) (b c-a d)^2 \log (c+d \tan (e+f x))}{d^3 f \left(c^2+d^2\right)^3}-\frac{\left(-\left(a^4 \left(3 c^2 d-d^3\right)\right)+4 a^3 b c \left(c^2-3 d^2\right)+6 a^2 b^2 d \left(3 c^2-d^2\right)-4 a b^3 c \left(c^2-3 d^2\right)-b^4 d \left(3 c^2-d^2\right)\right) \log (\cos (e+f x))}{f \left(c^2+d^2\right)^3}-\frac{x \left(-\left(a^4 \left(c^3-3 c d^2\right)\right)-4 a^3 b d \left(3 c^2-d^2\right)+6 a^2 b^2 c \left(c^2-3 d^2\right)+4 a b^3 d \left(3 c^2-d^2\right)-b^4 c \left(c^2-3 d^2\right)\right)}{\left(c^2+d^2\right)^3}-\frac{(b c-a d)^2 (a+b \tan (e+f x))^2}{2 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}+\frac{\left(2 a c d+b \left(c^2+3 d^2\right)\right) (b c-a d)^3}{d^3 f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}",1,"((I*b^4*c^13*d^2 + b^4*c^12*d^3 + (5*I)*b^4*c^11*d^4 - (4*I)*a^3*b*c^10*d^5 + (4*I)*a*b^3*c^10*d^5 + 5*b^4*c^10*d^5 + (3*I)*a^4*c^9*d^6 - 4*a^3*b*c^9*d^6 - (18*I)*a^2*b^2*c^9*d^6 + 4*a*b^3*c^9*d^6 + (13*I)*b^4*c^9*d^6 + 3*a^4*c^8*d^7 + (4*I)*a^3*b*c^8*d^7 - 18*a^2*b^2*c^8*d^7 - (4*I)*a*b^3*c^8*d^7 + 13*b^4*c^8*d^7 + (5*I)*a^4*c^7*d^8 + 4*a^3*b*c^7*d^8 - (30*I)*a^2*b^2*c^7*d^8 - 4*a*b^3*c^7*d^8 + (15*I)*b^4*c^7*d^8 + 5*a^4*c^6*d^9 + (20*I)*a^3*b*c^6*d^9 - 30*a^2*b^2*c^6*d^9 - (20*I)*a*b^3*c^6*d^9 + 15*b^4*c^6*d^9 + I*a^4*c^5*d^10 + 20*a^3*b*c^5*d^10 - (6*I)*a^2*b^2*c^5*d^10 - 20*a*b^3*c^5*d^10 + (6*I)*b^4*c^5*d^10 + a^4*c^4*d^11 + (12*I)*a^3*b*c^4*d^11 - 6*a^2*b^2*c^4*d^11 - (12*I)*a*b^3*c^4*d^11 + 6*b^4*c^4*d^11 - I*a^4*c^3*d^12 + 12*a^3*b*c^3*d^12 + (6*I)*a^2*b^2*c^3*d^12 - 12*a*b^3*c^3*d^12 - a^4*c^2*d^13 + 6*a^2*b^2*c^2*d^13)*(e + f*x)*Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^4)/(c^2*(c - I*d)^6*(c + I*d)^5*d^5*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^3) - (I*(b^4*c^6 + 3*b^4*c^4*d^2 - 4*a^3*b*c^3*d^3 + 4*a*b^3*c^3*d^3 + 3*a^4*c^2*d^4 - 18*a^2*b^2*c^2*d^4 + 6*b^4*c^2*d^4 + 12*a^3*b*c*d^5 - 12*a*b^3*c*d^5 - a^4*d^6 + 6*a^2*b^2*d^6)*ArcTan[Tan[e + f*x]]*Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^4)/(d^3*(c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^3) - (b^4*Cos[e + f*x]*Log[Cos[e + f*x]]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^4)/(d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^3) + ((b^4*c^6 + 3*b^4*c^4*d^2 - 4*a^3*b*c^3*d^3 + 4*a*b^3*c^3*d^3 + 3*a^4*c^2*d^4 - 18*a^2*b^2*c^2*d^4 + 6*b^4*c^2*d^4 + 12*a^3*b*c*d^5 - 12*a*b^3*c*d^5 - a^4*d^6 + 6*a^2*b^2*d^6)*Cos[e + f*x]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^4)/(2*d^3*(c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^3) + (Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])*(-2*b^4*c^7*d + 4*a*b^3*c^6*d^2 - 6*b^4*c^5*d^3 - 4*a^3*b*c^4*d^4 + 16*a*b^3*c^4*d^4 + 2*a^4*c^3*d^5 - 12*a^2*b^2*c^3*d^5 - 4*b^4*c^3*d^5 + 12*a*b^3*c^2*d^6 + 2*a^4*c*d^7 - 12*a^2*b^2*c*d^7 + 4*a^3*b*d^8 + a^4*c^6*d^2*(e + f*x) - 6*a^2*b^2*c^6*d^2*(e + f*x) + b^4*c^6*d^2*(e + f*x) + 12*a^3*b*c^5*d^3*(e + f*x) - 12*a*b^3*c^5*d^3*(e + f*x) - 2*a^4*c^4*d^4*(e + f*x) + 12*a^2*b^2*c^4*d^4*(e + f*x) - 2*b^4*c^4*d^4*(e + f*x) + 8*a^3*b*c^3*d^5*(e + f*x) - 8*a*b^3*c^3*d^5*(e + f*x) - 3*a^4*c^2*d^6*(e + f*x) + 18*a^2*b^2*c^2*d^6*(e + f*x) - 3*b^4*c^2*d^6*(e + f*x) - 4*a^3*b*c*d^7*(e + f*x) + 4*a*b^3*c*d^7*(e + f*x) + b^4*c^7*d*Cos[2*(e + f*x)] - 6*a^2*b^2*c^5*d^3*Cos[2*(e + f*x)] + 5*b^4*c^5*d^3*Cos[2*(e + f*x)] + 8*a^3*b*c^4*d^4*Cos[2*(e + f*x)] - 12*a*b^3*c^4*d^4*Cos[2*(e + f*x)] - 3*a^4*c^3*d^5*Cos[2*(e + f*x)] + 6*a^2*b^2*c^3*d^5*Cos[2*(e + f*x)] + 4*b^4*c^3*d^5*Cos[2*(e + f*x)] + 4*a^3*b*c^2*d^6*Cos[2*(e + f*x)] - 12*a*b^3*c^2*d^6*Cos[2*(e + f*x)] - 3*a^4*c*d^7*Cos[2*(e + f*x)] + 12*a^2*b^2*c*d^7*Cos[2*(e + f*x)] - 4*a^3*b*d^8*Cos[2*(e + f*x)] + a^4*c^6*d^2*(e + f*x)*Cos[2*(e + f*x)] - 6*a^2*b^2*c^6*d^2*(e + f*x)*Cos[2*(e + f*x)] + b^4*c^6*d^2*(e + f*x)*Cos[2*(e + f*x)] + 12*a^3*b*c^5*d^3*(e + f*x)*Cos[2*(e + f*x)] - 12*a*b^3*c^5*d^3*(e + f*x)*Cos[2*(e + f*x)] - 4*a^4*c^4*d^4*(e + f*x)*Cos[2*(e + f*x)] + 24*a^2*b^2*c^4*d^4*(e + f*x)*Cos[2*(e + f*x)] - 4*b^4*c^4*d^4*(e + f*x)*Cos[2*(e + f*x)] - 16*a^3*b*c^3*d^5*(e + f*x)*Cos[2*(e + f*x)] + 16*a*b^3*c^3*d^5*(e + f*x)*Cos[2*(e + f*x)] + 3*a^4*c^2*d^6*(e + f*x)*Cos[2*(e + f*x)] - 18*a^2*b^2*c^2*d^6*(e + f*x)*Cos[2*(e + f*x)] + 3*b^4*c^2*d^6*(e + f*x)*Cos[2*(e + f*x)] + 4*a^3*b*c*d^7*(e + f*x)*Cos[2*(e + f*x)] - 4*a*b^3*c*d^7*(e + f*x)*Cos[2*(e + f*x)] - b^4*c^8*Sin[2*(e + f*x)] + 6*a^2*b^2*c^6*d^2*Sin[2*(e + f*x)] - 5*b^4*c^6*d^2*Sin[2*(e + f*x)] - 8*a^3*b*c^5*d^3*Sin[2*(e + f*x)] + 12*a*b^3*c^5*d^3*Sin[2*(e + f*x)] + 3*a^4*c^4*d^4*Sin[2*(e + f*x)] - 6*a^2*b^2*c^4*d^4*Sin[2*(e + f*x)] - 4*b^4*c^4*d^4*Sin[2*(e + f*x)] - 4*a^3*b*c^3*d^5*Sin[2*(e + f*x)] + 12*a*b^3*c^3*d^5*Sin[2*(e + f*x)] + 3*a^4*c^2*d^6*Sin[2*(e + f*x)] - 12*a^2*b^2*c^2*d^6*Sin[2*(e + f*x)] + 4*a^3*b*c*d^7*Sin[2*(e + f*x)] + 2*a^4*c^5*d^3*(e + f*x)*Sin[2*(e + f*x)] - 12*a^2*b^2*c^5*d^3*(e + f*x)*Sin[2*(e + f*x)] + 2*b^4*c^5*d^3*(e + f*x)*Sin[2*(e + f*x)] + 24*a^3*b*c^4*d^4*(e + f*x)*Sin[2*(e + f*x)] - 24*a*b^3*c^4*d^4*(e + f*x)*Sin[2*(e + f*x)] - 6*a^4*c^3*d^5*(e + f*x)*Sin[2*(e + f*x)] + 36*a^2*b^2*c^3*d^5*(e + f*x)*Sin[2*(e + f*x)] - 6*b^4*c^3*d^5*(e + f*x)*Sin[2*(e + f*x)] - 8*a^3*b*c^2*d^6*(e + f*x)*Sin[2*(e + f*x)] + 8*a*b^3*c^2*d^6*(e + f*x)*Sin[2*(e + f*x)])*(a + b*Tan[e + f*x])^4)/(2*c*(c - I*d)^3*(c + I*d)^3*d^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^4*(c + d*Tan[e + f*x])^3)","C",1
1224,1,327,240,5.5444598,"\int \frac{(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^3,x]","\frac{2 b d \left(3 a^2-b^2\right) \left(\frac{d \left(2 c \log (c+d \tan (e+f x))-\frac{c^2+d^2}{c+d \tan (e+f x)}\right)}{\left(c^2+d^2\right)^2}-\frac{i \log (-\tan (e+f x)+i)}{2 (c+i d)^2}+\frac{i \log (\tan (e+f x)+i)}{2 (c-i d)^2}\right)+d \left(a^3 d-3 a^2 b c-3 a b^2 d+b^3 c\right) \left(\frac{d \left(\left(6 c^2-2 d^2\right) \log (c+d \tan (e+f x))-\frac{\left(c^2+d^2\right) \left(5 c^2+4 c d \tan (e+f x)+d^2\right)}{(c+d \tan (e+f x))^2}\right)}{\left(c^2+d^2\right)^3}+\frac{\log (-\tan (e+f x)+i)}{(d-i c)^3}+\frac{\log (\tan (e+f x)+i)}{(d+i c)^3}\right)-\frac{b^2 (a d+b c)}{(c+d \tan (e+f x))^2}-\frac{2 b^2 d (a+b \tan (e+f x))}{(c+d \tan (e+f x))^2}}{2 d^2 f}","-\frac{\left(a^2 \left(3 c^2-d^2\right)+8 a b c d-b^2 \left(c^2-3 d^2\right)\right) (b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^3}+\frac{x (a c+b d) \left(a^2 \left(c^2-3 d^2\right)+8 a b c d-b^2 \left(3 c^2-d^2\right)\right)}{\left(c^2+d^2\right)^3}-\frac{\left(4 a c d+b \left(c^2+5 d^2\right)\right) (b c-a d)^2}{2 d^2 f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{2 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}",1,"(-((b^2*(b*c + a*d))/(c + d*Tan[e + f*x])^2) - (2*b^2*d*(a + b*Tan[e + f*x]))/(c + d*Tan[e + f*x])^2 + 2*b*(3*a^2 - b^2)*d*(((-1/2*I)*Log[I - Tan[e + f*x]])/(c + I*d)^2 + ((I/2)*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (d*(2*c*Log[c + d*Tan[e + f*x]] - (c^2 + d^2)/(c + d*Tan[e + f*x])))/(c^2 + d^2)^2) + d*(-3*a^2*b*c + b^3*c + a^3*d - 3*a*b^2*d)*(Log[I - Tan[e + f*x]]/((-I)*c + d)^3 + Log[I + Tan[e + f*x]]/(I*c + d)^3 + (d*((6*c^2 - 2*d^2)*Log[c + d*Tan[e + f*x]] - ((c^2 + d^2)*(5*c^2 + d^2 + 4*c*d*Tan[e + f*x]))/(c + d*Tan[e + f*x])^2))/(c^2 + d^2)^3))/(2*d^2*f)","C",1
1225,1,292,221,4.2340739,"\int \frac{(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^3,x]","-\frac{(b c-a d) \left(\frac{2 \left(a^2 \left(3 c^2 d-d^3\right)-2 a b c \left(c^2-3 d^2\right)+b^2 d \left(d^2-3 c^2\right)\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right)^2}-\frac{2 (a d-b c) \left(2 a c d+b \left(d^2-c^2\right)\right)}{d \left(c^2+d^2\right) (c+d \tan (e+f x))}+\frac{(a+i b)^2 (d+i c)^3 \log (-\tan (e+f x)+i)}{\left(c^2+d^2\right)^2}+\frac{i (a-i b)^2 (c+i d) \log (\tan (e+f x)+i)}{(c-i d)^2}\right)+\frac{d^2 (a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^2}-\frac{b d (a+b \tan (e+f x))^2}{c+d \tan (e+f x)}}{2 f \left(c^2+d^2\right) (a d-b c)}","-\frac{\left(-\left(a^2 \left(3 c^2 d-d^3\right)\right)+2 a b c \left(c^2-3 d^2\right)+b^2 d \left(3 c^2-d^2\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^3}-\frac{x \left(-\left(a^2 \left(c^3-3 c d^2\right)\right)-2 a b d \left(3 c^2-d^2\right)+b^2 c \left(c^2-3 d^2\right)\right)}{\left(c^2+d^2\right)^3}-\frac{(b c-a d)^2}{2 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}+\frac{2 (a c+b d) (b c-a d)}{f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}",1,"-1/2*((d^2*(a + b*Tan[e + f*x])^3)/(c + d*Tan[e + f*x])^2 - (b*d*(a + b*Tan[e + f*x])^2)/(c + d*Tan[e + f*x]) + (b*c - a*d)*(((a + I*b)^2*(I*c + d)^3*Log[I - Tan[e + f*x]])/(c^2 + d^2)^2 + (I*(a - I*b)^2*(c + I*d)*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (2*(-2*a*b*c*(c^2 - 3*d^2) + b^2*d*(-3*c^2 + d^2) + a^2*(3*c^2*d - d^3))*Log[c + d*Tan[e + f*x]])/(c^2 + d^2)^2 - (2*(-(b*c) + a*d)*(2*a*c*d + b*(-c^2 + d^2)))/(d*(c^2 + d^2)*(c + d*Tan[e + f*x]))))/((-(b*c) + a*d)*(c^2 + d^2)*f)","C",1
1226,1,244,177,4.4379614,"\int \frac{a+b \tan (e+f x)}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^3,x]","\frac{(b c-a d) \left(\frac{d \left(\frac{\left(c^2+d^2\right) \left(5 c^2+4 c d \tan (e+f x)+d^2\right)}{(c+d \tan (e+f x))^2}+\left(2 d^2-6 c^2\right) \log (c+d \tan (e+f x))\right)}{\left(c^2+d^2\right)^3}+\frac{i \log (-\tan (e+f x)+i)}{(c+i d)^3}-\frac{\log (\tan (e+f x)+i)}{(d+i c)^3}\right)-b \left(\frac{2 d \left(\frac{c^2+d^2}{c+d \tan (e+f x)}-2 c \log (c+d \tan (e+f x))\right)}{\left(c^2+d^2\right)^2}+\frac{i \log (-\tan (e+f x)+i)}{(c+i d)^2}-\frac{i \log (\tan (e+f x)+i)}{(c-i d)^2}\right)}{2 d f}","\frac{b c-a d}{2 f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}-\frac{2 a c d-b \left(c^2-d^2\right)}{f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}+\frac{\left(a d \left(3 c^2-d^2\right)-b \left(c^3-3 c d^2\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^3}+\frac{x \left(a c^3-3 a c d^2+3 b c^2 d-b d^3\right)}{\left(c^2+d^2\right)^3}",1,"(-(b*((I*Log[I - Tan[e + f*x]])/(c + I*d)^2 - (I*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (2*d*(-2*c*Log[c + d*Tan[e + f*x]] + (c^2 + d^2)/(c + d*Tan[e + f*x])))/(c^2 + d^2)^2)) + (b*c - a*d)*((I*Log[I - Tan[e + f*x]])/(c + I*d)^3 - Log[I + Tan[e + f*x]]/(I*c + d)^3 + (d*((-6*c^2 + 2*d^2)*Log[c + d*Tan[e + f*x]] + ((c^2 + d^2)*(5*c^2 + d^2 + 4*c*d*Tan[e + f*x]))/(c + d*Tan[e + f*x])^2))/(c^2 + d^2)^3))/(2*d*f)","C",1
1227,1,409,286,5.495025,"\int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx","Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3),x]","\frac{\frac{-2 b d^2 \left(a^2+b^2\right) \left(a^2 d^2 \left(d^2-3 c^2\right)+8 a b c^3 d-b^2 \left(6 c^4+3 c^2 d^2+d^4\right)\right) \log (c+d \tan (e+f x))-2 b^5 \left(c^2+d^2\right)^3 \log (a+b \tan (e+f x))+(b c-a d)^3 \left(b d \left(\sqrt{-b^2}-a\right) \left(d^2-3 c^2\right)+a \sqrt{-b^2} c \left(c^2-3 d^2\right)+b^2 \left(c^3-3 c d^2\right)\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right)-(b c-a d)^3 \left(b d \left(a+\sqrt{-b^2}\right) \left(d^2-3 c^2\right)+a \sqrt{-b^2} c \left(c^2-3 d^2\right)-\left(b^2 \left(c^3-3 c d^2\right)\right)\right) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right)}{b \left(a^2+b^2\right) \left(c^2+d^2\right)^2 (b c-a d)^2}+\frac{2 d^2 \left(b \left(3 c^2+d^2\right)-2 a c d\right)}{\left(c^2+d^2\right) (a d-b c) (c+d \tan (e+f x))}-\frac{d^2}{(c+d \tan (e+f x))^2}}{2 f \left(c^2+d^2\right) (a d-b c)}","-\frac{x \left(b d \left(3 c^2-d^2\right)-a \left(c^3-3 c d^2\right)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)^3}+\frac{d^2 \left(-a^2 d^2 \left(3 c^2-d^2\right)+8 a b c^3 d-b^2 \left(6 c^4+3 c^2 d^2+d^4\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^3 (b c-a d)^3}+\frac{b^4 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right) (b c-a d)^3}-\frac{d^2 \left(2 a c d-b \left(3 c^2+d^2\right)\right)}{f \left(c^2+d^2\right)^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{d^2}{2 f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))^2}",1,"(((b*c - a*d)^3*(a*Sqrt[-b^2]*c*(c^2 - 3*d^2) + b*(-a + Sqrt[-b^2])*d*(-3*c^2 + d^2) + b^2*(c^3 - 3*c*d^2))*Log[Sqrt[-b^2] - b*Tan[e + f*x]] - 2*b^5*(c^2 + d^2)^3*Log[a + b*Tan[e + f*x]] - (b*c - a*d)^3*(a*Sqrt[-b^2]*c*(c^2 - 3*d^2) + b*(a + Sqrt[-b^2])*d*(-3*c^2 + d^2) - b^2*(c^3 - 3*c*d^2))*Log[Sqrt[-b^2] + b*Tan[e + f*x]] - 2*b*(a^2 + b^2)*d^2*(8*a*b*c^3*d + a^2*d^2*(-3*c^2 + d^2) - b^2*(6*c^4 + 3*c^2*d^2 + d^4))*Log[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)^2) - d^2/(c + d*Tan[e + f*x])^2 + (2*d^2*(-2*a*c*d + b*(3*c^2 + d^2)))/((-(b*c) + a*d)*(c^2 + d^2)*(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2)*f)","A",1
1228,1,833,457,7.2749049,"\int \frac{1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx","Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3),x]","-\frac{b^2}{\left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}-\frac{-\frac{d^2 \left(d a^2-b c a+3 b^2 d\right)-c \left(b d (b c-a d)-3 b^2 c d\right)}{2 (a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))^2}-\frac{-\frac{2 d^2 \left(a b (2 b c+a d) d^2+\frac{1}{2} \left(-d a^2+b c a-3 b^2 d\right) \left(2 a c d-2 b \left(c^2+d^2\right)\right)\right)-c \left(2 d (b c-a d)^2 (b c+a d)-2 b c d \left(\left(2 c^2+3 d^2\right) b^2+a^2 d^2\right)\right)}{(a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))}-\frac{-\frac{2 \left(-5 d a^2+2 b c a-3 b^2 d\right) \left(c^2+d^2\right)^2 \log (a+b \tan (e+f x)) b^5}{\left(a^2+b^2\right) (b c-a d)}+\frac{(b c-a d)^3 \left(2 a b c^3+3 a^2 d c^2-3 b^2 d c^2-6 a b d^2 c-a^2 d^3+b^2 d^3+\frac{b \left(-\left(\left(c^3-3 c d^2\right) a^2\right)+b \left(6 c^2 d-2 d^3\right) a+b^2 c \left(c^2-3 d^2\right)\right)}{\sqrt{-b^2}}\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right) b}{\left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{(b c-a d)^3 \left(2 a b c^3+3 a^2 d c^2-3 b^2 d c^2-6 a b d^2 c-a^2 d^3+b^2 d^3+\frac{\sqrt{-b^2} \left(-\left(\left(c^3-3 c d^2\right) a^2\right)+b \left(6 c^2 d-2 d^3\right) a+b^2 c \left(c^2-3 d^2\right)\right)}{b}\right) \log \left(b \tan (e+f x)+\sqrt{-b^2}\right) b}{\left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{2 \left(a^2+b^2\right) d^3 \left(10 b^2 c^4-10 a b d c^3+3 a^2 d^2 c^2+9 b^2 d^2 c^2-2 a b d^3 c-a^2 d^4+3 b^2 d^4\right) \log (c+d \tan (e+f x)) b}{(b c-a d) \left(c^2+d^2\right)}}{b (a d-b c) \left(c^2+d^2\right) f}}{2 (a d-b c) \left(c^2+d^2\right)}}{\left(a^2+b^2\right) (b c-a d)}","-\frac{d \left(a^2 d^2+b^2 \left(2 c^2+3 d^2\right)\right)}{2 f \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)^2 (c+d \tan (e+f x))^2}+\frac{d^3 \left(a^2 d^2 \left(3 c^2-d^2\right)-2 a b c d \left(5 c^2+d^2\right)+b^2 \left(10 c^4+9 c^2 d^2+3 d^4\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right)^3 (b c-a d)^4}-\frac{x \left(-\left(a^2 \left(c^3-3 c d^2\right)\right)+a b \left(6 c^2 d-2 d^3\right)+b^2 c \left(c^2-3 d^2\right)\right)}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)^3}-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}+\frac{b^4 \left(-5 a^2 d+2 a b c-3 b^2 d\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right)^2 (b c-a d)^4}+\frac{d \left(2 a^3 c d^3-2 a^2 b d^2 \left(2 c^2+d^2\right)+2 a b^2 c d^3-\left(b^3 \left(c^4+6 c^2 d^2+3 d^4\right)\right)\right)}{f \left(a^2+b^2\right) \left(c^2+d^2\right)^2 (b c-a d)^3 (c+d \tan (e+f x))}",1,"-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2)) - (-1/2*(d^2*(-(a*b*c) + a^2*d + 3*b^2*d) - c*(-3*b^2*c*d + b*d*(b*c - a*d)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-(((b*(b*c - a*d)^3*(2*a*b*c^3 + 3*a^2*c^2*d - 3*b^2*c^2*d - 6*a*b*c*d^2 - a^2*d^3 + b^2*d^3 + (b*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3)))/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b^5*(2*a*b*c - 5*a^2*d - 3*b^2*d)*(c^2 + d^2)^2*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + (b*(b*c - a*d)^3*(2*a*b*c^3 + 3*a^2*c^2*d - 3*b^2*c^2*d - 6*a*b*c*d^2 - a^2*d^3 + b^2*d^3 + (Sqrt[-b^2]*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)*d^3*(10*b^2*c^4 - 10*a*b*c^3*d + 3*a^2*c^2*d^2 + 9*b^2*c^2*d^2 - 2*a*b*c*d^3 - a^2*d^4 + 3*b^2*d^4)*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (2*d^2*(a*b*d^2*(2*b*c + a*d) + ((a*b*c - a^2*d - 3*b^2*d)*(2*a*c*d - 2*b*(c^2 + d^2)))/2) - c*(2*d*(b*c - a*d)^2*(b*c + a*d) - 2*b*c*d*(a^2*d^2 + b^2*(2*c^2 + 3*d^2))))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2)))/((a^2 + b^2)*(b*c - a*d))","A",1
1229,1,194,209,2.2289766,"\int (a+b \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{2 b \sqrt{c+d \tan (e+f x)} \left(45 a^2 d^2+b d (15 a d+b c) \tan (e+f x)+15 a b c d-b^2 \left(2 c^2+15 d^2\right)+3 b^2 d^2 \tan ^2(e+f x)\right)}{d^2}-15 i (a-i b)^3 \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+15 i (a+i b)^3 \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{15 f}","\frac{2 b \left(3 a^2-b^2\right) \sqrt{c+d \tan (e+f x)}}{f}-\frac{4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac{2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac{(-b+i a)^3 \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{(b+i a)^3 \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"((-15*I)*(a - I*b)^3*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (15*I)*(a + I*b)^3*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (2*b*Sqrt[c + d*Tan[e + f*x]]*(15*a*b*c*d + 45*a^2*d^2 - b^2*(2*c^2 + 15*d^2) + b*d*(b*c + 15*a*d)*Tan[e + f*x] + 3*b^2*d^2*Tan[e + f*x]^2))/d^2)/(15*f)","A",1
1230,1,149,157,0.559817,"\int (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]],x]","\frac{2 b \sqrt{c+d \tan (e+f x)} (6 a d+b c+b d \tan (e+f x))-3 i d (a-i b)^2 \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+3 i d (a+i b)^2 \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{3 d f}","\frac{4 a b \sqrt{c+d \tan (e+f x)}}{f}-\frac{i (a-i b)^2 \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{i (a+i b)^2 \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{2 b^2 (c+d \tan (e+f x))^{3/2}}{3 d f}",1,"((-3*I)*(a - I*b)^2*Sqrt[c - I*d]*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (3*I)*(a + I*b)^2*Sqrt[c + I*d]*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + 2*b*Sqrt[c + d*Tan[e + f*x]]*(b*c + 6*a*d + b*d*Tan[e + f*x]))/(3*d*f)","A",1
1231,1,120,122,0.1379271,"\int (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]],x]","\frac{-i (a-i b) \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+i (a+i b) \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)+2 b \sqrt{c+d \tan (e+f x)}}{f}","-\frac{(b+i a) \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{(-b+i a) \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{2 b \sqrt{c+d \tan (e+f x)}}{f}",1,"((-I)*(a - I*b)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + I*(a + I*b)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + 2*b*Sqrt[c + d*Tan[e + f*x]])/f","A",1
1232,1,158,170,0.3322145,"\int \frac{\sqrt{c+d \tan (e+f x)}}{a+b \tan (e+f x)} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x]),x]","\frac{(b-i a) \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+(b+i a) \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)-2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right)}","-\frac{2 \sqrt{b} \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right)}+\frac{\sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a)}-\frac{\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)}",1,"(((-I)*a + b)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (I*a + b)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*Sqrt[b]*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*f)","A",1
1233,1,276,231,2.0270368,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^2,x]","-\frac{\frac{\sqrt{b} \sqrt{b c-a d} \left(-3 a^2 d+4 a b c+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{a^2+b^2}+\frac{i \left((a-i b)^2 \sqrt{c+i d} (a d-b c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)+(a+i b)^2 \sqrt{c-i d} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)}{a^2+b^2}+\frac{b^2 (c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)}-b d \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) (b c-a d)}","-\frac{b \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) (a+b \tan (e+f x))}-\frac{\sqrt{b} \left(-3 a^2 d+4 a b c+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right)^2 \sqrt{b c-a d}}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2}+\frac{i \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2}",1,"-(((I*((a + I*b)^2*Sqrt[c - I*d]*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (a - I*b)^2*Sqrt[c + I*d]*(-(b*c) + a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]))/(a^2 + b^2) + (Sqrt[b]*Sqrt[b*c - a*d]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(a^2 + b^2) - b*d*Sqrt[c + d*Tan[e + f*x]] + (b^2*(c + d*Tan[e + f*x])^(3/2))/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f))","A",1
1234,1,747,342,6.3537402,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+b \tan (e+f x))^3} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^3,x]","-\frac{b^2 (c+d \tan (e+f x))^{3/2}}{2 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^2}-\frac{-\frac{b d \sqrt{c+d \tan (e+f x)}}{f (a+b \tan (e+f x))}-\frac{2 \left(-\frac{\left(\frac{1}{4} b^3 (b c-a d) (4 a c+b d)-a \left(\frac{3}{4} a b^2 d (b c-a d)-b^2 (b c-a d)^2\right)\right) \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))}-\frac{\frac{2 \sqrt{b c-a d} \left(\frac{1}{8} a^2 b^2 d (b c-a d) \left(-7 a^2 d+8 a b c+b^2 d\right)-a b^2 (b c-a d)^2 \left(a^2 (-d)+2 a b c+b^2 d\right)-\frac{1}{8} b^3 (b c-a d) \left(-8 a^3 c d+8 a^2 b c^2-9 a^2 b d^2+16 a b^2 c d-8 b^3 c^2-b^3 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} f \left(a^2+b^2\right) (a d-b c)}+\frac{\frac{i \sqrt{c-i d} \left(-b \left(a^3 c+3 a^2 b d-3 a b^2 c-b^3 d\right) (b c-a d)^2-i b \left(a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right) (b c-a d)^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (-c+i d)}-\frac{i \sqrt{c+i d} \left(-b (b c-a d)^2 \left(a^3 c+3 a^2 b d-3 a b^2 c-b^3 d\right)+i b (b c-a d)^2 \left(a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-c-i d)}}{a^2+b^2}}{\left(a^2+b^2\right) (b c-a d)}\right)}{b}}{2 \left(a^2+b^2\right) (b c-a d)}","-\frac{b \left(-7 a^2 d+8 a b c+b^2 d\right) \sqrt{c+d \tan (e+f x)}}{4 f \left(a^2+b^2\right)^2 (b c-a d) (a+b \tan (e+f x))}-\frac{b \sqrt{c+d \tan (e+f x)}}{2 f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}+\frac{\sqrt{b} \left(-15 a^4 d^2+40 a^3 b c d-6 a^2 b^2 \left(4 c^2-3 d^2\right)-24 a b^3 c d+b^4 \left(8 c^2+d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{4 f \left(a^2+b^2\right)^3 (b c-a d)^{3/2}}-\frac{\sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a)^3}+\frac{\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)^3}",1,"-1/2*(b^2*(c + d*Tan[e + f*x])^(3/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*Sqrt[c + d*Tan[e + f*x]])/(f*(a + b*Tan[e + f*x]))) - (2*(-((((I*Sqrt[c - I*d]*((-I)*b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) - b*(b*c - a*d)^2*(a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(I*b*(b*c - a*d)^2*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) - b*(b*c - a*d)^2*(a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*((a^2*b^2*d*(b*c - a*d)*(8*a*b*c - 7*a^2*d + b^2*d))/8 - a*b^2*(b*c - a*d)^2*(2*a*b*c - a^2*d + b^2*d) - (b^3*(b*c - a*d)*(8*a^2*b*c^2 - 8*b^3*c^2 - 8*a^3*c*d + 16*a*b^2*c*d - 9*a^2*b*d^2 - b^3*d^2))/8)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((b^3*(b*c - a*d)*(4*a*c + b*d))/4 - a*((3*a*b^2*d*(b*c - a*d))/4 - b^2*(b*c - a*d)^2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b)/(2*(a^2 + b^2)*(b*c - a*d))","B",1
1235,1,247,256,3.3174898,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2),x]","\frac{\frac{4 b^2 (8 a d-b c) (c+d \tan (e+f x))^{5/2}}{d}+10 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}+\frac{35}{3} d (-b+i a)^3 \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)-\frac{35}{3} d (b+i a)^3 \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)}{35 d f}","\frac{2 b \left(3 a^2-b^2\right) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \left(a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right) \sqrt{c+d \tan (e+f x)}}{f}-\frac{4 b^2 (b c-8 a d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac{2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac{(-b+i a)^3 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{(b+i a)^3 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}",1,"((4*b^2*(-(b*c) + 8*a*d)*(c + d*Tan[e + f*x])^(5/2))/d + 10*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2) - (35*(I*a + b)^3*d*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])))/3 + (35*(I*a - b)^3*d*(-3*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])))/3)/(35*d*f)","A",1
1236,1,202,195,1.5517406,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2),x]","\frac{5 i (a-i b)^2 \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)-5 i (a+i b)^2 \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)+\frac{6 b^2 (c+d \tan (e+f x))^{5/2}}{d}}{15 f}","\frac{2 \left(a^2 d+2 a b c-b^2 d\right) \sqrt{c+d \tan (e+f x)}}{f}+\frac{4 a b (c+d \tan (e+f x))^{3/2}}{3 f}-\frac{i (a-i b)^2 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{i (a+i b)^2 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}",1,"((6*b^2*(c + d*Tan[e + f*x])^(5/2))/d + (5*I)*(a - I*b)^2*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])) - (5*I)*(a + I*b)^2*(-3*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])))/(15*f)","A",1
1237,1,140,150,0.5373891,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 \sqrt{c+d \tan (e+f x)} (3 a d+4 b c+b d \tan (e+f x))-3 i (a-i b) (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+3 i (a+i b) (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{3 f}","\frac{2 (a d+b c) \sqrt{c+d \tan (e+f x)}}{f}-\frac{(b+i a) (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{(-b+i a) (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{2 b (c+d \tan (e+f x))^{3/2}}{3 f}",1,"((-3*I)*(a - I*b)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (3*I)*(a + I*b)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + 2*Sqrt[c + d*Tan[e + f*x]]*(4*b*c + 3*a*d + b*d*Tan[e + f*x]))/(3*f)","A",1
1238,1,168,170,0.3947672,"\int \frac{(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x]),x]","\frac{\sqrt{b} (b-i a) (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+\sqrt{b} (b+i a) (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)-2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} f \left(a^2+b^2\right)}","-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} f \left(a^2+b^2\right)}+\frac{(c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a)}-\frac{(c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)}",1,"(Sqrt[b]*((-I)*a + b)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[b]*(I*a + b)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*f)","A",1
1239,1,316,239,3.2479484,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^2,x]","\frac{-\frac{4 \left(\frac{3}{4} b^{3/2} (b c-a d)^{3/2} \left(a^2 (-d)+4 a b c+3 b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)+\frac{3}{4} i b^2 (a+i b)^2 (c-i d)^{3/2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)-\frac{3}{4} i b^2 (a-i b)^2 (c+i d)^{3/2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)}{b^2 \left(a^2+b^2\right)}-\frac{3 b^2 (c+d \tan (e+f x))^{5/2}}{a+b \tan (e+f x)}+3 d (b c-a d) \sqrt{c+d \tan (e+f x)}+3 b d (c+d \tan (e+f x))^{3/2}}{3 f \left(a^2+b^2\right) (b c-a d)}","-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) (a+b \tan (e+f x))}-\frac{\sqrt{b c-a d} \left(a^2 (-d)+4 a b c+3 b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} f \left(a^2+b^2\right)^2}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2}",1,"((-4*(((3*I)/4)*(a + I*b)^2*b^2*(c - I*d)^(3/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] - ((3*I)/4)*(a - I*b)^2*b^2*(c + I*d)^(3/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (3*b^(3/2)*(b*c - a*d)^(3/2)*(4*a*b*c - a^2*d + 3*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/4))/(b^2*(a^2 + b^2)) + 3*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]] + 3*b*d*(c + d*Tan[e + f*x])^(3/2) - (3*b^2*(c + d*Tan[e + f*x])^(5/2))/(a + b*Tan[e + f*x]))/(3*(a^2 + b^2)*(b*c - a*d)*f)","A",1
1240,1,2093,341,6.3399209,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{\left(-3 a^2 d+8 a b c+5 b^2 d\right) \sqrt{c+d \tan (e+f x)}}{4 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{2 f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}+\frac{\left(-3 a^4 d^2+24 a^3 b c d-2 a^2 b^2 \left(12 c^2-13 d^2\right)-40 a b^3 c d+b^4 \left(8 c^2-3 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{4 \sqrt{b} f \left(a^2+b^2\right)^3 \sqrt{b c-a d}}-\frac{(c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a)^3}+\frac{(c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)^3}",1,"-1/2*(b^2*(c + d*Tan[e + f*x])^(5/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*(c + d*Tan[e + f*x])^(3/2))/(f*(a + b*Tan[e + f*x]))) + (2*(-1/2*(b*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + b*Tan[e + f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) - I*(a*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - b*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) + I*(a*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - b*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*(-(a*b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)) + (a^2*d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2 + b^2*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b))/b)/(2*(a^2 + b^2)*(b*c - a*d))","B",1
1241,1,413,322,6.1960174,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2),x]","\frac{2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac{2 \left(\frac{i \left(\frac{9}{2} a d \left(a^2-3 b^2\right)-\frac{9}{2} i b d \left(3 a^2-b^2\right)\right) \left(\frac{2}{5} (c+d \tan (e+f x))^{5/2}+(c-i d) \left(\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left(2 \sqrt{c+d \tan (e+f x)}+\frac{2 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{-c+i d}\right)\right)\right)}{2 f}-\frac{i \left(\frac{9}{2} a d \left(a^2-3 b^2\right)+\frac{9}{2} i b d \left(3 a^2-b^2\right)\right) \left(\frac{2}{5} (c+d \tan (e+f x))^{5/2}+(c+i d) \left(\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left(2 \sqrt{c+d \tan (e+f x)}+\frac{2 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{-c-i d}\right)\right)\right)}{2 f}-\frac{2 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{7 d f}\right)}{9 d}","\frac{2 b \left(3 a^2-b^2\right) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 \left(2 a^3 c d+3 a^2 b \left(c^2-d^2\right)-6 a b^2 c d-b^3 \left(c^2-d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 \left(a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac{4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac{(b+i a)^3 (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}-\frac{(-b+i a)^3 (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}",1,"(2*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(7/2))/(9*d*f) + (2*((-2*b^2*(b*c - 10*a*d)*(c + d*Tan[e + f*x])^(7/2))/(7*d*f) + ((I/2)*((9*a*(a^2 - 3*b^2)*d)/2 - ((9*I)/2)*b*(3*a^2 - b^2)*d)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (c - I*d)*((2*(c + d*Tan[e + f*x])^(3/2))/3 + (c - I*d)*((2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(-c + I*d) + 2*Sqrt[c + d*Tan[e + f*x]]))))/f - ((I/2)*((9*a*(a^2 - 3*b^2)*d)/2 + ((9*I)/2)*b*(3*a^2 - b^2)*d)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (c + I*d)*((2*(c + d*Tan[e + f*x])^(3/2))/3 + (c + I*d)*((2*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(-c - I*d) + 2*Sqrt[c + d*Tan[e + f*x]]))))/f))/(9*d)","A",1
1242,1,262,231,2.0423803,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2),x]","\frac{7 i (a-i b)^2 \left(\frac{2}{5} (c+d \tan (e+f x))^{5/2}+\frac{2}{3} (c-i d) \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)\right)-7 i (a+i b)^2 \left(\frac{2}{5} (c+d \tan (e+f x))^{5/2}+\frac{2}{3} (c+i d) \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)\right)+\frac{4 b^2 (c+d \tan (e+f x))^{7/2}}{d}}{14 f}","\frac{2 \left(a^2 d+2 a b c-b^2 d\right) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{4 a b (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{4 (a d+b c) (a c-b d) \sqrt{c+d \tan (e+f x)}}{f}-\frac{i (a-i b)^2 (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{i (a+i b)^2 (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{2 b^2 (c+d \tan (e+f x))^{7/2}}{7 d f}",1,"((4*b^2*(c + d*Tan[e + f*x])^(7/2))/d + (7*I)*(a - I*b)^2*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c - I*d)*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])))/3) - (7*I)*(a + I*b)^2*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c + I*d)*(-3*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])))/3))/(14*f)","A",1
1243,1,233,188,1.0833704,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2),x]","\frac{i \left((a-i b) \left(\frac{2}{5} (c+d \tan (e+f x))^{5/2}+\frac{2}{3} (c-i d) \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)\right)-(a+i b) \left(\frac{2}{5} (c+d \tan (e+f x))^{5/2}+\frac{2}{3} (c+i d) \left(\sqrt{c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)\right)\right)}{2 f}","\frac{2 \left(2 a c d+b \left(c^2-d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (a d+b c) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac{(b+i a) (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{(-b+i a) (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}+\frac{2 b (c+d \tan (e+f x))^{5/2}}{5 f}",1,"((I/2)*((a - I*b)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c - I*d)*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])))/3) - (a + I*b)*((2*(c + d*Tan[e + f*x])^(5/2))/5 + (2*(c + I*d)*(-3*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f*x])))/3)))/f","A",1
1244,1,199,195,0.5790217,"\int \frac{(c+d \tan (e+f x))^{5/2}}{a+b \tan (e+f x)} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x]),x]","\frac{2 \sqrt{b} d^2 \left(a^2+b^2\right) \sqrt{c+d \tan (e+f x)}+b^{3/2} (b-i a) (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+b^{3/2} (b+i a) (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)-2 (b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{3/2} f \left(a^2+b^2\right)}","-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{3/2} f \left(a^2+b^2\right)}+\frac{(c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a)}-\frac{(c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)}+\frac{2 d^2 \sqrt{c+d \tan (e+f x)}}{b f}",1,"(b^(3/2)*((-I)*a + b)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + b^(3/2)*(I*a + b)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + 2*Sqrt[b]*(a^2 + b^2)*d^2*Sqrt[c + d*Tan[e + f*x]])/(b^(3/2)*(a^2 + b^2)*f)","A",1
1245,1,333,243,5.056864,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^2} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^2,x]","\frac{-\frac{(b c-a d)^{5/2} \left(a^2 d+4 a b c+5 b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)+i b^{3/2} (a+i b)^2 (c-i d)^{5/2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+i b^{3/2} (b+i a)^2 (c+i d)^{5/2} (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{b^{3/2} \left(a^2+b^2\right)}-\frac{b^2 (c+d \tan (e+f x))^{7/2}}{a+b \tan (e+f x)}+d (b c-a d) (c+d \tan (e+f x))^{3/2}+\frac{d (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b}+b d (c+d \tan (e+f x))^{5/2}}{f \left(a^2+b^2\right) (b c-a d)}","-\frac{(b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}-\frac{(b c-a d)^{3/2} \left(a^2 d+4 a b c+5 b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{3/2} f \left(a^2+b^2\right)^2}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2}",1,"(-((I*(a + I*b)^2*b^(3/2)*(c - I*d)^(5/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + I*b^(3/2)*(I*a + b)^2*(c + I*d)^(5/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (b*c - a*d)^(5/2)*(4*a*b*c + a^2*d + 5*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2))) + (d*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/b + d*(b*c - a*d)*(c + d*Tan[e + f*x])^(3/2) + b*d*(c + d*Tan[e + f*x])^(5/2) - (b^2*(c + d*Tan[e + f*x])^(7/2))/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f)","A",1
1246,1,2946,355,6.4700535,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{(b c-a d) \left(a^2 d+8 a b c+9 b^2 d\right) \sqrt{c+d \tan (e+f x)}}{4 b f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{(b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{2 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}+\frac{\sqrt{b c-a d} \left(a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left(4 c^2-3 d^2\right)-56 a b^3 c d+b^4 \left(8 c^2-15 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{4 b^{3/2} f \left(a^2+b^2\right)^3}-\frac{(c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a)^3}+\frac{(c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)^3}",1,"-1/2*(b^2*(c + d*Tan[e + f*x])^(7/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*(c + d*Tan[e + f*x])^(5/2))/(f*(a + b*Tan[e + f*x]))) + (2*((-3*b*d*(b*c - a*d)*(c + d*Tan[e + f*x])^(3/2))/(2*f*(a + b*Tan[e + f*x])) + (2*((-3*b*d*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(4*f*(a + b*Tan[e + f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) + a*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2) - I*(a*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) - b*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(b*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) + a*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2) + I*(a*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) - b*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*(-(a*b*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16)) + (a^2*d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2 + b^2*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))))/b))/b))/(3*b))/(2*(a^2 + b^2)*(b*c - a*d))","B",1
1247,1,235,248,3.3622432,"\int \frac{(a+b \tan (e+f x))^4}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^4/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{2 b^2 \left(87 a^2 d^2-40 a b c d+b^2 \left(8 c^2-15 d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{d^2}+\frac{4 b^3 (7 a d-2 b c) \tan (e+f x) \sqrt{c+d \tan (e+f x)}}{d}+6 b^2 (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}-\frac{15 i d (a-i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}+\frac{15 i d (a+i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}}{15 d f}","-\frac{2 b^2 \left(-87 a^2 d^2+40 a b c d-\left(b^2 \left(8 c^2-15 d^2\right)\right)\right) \sqrt{c+d \tan (e+f x)}}{15 d^3 f}-\frac{4 b^3 (2 b c-7 a d) \tan (e+f x) \sqrt{c+d \tan (e+f x)}}{15 d^2 f}+\frac{2 b^2 (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}{5 d f}-\frac{i (a-i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}+\frac{i (a+i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}",1,"(((-15*I)*(a - I*b)^4*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((15*I)*(a + I*b)^4*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*b^2*(-40*a*b*c*d + 87*a^2*d^2 + b^2*(8*c^2 - 15*d^2))*Sqrt[c + d*Tan[e + f*x]])/d^2 + (4*b^3*(-2*b*c + 7*a*d)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/d + 6*b^2*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(15*d*f)","A",1
1248,1,178,178,1.0591202,"\int \frac{(a+b \tan (e+f x))^3}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/Sqrt[c + d*Tan[e + f*x]],x]","\frac{2 \left(\frac{2 b^2 (4 a d-b c) \sqrt{c+d \tan (e+f x)}}{d}+b^2 (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}-\frac{3 i d (a-i b)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{2 \sqrt{c-i d}}+\frac{3 i d (a+i b)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 \sqrt{c+i d}}\right)}{3 d f}","-\frac{4 b^2 (b c-4 a d) \sqrt{c+d \tan (e+f x)}}{3 d^2 f}+\frac{2 b^2 (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}{3 d f}-\frac{(-b+i a)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}+\frac{(b+i a)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}",1,"(2*((((-3*I)/2)*(a - I*b)^3*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (((3*I)/2)*(a + I*b)^3*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*b^2*(-(b*c) + 4*a*d)*Sqrt[c + d*Tan[e + f*x]])/d + b^2*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(3*d*f)","A",1
1249,1,129,134,0.2221436,"\int \frac{(a+b \tan (e+f x))^2}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/Sqrt[c + d*Tan[e + f*x]],x]","\frac{-\frac{i (a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}+\frac{i (a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}+\frac{2 b^2 \sqrt{c+d \tan (e+f x)}}{d}}{f}","-\frac{i (a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}+\frac{i (a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}+\frac{2 b^2 \sqrt{c+d \tan (e+f x)}}{d f}",1,"(((-I)*(a - I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*(a + I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*b^2*Sqrt[c + d*Tan[e + f*x]])/d)/f","A",1
1250,1,101,102,0.1317136,"\int \frac{a+b \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])/Sqrt[c + d*Tan[e + f*x]],x]","\frac{i \left(\frac{(a+i b) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}-\frac{(a-i b) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}\right)}{f}","\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}-\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}",1,"(I*(-(((a - I*b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d]) + ((a + I*b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]))/f","A",1
1251,1,158,170,0.4139379,"\int \frac{1}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{-\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{(b-i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}+\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}}{f \left(a^2+b^2\right)}","-\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right) \sqrt{b c-a d}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a) \sqrt{c-i d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a) \sqrt{c+i d}}",1,"((((-I)*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/((a^2 + b^2)*f)","A",1
1252,1,258,244,2.3879049,"\int \frac{1}{(a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{-\frac{i \left(\frac{(a-i b)^2 (a d-b c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}+\frac{(a+i b)^2 (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}\right)}{a^2+b^2}+\frac{b^{3/2} \left(5 a^2 d-4 a b c+b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\left(a^2+b^2\right) \sqrt{b c-a d}}-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{a+b \tan (e+f x)}}{f \left(a^2+b^2\right) (b c-a d)}","-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))}-\frac{b^{3/2} \left(-5 a^2 d+4 a b c-b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right)^2 (b c-a d)^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2 \sqrt{c-i d}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2 \sqrt{c+i d}}",1,"(((-I)*(((a + I*b)^2*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((a - I*b)^2*(-(b*c) + a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]))/(a^2 + b^2) + (b^(3/2)*(-4*a*b*c + 5*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*d]) - (b^2*Sqrt[c + d*Tan[e + f*x]])/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f)","A",1
1253,1,415,317,6.3452649,"\int \frac{(a+b \tan (e+f x))^4}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 b^2 (a+b \tan (e+f x))^2}{3 d f \sqrt{c+d \tan (e+f x)}}+\frac{2 \left(-\frac{2 b^2 (2 b c-5 a d) (a+b \tan (e+f x))}{d f \sqrt{c+d \tan (e+f x)}}+\frac{-\frac{2 \left(29 a^2 b^2 d^2-28 a b^3 c d+8 b^4 c^2-3 b^4 d^2\right)}{d \sqrt{c+d \tan (e+f x)}}+\frac{2 \left(\frac{\left(\frac{3}{2} d^4 \left(a^4-6 a^2 b^2+b^4\right)-6 a b c d^3 (a-b) (a+b)\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{(-d+i c) \sqrt{c+d \tan (e+f x)}}-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{(d+i c) \sqrt{c+d \tan (e+f x)}}\right)}{d}+6 a b d^2 (a-b) (a+b) \left(\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}\right)\right)}{d}}{2 d f}\right)}{3 d}","-\frac{2 b \left(-6 a^3 d^3+15 a^2 b c d^2-12 a b^2 d \left(2 c^2+d^2\right)+b^3 \left(8 c^3+5 c d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{3 d^3 f \left(c^2+d^2\right)}-\frac{2 b^2 \left(3 a d (2 b c-a d)-b^2 \left(4 c^2+d^2\right)\right) \tan (e+f x) \sqrt{c+d \tan (e+f x)}}{3 d^2 f \left(c^2+d^2\right)}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))^2}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}+\frac{i (a+i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}",1,"(2*b^2*(a + b*Tan[e + f*x])^2)/(3*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*((-2*b^2*(2*b*c - 5*a*d)*(a + b*Tan[e + f*x]))/(d*f*Sqrt[c + d*Tan[e + f*x]]) + ((-2*(8*b^4*c^2 - 28*a*b^3*c*d + 29*a^2*b^2*d^2 - 3*b^4*d^2))/(d*Sqrt[c + d*Tan[e + f*x]]) + (2*(6*a*(a - b)*b*(a + b)*d^2*(((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]) + ((-6*a*(a - b)*b*(a + b)*c*d^3 + (3*(a^4 - 6*a^2*b^2 + b^4)*d^4)/2)*(-(Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*Sqrt[c + d*Tan[e + f*x]])) + Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)]/((I*c - d)*Sqrt[c + d*Tan[e + f*x]])))/d))/d)/(2*d*f)))/(3*d)","C",1
1254,1,287,216,1.8638271,"\int \frac{(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(3/2),x]","\frac{-i b \left(3 a^2-b^2\right) \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}\right)+\frac{\left(a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right) \left((d-i c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)+(d+i c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)\right)}{\left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}+\frac{4 b^2 (b c-2 a d)}{d \sqrt{c+d \tan (e+f x)}}+\frac{2 b^2 (a+b \tan (e+f x))}{\sqrt{c+d \tan (e+f x)}}}{d f}","-\frac{2 b \left(a d (2 b c-a d)-b^2 \left(2 c^2+d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{d^2 f \left(c^2+d^2\right)}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{(-b+i a)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}+\frac{(b+i a)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}",1,"((-I)*b*(3*a^2 - b^2)*(ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/Sqrt[c - I*d] - ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]/Sqrt[c + I*d]) + (4*b^2*(b*c - 2*a*d))/(d*Sqrt[c + d*Tan[e + f*x]]) + ((3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*(((-I)*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] + (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)]))/((c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]]) + (2*b^2*(a + b*Tan[e + f*x]))/Sqrt[c + d*Tan[e + f*x]])/(d*f)","C",1
1255,1,124,150,0.1678041,"\int \frac{(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2),x]","\frac{-\frac{(a-i b)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{d+i c}+\frac{(a+i b)^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{-d+i c}-\frac{2 b^2}{d}}{f \sqrt{c+d \tan (e+f x)}}","-\frac{2 (b c-a d)^2}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}+\frac{i (a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}",1,"((-2*b^2)/d - ((a - I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(I*c + d) + ((a + I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(I*c - d))/(f*Sqrt[c + d*Tan[e + f*x]])","C",1
1256,1,113,138,0.1951045,"\int \frac{a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2),x]","\frac{i \left(\frac{(a-i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{c-i d}-\frac{(a+i b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{c+i d}\right)}{f \sqrt{c+d \tan (e+f x)}}","\frac{2 (b c-a d)}{f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}+\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}",1,"(I*(((a - I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(c - I*d) - ((a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d)))/(f*Sqrt[c + d*Tan[e + f*x]])","C",1
1257,1,247,211,2.0849594,"\int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{-\frac{i \left(\frac{(a+i b) (c+i d) (a d-b c) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}+\frac{(a-i b) (c-i d) (b c-a d) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}\right)}{a^2+b^2}+\frac{2 b^{5/2} \left(c^2+d^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\left(a^2+b^2\right) \sqrt{b c-a d}}-\frac{2 d^2}{\sqrt{c+d \tan (e+f x)}}}{f \left(c^2+d^2\right) (a d-b c)}","-\frac{2 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right) (b c-a d)^{3/2}}+\frac{2 d^2}{f \left(c^2+d^2\right) (b c-a d) \sqrt{c+d \tan (e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a) (c-i d)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a) (c+i d)^{3/2}}",1,"(((-I)*(((a + I*b)*(c + I*d)*(-(b*c) + a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((a - I*b)*(c - I*d)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]))/(a^2 + b^2) + (2*b^(5/2)*(c^2 + d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*d]) - (2*d^2)/Sqrt[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f)","A",1
1258,1,628,314,6.2510563,"\int \frac{1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{-\frac{2 \left(\frac{1}{2} d^2 \left(2 a^2 d-2 a b c+3 b^2 d\right)-c \left(b d (b c-a d)-\frac{3}{2} b^2 c d\right)\right)}{f \left(c^2+d^2\right) (a d-b c) \sqrt{c+d \tan (e+f x)}}-\frac{2 \left(\frac{2 \sqrt{b c-a d} \left(\frac{1}{4} a^2 b d \left(2 a^2 d^2+b^2 \left(c^2+3 d^2\right)\right)+\frac{1}{4} b^2 \left(-2 a^3 c d^2+4 a^2 b d \left(c^2+d^2\right)-2 a b^2 c \left(c^2+2 d^2\right)+3 b^3 d \left(c^2+d^2\right)\right)-\frac{1}{2} a b (b c-a d)^2 (a d+b c)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} f \left(a^2+b^2\right) (a d-b c)}+\frac{\frac{i \sqrt{c-i d} \left(-\frac{1}{2} \left(a^2 c-2 a b d-b^2 c\right) (b c-a d)^2-\frac{1}{2} i \left(a^2 d+2 a b c-b^2 d\right) (b c-a d)^2\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (-c+i d)}-\frac{i \sqrt{c+i d} \left(-\frac{1}{2} (b c-a d)^2 \left(a^2 c-2 a b d-b^2 c\right)+\frac{1}{2} i (b c-a d)^2 \left(a^2 d+2 a b c-b^2 d\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-c-i d)}}{a^2+b^2}\right)}{\left(c^2+d^2\right) (a d-b c)}}{\left(a^2+b^2\right) (b c-a d)}","-\frac{d \left(2 a^2 d^2+b^2 \left(c^2+3 d^2\right)\right)}{f \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{b^{5/2} \left(-7 a^2 d+4 a b c-3 b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right)^2 (b c-a d)^{5/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2 (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2 (c+i d)^{3/2}}",1,"-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])) - ((-2*(((I*Sqrt[c - I*d]*(-1/2*((b*c - a*d)^2*(a^2*c - b^2*c - 2*a*b*d)) - (I/2)*(b*c - a*d)^2*(2*a*b*c + a^2*d - b^2*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(-1/2*((b*c - a*d)^2*(a^2*c - b^2*c - 2*a*b*d)) + (I/2)*(b*c - a*d)^2*(2*a*b*c + a^2*d - b^2*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*(-1/2*(a*b*(b*c - a*d)^2*(b*c + a*d)) + (b^2*(-2*a^3*c*d^2 + 4*a^2*b*d*(c^2 + d^2) + 3*b^3*d*(c^2 + d^2) - 2*a*b^2*c*(c^2 + 2*d^2)))/4 + (a^2*b*d*(2*a^2*d^2 + b^2*(c^2 + 3*d^2)))/4)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*(-(b*c) + a*d)*f)))/((-(b*c) + a*d)*(c^2 + d^2)) - (2*((d^2*(-2*a*b*c + 2*a^2*d + 3*b^2*d))/2 - c*((-3*b^2*c*d)/2 + b*d*(b*c - a*d))))/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/((a^2 + b^2)*(b*c - a*d))","A",1
1259,1,368,290,3.6196709,"\int \frac{(a+b \tan (e+f x))^4}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{-2 b^2 (c-i d) (c+i d) \left(9 a^2 d^2-20 a b c d+b^2 \left(8 c^2+d^2\right)\right)-12 a b d^2 \left(a^2-b^2\right) (c+d \tan (e+f x)) \left(i (c+i d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)-(d+i c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)\right)+d^2 \left(a^4 d-4 a^3 b c-6 a^2 b^2 d+4 a b^3 c+b^4 d\right) \left((d-i c) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)+(d+i c) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)\right)-6 b^2 d^2 (c-i d) (c+i d) (a+b \tan (e+f x))^2-12 b^2 d (c-i d) (c+i d) (2 b c-3 a d) (a+b \tan (e+f x))}{3 d^3 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}","-\frac{2 b^2 \left(a d (2 b c-a d)-b^2 \left(4 c^2+3 d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{3 d^3 f \left(c^2+d^2\right)}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}+\frac{4 (b c-a d)^3 \left(3 a c d+2 b c^2+5 b d^2\right)}{3 d^3 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}+\frac{i (a+i b)^4 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{5/2}}",1,"-1/3*(-2*b^2*(c - I*d)*(c + I*d)*(-20*a*b*c*d + 9*a^2*d^2 + b^2*(8*c^2 + d^2)) + d^2*(-4*a^3*b*c + 4*a*b^3*c + a^4*d - 6*a^2*b^2*d + b^4*d)*(((-I)*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)] + (I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]) - 12*b^2*(c - I*d)*(c + I*d)*d*(2*b*c - 3*a*d)*(a + b*Tan[e + f*x]) - 6*b^2*(c - I*d)*(c + I*d)*d^2*(a + b*Tan[e + f*x])^2 - 12*a*b*(a^2 - b^2)*d^2*(I*(c + I*d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])*(c + d*Tan[e + f*x]))/(d^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2))","C",1
1260,1,284,219,1.5583363,"\int \frac{(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{-3 b d \left(3 a^2-b^2\right) (c+d \tan (e+f x)) \left(i (c+i d) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)-(d+i c) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)\right)-d \left(a^3 d-3 a^2 b c-3 a b^2 d+b^3 c\right) \left(i (c+i d) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)-(d+i c) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)\right)+6 b^2 d (c-i d) (c+i d) (a+b \tan (e+f x))+4 b^3 c \left(c^2+d^2\right)}{3 d^2 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}","-\frac{4 (b c-a d)^2 \left(3 a c d+b \left(c^2+4 d^2\right)\right)}{3 d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{(-b+i a)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{5/2}}+\frac{(b+i a)^3 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}",1,"-1/3*(4*b^3*c*(c^2 + d^2) - d*(-3*a^2*b*c + b^3*c + a^3*d - 3*a*b^2*d)*(I*(c + I*d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]) + 6*b^2*(c - I*d)*(c + I*d)*d*(a + b*Tan[e + f*x]) - 3*b*(3*a^2 - b^2)*d*(I*(c + I*d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])*(c + d*Tan[e + f*x]))/(d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2))","C",1
1261,1,127,195,0.210218,"\int \frac{(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(5/2),x]","\frac{-\frac{(a-i b)^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{d+i c}+\frac{(a+i b)^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{-d+i c}-\frac{2 b^2}{d}}{3 f (c+d \tan (e+f x))^{3/2}}","\frac{4 (b c-a d) (a c+b d)}{f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 (b c-a d)^2}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i (a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}+\frac{i (a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{5/2}}",1,"((-2*b^2)/d - ((a - I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(I*c + d) + ((a + I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(I*c - d))/(3*f*(c + d*Tan[e + f*x])^(3/2))","C",1
1262,1,115,186,0.1734122,"\int \frac{a+b \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{i \left(\frac{(a+i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{c+i d}-\frac{(a-i b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{c-i d}\right)}{3 f (c+d \tan (e+f x))^{3/2}}","\frac{2 (b c-a d)}{3 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(2 a c d-b \left(c^2-d^2\right)\right)}{f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}+\frac{(-b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{5/2}}",1,"((-1/3*I)*(-(((a - I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(c - I*d)) + ((a + I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d)))/(f*(c + d*Tan[e + f*x])^(3/2))","C",1
1263,1,323,272,5.2164222,"\int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\frac{3 \left(-\frac{2 b^{7/2} \left(c^2+d^2\right)^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b c-a d}}+\frac{(b-i a) (c+i d)^2 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}+\frac{(b+i a) (c-i d)^2 (b c-a d)^2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right) (a d-b c)}-\frac{6 d^2 \left(b \left(3 c^2+d^2\right)-2 a c d\right)}{\left(c^2+d^2\right) (b c-a d) \sqrt{c+d \tan (e+f x)}}-\frac{2 d^2}{(c+d \tan (e+f x))^{3/2}}}{3 f \left(c^2+d^2\right) (a d-b c)}","-\frac{2 b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right) (b c-a d)^{5/2}}-\frac{2 d^2 \left(2 a c d-b \left(3 c^2+d^2\right)\right)}{f \left(c^2+d^2\right)^2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}+\frac{2 d^2}{3 f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (b+i a) (c-i d)^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a) (c+i d)^{5/2}}",1,"((3*((((-I)*a + b)*(c + I*d)^2*(b*c - a*d)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((I*a + b)*(c - I*d)^2*(b*c - a*d)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*b^(7/2)*(c^2 + d^2)^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d]))/((a^2 + b^2)*(-(b*c) + a*d)*(c^2 + d^2)) - (2*d^2)/(c + d*Tan[e + f*x])^(3/2) - (6*d^2*(-2*a*c*d + b*(3*c^2 + d^2)))/((b*c - a*d)*(c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]]))/(3*(-(b*c) + a*d)*(c^2 + d^2)*f)","A",1
1264,1,2536,425,6.2683756,"\int \frac{1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)),x]","\text{Result too large to show}","-\frac{d \left(2 a^2 d^2+b^2 \left(3 c^2+5 d^2\right)\right)}{3 f \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)^2 (c+d \tan (e+f x))^{3/2}}-\frac{b^2}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}-\frac{b^{7/2} \left(-9 a^2 d+4 a b c-5 b^2 d\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{f \left(a^2+b^2\right)^2 (b c-a d)^{7/2}}+\frac{d \left(4 a^3 c d^3-4 a^2 b d^2 \left(2 c^2+d^2\right)+4 a b^2 c d^3-\left(b^3 \left(c^4+10 c^2 d^2+5 d^4\right)\right)\right)}{f \left(a^2+b^2\right) \left(c^2+d^2\right)^2 (b c-a d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2 (c-i d)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2 (c+i d)^{5/2}}",1,"-(b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2))) - ((-2*((d^2*(-2*a*b*c + 2*a^2*d + 5*b^2*d))/2 - c*((-5*b^2*c*d)/2 + b*d*(b*c - a*d))))/(3*(-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*((-2*(((I*Sqrt[c - I*d]*((b*(-(b*c) + a*d)*((-3*c*(b*c - a*d)^2*(b*c + a*d))/2 - (3*d*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (3*b*d*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 + a*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (a*d*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 - (b*((-3*d^2*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)))/2) - I*((a*(-(b*c) + a*d)*((-3*c*(b*c - a*d)^2*(b*c + a*d))/2 - (3*d*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (3*b*d*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 - b*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (a*d*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 - (b*((-3*d^2*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*((b*(-(b*c) + a*d)*((-3*c*(b*c - a*d)^2*(b*c + a*d))/2 - (3*d*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (3*b*d*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 + a*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (a*d*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 - (b*((-3*d^2*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)))/2) + I*((a*(-(b*c) + a*d)*((-3*c*(b*c - a*d)^2*(b*c + a*d))/2 - (3*d*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (3*b*d*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 - b*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (a*d*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2 - (b*((-3*d^2*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f))/(a^2 + b^2) + (2*Sqrt[b*c - a*d]*(-1/2*(a*b*(-(b*c) + a*d)*((-3*c*(b*c - a*d)^2*(b*c + a*d))/2 - (3*d*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (3*b*d*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)) + (a^2*b*((-3*d^2*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)))/2 + b^2*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - (a*d*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4))/2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*(-(b*c) + a*d)*f)))/((-(b*c) + a*d)*(c^2 + d^2)) - (2*((-3*d^2*(2*a^3*c*d^2 - 4*a^2*b*d*(c^2 + d^2) - 5*b^3*d*(c^2 + d^2) + 2*a*b^2*c*(c^2 + 2*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/2 - (3*b*c*(2*a^2*d^3 + b^2*(3*c^2*d + 5*d^3)))/4)))/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])))/(3*(-(b*c) + a*d)*(c^2 + d^2)))/((a^2 + b^2)*(b*c - a*d))","B",1
1265,1,565,337,5.9690741,"\int (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]],x]","\frac{-\frac{b^{5/2} \sqrt{c-\frac{a d}{b}} \left(-15 a^2 d^2-10 a b c d+b^2 \left(c^2+8 d^2\right)\right) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right)}{\sqrt{d} \sqrt{c+d \tan (e+f x)}}+\frac{4 b d \left(b \left(a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right)+\sqrt{-b^2} \left(a^3 c-3 a^2 b d-3 a b^2 c+b^3 d\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{4 b d \left(b \left(a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right)-\sqrt{-b^2} \left(a^3 c-3 a^2 b d-3 a b^2 c+b^3 d\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+\sqrt{-b^2}} \sqrt{\frac{\sqrt{-b^2} d}{b}+c}}+2 b^4 \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}+b^3 (9 a d-b c) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^2 d f}","\frac{\sqrt{b} \left(15 a^2 d^2+10 a b c d-\left(b^2 \left(c^2+8 d^2\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{4 d^{3/2} f}+\frac{b^2 \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}-\frac{b (b c-9 a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d f}-\frac{i (a-i b)^{5/2} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i (a+i b)^{5/2} \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}",1,"((4*b*d*(b*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d) + Sqrt[-b^2]*(a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d))*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (4*b*d*(b*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d) - Sqrt[-b^2]*(a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + b^3*(-(b*c) + 9*a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + 2*b^4*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2) - (b^(5/2)*Sqrt[c - (a*d)/b]*(-10*a*b*c*d - 15*a^2*d^2 + b^2*(c^2 + 8*d^2))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(4*b^2*d*f)","A",1
1266,1,1518,258,6.115598,"\int (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]],x]","-\frac{i (-a-i b) \left(-\frac{2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(\frac{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{2 \sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)}\right) \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}{\sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-(-a-i b) \left(-\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{b^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 (-c-i d) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}\right)\right)}{2 f}-\frac{i (i b-a) \left(\frac{2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2} \left(\frac{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{2 \sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)}\right)}{\sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-(i b-a) \left(\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{i d-c} \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{i b-a}}\right)\right)}{2 f}","\frac{b \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}-\frac{i (a-i b)^{3/2} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i (a+i b)^{3/2} \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{\sqrt{b} (3 a d+b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{d} f}",1,"((-1/2*I)*(-a - I*b)*(-((-a - I*b)*((-2*(-c - I*d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d]) - (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) - (2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])))/f - ((I/2)*(-a + I*b)*(-((-a + I*b)*((-2*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I*b] + (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) + (2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])))/f","B",1
1267,1,261,218,2.1983209,"\int \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]],x]","\frac{i \sqrt{-a+i b} \sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)+i \sqrt{a+i b} \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)+\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{c+d \tan (e+f x)}}}{f}","-\frac{i \sqrt{a-i b} \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i \sqrt{a+i b} \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{f}",1,"(I*Sqrt[-a + I*b]*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])] + I*Sqrt[a + I*b]*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])] + (2*Sqrt[d]*Sqrt[b*c - a*d]*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/Sqrt[c + d*Tan[e + f*x]])/f","A",1
1268,1,167,163,0.2246451,"\int \frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{a+b \tan (e+f x)}} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]],x]","-\frac{i \left(\frac{\sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b}}-\frac{\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b}}\right)}{f}","\frac{i \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a+i b}}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a-i b}}",1,"((-I)*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I*b] - (Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a + I*b]))/f","A",1
1269,1,224,206,1.6880911,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^(3/2),x]","\frac{\frac{i \sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{3/2}}+\frac{\frac{(b+i a) \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{3/2}}-\frac{2 b \sqrt{c+d \tan (e+f x)}}{(a+i b) \sqrt{a+b \tan (e+f x)}}}{a-i b}}{f}","-\frac{2 b \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) \sqrt{a+b \tan (e+f x)}}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{3/2}}+\frac{i \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{3/2}}",1,"((I*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + (((I*a + b)*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(3/2) - (2*b*Sqrt[c + d*Tan[e + f*x]])/((a + I*b)*Sqrt[a + b*Tan[e + f*x]]))/(a - I*b))/f","A",1
1270,1,266,280,4.7792285,"\int \frac{\sqrt{c+d \tan (e+f x)}}{(a+b \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^(5/2),x]","\frac{\frac{2 b \sqrt{c+d \tan (e+f x)} \left(-6 a^3 d+b \left(-5 a^2 d+6 a b c+b^2 d\right) \tan (e+f x)+7 a^2 b c+b^3 c\right)}{\left(a^2+b^2\right)^2 (a d-b c) (a+b \tan (e+f x))^{3/2}}-\frac{3 i \sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{5/2}}+\frac{3 i \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{5/2}}}{3 f}","-\frac{2 b \left(-5 a^2 d+6 a b c+b^2 d\right) \sqrt{c+d \tan (e+f x)}}{3 f \left(a^2+b^2\right)^2 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{2 b \sqrt{c+d \tan (e+f x)}}{3 f \left(a^2+b^2\right) (a+b \tan (e+f x))^{3/2}}-\frac{i \sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{5/2}}+\frac{i \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{5/2}}",1,"(((-3*I)*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(5/2) + ((3*I)*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(5/2) + (2*b*Sqrt[c + d*Tan[e + f*x]]*(7*a^2*b*c + b^3*c - 6*a^3*d + b*(6*a*b*c - 5*a^2*d + b^2*d)*Tan[e + f*x]))/((a^2 + b^2)^2*(-(b*c) + a*d)*(a + b*Tan[e + f*x])^(3/2)))/(3*f)","A",1
1271,1,2566,330,6.1404122,"\int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{\left(3 a^2 d^2+18 a b c d+b^2 \left(3 c^2-8 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{4 \sqrt{b} \sqrt{d} f}+\frac{b \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac{(5 a d+3 b c) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 f}-\frac{i (a-i b)^{3/2} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i (a+i b)^{3/2} (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}",1,"((-1/2*I)*(-a - I*b)*(-((-a - I*b)*(-((-c - I*d)*((-2*(-c - I*d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d]) - (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) - (2*d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]))) - (2*(b*c - a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(5/2)*((3*Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(8*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(5/2)) + (3/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2) + (1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(-1))/4))/(b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])))/f - ((I/2)*(-a + I*b)*(-((-a + I*b)*(-((-c + I*d)*((-2*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I*b] + (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) + (2*d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]))) + (2*(b*c - a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(5/2)*((3*Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(8*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(5/2)) + (3/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2) + (1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(-1))/4))/(b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])))/f","B",1
1272,1,1526,258,6.0855574,"\int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2),x]","-\frac{i (-a-i b) \left(-\frac{2 d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(\frac{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{2 \sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)}\right) \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-(-c-i d) \left(-\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{b^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 (-c-i d) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}\right)\right)}{2 f}-\frac{i (i b-a) \left(\frac{2 d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2} \left(\frac{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{2 \sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}+\frac{1}{2 \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)}\right)}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-(i d-c) \left(\frac{2 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{i d-c} \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{i b-a}}\right)\right)}{2 f}","\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{f}-\frac{i \sqrt{a-i b} (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i \sqrt{a+i b} (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{b} f}",1,"((-1/2*I)*(-a - I*b)*(-((-c - I*d)*((-2*(-c - I*d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d]) - (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) - (2*d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])))/f - ((I/2)*(-a + I*b)*(-((-c + I*d)*((-2*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I*b] + (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) + (2*d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])))/f","B",1
1273,1,292,218,1.7394065,"\int \frac{(c+d \tan (e+f x))^{3/2}}{\sqrt{a+b \tan (e+f x)}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/Sqrt[a + b*Tan[e + f*x]],x]","\frac{\frac{2 d^{3/2} \sqrt{a+i b} \sqrt{b c-a d} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d}}\right)+i b (c+i d)^{3/2} \sqrt{c+d \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{b \sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}+\frac{i (-c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b}}}{f}","\frac{2 d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{b} f}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a-i b}}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a+i b}}",1,"((I*(-c + I*d)^(3/2)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I*b] + (I*b*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])]*Sqrt[c + d*Tan[e + f*x]] + 2*Sqrt[a + I*b]*d^(3/2)*Sqrt[b*c - a*d]*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(Sqrt[a + I*b]*b*Sqrt[c + d*Tan[e + f*x]]))/f","A",1
1274,1,231,213,2.0234259,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{3/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(3/2),x]","\frac{\frac{\frac{2 (a d-b c) \sqrt{c+d \tan (e+f x)}}{(a+i b) \sqrt{a+b \tan (e+f x)}}+\frac{(b+i a) (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{3/2}}}{a-i b}-\frac{i (-c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{3/2}}}{f}","-\frac{2 (b c-a d) \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) \sqrt{a+b \tan (e+f x)}}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{3/2}}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{3/2}}",1,"(((-I)*(-c + I*d)^(3/2)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + (((I*a + b)*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(3/2) + (2*(-(b*c) + a*d)*Sqrt[c + d*Tan[e + f*x]])/((a + I*b)*Sqrt[a + b*Tan[e + f*x]]))/(a - I*b))/f","A",1
1275,1,264,277,3.4406613,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{5/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(5/2),x]","\frac{-\frac{2 \sqrt{c+d \tan (e+f x)} \left(-3 a^3 d+2 b \left(a^2 (-d)+3 a b c+2 b^2 d\right) \tan (e+f x)+7 a^2 b c+3 a b^2 d+b^3 c\right)}{\left(a^2+b^2\right)^2 (a+b \tan (e+f x))^{3/2}}+\frac{3 i (-c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{5/2}}+\frac{3 i (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{5/2}}}{3 f}","-\frac{4 \left(a^2 (-d)+3 a b c+2 b^2 d\right) \sqrt{c+d \tan (e+f x)}}{3 f \left(a^2+b^2\right)^2 \sqrt{a+b \tan (e+f x)}}-\frac{2 (b c-a d) \sqrt{c+d \tan (e+f x)}}{3 f \left(a^2+b^2\right) (a+b \tan (e+f x))^{3/2}}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{5/2}}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{5/2}}",1,"(((3*I)*(-c + I*d)^(3/2)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(5/2) + ((3*I)*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(5/2) - (2*Sqrt[c + d*Tan[e + f*x]]*(7*a^2*b*c + b^3*c - 3*a^3*d + 3*a*b^2*d + 2*b*(3*a*b*c - a^2*d + 2*b^2*d)*Tan[e + f*x]))/((a^2 + b^2)^2*(a + b*Tan[e + f*x])^(3/2)))/(3*f)","A",1
1276,1,498,391,6.2710142,"\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(7/2),x]","\frac{b (c+d \tan (e+f x))^{5/2}}{5 f (-b+i a) (b c-a d) (a+b \tan (e+f x))^{5/2}}-\frac{b (c+d \tan (e+f x))^{5/2}}{5 f (b+i a) (b c-a d) (a+b \tan (e+f x))^{5/2}}-\frac{\frac{(c+d \tan (e+f x))^{3/2}}{(a-i b) (a+b \tan (e+f x))^{3/2}}+\frac{3 (c-i d) \left(\frac{\sqrt{c+d \tan (e+f x)}}{(a-i b) \sqrt{a+b \tan (e+f x)}}+\frac{\sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{3/2}}\right)}{a-i b}}{3 f (b+i a)}+\frac{\frac{(c+d \tan (e+f x))^{3/2}}{(a+i b) (a+b \tan (e+f x))^{3/2}}-\frac{3 (c+i d) \left(\frac{\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{3/2}}-\frac{\sqrt{c+d \tan (e+f x)}}{(a+i b) \sqrt{a+b \tan (e+f x)}}\right)}{a+i b}}{3 f (-b+i a)}","-\frac{4 \left(-2 a^2 d+5 a b c+3 b^2 d\right) \sqrt{c+d \tan (e+f x)}}{15 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))^{3/2}}-\frac{2 (b c-a d) \sqrt{c+d \tan (e+f x)}}{5 f \left(a^2+b^2\right) (a+b \tan (e+f x))^{5/2}}+\frac{2 \left(-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left(45 c^2-49 d^2\right)-70 a b^3 c d+3 b^4 \left(5 c^2-d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{15 f \left(a^2+b^2\right)^3 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{7/2}}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{7/2}}",1,"(b*(c + d*Tan[e + f*x])^(5/2))/(5*(I*a - b)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(5/2)) - (b*(c + d*Tan[e + f*x])^(5/2))/(5*(I*a + b)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(5/2)) - ((c + d*Tan[e + f*x])^(3/2)/((a - I*b)*(a + b*Tan[e + f*x])^(3/2)) + (3*(c - I*d)*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + Sqrt[c + d*Tan[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e + f*x]])))/(a - I*b))/(3*(I*a + b)*f) + ((c + d*Tan[e + f*x])^(3/2)/((a + I*b)*(a + b*Tan[e + f*x])^(3/2)) - (3*(c + I*d)*((Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(3/2) - Sqrt[c + d*Tan[e + f*x]]/((a + I*b)*Sqrt[a + b*Tan[e + f*x]])))/(a + I*b))/(3*(I*a - b)*f)","A",1
1277,1,773,429,8.1956124,"\int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2),x]","\frac{\frac{\frac{3 d \left(-a^2 d^2+14 a b c d+b^2 \left(11 c^2-8 d^2\right)\right) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 f}+\frac{-\frac{6 b d^2 \left(\sqrt{-b^2} \left(-\left(a^2 \left(c^3-3 c d^2\right)\right)+a b \left(6 c^2 d-2 d^3\right)+b^2 c \left(c^2-3 d^2\right)\right)-b \left(a^2 \left(3 c^2 d-d^3\right)+2 a b c \left(c^2-3 d^2\right)-b^2 d \left(3 c^2-d^2\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{6 b d^2 \left(b \left(a^2 \left(3 c^2 d-d^3\right)+2 a b c \left(c^2-3 d^2\right)-b^2 d \left(3 c^2-d^2\right)\right)+\sqrt{-b^2} \left(-\left(a^2 \left(c^3-3 c d^2\right)\right)+a b \left(6 c^2 d-2 d^3\right)+b^2 c \left(c^2-3 d^2\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+\sqrt{-b^2}} \sqrt{\frac{\sqrt{-b^2} d}{b}+c}}+\frac{3 \sqrt{b} d^{3/2} \sqrt{c-\frac{a d}{b}} \left(-a^3 d^3+15 a^2 b c d^2+3 a b^2 d \left(15 c^2-8 d^2\right)+5 b^3 \left(c^3-8 c d^2\right)\right) \sqrt{\frac{b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right)}{4 \sqrt{c+d \tan (e+f x)}}}{b d f}}{2 d}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{4 f}}{3 b}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}","\frac{\left(-a^2 d^2+14 a b c d+b^2 \left(11 c^2-8 d^2\right)\right) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{8 b f}+\frac{\left(-a^3 d^3+15 a^2 b c d^2+3 a b^2 d \left(15 c^2-8 d^2\right)+5 b^3 \left(c^3-8 c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{8 b^{3/2} \sqrt{d} f}+\frac{d^2 (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 b f}+\frac{d (13 b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 b f}-\frac{i (a-i b)^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i (a+i b)^{3/2} (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}",1,"(d^2*(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]])/(3*b*f) + ((d*(13*b*c - a*d)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(4*f) + ((3*d*(14*a*b*c*d - a^2*d^2 + b^2*(11*c^2 - 8*d^2))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*f) + ((-6*b*d^2*(Sqrt[-b^2]*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3)) - b*(2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*c^2 - d^2) + a^2*(3*c^2*d - d^3)))*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (6*b*d^2*(Sqrt[-b^2]*(b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3)) + b*(2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*c^2 - d^2) + a^2*(3*c^2*d - d^3)))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + (3*Sqrt[b]*d^(3/2)*Sqrt[c - (a*d)/b]*(15*a^2*b*c*d^2 - a^3*d^3 + 3*a*b^2*d*(15*c^2 - 8*d^2) + 5*b^3*(c^3 - 8*c*d^2))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*c + b*d*Tan[e + f*x])/(b*c - a*d)])/(4*Sqrt[c + d*Tan[e + f*x]]))/(b*d*f))/(2*d))/(3*b)","A",1
1278,1,550,339,6.6037549,"\int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2),x]","\frac{\frac{\sqrt{d} \sqrt{c-\frac{a d}{b}} \left(-a^2 d^2+10 a b c d+b^2 \left(15 c^2-8 d^2\right)\right) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right)}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}+\frac{4 \left(b d \left(\sqrt{-b^2}-a\right) \left(d^2-3 c^2\right)+a \sqrt{-b^2} c \left(c^2-3 d^2\right)+b^2 \left(c^3-3 c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{4 \left(-b d \left(a+\sqrt{-b^2}\right) \left(d^2-3 c^2\right)-a \sqrt{-b^2} c \left(c^2-3 d^2\right)+b^2 \left(c^3-3 c d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+\sqrt{-b^2}} \sqrt{\frac{\sqrt{-b^2} d}{b}+c}}+2 d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}+d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}","\frac{\sqrt{d} \left(-a^2 d^2+10 a b c d+b^2 \left(15 c^2-8 d^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{4 b^{3/2} f}+\frac{d^2 (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 b f}+\frac{d (9 b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b f}-\frac{i \sqrt{a-i b} (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}+\frac{i \sqrt{a+i b} (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f}",1,"((4*(a*Sqrt[-b^2]*c*(c^2 - 3*d^2) + b*(-a + Sqrt[-b^2])*d*(-3*c^2 + d^2) + b^2*(c^3 - 3*c*d^2))*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (4*(-(a*Sqrt[-b^2]*c*(c^2 - 3*d^2)) - b*(a + Sqrt[-b^2])*d*(-3*c^2 + d^2) + b^2*(c^3 - 3*c*d^2))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + d*(9*b*c - a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + 2*d^2*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]] + (Sqrt[d]*Sqrt[c - (a*d)/b]*(10*a*b*c*d - a^2*d^2 + b^2*(15*c^2 - 8*d^2))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]]))/(4*b*f)","A",1
1279,1,432,264,2.8799194,"\int \frac{(c+d \tan (e+f x))^{5/2}}{\sqrt{a+b \tan (e+f x)}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/Sqrt[a + b*Tan[e + f*x]],x]","\frac{\frac{b \left(\sqrt{-b^2} c \left(c^2-3 d^2\right)-b d \left(d^2-3 c^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}+\frac{b \left(\sqrt{-b^2} c \left(c^2-3 d^2\right)+b d \left(d^2-3 c^2\right)\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+\sqrt{-b^2}} \sqrt{\frac{\sqrt{-b^2} d}{b}+c}}+\frac{\sqrt{b} d^{3/2} (5 b c-a d) \sqrt{c-\frac{a d}{b}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right)}{\sqrt{c+d \tan (e+f x)}}+b d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 f}","\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{b^{3/2} f}+\frac{d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a-i b}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a+i b}}",1,"((b*(Sqrt[-b^2]*c*(c^2 - 3*d^2) - b*d*(-3*c^2 + d^2))*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) + (b*(Sqrt[-b^2]*c*(c^2 - 3*d^2) + b*d*(-3*c^2 + d^2))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + b*d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + (Sqrt[b]*d^(3/2)*(5*b*c - a*d)*Sqrt[c - (a*d)/b]*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/Sqrt[c + d*Tan[e + f*x]])/(b^2*f)","A",1
1280,1,1187,273,6.2042948,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{3/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(3/2),x]","-\frac{i (-c-i d) \left(-\frac{2 d \sqrt{c+d \tan (e+f x)} \left(1-\frac{\sqrt{b} \sqrt{d} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right) \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}\right) \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{a+b \tan (e+f x)} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}-(-c-i d) \left(\frac{2 \sqrt{c+d \tan (e+f x)}}{(-a-i b) \sqrt{a+b \tan (e+f x)}}-\frac{2 \sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a-i b) \sqrt{a+i b}}\right)\right)}{2 f}-\frac{i (i d-c) \left(\frac{2 d \sqrt{c+d \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2} \left(1-\frac{\sqrt{b} \sqrt{d} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right) \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}\right)}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{a+b \tan (e+f x)} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}-(i d-c) \left(\frac{2 \sqrt{c+d \tan (e+f x)}}{(a-i b) \sqrt{a+b \tan (e+f x)}}-\frac{2 \sqrt{i d-c} \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{(a-i b) \sqrt{i b-a}}\right)\right)}{2 f}","-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b f \left(a^2+b^2\right) \sqrt{a+b \tan (e+f x)}}+\frac{2 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{b^{3/2} f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{3/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{3/2}}",1,"((-1/2*I)*(-c - I*d)*(-((-c - I*d)*((-2*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((-a - I*b)*Sqrt[a + I*b]) + (2*Sqrt[c + d*Tan[e + f*x]])/((-a - I*b)*Sqrt[a + b*Tan[e + f*x]]))) - (2*d*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*(1 - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[a + b*Tan[e + f*x]]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/f - ((I/2)*(-c + I*d)*(-((-c + I*d)*((-2*Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)*Sqrt[-a + I*b]) + (2*Sqrt[c + d*Tan[e + f*x]])/((a - I*b)*Sqrt[a + b*Tan[e + f*x]]))) + (2*d*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)*(1 - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*Sqrt[b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[a + b*Tan[e + f*x]]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/f","B",1
1281,1,350,292,5.8327278,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{5/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(5/2),x]","\frac{\frac{(d+i c) \left(\frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{3/2}}+3 (c-i d) \left(\frac{\sqrt{c+d \tan (e+f x)}}{(a-i b) \sqrt{a+b \tan (e+f x)}}+\frac{\sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{3/2}}\right)\right)}{a-i b}-(-d+i c) \left(\frac{\sqrt{c+d \tan (e+f x)} ((a d+3 b c+4 i b d) \tan (e+f x)+4 a c+3 i a d+i b c)}{(a+i b)^2 (a+b \tan (e+f x))^{3/2}}-\frac{3 (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{5/2}}\right)}{3 f}","-\frac{2 (b c-a d) \left(a^2 d+6 a b c+7 b^2 d\right) \sqrt{c+d \tan (e+f x)}}{3 b f \left(a^2+b^2\right)^2 \sqrt{a+b \tan (e+f x)}}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{3 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^{3/2}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{5/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{5/2}}",1,"(-((I*c - d)*((-3*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(5/2) + (Sqrt[c + d*Tan[e + f*x]]*(4*a*c + I*b*c + (3*I)*a*d + (3*b*c + a*d + (4*I)*b*d)*Tan[e + f*x]))/((a + I*b)^2*(a + b*Tan[e + f*x])^(3/2)))) + ((I*c + d)*((c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(3/2) + 3*(c - I*d)*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + Sqrt[c + d*Tan[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e + f*x]]))))/(a - I*b))/(3*f)","A",1
1282,1,490,398,6.2642353,"\int \frac{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(7/2),x]","\frac{(c+d \tan (e+f x))^{5/2}}{5 f (-b+i a) (a+b \tan (e+f x))^{5/2}}-\frac{(c+d \tan (e+f x))^{5/2}}{5 f (b+i a) (a+b \tan (e+f x))^{5/2}}+\frac{(d+i c) \left(\frac{(c+d \tan (e+f x))^{3/2}}{(a-i b) (a+b \tan (e+f x))^{3/2}}+\frac{3 (c-i d) \left(\frac{\sqrt{c+d \tan (e+f x)}}{(a-i b) \sqrt{a+b \tan (e+f x)}}+\frac{\sqrt{-c+i d} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-a+i b)^{3/2}}\right)}{a-i b}\right)}{3 f (a-i b)}-\frac{(-d+i c) \left(\frac{(c+d \tan (e+f x))^{3/2}}{(a+i b) (a+b \tan (e+f x))^{3/2}}-\frac{3 (c+i d) \left(\frac{\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(a+i b)^{3/2}}-\frac{\sqrt{c+d \tan (e+f x)}}{(a+i b) \sqrt{a+b \tan (e+f x)}}\right)}{a+i b}\right)}{3 f (a+i b)}","-\frac{2 (b c-a d) \left(a^2 d+10 a b c+11 b^2 d\right) \sqrt{c+d \tan (e+f x)}}{15 b f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))^{3/2}}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^{5/2}}+\frac{2 \left(2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left(15 c^2-13 d^2\right)-100 a b^3 c d+b^4 \left(15 c^2-23 d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{15 b f \left(a^2+b^2\right)^3 \sqrt{a+b \tan (e+f x)}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{7/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{7/2}}",1,"(c + d*Tan[e + f*x])^(5/2)/(5*(I*a - b)*f*(a + b*Tan[e + f*x])^(5/2)) - (c + d*Tan[e + f*x])^(5/2)/(5*(I*a + b)*f*(a + b*Tan[e + f*x])^(5/2)) + ((I*c + d)*((c + d*Tan[e + f*x])^(3/2)/((a - I*b)*(a + b*Tan[e + f*x])^(3/2)) + (3*(c - I*d)*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + Sqrt[c + d*Tan[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e + f*x]])))/(a - I*b)))/(3*(a - I*b)*f) - ((I*c - d)*((c + d*Tan[e + f*x])^(3/2)/((a + I*b)*(a + b*Tan[e + f*x])^(3/2)) - (3*(c + I*d)*((Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(3/2) - Sqrt[c + d*Tan[e + f*x]]/((a + I*b)*Sqrt[a + b*Tan[e + f*x]])))/(a + I*b)))/(3*(a + I*b)*f)","A",1
1283,1,438,264,3.3343916,"\int \frac{(a+b \tan (e+f x))^{5/2}}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^(5/2)/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{d \left(3 a^2 b^2+a \sqrt{-b^2} \left(a^2-3 b^2\right)-b^4\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}+\frac{d \left(a^3 \sqrt{-b^2}-3 a^2 b^2+3 a \left(-b^2\right)^{3/2}+b^4\right) \tanh ^{-1}\left(\frac{\sqrt{\frac{\sqrt{-b^2} d}{b}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+\sqrt{-b^2}} \sqrt{\frac{\sqrt{-b^2} d}{b}+c}}-\frac{b^{5/2} (b c-5 a d) \sqrt{c-\frac{a d}{b}} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right)}{\sqrt{d} \sqrt{c+d \tan (e+f x)}}+b^3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b d f}","-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{d^{3/2} f}+\frac{b^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{d f}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{c-i d}}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{c+i d}}",1,"(((3*a^2*b^2 - b^4 + a*Sqrt[-b^2]*(a^2 - 3*b^2))*d*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) + ((-3*a^2*b^2 + b^4 + a^3*Sqrt[-b^2] + 3*a*(-b^2)^(3/2))*d*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + b^3*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] - (b^(5/2)*(b*c - 5*a*d)*Sqrt[c - (a*d)/b]*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b*d*f)","A",1
1284,1,294,218,2.03903,"\int \frac{(a+b \tan (e+f x))^{3/2}}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{i (-a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-c+i d}}+\frac{i \sqrt{d} (a+i b)^{3/2} \sqrt{c+d \tan (e+f x)} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)+2 b \sqrt{c+i d} \sqrt{b c-a d} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{d} \sqrt{c+i d} \sqrt{c+d \tan (e+f x)}}}{f}","\frac{2 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{d} f}-\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{c-i d}}+\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{c+i d}}",1,"((I*(-a + I*b)^(3/2)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-c + I*d] + (I*(a + I*b)^(3/2)*Sqrt[d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])]*Sqrt[c + d*Tan[e + f*x]] + 2*b*Sqrt[c + I*d]*Sqrt[b*c - a*d]*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(Sqrt[c + I*d]*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/f","A",1
1285,1,167,163,0.1869237,"\int \frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]],x]","-\frac{i \left(\frac{\sqrt{-a+i b} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-c+i d}}-\frac{\sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{c+i d}}\right)}{f}","\frac{i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{c+i d}}-\frac{i \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{c-i d}}",1,"((-I)*((Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-c + I*d] - (Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[c + I*d]))/f","A",1
1286,1,166,163,0.1687047,"\int \frac{1}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{i \left(\frac{\tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}\right)}{f}","\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a+i b} \sqrt{c+i d}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a-i b} \sqrt{c-i d}}",1,"(I*(ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])]/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])]/(Sqrt[a + I*b]*Sqrt[c + I*d])))/f","A",1
1287,1,232,218,1.5946637,"\int \frac{1}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\frac{2 b^2 \sqrt{c+d \tan (e+f x)}}{(a d-b c) \sqrt{a+b \tan (e+f x)}}+\frac{i (a+i b) \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}+\frac{(b+i a) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}}{f \left(a^2+b^2\right)}","-\frac{2 b^2 \sqrt{c+d \tan (e+f x)}}{f \left(a^2+b^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{3/2} \sqrt{c-i d}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{3/2} \sqrt{c+i d}}",1,"((I*(a + I*b)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((I*a + b)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d]) + (2*b^2*Sqrt[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*Sqrt[a + b*Tan[e + f*x]]))/((a^2 + b^2)*f)","A",1
1288,1,308,295,2.2262657,"\int \frac{1}{(a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\frac{4 b^2 \left(4 a^2 d-3 a b c+b^2 d\right) \sqrt{c+d \tan (e+f x)}}{(b c-a d)^2 \sqrt{a+b \tan (e+f x)}}+\frac{2 b^2 \left(a^2+b^2\right) \sqrt{c+d \tan (e+f x)}}{(a d-b c) (a+b \tan (e+f x))^{3/2}}+3 i \left(\frac{(a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(a+i b)^2 \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}\right)}{3 f \left(a^2+b^2\right)^2}","-\frac{4 b^2 \left(-4 a^2 d+3 a b c-b^2 d\right) \sqrt{c+d \tan (e+f x)}}{3 f \left(a^2+b^2\right)^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}-\frac{2 b^2 \sqrt{c+d \tan (e+f x)}}{3 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{5/2} \sqrt{c-i d}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{5/2} \sqrt{c+i d}}",1,"((3*I)*(((a + I*b)^2*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((a - I*b)^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])) + (2*b^2*(a^2 + b^2)*Sqrt[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*(a + b*Tan[e + f*x])^(3/2)) + (4*b^2*(-3*a*b*c + 4*a^2*d + b^2*d)*Sqrt[c + d*Tan[e + f*x]])/((b*c - a*d)^2*Sqrt[a + b*Tan[e + f*x]]))/(3*(a^2 + b^2)^2*f)","A",1
1289,1,1877,356,6.355027,"\int \frac{(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(3/2),x]","-\frac{i (-a-i b) \left(-\frac{2 b \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}\right) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} (a+b \tan (e+f x))^{5/2}}{5 (b c-a d) \sqrt{c+d \tan (e+f x)}}-(-a-i b) \left(-\frac{2 (b c-a d) \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{3/2} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right) \left(-\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right) \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}\right) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^2}{b d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-(-a-i b) \left(\frac{2 \sqrt{a+b \tan (e+f x)}}{(-c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c-i d) \sqrt{c+i d}}\right)\right)\right)}{2 f}-\frac{i (i b-a) \left(\frac{2 b \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{3/2} \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}\right) (a+b \tan (e+f x))^{5/2} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}{5 (b c-a d) \sqrt{c+d \tan (e+f x)}}-(i b-a) \left(\frac{2 (b c-a d) \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{3/2} \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^2 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right) \left(-\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right) \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}\right)}{b d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-(i b-a) \left(\frac{2 \sqrt{a+b \tan (e+f x)}}{(c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{i b-a} \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{(c-i d) \sqrt{i d-c}}\right)\right)\right)}{2 f}","-\frac{b^{5/2} (3 b c-7 a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{d^{5/2} f}-\frac{b \left(2 a d (2 b c-a d)-b^2 \left(3 c^2+d^2\right)\right) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{d^2 f \left(c^2+d^2\right)}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}+\frac{i (a+i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{3/2}}",1,"((-1/2*I)*(-a - I*b)*((-2*b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, -((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f*x])^(5/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(5*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]]) - (-a - I*b)*(-((-a - I*b)*((-2*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((-c - I*d)*Sqrt[c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((-c - I*d)*Sqrt[c + d*Tan[e + f*x]]))) - (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]))))/f - ((I/2)*(-a + I*b)*((2*b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, -((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f*x])^(5/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(5*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]]) - (-a + I*b)*(-((-a + I*b)*((-2*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)*Sqrt[-c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((c - I*d)*Sqrt[c + d*Tan[e + f*x]]))) + (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]))))/f","C",0
1290,1,1503,273,6.1767598,"\int \frac{(a+b \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(3/2),x]","-\frac{i (-a-i b) \left(-\frac{2 (b c-a d) \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{3/2} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right) \left(-\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right) \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}\right) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^2}{b d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-(-a-i b) \left(\frac{2 \sqrt{a+b \tan (e+f x)}}{(-c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c-i d) \sqrt{c+i d}}\right)\right)}{2 f}-\frac{i (i b-a) \left(\frac{2 (b c-a d) \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{3/2} \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^2 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right) \left(-\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right)}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right) \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}\right)}{b d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}-(i b-a) \left(\frac{2 \sqrt{a+b \tan (e+f x)}}{(c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{i b-a} \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{(c-i d) \sqrt{i d-c}}\right)\right)}{2 f}","\frac{2 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{d^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{3/2}}",1,"((-1/2*I)*(-a - I*b)*(-((-a - I*b)*((-2*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((-c - I*d)*Sqrt[c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((-c - I*d)*Sqrt[c + d*Tan[e + f*x]]))) - (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/f - ((I/2)*(-a + I*b)*(-((-a + I*b)*((-2*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)*Sqrt[-c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((c - I*d)*Sqrt[c + d*Tan[e + f*x]]))) + (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/f","B",1
1291,1,231,213,2.026443,"\int \frac{(a+b \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(3/2),x]","\frac{\frac{\frac{2 (b c-a d) \sqrt{a+b \tan (e+f x)}}{(c+i d) \sqrt{c+d \tan (e+f x)}}+\frac{(a+i b)^{3/2} (d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(c+i d)^{3/2}}}{c-i d}-\frac{i (-a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c+i d)^{3/2}}}{f}","\frac{2 (b c-a d) \sqrt{a+b \tan (e+f x)}}{f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}+\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{3/2}}",1,"(((-I)*(-a + I*b)^(3/2)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-c + I*d)^(3/2) + (((a + I*b)^(3/2)*(I*c + d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(c + I*d)^(3/2) + (2*(b*c - a*d)*Sqrt[a + b*Tan[e + f*x]])/((c + I*d)*Sqrt[c + d*Tan[e + f*x]]))/(c - I*d))/f","A",1
1292,1,224,206,1.6156813,"\int \frac{\sqrt{a+b \tan (e+f x)}}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(3/2),x]","\frac{\frac{i \sqrt{-a+i b} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c+i d)^{3/2}}+\frac{\frac{\sqrt{a+i b} (d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(c+i d)^{3/2}}-\frac{2 d \sqrt{a+b \tan (e+f x)}}{(c+i d) \sqrt{c+d \tan (e+f x)}}}{c-i d}}{f}","-\frac{2 d \sqrt{a+b \tan (e+f x)}}{f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{i \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{3/2}}+\frac{i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{3/2}}",1,"((I*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-c + I*d)^(3/2) + ((Sqrt[a + I*b]*(I*c + d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(c + I*d)^(3/2) - (2*d*Sqrt[a + b*Tan[e + f*x]])/((c + I*d)*Sqrt[c + d*Tan[e + f*x]]))/(c - I*d))/f","A",1
1293,1,243,218,1.1087621,"\int \frac{1}{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{\frac{2 d^2 \sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}+(b c-a d) \left(\frac{i (c+i d) \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}+\frac{(d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}\right)}{f \left(c^2+d^2\right) (a d-b c)}","\frac{2 d^2 \sqrt{a+b \tan (e+f x)}}{f \left(c^2+d^2\right) (b c-a d) \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a-i b} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a+i b} (c+i d)^{3/2}}",1,"-(((b*c - a*d)*((I*(c + I*d)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((I*c + d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])) + (2*d^2*Sqrt[a + b*Tan[e + f*x]])/Sqrt[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f))","A",1
1294,1,349,301,4.8562376,"\int \frac{1}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{\frac{2 d \left(a^2 d^2+b^2 \left(c^2+2 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{\left(c^2+d^2\right) (b c-a d) \sqrt{c+d \tan (e+f x)}}+\frac{2 b^2}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{(b c-a d)^2 \left(\frac{i (a+i b) (c+i d) \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}+\frac{(b+i a) (c-i d) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}\right)}{\left(c^2+d^2\right) (a d-b c)}}{f \left(a^2+b^2\right) (b c-a d)}","-\frac{2 d \left(a^2 d^2+b^2 \left(c^2+2 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{f \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2}{f \left(a^2+b^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{3/2} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{3/2} (c+i d)^{3/2}}",1,"-((((b*c - a*d)^2*((I*(a + I*b)*(c + I*d)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((I*a + b)*(c - I*d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])))/((-(b*c) + a*d)*(c^2 + d^2)) + (2*b^2)/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + (2*d*(a^2*d^2 + b^2*(c^2 + 2*d^2))*Sqrt[a + b*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]]))/((a^2 + b^2)*(b*c - a*d)*f))","A",1
1295,1,601,417,6.4899266,"\int \frac{1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{2 b^2}{3 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 \left(-\frac{2 \left(\frac{1}{2} b^2 \left(4 b^2 d-3 a (b c-a d)\right)-a \left(\frac{3}{2} b^2 (b c-a d)-2 a b^2 d\right)\right)}{f \left(a^2+b^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{2 \left(-\frac{2 \left(\frac{1}{4} d^2 \left(3 a^4 d^2-6 a^3 b c d+a^2 b^2 \left(3 c^2+17 d^2\right)-6 a b^3 c d-b^4 \left(3 c^2-8 d^2\right)\right)-c \left(b^2 c d \left(-5 a^2 d+3 a b c-2 b^2 d\right)-\frac{3}{2} a b d (b c-a d)^2\right)\right) \sqrt{a+b \tan (e+f x)}}{f \left(c^2+d^2\right) (a d-b c) \sqrt{c+d \tan (e+f x)}}-\frac{3 (b c-a d)^3 \left(\frac{(a-i b)^2 (d+i c) \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(a+i b)^2 (-d+i c) \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}\right)}{4 f \left(c^2+d^2\right) (a d-b c)}\right)}{\left(a^2+b^2\right) (b c-a d)}\right)}{3 \left(a^2+b^2\right) (b c-a d)}","-\frac{4 b^2 \left(-5 a^2 d+3 a b c-2 b^2 d\right)}{3 f \left(a^2+b^2\right)^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2}{3 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}+\frac{2 d \left(3 a^4 d^3+a^2 b^2 d \left(11 c^2+17 d^2\right)-6 a b^3 c \left(c^2+d^2\right)+b^4 d \left(5 c^2+8 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(a^2+b^2\right)^2 \left(c^2+d^2\right) (b c-a d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{5/2} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{5/2} (c+i d)^{3/2}}",1,"(-2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]) - (2*((-2*((b^2*(4*b^2*d - 3*a*(b*c - a*d)))/2 - a*(-2*a*b^2*d + (3*b^2*(b*c - a*d))/2)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - (2*((-3*(b*c - a*d)^3*(((a + I*b)^2*(I*c - d)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((a - I*b)^2*(I*c + d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])))/(4*(-(b*c) + a*d)*(c^2 + d^2)*f) - (2*(-(c*((-3*a*b*d*(b*c - a*d)^2)/2 + b^2*c*d*(3*a*b*c - 5*a^2*d - 2*b^2*d))) + (d^2*(-6*a^3*b*c*d - 6*a*b^3*c*d + 3*a^4*d^2 - b^4*(3*c^2 - 8*d^2) + a^2*b^2*(3*c^2 + 17*d^2)))/4)*Sqrt[a + b*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])))/((a^2 + b^2)*(b*c - a*d))))/(3*(a^2 + b^2)*(b*c - a*d))","A",1
1296,1,2261,470,6.755469,"\int \frac{(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(9/2)/(c + d*Tan[e + f*x])^(5/2),x]","\text{Result too large to show}","\frac{b \left(4 a^3 c d^3-4 a^2 b d^2 \left(c^2-2 d^2\right)-4 a b^2 c d \left(c^2+4 d^2\right)+b^3 \left(5 c^4+10 c^2 d^2+d^4\right)\right) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{d^3 f \left(c^2+d^2\right)^2}-\frac{b^{7/2} (5 b c-9 a d) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{d^{7/2} f}-\frac{2 (b c-a d)^2 \left(6 a c d+5 b c^2+11 b d^2\right) (a+b \tan (e+f x))^{3/2}}{3 d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i (a-i b)^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}+\frac{i (a+i b)^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{5/2}}",1,"((-1/2*I)*(-a - I*b)*((-2*b^2*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(5/2)*Hypergeometric2F1[5/2, 7/2, 9/2, -((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f*x])^(7/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(7*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]]) - (-a - I*b)*((2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(5/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^3*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2*((b^2*d^2*(a + b*Tan[e + f*x])^2)/(3*(b*c - a*d)^2*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2) - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^3*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) - (-a - I*b)*((-2*(a + b*Tan[e + f*x])^(3/2))/(3*(c + I*d)*(c + d*Tan[e + f*x])^(3/2)) + ((a + I*b)*((-2*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((-c - I*d)*Sqrt[c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((-c - I*d)*Sqrt[c + d*Tan[e + f*x]])))/(c + I*d)))))/f - ((I/2)*(-a + I*b)*((2*b^2*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(5/2)*Hypergeometric2F1[5/2, 7/2, 9/2, -((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f*x])^(7/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(7*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]]) - (-a + I*b)*((-2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(5/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^3*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2*((b^2*d^2*(a + b*Tan[e + f*x])^2)/(3*(b*c - a*d)^2*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2) - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^3*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) - (-a + I*b)*((-2*(a + b*Tan[e + f*x])^(3/2))/(3*(-c + I*d)*(c + d*Tan[e + f*x])^(3/2)) + ((-a + I*b)*((-2*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)*Sqrt[-c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((c - I*d)*Sqrt[c + d*Tan[e + f*x]])))/(-c + I*d)))))/f","C",0
1297,1,1883,347,6.4452199,"\int \frac{(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{i (-a-i b) \left(\frac{2 (b c-a d) \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{5/2} \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^3 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)^2 \left(\frac{b^2 d^2 (a+b \tan (e+f x))^2}{3 (b c-a d)^2 \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^2 \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)^2}-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right) \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}-\frac{\sqrt{b} \sqrt{d} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right) \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}\right)}{b d^3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}-(-a-i b) \left(\frac{(a+i b) \left(\frac{2 \sqrt{a+b \tan (e+f x)}}{(-c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c-i d) \sqrt{c+i d}}\right)}{c+i d}-\frac{2 (a+b \tan (e+f x))^{3/2}}{3 (c+i d) (c+d \tan (e+f x))^{3/2}}\right)\right)}{2 f}-\frac{i (i b-a) \left(-\frac{2 (b c-a d) \left(\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right)^{5/2} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)^2 \left(\frac{b^2 d^2 (a+b \tan (e+f x))^2}{3 (b c-a d)^2 \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^2 \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)^2}-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right) \left(-\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}-1\right)}-\frac{\sqrt{b} \sqrt{d} \sinh ^{-1}\left(\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}}\right) \sqrt{a+b \tan (e+f x)}}{\sqrt{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1}}\right) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)^3}{b d^3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(\frac{b d (a+b \tan (e+f x))}{(b c-a d) \left(\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right)}+1\right)^{3/2}}-(i b-a) \left(\frac{(i b-a) \left(\frac{2 \sqrt{a+b \tan (e+f x)}}{(c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{i b-a} \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{(c-i d) \sqrt{i d-c}}\right)}{i d-c}-\frac{2 (a+b \tan (e+f x))^{3/2}}{3 (i d-c) (c+d \tan (e+f x))^{3/2}}\right)\right)}{2 f}","\frac{2 b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{d^{5/2} f}-\frac{2 (b c-a d)^2 \left(2 a c d+b \left(c^2+3 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i (a-i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}+\frac{i (a+i b)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{5/2}}",1,"((-1/2*I)*(-a - I*b)*((2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(5/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^3*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2*((b^2*d^2*(a + b*Tan[e + f*x])^2)/(3*(b*c - a*d)^2*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2) - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^3*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) - (-a - I*b)*((-2*(a + b*Tan[e + f*x])^(3/2))/(3*(c + I*d)*(c + d*Tan[e + f*x])^(3/2)) + ((a + I*b)*((-2*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((-c - I*d)*Sqrt[c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((-c - I*d)*Sqrt[c + d*Tan[e + f*x]])))/(c + I*d))))/f - ((I/2)*(-a + I*b)*((-2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(5/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^3*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2*((b^2*d^2*(a + b*Tan[e + f*x])^2)/(3*(b*c - a*d)^2*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2) - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^3*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/2)) - (-a + I*b)*((-2*(a + b*Tan[e + f*x])^(3/2))/(3*(-c + I*d)*(c + d*Tan[e + f*x])^(3/2)) + ((-a + I*b)*((-2*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)*Sqrt[-c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((c - I*d)*Sqrt[c + d*Tan[e + f*x]])))/(-c + I*d))))/f","B",1
1298,1,350,292,5.7932573,"\int \frac{(a+b \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(5/2),x]","\frac{\frac{(b+i a) \left(\frac{(a+b \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{3/2}}+3 (a-i b) \left(\frac{\sqrt{a+b \tan (e+f x)}}{(c-i d) \sqrt{c+d \tan (e+f x)}}+\frac{\sqrt{-a+i b} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c+i d)^{3/2}}\right)\right)}{c-i d}-(-b+i a) \left(\frac{\sqrt{a+b \tan (e+f x)} ((3 a d+b (c+4 i d)) \tan (e+f x)+4 a c+i a d+3 i b c)}{(c+i d)^2 (c+d \tan (e+f x))^{3/2}}-\frac{3 (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(c+i d)^{5/2}}\right)}{3 f}","\frac{2 (b c-a d) \left(6 a c d+b \left(c^2+7 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{3 d f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{5/2}}",1,"(-((I*a - b)*((-3*(a + I*b)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(c + I*d)^(5/2) + (Sqrt[a + b*Tan[e + f*x]]*(4*a*c + (3*I)*b*c + I*a*d + (b*(c + (4*I)*d) + 3*a*d)*Tan[e + f*x]))/((c + I*d)^2*(c + d*Tan[e + f*x])^(3/2)))) + ((I*a + b)*((a + b*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(3/2) + 3*(a - I*b)*((Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-c + I*d)^(3/2) + Sqrt[a + b*Tan[e + f*x]]/((c - I*d)*Sqrt[c + d*Tan[e + f*x]]))))/(c - I*d))/(3*f)","A",1
1299,1,264,276,3.0669447,"\int \frac{(a+b \tan (e+f x))^{3/2}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(5/2),x]","\frac{-\frac{2 \sqrt{a+b \tan (e+f x)} \left(2 d \left(3 a c d-b c^2+2 b d^2\right) \tan (e+f x)+7 a c^2 d+a d^3-3 b c^3+3 b c d^2\right)}{\left(c^2+d^2\right)^2 (c+d \tan (e+f x))^{3/2}}+\frac{3 i (-a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c+i d)^{5/2}}+\frac{3 i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(c+i d)^{5/2}}}{3 f}","\frac{4 \left(-3 a c d+b c^2-2 b d^2\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}+\frac{2 (b c-a d) \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i (a-i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}+\frac{i (a+i b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{5/2}}",1,"(((3*I)*(-a + I*b)^(3/2)*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-c + I*d)^(5/2) + ((3*I)*(a + I*b)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(c + I*d)^(5/2) - (2*Sqrt[a + b*Tan[e + f*x]]*(-3*b*c^3 + 7*a*c^2*d + 3*b*c*d^2 + a*d^3 + 2*d*(-(b*c^2) + 3*a*c*d + 2*b*d^2)*Tan[e + f*x]))/((c^2 + d^2)^2*(c + d*Tan[e + f*x])^(3/2)))/(3*f)","A",1
1300,1,266,283,4.8573589,"\int \frac{\sqrt{a+b \tan (e+f x)}}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(5/2),x]","\frac{\frac{2 d \sqrt{a+b \tan (e+f x)} \left(d \left(6 a c d+b \left(d^2-5 c^2\right)\right) \tan (e+f x)+a d \left(7 c^2+d^2\right)-6 b c^3\right)}{\left(c^2+d^2\right)^2 (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac{3 i \sqrt{-a+i b} \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(-c+i d)^{5/2}}+\frac{3 i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{(c+i d)^{5/2}}}{3 f}","\frac{2 d \left(6 a c d-b \left(5 c^2-d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right)^2 (b c-a d) \sqrt{c+d \tan (e+f x)}}-\frac{2 d \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{i \sqrt{a-i b} \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c-i d)^{5/2}}+\frac{i \sqrt{a+i b} \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (c+i d)^{5/2}}",1,"(((-3*I)*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-c + I*d)^(5/2) + ((3*I)*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(c + I*d)^(5/2) + (2*d*Sqrt[a + b*Tan[e + f*x]]*(-6*b*c^3 + a*d*(7*c^2 + d^2) + d*(6*a*c*d + b*(-5*c^2 + d^2))*Tan[e + f*x]))/((b*c - a*d)*(c^2 + d^2)^2*(c + d*Tan[e + f*x])^(3/2)))/(3*f)","A",1
1301,1,316,295,2.0862642,"\int \frac{1}{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\frac{4 d^2 \left(b \left(4 c^2+d^2\right)-3 a c d\right) \sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}+\frac{2 d^2 \left(c^2+d^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)}}{(c+d \tan (e+f x))^{3/2}}+3 i (b c-a d)^2 \left(\frac{(c-i d)^2 \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(c+i d)^2 \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}\right)}{3 f \left(c^2+d^2\right)^2 (b c-a d)^2}","-\frac{4 d^2 \left(3 a c d-b \left(4 c^2+d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right)^2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}+\frac{2 d^2 \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))^{3/2}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a-i b} (c-i d)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f \sqrt{a+i b} (c+i d)^{5/2}}",1,"((3*I)*(b*c - a*d)^2*(((c + I*d)^2*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((c - I*d)^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])) + (2*d^2*(b*c - a*d)*(c^2 + d^2)*Sqrt[a + b*Tan[e + f*x]])/(c + d*Tan[e + f*x])^(3/2) + (4*d^2*(-3*a*c*d + b*(4*c^2 + d^2))*Sqrt[a + b*Tan[e + f*x]])/Sqrt[c + d*Tan[e + f*x]])/(3*(b*c - a*d)^2*(c^2 + d^2)^2*f)","A",1
1302,1,610,433,6.5228448,"\int \frac{1}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)),x]","-\frac{2 b^2}{f \left(a^2+b^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{2 \left(\frac{1}{2} d^2 \left(a^2 d-a b c+4 b^2 d\right)-c \left(\frac{1}{2} b d (b c-a d)-2 b^2 c d\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(c^2+d^2\right) (a d-b c) (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{2 \left(\frac{1}{4} d^2 \left(-3 a^3 c d^2+a^2 b d \left(6 c^2+5 d^2\right)-3 a b^2 c \left(c^2+2 d^2\right)+b^3 d \left(9 c^2+8 d^2\right)\right)-c \left(\frac{3}{4} d (b c-a d)^2 (a d+b c)-\frac{1}{2} b c \left(a^2 d^3+b^2 \left(3 c^2 d+4 d^3\right)\right)\right)\right) \sqrt{a+b \tan (e+f x)}}{f \left(c^2+d^2\right) (a d-b c) \sqrt{c+d \tan (e+f x)}}+\frac{3 (b c-a d)^3 \left(\frac{(b+i a) (c-i d)^2 \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(-b+i a) (c+i d)^2 \tanh ^{-1}\left(\frac{\sqrt{-c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{-a+i b} \sqrt{-c+i d}}\right)}{4 f \left(c^2+d^2\right) (a d-b c)}\right)}{3 \left(c^2+d^2\right) (a d-b c)}\right)}{\left(a^2+b^2\right) (b c-a d)}","-\frac{2 d \left(a^2 d^2+b^2 \left(3 c^2+4 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(a^2+b^2\right) \left(c^2+d^2\right) (b c-a d)^2 (c+d \tan (e+f x))^{3/2}}-\frac{2 b^2}{f \left(a^2+b^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(6 a^3 c d^4-a^2 b d^3 \left(11 c^2+5 d^2\right)+6 a b^2 c d^4-b^3 \left(3 c^4 d+17 c^2 d^3+8 d^5\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(a^2+b^2\right) \left(c^2+d^2\right)^2 (b c-a d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{3/2} (c-i d)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{3/2} (c+i d)^{5/2}}",1,"(-2*b^2)/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) - (2*((-2*((d^2*(-(a*b*c) + a^2*d + 4*b^2*d))/2 - c*(-2*b^2*c*d + (b*d*(b*c - a*d))/2))*Sqrt[a + b*Tan[e + f*x]])/(3*(-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*((3*(b*c - a*d)^3*(((I*a - b)*(c + I*d)^2*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((I*a + b)*(c - I*d)^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])))/(4*(-(b*c) + a*d)*(c^2 + d^2)*f) - (2*((d^2*(-3*a^3*c*d^2 - 3*a*b^2*c*(c^2 + 2*d^2) + a^2*b*d*(6*c^2 + 5*d^2) + b^3*d*(9*c^2 + 8*d^2)))/4 - c*((3*d*(b*c - a*d)^2*(b*c + a*d))/4 - (b*c*(a^2*d^3 + b^2*(3*c^2*d + 4*d^3)))/2))*Sqrt[a + b*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])))/(3*(-(b*c) + a*d)*(c^2 + d^2))))/((a^2 + b^2)*(b*c - a*d))","A",1
1303,1,1050,596,6.6717885,"\int \frac{1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)),x]","-\frac{2 b^2}{3 \left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{2 \left(-\frac{3}{2} \left(-d a^2+b c a-2 b^2 d\right) b^2-a \left(\frac{3}{2} b^2 (b c-a d)-3 a b^2 d\right)\right)}{\left(a^2+b^2\right) (b c-a d) f \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{2 \sqrt{a+b \tan (e+f x)} \left(-\frac{3}{4} \left(-d^2 a^4+2 b c d a^3-b^2 \left(c^2+15 d^2\right) a^2+6 b^3 c d a+b^4 \left(c^2-8 d^2\right)\right) d^2-c \left(6 b^2 c d \left(-2 d a^2+b c a-b^2 d\right)-\frac{3}{2} a b d (b c-a d)^2\right)\right)}{3 (a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{9 i \left(\frac{(a-i b)^2 \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right) (c-i d)^2}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(a+i b)^2 (c+i d)^2 \tanh ^{-1}\left(\frac{\sqrt{i d-c} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right)}{\sqrt{i b-a} \sqrt{i d-c}}\right) (b c-a d)^4}{8 (a d-b c) \left(c^2+d^2\right) f}-\frac{2 \left(d^2 \left(\left(\frac{b c}{2}-\frac{3 a d}{2}\right) \left(6 b^2 c d \left(-2 d a^2+b c a-b^2 d\right)-\frac{3}{2} a b d (b c-a d)^2\right)-\frac{3}{4} \left(b d^2-\frac{3}{2} c (a d-b c)\right) \left(-d^2 a^4+2 b c d a^3-b^2 \left(c^2+15 d^2\right) a^2+6 b^3 c d a+b^4 \left(c^2-8 d^2\right)\right)\right)-c \left(\frac{3}{2} d (a d-b c) \left(6 b^2 \left(-2 d a^2+b c a-b^2 d\right) d^2-\frac{3}{4} \left(-d^2 a^4+2 b c d a^3-b^2 \left(c^2+15 d^2\right) a^2+6 b^3 c d a+b^4 \left(c^2-8 d^2\right)\right) d+\frac{3}{2} a b c (b c-a d)^2\right)-b c \left(-\frac{3}{4} \left(-d^2 a^4+2 b c d a^3-b^2 \left(c^2+15 d^2\right) a^2+6 b^3 c d a+b^4 \left(c^2-8 d^2\right)\right) d^2-c \left(6 b^2 c d \left(-2 d a^2+b c a-b^2 d\right)-\frac{3}{2} a b d (b c-a d)^2\right)\right)\right)\right) \sqrt{a+b \tan (e+f x)}}{(a d-b c) \left(c^2+d^2\right) f \sqrt{c+d \tan (e+f x)}}\right)}{3 (a d-b c) \left(c^2+d^2\right)}\right)}{\left(a^2+b^2\right) (b c-a d)}\right)}{3 \left(a^2+b^2\right) (b c-a d)}","-\frac{4 b^2 \left(-2 a^2 d+a b c-b^2 d\right)}{f \left(a^2+b^2\right)^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}-\frac{2 b^2}{3 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}+\frac{2 d \left(a^4 d^3+a^2 b^2 d \left(13 c^2+15 d^2\right)-6 a b^3 c \left(c^2+d^2\right)+b^4 d \left(7 c^2+8 d^2\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(a^2+b^2\right)^2 \left(c^2+d^2\right) (b c-a d)^3 (c+d \tan (e+f x))^{3/2}}-\frac{4 d \left(3 a^5 c d^4-a^4 b d^3 \left(7 c^2+4 d^2\right)+6 a^3 b^2 c d^4-a^2 b^3 d \left(7 c^4+28 c^2 d^2+15 d^4\right)+3 a b^4 c \left(c^4+2 c^2 d^2+2 d^4\right)-b^5 d \left(4 c^4+15 c^2 d^2+8 d^4\right)\right) \sqrt{a+b \tan (e+f x)}}{3 f \left(a^2+b^2\right)^2 \left(c^2+d^2\right)^2 (b c-a d)^4 \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left(\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a-i b)^{5/2} (c-i d)^{5/2}}+\frac{i \tanh ^{-1}\left(\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right)}{f (a+i b)^{5/2} (c+i d)^{5/2}}",1,"(-2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)) - (2*((-2*((-3*b^2*(a*b*c - a^2*d - 2*b^2*d))/2 - a*(-3*a*b^2*d + (3*b^2*(b*c - a*d))/2)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) - (2*((-2*(-(c*((-3*a*b*d*(b*c - a*d)^2)/2 + 6*b^2*c*d*(a*b*c - 2*a^2*d - b^2*d))) - (3*d^2*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/4)*Sqrt[a + b*Tan[e + f*x]])/(3*(-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*((((-9*I)/8)*(b*c - a*d)^4*(((a + I*b)^2*(c + I*d)^2*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + I*b]*Sqrt[-c + I*d]) + ((a - I*b)^2*(c - I*d)^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d])))/((-(b*c) + a*d)*(c^2 + d^2)*f) - (2*(d^2*(((b*c)/2 - (3*a*d)/2)*((-3*a*b*d*(b*c - a*d)^2)/2 + 6*b^2*c*d*(a*b*c - 2*a^2*d - b^2*d)) - (3*(b*d^2 - (3*c*(-(b*c) + a*d))/2)*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/4) - c*((3*d*(-(b*c) + a*d)*((3*a*b*c*(b*c - a*d)^2)/2 + 6*b^2*d^2*(a*b*c - 2*a^2*d - b^2*d) - (3*d*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/4))/2 - b*c*(-(c*((-3*a*b*d*(b*c - a*d)^2)/2 + 6*b^2*c*d*(a*b*c - 2*a^2*d - b^2*d))) - (3*d^2*(2*a^3*b*c*d + 6*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 8*d^2) - a^2*b^2*(c^2 + 15*d^2)))/4)))*Sqrt[a + b*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])))/(3*(-(b*c) + a*d)*(c^2 + d^2))))/((a^2 + b^2)*(b*c - a*d))))/(3*(a^2 + b^2)*(b*c - a*d))","A",1
1304,0,0,257,2.0961042,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n,x]","\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx","\frac{(a+b \tan (e+f x))^{m+1} (c+d \tan (e+f x))^n \left(\frac{b (c+d \tan (e+f x))}{b c-a d}\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a)}-\frac{(a+b \tan (e+f x))^{m+1} (c+d \tan (e+f x))^n \left(\frac{b (c+d \tan (e+f x))}{b c-a d}\right)^{-n} F_1\left(m+1;-n,1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a)}",1,"Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x]","F",-1
1305,1,189,214,2.0694208,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3,x]","\frac{(a+b \tan (e+f x))^{m+1} \left(\frac{2 d^2 (b c (2 m+5)-a d)}{b (m+1)}-\frac{i b (m+2) (c-i d)^3 \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{(m+1) (a-i b)}+\frac{i b (m+2) (c+i d)^3 \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{(m+1) (a+i b)}+2 d^2 (c+d \tan (e+f x))\right)}{2 b f (m+2)}","\frac{d^2 (3 b c-a d) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+1)}+\frac{d^3 (a+b \tan (e+f x))^{m+2}}{b^2 f (m+2)}+\frac{(d+i c)^3 (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (a-i b)}-\frac{(-d+i c)^3 (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (a+i b)}",1,"((a + b*Tan[e + f*x])^(1 + m)*((2*d^2*(-(a*d) + b*c*(5 + 2*m)))/(b*(1 + m)) - (I*b*(c - I*d)^3*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/((a - I*b)*(1 + m)) + (I*b*(c + I*d)^3*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/((a + I*b)*(1 + m)) + 2*d^2*(c + d*Tan[e + f*x])))/(2*b*f*(2 + m))","A",1
1306,1,135,176,0.2561223,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2,x]","\frac{(a+b \tan (e+f x))^{m+1} \left(-\frac{i (c-i d)^2 \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{a-i b}+\frac{i (c+i d)^2 \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{a+i b}+\frac{2 d^2}{b}\right)}{2 f (m+1)}","\frac{(c-i d)^2 (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a)}-\frac{(c+i d)^2 (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a)}+\frac{d^2 (a+b \tan (e+f x))^{m+1}}{b f (m+1)}",1,"(((2*d^2)/b - (I*(c - I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/(a - I*b) + (I*(c + I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/(a + I*b))*(a + b*Tan[e + f*x])^(1 + m))/(2*f*(1 + m))","A",1
1307,1,120,143,0.2208103,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x]),x]","\frac{i (a+b \tan (e+f x))^{m+1} \left(\frac{(c+i d) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{a+i b}-\frac{(c-i d) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{a-i b}\right)}{2 f (m+1)}","\frac{(c-i d) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a)}+\frac{(-d+i c) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (a+i b)}",1,"((I/2)*(-(((c - I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/(a - I*b)) + ((c + I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/(a + I*b))*(a + b*Tan[e + f*x])^(1 + m))/(f*(1 + m))","A",1
1308,1,118,167,0.164284,"\int (a+b \tan (e+f x))^m \, dx","Integrate[(a + b*Tan[e + f*x])^m,x]","\frac{(a+b \tan (e+f x))^{m+1} \left((a+i b) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)-(a-i b) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)\right)}{2 f (m+1) (a+i b) (b+i a)}","\frac{b (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} f (m+1) \left(a-\sqrt{-b^2}\right)}-\frac{b (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+\sqrt{-b^2}}\right)}{2 \sqrt{-b^2} f (m+1) \left(a+\sqrt{-b^2}\right)}",1,"(((a + I*b)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)] - (a - I*b)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*(I*a + b)*f*(1 + m))","C",1
1309,1,178,223,0.5529941,"\int \frac{(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x]),x]","\frac{(a+b \tan (e+f x))^{m+1} \left(-\frac{2 d^2 \, _2F_1\left(1,m+1;m+2;\frac{d (a+b \tan (e+f x))}{a d-b c}\right)}{\left(c^2+d^2\right) (a d-b c)}+\frac{\, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{(b+i a) (c-i d)}+\frac{i \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{(a+i b) (c+i d)}\right)}{2 f (m+1)}","\frac{d^2 (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right)}{f (m+1) \left(c^2+d^2\right) (b c-a d)}+\frac{(a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) (c-i d)}-\frac{(a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (a+i b) (-d+i c)}",1,"((Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]/((I*a + b)*(c - I*d)) + (I*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/((a + I*b)*(c + I*d)) - (2*d^2*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)])/((-(b*c) + a*d)*(c^2 + d^2)))*(a + b*Tan[e + f*x])^(1 + m))/(2*f*(1 + m))","A",1
1310,1,266,301,4.362065,"\int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^2} \, dx","Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^2,x]","\frac{(a+b \tan (e+f x))^{m+1} \left(-\frac{2 d^2 \left(2 a c d+b c^2 (m-2)+b d^2 m\right) \, _2F_1\left(1,m+1;m+2;\frac{d (a+b \tan (e+f x))}{a d-b c}\right)}{(m+1) \left(c^2+d^2\right) (a d-b c)}-\frac{i \left(\frac{(c-i d)^2 (b c-a d) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{a+i b}+\frac{(c+i d)^2 (a d-b c) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{a-i b}\right)}{(m+1) \left(c^2+d^2\right)}-\frac{2 d^2}{c+d \tan (e+f x)}\right)}{2 f \left(c^2+d^2\right) (a d-b c)}","-\frac{d^2 \left(2 a c d-b \left(c^2 (2-m)-d^2 m\right)\right) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right)}{f (m+1) \left(c^2+d^2\right)^2 (b c-a d)^2}+\frac{d^2 (a+b \tan (e+f x))^{m+1}}{f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))}+\frac{(a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) (c-i d)^2}-\frac{(a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a) (c+i d)^2}",1,"((a + b*Tan[e + f*x])^(1 + m)*(((-I)*(((c + I*d)^2*(-(b*c) + a*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/(a - I*b) + ((c - I*d)^2*(b*c - a*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/(a + I*b)))/((c^2 + d^2)*(1 + m)) - (2*d^2*(2*a*c*d + b*c^2*(-2 + m) + b*d^2*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)])/((-(b*c) + a*d)*(c^2 + d^2)*(1 + m)) - (2*d^2)/(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2)*f)","A",1
1311,1,670,455,6.2846238,"\int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx","Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^3,x]","-\frac{d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left(c^2+d^2\right) (a d-b c) (c+d \tan (e+f x))^2}-\frac{-\frac{\left(d^2 \left(2 c (b c-a d)+b d^2 (1-m)\right)-c \left(-2 d^2 (b c-a d)-b c d^2 (1-m)\right)\right) (a+b \tan (e+f x))^{m+1}}{f \left(c^2+d^2\right) (a d-b c) (c+d \tan (e+f x))}-\frac{-\frac{\left(b c^2 d^2 m \left(4 a c d-b c^2 (5-m)-b d^2 (1-m)\right)+d^2 \left(-\left(-2 a c d+2 b c^2+b d^2 (1-m)\right) \left(a c d-b \left(c^2-d^2 m\right)\right)-\left(d^2 (2 a d-b c (3-m)) (a d-b c (m+1))\right)\right)+4 c^2 d^2 (b c-a d)^2\right) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{d (a+b \tan (e+f x))}{a d-b c}\right)}{f (m+1) \left(c^2+d^2\right) (a d-b c)}+\frac{\frac{i \left(2 c \left(c^2-3 d^2\right) (b c-a d)^2-2 i d \left(3 c^2-d^2\right) (b c-a d)^2\right) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{-i a-i b \tan (e+f x)}{b-i a}\right)}{2 f (m+1) (a+i b)}-\frac{i \left(2 c \left(c^2-3 d^2\right) (b c-a d)^2+2 i d \left(3 c^2-d^2\right) (b c-a d)^2\right) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{i a+i b \tan (e+f x)}{-i a-b}\right)}{2 f (m+1) (a-i b)}}{c^2+d^2}}{\left(c^2+d^2\right) (a d-b c)}}{2 \left(c^2+d^2\right) (a d-b c)}","\frac{d^2 \left(2 a^2 d^2 \left(3 c^2-d^2\right)-4 a b c d \left(c^2 (3-m)-d^2 (m+1)\right)-\left(b^2 \left(-\left(c^4 \left(m^2-5 m+6\right)\right)+2 c^2 d^2 \left(-m^2+3 m+1\right)+d^4 (1-m) m\right)\right)\right) (a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right)}{2 f (m+1) \left(c^2+d^2\right)^3 (b c-a d)^3}-\frac{d^2 \left(4 a c d-b \left(c^2 (5-m)+d^2 (1-m)\right)\right) (a+b \tan (e+f x))^{m+1}}{2 f \left(c^2+d^2\right)^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{d^2 (a+b \tan (e+f x))^{m+1}}{2 f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))^2}+\frac{(a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) (c-i d)^3}+\frac{(a+b \tan (e+f x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (a+i b) (-d+i c)^3}",1,"-1/2*(d^2*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-(((d^2*(2*c*(b*c - a*d) + b*d^2*(1 - m)) - c*(-2*d^2*(b*c - a*d) - b*c*d^2*(1 - m)))*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))) - (-(((4*c^2*d^2*(b*c - a*d)^2 + b*c^2*d^2*(4*a*c*d - b*d^2*(1 - m) - b*c^2*(5 - m))*m + d^2*(-(d^2*(2*a*d - b*c*(3 - m))*(a*d - b*c*(1 + m))) - (2*b*c^2 - 2*a*c*d + b*d^2*(1 - m))*(a*c*d - b*(c^2 - d^2*m))))*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)]*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(1 + m))) + (((I/2)*(2*c*(b*c - a*d)^2*(c^2 - 3*d^2) - (2*I)*d*(b*c - a*d)^2*(3*c^2 - d^2))*Hypergeometric2F1[1, 1 + m, 2 + m, ((-I)*a - I*b*Tan[e + f*x])/((-I)*a + b)]*(a + b*Tan[e + f*x])^(1 + m))/((a + I*b)*f*(1 + m)) - ((I/2)*(2*c*(b*c - a*d)^2*(c^2 - 3*d^2) + (2*I)*d*(b*c - a*d)^2*(3*c^2 - d^2))*Hypergeometric2F1[1, 1 + m, 2 + m, -((I*a + I*b*Tan[e + f*x])/((-I)*a - b))]*(a + b*Tan[e + f*x])^(1 + m))/((a - I*b)*f*(1 + m)))/(c^2 + d^2))/((-(b*c) + a*d)*(c^2 + d^2)))/(2*(-(b*c) + a*d)*(c^2 + d^2))","A",1
1312,0,0,283,7.5864938,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2),x]","\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^{3/2} \, dx","\frac{(b c-a d) \sqrt{c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} F_1\left(m+1;-\frac{3}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 b f (m+1) (b+i a) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-\frac{(b c-a d) \sqrt{c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} F_1\left(m+1;-\frac{3}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 b f (m+1) (-b+i a) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}",1,"Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2), x]","F",-1
1313,0,0,261,0.8002215,"\int (a+b \tan (e+f x))^m \sqrt{c+d \tan (e+f x)} \, dx","Integrate[(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]],x]","\int (a+b \tan (e+f x))^m \sqrt{c+d \tan (e+f x)} \, dx","\frac{\sqrt{c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} F_1\left(m+1;-\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}-\frac{\sqrt{c+d \tan (e+f x)} (a+b \tan (e+f x))^{m+1} F_1\left(m+1;-\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a) \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}}}",1,"Integrate[(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]], x]","F",-1
1314,0,0,261,5.0111873,"\int \frac{(a+b \tan (e+f x))^m}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(a + b*Tan[e + f*x])^m/Sqrt[c + d*Tan[e + f*x]],x]","\int \frac{(a+b \tan (e+f x))^m}{\sqrt{c+d \tan (e+f x)}} \, dx","\frac{(a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left(m+1;\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) \sqrt{c+d \tan (e+f x)}}-\frac{(a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left(m+1;\frac{1}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a) \sqrt{c+d \tan (e+f x)}}",1,"Integrate[(a + b*Tan[e + f*x])^m/Sqrt[c + d*Tan[e + f*x]], x]","F",-1
1315,0,0,283,8.8417546,"\int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2),x]","\int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx","\frac{b (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left(m+1;\frac{3}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) (b c-a d) \sqrt{c+d \tan (e+f x)}}-\frac{b (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left(m+1;\frac{3}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a) (b c-a d) \sqrt{c+d \tan (e+f x)}}",1,"Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2), x]","F",-1
1316,0,0,287,20.7731844,"\int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2),x]","\int \frac{(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{5/2}} \, dx","\frac{b^2 (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left(m+1;\frac{5}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a-i b}\right)}{2 f (m+1) (b+i a) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{b^2 (a+b \tan (e+f x))^{m+1} \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} F_1\left(m+1;\frac{5}{2},1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d},\frac{a+b \tan (e+f x)}{a+i b}\right)}{2 f (m+1) (-b+i a) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}",1,"Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2), x]","F",-1
1317,0,0,99,5.8481804,"\int \left(c (d \tan (e+f x))^p\right)^n (a+i a \tan (e+f x))^m \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^m,x]","\int \left(c (d \tan (e+f x))^p\right)^n (a+i a \tan (e+f x))^m \, dx","\frac{\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m F_1(n p+1;1-m,1;n p+2;-i \tan (e+f x),i \tan (e+f x)) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}",1,"Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^m, x]","F",-1
1318,1,981,132,9.6301744,"\int \left(c (d \tan (e+f x))^p\right)^n (a+i a \tan (e+f x))^3 \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3,x]","\frac{i 2^{2-n p} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{n p} \cos ^3(e+f x) \left(2^{n p} \, _2F_1\left(1,n p;n p+1;-\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)-\left(1+e^{2 i (e+f x)}\right)^{n p} \, _2F_1\left(n p,n p;n p+1;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)\right) \tan ^{-n p}(e+f x) (i \tan (e+f x) a+a)^3 \left(c (d \tan (e+f x))^p\right)^n}{\left(e^{i e}+e^{3 i e}\right) f n p (\cos (f x)+i \sin (f x))^3}-\frac{4 i e^{-3 i e} \left(-1+e^{2 i (e+f x)}\right)^{n p} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{n p} \left(\frac{-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{-n p} \cos ^3(e+f x) \left(-\frac{\, _2F_1\left(1,n p;n p+1;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{-n p}}{n p}-\frac{\left(1+e^{2 i e}\right) \left(-1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,n p+1;n p+2;\frac{1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right) \left(1+e^{2 i (e+f x)}\right)^{-n p-1}}{n p+1}+\frac{2^{-n p} \, _2F_1\left(n p,n p;n p+1;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)}{n p}\right) \tan ^{-n p}(e+f x) (i \tan (e+f x) a+a)^3 \left(c (d \tan (e+f x))^p\right)^n}{\left(1+e^{2 i e}\right) f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\frac{(\cos (2 e)+3 i \sin (2 e)-1) \left(\frac{1}{2} i \cos (3 e)+\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{n p+1}+\frac{\sec (e+f x) \left(\frac{1}{2} i \cos (3 e)+\frac{1}{2} \sin (3 e)\right) (-\cos (e-f x)+\cos (e+f x)-3 i \sin (e-f x)+3 i \sin (e+f x)) \sec ^2(e)}{n p+1}\right) (i \tan (e+f x) a+a)^3 \left(c (d \tan (e+f x))^p\right)^n}{f (\cos (f x)+i \sin (f x))^3}+\frac{\cos ^3(e+f x) \left(\frac{(-2 n p+\cos (2 e)-3) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{(n p+1) (n p+2)}+\frac{(\cos (e+f x)-\cos (e-f x)) \sec (e+f x) \left(-\frac{1}{2} i \cos (3 e)-\frac{1}{2} \sin (3 e)\right) \sec ^2(e)}{n p+1}+\frac{\sec ^2(e+f x) (-i \cos (3 e)-\sin (3 e))}{n p+2}\right) (i \tan (e+f x) a+a)^3 \left(c (d \tan (e+f x))^p\right)^n}{f (\cos (f x)+i \sin (f x))^3}","\frac{4 a^3 \tan (e+f x) \, _2F_1(1,n p+1;n p+2;i \tan (e+f x)) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}-\frac{i a^3 \tan ^2(e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+2)}-\frac{3 a^3 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}",1,"(Cos[e + f*x]^3*((Sec[e + f*x]^2*((-I)*Cos[3*e] - Sin[3*e]))/(2 + n*p) + ((-3 - 2*n*p + Cos[2*e])*Sec[e]^2*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/((1 + n*p)*(2 + n*p)) + ((-Cos[e - f*x] + Cos[e + f*x])*Sec[e]^2*Sec[e + f*x]*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/(1 + n*p))*(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*((Sec[e]^2*(-1 + Cos[2*e] + (3*I)*Sin[2*e])*((I/2)*Cos[3*e] + Sin[3*e]/2))/(1 + n*p) + (Sec[e]^2*Sec[e + f*x]*((I/2)*Cos[3*e] + Sin[3*e]/2)*(-Cos[e - f*x] + Cos[e + f*x] - (3*I)*Sin[e - f*x] + (3*I)*Sin[e + f*x]))/(1 + n*p))*(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (I*2^(2 - n*p)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(n*p)*Cos[e + f*x]^3*(2^(n*p)*Hypergeometric2F1[1, n*p, 1 + n*p, -((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))] - (1 + E^((2*I)*(e + f*x)))^(n*p)*Hypergeometric2F1[n*p, n*p, 1 + n*p, (1 - E^((2*I)*(e + f*x)))/2])*(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3)/((E^(I*e) + E^((3*I)*e))*f*n*p*(Cos[f*x] + I*Sin[f*x])^3*Tan[e + f*x]^(n*p)) - ((4*I)*(-1 + E^((2*I)*(e + f*x)))^(n*p)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(n*p)*Cos[e + f*x]^3*(-(Hypergeometric2F1[1, n*p, 1 + n*p, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))]/((1 + E^((2*I)*(e + f*x)))^(n*p)*n*p)) - ((1 + E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(-1 - n*p)*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/(1 + n*p) + Hypergeometric2F1[n*p, n*p, 1 + n*p, (1 - E^((2*I)*(e + f*x)))/2]/(2^(n*p)*n*p))*(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*(1 + E^((2*I)*e))*((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^(n*p)*f*(Cos[f*x] + I*Sin[f*x])^3*Tan[e + f*x]^(n*p))","B",0
1319,1,185,93,2.2960015,"\int \left(c (d \tan (e+f x))^p\right)^n (a+i a \tan (e+f x))^2 \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^2,x]","\frac{a^2 e^{-2 i e} 2^{-n p} \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{n p+1} (\cos (e+f x)+i \sin (e+f x))^2 \left(-2^{n p}+\left(1+e^{2 i (e+f x)}\right)^{n p+1} \, _2F_1\left(n p+1,n p+1;n p+2;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right)\right) \tan ^{-n p}(e+f x) \left(c (d \tan (e+f x))^p\right)^n}{(f n p+f) (\cos (f x)+i \sin (f x))^2}","-\frac{a^2 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}+\frac{2 a^2 \tan (e+f x) \, _2F_1(1,n p+1;n p+2;i \tan (e+f x)) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}",1,"(a^2*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(1 + n*p)*(-2^(n*p) + (1 + E^((2*I)*(e + f*x)))^(1 + n*p)*Hypergeometric2F1[1 + n*p, 1 + n*p, 2 + n*p, (1 - E^((2*I)*(e + f*x)))/2])*(Cos[e + f*x] + I*Sin[e + f*x])^2*(c*(d*Tan[e + f*x])^p)^n)/(2^(n*p)*E^((2*I)*e)*(f + f*n*p)*(Cos[f*x] + I*Sin[f*x])^2*Tan[e + f*x]^(n*p))","A",1
1320,1,173,54,1.027267,"\int \left(c (d \tan (e+f x))^p\right)^n (a+i a \tan (e+f x)) \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x]),x]","\frac{a e^{-i e} 2^{-n p-1} \cos (e+f x) (1+i \tan (e+f x)) (\cos (f x)-i \sin (f x)) \left(-\frac{i \left(-1+e^{2 i (e+f x)}\right)}{1+e^{2 i (e+f x)}}\right)^{n p+1} \left(1+e^{2 i (e+f x)}\right)^{n p+1} \, _2F_1\left(n p+1,n p+1;n p+2;\frac{1}{2} \left(1-e^{2 i (e+f x)}\right)\right) \tan ^{-n p}(e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f n p+f}","\frac{a \tan (e+f x) \, _2F_1(1,n p+1;n p+2;i \tan (e+f x)) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}",1,"(2^(-1 - n*p)*a*(((-I)*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x))))^(1 + n*p)*(1 + E^((2*I)*(e + f*x)))^(1 + n*p)*Cos[e + f*x]*Hypergeometric2F1[1 + n*p, 1 + n*p, 2 + n*p, (1 - E^((2*I)*(e + f*x)))/2]*(Cos[f*x] - I*Sin[f*x])*(1 + I*Tan[e + f*x])*(c*(d*Tan[e + f*x])^p)^n)/(E^(I*e)*(f + f*n*p)*Tan[e + f*x]^(n*p))","B",1
1321,0,0,134,22.4656293,"\int \frac{\left(c (d \tan (e+f x))^p\right)^n}{a+i a \tan (e+f x)} \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n/(a + I*a*Tan[e + f*x]),x]","\int \frac{\left(c (d \tan (e+f x))^p\right)^n}{a+i a \tan (e+f x)} \, dx","\frac{\tan (e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{a f (n p+1)}-\frac{i \tan ^2(e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{a f (n p+2)}",1,"Integrate[(c*(d*Tan[e + f*x])^p)^n/(a + I*a*Tan[e + f*x]), x]","F",-1
1322,0,0,227,6.3169546,"\int \frac{\left(c (d \tan (e+f x))^p\right)^n}{(a+i a \tan (e+f x))^2} \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n/(a + I*a*Tan[e + f*x])^2,x]","\int \frac{\left(c (d \tan (e+f x))^p\right)^n}{(a+i a \tan (e+f x))^2} \, dx","\frac{\left(2 n^2 p^2-4 n p+1\right) \tan (e+f x) \, _2F_1(1,n p+1;n p+2;-i \tan (e+f x)) \left(c (d \tan (e+f x))^p\right)^n}{8 a^2 f (n p+1)}+\frac{\tan (e+f x) \, _2F_1(1,n p+1;n p+2;i \tan (e+f x)) \left(c (d \tan (e+f x))^p\right)^n}{8 a^2 f (n p+1)}+\frac{(2-n p) \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n}{4 a^2 f (1+i \tan (e+f x))}+\frac{\tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n}{4 a^2 f (1+i \tan (e+f x))^2}",1,"Integrate[(c*(d*Tan[e + f*x])^p)^n/(a + I*a*Tan[e + f*x])^2, x]","F",-1
1323,0,0,201,1.655044,"\int \left(c (d \tan (e+f x))^p\right)^n (a+b \tan (e+f x))^m \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^m,x]","\int \left(c (d \tan (e+f x))^p\right)^n (a+b \tan (e+f x))^m \, dx","\frac{\tan (e+f x) (a+b \tan (e+f x))^m \left(\frac{b \tan (e+f x)}{a}+1\right)^{-m} \left(c (d \tan (e+f x))^p\right)^n F_1\left(n p+1;-m,1;n p+2;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right)}{2 f (n p+1)}+\frac{\tan (e+f x) (a+b \tan (e+f x))^m \left(\frac{b \tan (e+f x)}{a}+1\right)^{-m} \left(c (d \tan (e+f x))^p\right)^n F_1\left(n p+1;-m,1;n p+2;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right)}{2 f (n p+1)}",1,"Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^m, x]","F",-1
1324,1,163,219,0.9414388,"\int \left(c (d \tan (e+f x))^p\right)^n (a+b \tan (e+f x))^3 \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^3,x]","\frac{\tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \left(a \left(a^2-3 b^2\right) (n p+2) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)+b \left(\left(3 a^2-b^2\right) (n p+1) \tan (e+f x) \, _2F_1\left(1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right)+b (3 a (n p+2)+b (n p+1) \tan (e+f x))\right)\right)}{f (n p+1) (n p+2)}","\frac{b \left(3 a^2-b^2\right) \tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+2)}+\frac{a \left(a^2-3 b^2\right) \tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}+\frac{3 a b^2 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}+\frac{b^3 \tan ^2(e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+2)}",1,"(Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a*(a^2 - 3*b^2)*(2 + n*p)*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2] + b*((3*a^2 - b^2)*(1 + n*p)*Hypergeometric2F1[1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x] + b*(3*a*(2 + n*p) + b*(1 + n*p)*Tan[e + f*x]))))/(f*(1 + n*p)*(2 + n*p))","A",1
1325,1,136,171,0.5702029,"\int \left(c (d \tan (e+f x))^p\right)^n (a+b \tan (e+f x))^2 \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^2,x]","\frac{\tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \left(\left(a^2-b^2\right) (n p+2) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)+b \left(2 a (n p+1) \tan (e+f x) \, _2F_1\left(1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right)+b (n p+2)\right)\right)}{f (n p+1) (n p+2)}","\frac{\left(a^2-b^2\right) \tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}+\frac{2 a b \tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+2)}+\frac{b^2 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}",1,"(Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*((a^2 - b^2)*(2 + n*p)*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2] + b*(b*(2 + n*p) + 2*a*(1 + n*p)*Hypergeometric2F1[1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(f*(1 + n*p)*(2 + n*p))","A",1
1326,1,117,127,0.3137764,"\int \left(c (d \tan (e+f x))^p\right)^n (a+b \tan (e+f x)) \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x]),x]","\frac{\tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \left(a (n p+2) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)+b (n p+1) \tan (e+f x) \, _2F_1\left(1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right)\right)}{f (n p+1) (n p+2)}","\frac{a \tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+1)}+\frac{b \tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f (n p+2)}",1,"(Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a*(2 + n*p)*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2] + b*(1 + n*p)*Hypergeometric2F1[1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]))/(f*(1 + n*p)*(2 + n*p))","A",1
1327,1,166,216,0.7415225,"\int \frac{\left(c (d \tan (e+f x))^p\right)^n}{a+b \tan (e+f x)} \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n/(a + b*Tan[e + f*x]),x]","\frac{\tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \left(a^2 (n p+2) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)+b \left(b (n p+2) \, _2F_1\left(1,n p+1;n p+2;-\frac{b \tan (e+f x)}{a}\right)-a (n p+1) \tan (e+f x) \, _2F_1\left(1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right)\right)\right)}{a f \left(a^2+b^2\right) (n p+1) (n p+2)}","-\frac{b \tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f \left(a^2+b^2\right) (n p+2)}+\frac{a \tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f \left(a^2+b^2\right) (n p+1)}+\frac{b^2 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \, _2F_1\left(1,n p+1;n p+2;-\frac{b \tan (e+f x)}{a}\right)}{a f \left(a^2+b^2\right) (n p+1)}",1,"(Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a^2*(2 + n*p)*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2] + b*(b*(2 + n*p)*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, -((b*Tan[e + f*x])/a)] - a*(1 + n*p)*Hypergeometric2F1[1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x])))/(a*(a^2 + b^2)*f*(1 + n*p)*(2 + n*p))","A",1
1328,1,231,293,1.9686345,"\int \frac{\left(c (d \tan (e+f x))^p\right)^n}{(a+b \tan (e+f x))^2} \, dx","Integrate[(c*(d*Tan[e + f*x])^p)^n/(a + b*Tan[e + f*x])^2,x]","\frac{\tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \left(\frac{a \left(\left(a^2-b^2\right) (n p+2) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)-2 a b (n p+1) \tan (e+f x) \, _2F_1\left(1,\frac{n p}{2}+1;\frac{n p}{2}+2;-\tan ^2(e+f x)\right)\right)}{\left(a^2+b^2\right) (n p+1) (n p+2)}-\frac{b^2 \left(a^2 (n p-2)+b^2 n p\right) \, _2F_1\left(1,n p+1;n p+2;-\frac{b \tan (e+f x)}{a}\right)}{a \left(a^2+b^2\right) (n p+1)}+\frac{b^2}{a+b \tan (e+f x)}\right)}{a f \left(a^2+b^2\right)}","-\frac{2 a b \tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f \left(a^2+b^2\right)^2 (n p+2)}+\frac{\left(a^2-b^2\right) \tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(c (d \tan (e+f x))^p\right)^n}{f \left(a^2+b^2\right)^2 (n p+1)}+\frac{2 b^2 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \, _2F_1\left(1,n p+1;n p+2;-\frac{b \tan (e+f x)}{a}\right)}{f \left(a^2+b^2\right)^2 (n p+1)}+\frac{b^2 \tan (e+f x) \left(c (d \tan (e+f x))^p\right)^n \, _2F_1\left(2,n p+1;n p+2;-\frac{b \tan (e+f x)}{a}\right)}{a^2 f \left(a^2+b^2\right) (n p+1)}",1,"(Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(-((b^2*(b^2*n*p + a^2*(-2 + n*p))*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, -((b*Tan[e + f*x])/a)])/(a*(a^2 + b^2)*(1 + n*p))) + b^2/(a + b*Tan[e + f*x]) + (a*((a^2 - b^2)*(2 + n*p)*Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2] - 2*a*b*(1 + n*p)*Hypergeometric2F1[1, 1 + (n*p)/2, 2 + (n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]))/((a^2 + b^2)*(1 + n*p)*(2 + n*p))))/(a*(a^2 + b^2)*f)","A",1