1,1,86,87,0.5981226,"\int \tan (c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{(6 b C-6 a B) \tan ^{-1}(\tan (c+d x))+3 (a C+b B) \tan ^2(c+d x)+6 (a B-b C) \tan (c+d x)+6 (a C+b B) \log (\cos (c+d x))+2 b C \tan ^3(c+d x)}{6 d}","\frac{(a C+b B) \tan ^2(c+d x)}{2 d}+\frac{(a B-b C) \tan (c+d x)}{d}+\frac{(a C+b B) \log (\cos (c+d x))}{d}-x (a B-b C)+\frac{b C \tan ^3(c+d x)}{3 d}",1,"((-6*a*B + 6*b*C)*ArcTan[Tan[c + d*x]] + 6*(b*B + a*C)*Log[Cos[c + d*x]] + 6*(a*B - b*C)*Tan[c + d*x] + 3*(b*B + a*C)*Tan[c + d*x]^2 + 2*b*C*Tan[c + d*x]^3)/(6*d)","A",1
2,1,67,66,0.3201811,"\int (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-2 (a C+b B) \tan ^{-1}(\tan (c+d x))+2 (a C+b B) \tan (c+d x)+2 (b C-a B) \log (\cos (c+d x))+b C \tan ^2(c+d x)}{2 d}","-\frac{(a B-b C) \log (\cos (c+d x))}{d}-x (a C+b B)+\frac{C (a+b \tan (c+d x))^2}{2 b d}+\frac{b B \tan (c+d x)}{d}",1,"(-2*(b*B + a*C)*ArcTan[Tan[c + d*x]] + 2*(-(a*B) + b*C)*Log[Cos[c + d*x]] + 2*(b*B + a*C)*Tan[c + d*x] + b*C*Tan[c + d*x]^2)/(2*d)","A",1
3,1,59,42,0.056121,"\int \cot (c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","a B x-\frac{a C \log (\cos (c+d x))}{d}-\frac{b B \log (\cos (c+d x))}{d}-\frac{b C \tan ^{-1}(\tan (c+d x))}{d}+\frac{b C \tan (c+d x)}{d}","-\frac{(a C+b B) \log (\cos (c+d x))}{d}+x (a B-b C)+\frac{b C \tan (c+d x)}{d}",1,"a*B*x - (b*C*ArcTan[Tan[c + d*x]])/d - (b*B*Log[Cos[c + d*x]])/d - (a*C*Log[Cos[c + d*x]])/d + (b*C*Tan[c + d*x])/d","A",1
4,1,44,37,0.0717879,"\int \cot ^2(c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{a B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+a C x+b B x-\frac{b C \log (\cos (c+d x))}{d}","x (a C+b B)+\frac{a B \log (\sin (c+d x))}{d}-\frac{b C \log (\cos (c+d x))}{d}",1,"b*B*x + a*C*x - (b*C*Log[Cos[c + d*x]])/d + (a*B*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","A",1
5,1,78,43,0.1593445,"\int \cot ^3(c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","-\frac{a B \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}+\frac{a C (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{b B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b C x","\frac{(a C+b B) \log (\sin (c+d x))}{d}-(x (a B-b C))-\frac{a B \cot (c+d x)}{d}",1,"b*C*x - (a*B*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d + (b*B*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d + (a*C*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
6,1,77,66,0.4692765,"\int \cot ^4(c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","-\frac{2 (a C+b B) \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)+2 (a B-b C) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a B \cot ^2(c+d x)}{2 d}","-\frac{(a C+b B) \cot (c+d x)}{d}-\frac{(a B-b C) \log (\sin (c+d x))}{d}-x (a C+b B)-\frac{a B \cot ^2(c+d x)}{2 d}",1,"-1/2*(a*B*Cot[c + d*x]^2 + 2*(b*B + a*C)*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 2*(a*B - b*C)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d","C",1
7,1,101,87,1.029146,"\int \cot ^5(c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","-\frac{3 (a C+b B) \left(\cot ^2(c+d x)+2 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right)+2 a B \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)+6 b C \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{6 d}","-\frac{(a C+b B) \cot ^2(c+d x)}{2 d}+\frac{(a B-b C) \cot (c+d x)}{d}-\frac{(a C+b B) \log (\sin (c+d x))}{d}+x (a B-b C)-\frac{a B \cot ^3(c+d x)}{3 d}",1,"-1/6*(2*a*B*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 6*b*C*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 3*(b*B + a*C)*(Cot[c + d*x]^2 + 2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
8,1,100,108,1.1465394,"\int \cot ^6(c+d x) (a+b \tan (c+d x)) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","-\frac{4 (a C+b B) \cot ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(c+d x)\right)+3 \left((2 b C-2 a B) \cot ^2(c+d x)-4 (a B-b C) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a B \cot ^4(c+d x)\right)}{12 d}","-\frac{(a C+b B) \cot ^3(c+d x)}{3 d}+\frac{(a B-b C) \cot ^2(c+d x)}{2 d}+\frac{(a C+b B) \cot (c+d x)}{d}+\frac{(a B-b C) \log (\sin (c+d x))}{d}+x (a C+b B)-\frac{a B \cot ^4(c+d x)}{4 d}",1,"-1/12*(4*(b*B + a*C)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + 3*((-2*a*B + 2*b*C)*Cot[c + d*x]^2 + a*B*Cot[c + d*x]^4 - 4*(a*B - b*C)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d","C",1
9,1,221,148,6.2526154,"\int \tan (c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Tan[c + d*x]*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{C \tan (c+d x) (a+b \tan (c+d x))^3}{4 b d}+\frac{\frac{(4 b B-a C) (a+b \tan (c+d x))^3}{3 b d}+\frac{2 \left((b B-a C) \left(-i (a-i b)^2 \log (\tan (c+d x)+i)+i (a+i b)^2 \log (-\tan (c+d x)+i)-2 b^2 \tan (c+d x)\right)-C \left(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)\right)\right)}{d}}{4 b}","\frac{\left(a^2 C+2 a b B-b^2 C\right) \log (\cos (c+d x))}{d}-x \left(a^2 B-2 a b C-b^2 B\right)+\frac{(4 b B-a C) (a+b \tan (c+d x))^3}{12 b^2 d}-\frac{b (a C+b B) \tan (c+d x)}{d}+\frac{C \tan (c+d x) (a+b \tan (c+d x))^3}{4 b d}-\frac{C (a+b \tan (c+d x))^2}{2 d}",1,"(C*Tan[c + d*x]*(a + b*Tan[c + d*x])^3)/(4*b*d) + (((4*b*B - a*C)*(a + b*Tan[c + d*x])^3)/(3*b*d) + (2*((b*B - a*C)*(I*(a + I*b)^2*Log[I - Tan[c + d*x]] - I*(a - I*b)^2*Log[I + Tan[c + d*x]] - 2*b^2*Tan[c + d*x]) - C*((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)))/d)/(4*b)","C",1
10,1,172,112,1.8471616,"\int (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{3 (a B+b C) \left(-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)\right)+3 B \left(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)\right)+2 C (a+b \tan (c+d x))^3}{6 b d}","-\frac{\left(a^2 B-2 a b C-b^2 B\right) \log (\cos (c+d x))}{d}-x \left(a^2 C+2 a b B-b^2 C\right)+\frac{b (a B-b C) \tan (c+d x)}{d}+\frac{B (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 b d}",1,"(2*C*(a + b*Tan[c + d*x])^3 + 3*(a*B + b*C)*(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]) + 3*B*((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))/(6*b*d)","C",1
11,1,96,87,0.4701303,"\int \cot (c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{2 b (2 a C+b B) \tan (c+d x)+(a-i b)^2 (C+i B) \log (\tan (c+d x)+i)+(a+i b)^2 (C-i B) \log (-\tan (c+d x)+i)+b^2 C \tan ^2(c+d x)}{2 d}","-\frac{\left(a^2 C+2 a b B-b^2 C\right) \log (\cos (c+d x))}{d}+x \left(a^2 B-2 a b C-b^2 B\right)+\frac{b (a C+b B) \tan (c+d x)}{d}+\frac{C (a+b \tan (c+d x))^2}{2 d}",1,"((a + I*b)^2*((-I)*B + C)*Log[I - Tan[c + d*x]] + (a - I*b)^2*(I*B + C)*Log[I + Tan[c + d*x]] + 2*b*(b*B + 2*a*C)*Tan[c + d*x] + b^2*C*Tan[c + d*x]^2)/(2*d)","C",1
12,1,91,70,0.2835487,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","-\frac{-2 a^2 B \log (\tan (c+d x))+(a+i b)^2 (B+i C) \log (-\tan (c+d x)+i)+(a-i b)^2 (B-i C) \log (\tan (c+d x)+i)-2 b^2 C \tan (c+d x)}{2 d}","x \left(a^2 C+2 a b B-b^2 C\right)+\frac{a^2 B \log (\sin (c+d x))}{d}-\frac{b (2 a C+b B) \log (\cos (c+d x))}{d}+\frac{b^2 C \tan (c+d x)}{d}",1,"-1/2*((a + I*b)^2*(B + I*C)*Log[I - Tan[c + d*x]] - 2*a^2*B*Log[Tan[c + d*x]] + (a - I*b)^2*(B - I*C)*Log[I + Tan[c + d*x]] - 2*b^2*C*Tan[c + d*x])/d","C",1
13,1,100,72,0.251673,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-2 a^2 B \cot (c+d x)+2 a (a C+2 b B) \log (\tan (c+d x))+i (a+i b)^2 (B+i C) \log (-\tan (c+d x)+i)-(a-i b)^2 (C+i B) \log (\tan (c+d x)+i)}{2 d}","-x \left(a^2 B-2 a b C-b^2 B\right)-\frac{a^2 B \cot (c+d x)}{d}+\frac{a (a C+2 b B) \log (\sin (c+d x))}{d}-\frac{b^2 C \log (\cos (c+d x))}{d}",1,"(-2*a^2*B*Cot[c + d*x] + I*(a + I*b)^2*(B + I*C)*Log[I - Tan[c + d*x]] + 2*a*(2*b*B + a*C)*Log[Tan[c + d*x]] - (a - I*b)^2*(I*B + C)*Log[I + Tan[c + d*x]])/(2*d)","C",1
14,1,123,88,0.3527426,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-2 \left(a^2 B-2 a b C-b^2 B\right) \log (\tan (c+d x))-a^2 B \cot ^2(c+d x)-2 a (a C+2 b B) \cot (c+d x)+(a-i b)^2 (B-i C) \log (\tan (c+d x)+i)+(a+i b)^2 (B+i C) \log (-\tan (c+d x)+i)}{2 d}","-\frac{\left(a^2 B-2 a b C-b^2 B\right) \log (\sin (c+d x))}{d}-\frac{a^2 B \cot ^2(c+d x)}{2 d}+x \left(b^2 C-a (a C+2 b B)\right)-\frac{a (a C+2 b B) \cot (c+d x)}{d}",1,"(-2*a*(2*b*B + a*C)*Cot[c + d*x] - a^2*B*Cot[c + d*x]^2 + (a + I*b)^2*(B + I*C)*Log[I - Tan[c + d*x]] - 2*(a^2*B - b^2*B - 2*a*b*C)*Log[Tan[c + d*x]] + (a - I*b)^2*(B - I*C)*Log[I + Tan[c + d*x]])/(2*d)","C",1
15,1,152,118,1.174684,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{6 \left(a^2 B-2 a b C-b^2 B\right) \cot (c+d x)-6 \left(a^2 C+2 a b B-b^2 C\right) \log (\tan (c+d x))-2 a^2 B \cot ^3(c+d x)-3 a (a C+2 b B) \cot ^2(c+d x)+3 (a+i b)^2 (C-i B) \log (-\tan (c+d x)+i)+3 (a-i b)^2 (C+i B) \log (\tan (c+d x)+i)}{6 d}","\frac{\left(a^2 B-2 a b C-b^2 B\right) \cot (c+d x)}{d}+x \left(a^2 B-2 a b C-b^2 B\right)-\frac{a^2 B \cot ^3(c+d x)}{3 d}+\frac{\left(b^2 C-a (a C+2 b B)\right) \log (\sin (c+d x))}{d}-\frac{a (a C+2 b B) \cot ^2(c+d x)}{2 d}",1,"(6*(a^2*B - b^2*B - 2*a*b*C)*Cot[c + d*x] - 3*a*(2*b*B + a*C)*Cot[c + d*x]^2 - 2*a^2*B*Cot[c + d*x]^3 + 3*(a + I*b)^2*((-I)*B + C)*Log[I - Tan[c + d*x]] - 6*(2*a*b*B + a^2*C - b^2*C)*Log[Tan[c + d*x]] + 3*(a - I*b)^2*(I*B + C)*Log[I + Tan[c + d*x]])/(6*d)","C",1
16,1,180,151,2.9306214,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{6 \left(a^2 B-2 a b C-b^2 B\right) \cot ^2(c+d x)+12 \left(a^2 C+2 a b B-b^2 C\right) \cot (c+d x)-6 \left(\left(-2 a^2 B+4 a b C+2 b^2 B\right) \log (\tan (c+d x))+(a-i b)^2 (B-i C) \log (\tan (c+d x)+i)+(a+i b)^2 (B+i C) \log (-\tan (c+d x)+i)\right)-3 a^2 B \cot ^4(c+d x)-4 a (a C+2 b B) \cot ^3(c+d x)}{12 d}","\frac{\left(a^2 B-2 a b C-b^2 B\right) \cot ^2(c+d x)}{2 d}+\frac{\left(a^2 B-2 a b C-b^2 B\right) \log (\sin (c+d x))}{d}+x \left(a^2 C+2 a b B-b^2 C\right)-\frac{a^2 B \cot ^4(c+d x)}{4 d}-\frac{\left(b^2 C-a (a C+2 b B)\right) \cot (c+d x)}{d}-\frac{a (a C+2 b B) \cot ^3(c+d x)}{3 d}",1,"(12*(2*a*b*B + a^2*C - b^2*C)*Cot[c + d*x] + 6*(a^2*B - b^2*B - 2*a*b*C)*Cot[c + d*x]^2 - 4*a*(2*b*B + a*C)*Cot[c + d*x]^3 - 3*a^2*B*Cot[c + d*x]^4 - 6*((a + I*b)^2*(B + I*C)*Log[I - Tan[c + d*x]] + (-2*a^2*B + 2*b^2*B + 4*a*b*C)*Log[Tan[c + d*x]] + (a - I*b)^2*(B - I*C)*Log[I + Tan[c + d*x]]))/(12*d)","C",1
17,1,209,165,1.6287913,"\int (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-12 b^2 B \left(b^2-6 a^2\right) \tan (c+d x)+24 a b^3 B \tan ^2(c+d x)-6 (a B+b C) \left(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)\right)+6 i B (a-i b)^4 \log (\tan (c+d x)+i)-6 i B (a+i b)^4 \log (-\tan (c+d x)+i)+3 C (a+b \tan (c+d x))^4+4 b^4 B \tan ^3(c+d x)}{12 b d}","\frac{b \left(a^2 B-2 a b C-b^2 B\right) \tan (c+d x)}{d}-\frac{\left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right) \log (\cos (c+d x))}{d}-x \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right)+\frac{(a B-b C) (a+b \tan (c+d x))^2}{2 d}+\frac{B (a+b \tan (c+d x))^3}{3 d}+\frac{C (a+b \tan (c+d x))^4}{4 b d}",1,"((-6*I)*(a + I*b)^4*B*Log[I - Tan[c + d*x]] + (6*I)*(a - I*b)^4*B*Log[I + Tan[c + d*x]] - 12*b^2*(-6*a^2 + b^2)*B*Tan[c + d*x] + 24*a*b^3*B*Tan[c + d*x]^2 + 4*b^4*B*Tan[c + d*x]^3 + 3*C*(a + b*Tan[c + d*x])^4 - 6*(a*B + b*C)*((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))/(12*b*d)","C",1
18,1,130,140,1.0700221,"\int \cot (c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{6 b \left(3 a^2 C+3 a b B-b^2 C\right) \tan (c+d x)+3 b^2 (3 a C+b B) \tan ^2(c+d x)+3 (a-i b)^3 (C+i B) \log (\tan (c+d x)+i)+3 (a+i b)^3 (C-i B) \log (-\tan (c+d x)+i)+2 b^3 C \tan ^3(c+d x)}{6 d}","\frac{b \left(a^2 C+2 a b B-b^2 C\right) \tan (c+d x)}{d}-\frac{\left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \log (\cos (c+d x))}{d}+x \left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right)+\frac{(a C+b B) (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 d}",1,"(3*(a + I*b)^3*((-I)*B + C)*Log[I - Tan[c + d*x]] + 3*(a - I*b)^3*(I*B + C)*Log[I + Tan[c + d*x]] + 6*b*(3*a*b*B + 3*a^2*C - b^2*C)*Tan[c + d*x] + 3*b^2*(b*B + 3*a*C)*Tan[c + d*x]^2 + 2*b^3*C*Tan[c + d*x]^3)/(6*d)","C",1
19,1,113,117,0.4705169,"\int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{2 a^3 B \log (\tan (c+d x))+2 b^2 (3 a C+b B) \tan (c+d x)-(a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)-(a-i b)^3 (B-i C) \log (\tan (c+d x)+i)+b^3 C \tan ^2(c+d x)}{2 d}","\frac{a^3 B \log (\sin (c+d x))}{d}-\frac{b \left(3 a^2 C+3 a b B-b^2 C\right) \log (\cos (c+d x))}{d}+x \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right)+\frac{b^2 (2 a C+b B) \tan (c+d x)}{d}+\frac{b C (a+b \tan (c+d x))^2}{2 d}",1,"(-((a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]]) + 2*a^3*B*Log[Tan[c + d*x]] - (a - I*b)^3*(B - I*C)*Log[I + Tan[c + d*x]] + 2*b^2*(b*B + 3*a*C)*Tan[c + d*x] + b^3*C*Tan[c + d*x]^2)/(2*d)","C",1
20,1,113,119,0.492547,"\int \cot ^3(c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-2 a^3 B \cot (c+d x)+2 a^2 (a C+3 b B) \log (\tan (c+d x))+i (a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)+(b+i a)^3 (B-i C) \log (\tan (c+d x)+i)+2 b^3 C \tan (c+d x)}{2 d}","\frac{a^2 (a C+3 b B) \log (\sin (c+d x))}{d}-x \left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right)+\frac{b^2 (a B+b C) \tan (c+d x)}{d}-\frac{b^2 (3 a C+b B) \log (\cos (c+d x))}{d}-\frac{a B \cot (c+d x) (a+b \tan (c+d x))^2}{d}",1,"(-2*a^3*B*Cot[c + d*x] + I*(a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]] + 2*a^2*(3*b*B + a*C)*Log[Tan[c + d*x]] + (I*a + b)^3*(B - I*C)*Log[I + Tan[c + d*x]] + 2*b^3*C*Tan[c + d*x])/(2*d)","C",1
21,1,126,127,0.4621287,"\int \cot ^4(c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{a^3 (-B) \cot ^2(c+d x)-2 a \left(a^2 B-3 a b C-3 b^2 B\right) \log (\tan (c+d x))-2 a^2 (a C+3 b B) \cot (c+d x)+(a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)+(a-i b)^3 (B-i C) \log (\tan (c+d x)+i)}{2 d}","-\frac{a \left(a^2 B-3 a b C-3 b^2 B\right) \log (\sin (c+d x))}{d}-\frac{a^2 (a C+2 b B) \cot (c+d x)}{d}-x \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right)-\frac{a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{b^3 C \log (\cos (c+d x))}{d}",1,"(-2*a^2*(3*b*B + a*C)*Cot[c + d*x] - a^3*B*Cot[c + d*x]^2 + (a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]] - 2*a*(a^2*B - 3*b^2*B - 3*a*b*C)*Log[Tan[c + d*x]] + (a - I*b)^3*(B - I*C)*Log[I + Tan[c + d*x]])/(2*d)","C",1
22,1,164,154,1.2750696,"\int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-2 a^3 B \cot ^3(c+d x)+6 a \left(a^2 B-3 a b C-3 b^2 B\right) \cot (c+d x)-3 a^2 (a C+3 b B) \cot ^2(c+d x)-6 \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \log (\tan (c+d x))+3 (a-i b)^3 (C+i B) \log (\tan (c+d x)+i)+3 (a+i b)^3 (C-i B) \log (-\tan (c+d x)+i)}{6 d}","\frac{a \left(3 a^2 B-9 a b C-8 b^2 B\right) \cot (c+d x)}{3 d}-\frac{a^2 (3 a C+5 b B) \cot ^2(c+d x)}{6 d}-\frac{\left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \log (\sin (c+d x))}{d}+x \left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right)-\frac{a B \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}",1,"(6*a*(a^2*B - 3*b^2*B - 3*a*b*C)*Cot[c + d*x] - 3*a^2*(3*b*B + a*C)*Cot[c + d*x]^2 - 2*a^3*B*Cot[c + d*x]^3 + 3*(a + I*b)^3*((-I)*B + C)*Log[I - Tan[c + d*x]] - 6*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Tan[c + d*x]] + 3*(a - I*b)^3*(I*B + C)*Log[I + Tan[c + d*x]])/(6*d)","C",1
23,1,199,191,0.7901327,"\int \cot ^6(c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-3 a^3 B \cot ^4(c+d x)+6 a \left(a^2 B-3 a b C-3 b^2 B\right) \cot ^2(c+d x)-4 a^2 (a C+3 b B) \cot ^3(c+d x)+12 \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \cot (c+d x)+12 \left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right) \log (\tan (c+d x))-6 (a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)-6 (a-i b)^3 (B-i C) \log (\tan (c+d x)+i)}{12 d}","\frac{a \left(2 a^2 B-6 a b C-5 b^2 B\right) \cot ^2(c+d x)}{4 d}-\frac{a^2 (2 a C+3 b B) \cot ^3(c+d x)}{6 d}+\frac{\left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \cot (c+d x)}{d}+\frac{\left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right) \log (\sin (c+d x))}{d}+x \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right)-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}",1,"(12*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x] + 6*a*(a^2*B - 3*b^2*B - 3*a*b*C)*Cot[c + d*x]^2 - 4*a^2*(3*b*B + a*C)*Cot[c + d*x]^3 - 3*a^3*B*Cot[c + d*x]^4 - 6*(a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]] + 12*(a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Log[Tan[c + d*x]] - 6*(a - I*b)^3*(B - I*C)*Log[I + Tan[c + d*x]])/(12*d)","C",1
24,1,237,233,1.2096784,"\int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left(B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{-12 a^3 B \cot ^5(c+d x)+20 a \left(a^2 B-3 a b C-3 b^2 B\right) \cot ^3(c+d x)-15 a^2 (a C+3 b B) \cot ^4(c+d x)+30 \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \cot ^2(c+d x)-60 \left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right) \cot (c+d x)+60 \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \log (\tan (c+d x))+30 i (a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)+30 (b+i a)^3 (B-i C) \log (\tan (c+d x)+i)}{60 d}","\frac{a \left(5 a^2 B-15 a b C-12 b^2 B\right) \cot ^3(c+d x)}{15 d}-\frac{a^2 (5 a C+7 b B) \cot ^4(c+d x)}{20 d}+\frac{\left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \cot ^2(c+d x)}{2 d}-\frac{\left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right) \cot (c+d x)}{d}+\frac{\left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \log (\sin (c+d x))}{d}-x \left(a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right)-\frac{a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}",1,"(-60*(a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Cot[c + d*x] + 30*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x]^2 + 20*a*(a^2*B - 3*b^2*B - 3*a*b*C)*Cot[c + d*x]^3 - 15*a^2*(3*b*B + a*C)*Cot[c + d*x]^4 - 12*a^3*B*Cot[c + d*x]^5 + (30*I)*(a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]] + 60*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Tan[c + d*x]] + 30*(I*a + b)^3*(B - I*C)*Log[I + Tan[c + d*x]])/(60*d)","C",1
25,1,138,127,1.4504435,"\int \frac{\tan ^2(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 a^3 (a C-b B) \log (a+b \tan (c+d x))}{b^2 \left(a^2+b^2\right)}+\frac{2 (b B-a C) \tan (c+d x)}{b}-\frac{b (B+i C) \log (-\tan (c+d x)+i)}{a+i b}-\frac{b (B-i C) \log (\tan (c+d x)+i)}{a-i b}+C \tan ^2(c+d x)}{2 b d}","\frac{(a B+b C) \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{x (b B-a C)}{a^2+b^2}-\frac{a^3 (b B-a C) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)}+\frac{(b B-a C) \tan (c+d x)}{b^2 d}+\frac{C \tan ^2(c+d x)}{2 b d}",1,"(-((b*(B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)) - (b*(B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*a^3*(-(b*B) + a*C)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)) + (2*(b*B - a*C)*Tan[c + d*x])/b + C*Tan[c + d*x]^2)/(2*b*d)","C",1
26,1,118,101,0.7004104,"\int \frac{\tan (c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{a+b \tan (c+d x)} \, dx","Integrate[(Tan[c + d*x]*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 a^2 (b B-a C) \log (a+b \tan (c+d x))}{b^2 \left(a^2+b^2\right)}+\frac{i (B+i C) \log (-\tan (c+d x)+i)}{a+i b}-\frac{(C+i B) \log (\tan (c+d x)+i)}{a-i b}+\frac{2 C \tan (c+d x)}{b}}{2 d}","\frac{a^2 (b B-a C) \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)}-\frac{(b B-a C) \log (\cos (c+d x))}{d \left(a^2+b^2\right)}-\frac{x (a B+b C)}{a^2+b^2}+\frac{C \tan (c+d x)}{b d}",1,"((I*(B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b) - ((I*B + C)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*a^2*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)) + (2*C*Tan[c + d*x])/b)/(2*d)","C",1
27,1,98,85,0.1976516,"\int \frac{B \tan (c+d x)+C \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]),x]","\frac{b (a-i b) (B+i C) \log (-\tan (c+d x)+i)+b (a+i b) (B-i C) \log (\tan (c+d x)+i)+2 a (a C-b B) \log (a+b \tan (c+d x))}{2 b d \left(a^2+b^2\right)}","-\frac{a (b B-a C) \log (a+b \tan (c+d x))}{b d \left(a^2+b^2\right)}-\frac{(a B+b C) \log (\cos (c+d x))}{d \left(a^2+b^2\right)}+\frac{x (b B-a C)}{a^2+b^2}",1,"((a - I*b)*b*(B + I*C)*Log[I - Tan[c + d*x]] + (a + I*b)*b*(B - I*C)*Log[I + Tan[c + d*x]] + 2*a*(-(b*B) + a*C)*Log[a + b*Tan[c + d*x]])/(2*b*(a^2 + b^2)*d)","C",1
28,1,67,58,0.1260496,"\int \frac{\cot (c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]","\frac{(b B-a C) \left(2 \log (a \cot (c+d x)+b)-\log \left(\csc ^2(c+d x)\right)\right)-2 (a B+b C) \tan ^{-1}(\cot (c+d x))}{2 d \left(a^2+b^2\right)}","\frac{(b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}+\frac{x (a B+b C)}{a^2+b^2}",1,"(-2*(a*B + b*C)*ArcTan[Cot[c + d*x]] + (b*B - a*C)*(2*Log[b + a*Cot[c + d*x]] - Log[Csc[c + d*x]^2]))/(2*(a^2 + b^2)*d)","A",1
29,1,113,80,0.3658536,"\int \frac{\cot ^2(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]","-\frac{\frac{2 b (b B-a C) \log (a+b \tan (c+d x))}{a \left(a^2+b^2\right)}+\frac{(B+i C) \log (-\tan (c+d x)+i)}{a+i b}+\frac{(B-i C) \log (\tan (c+d x)+i)}{a-i b}-\frac{2 B \log (\tan (c+d x))}{a}}{2 d}","-\frac{b (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left(a^2+b^2\right)}-\frac{x (b B-a C)}{a^2+b^2}+\frac{B \log (\sin (c+d x))}{a d}",1,"-1/2*(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*B*Log[Tan[c + d*x]])/a + ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)))/d","C",1
30,1,138,103,0.8876887,"\int \frac{\cot ^3(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 b^2 (b B-a C) \log (a+b \tan (c+d x))}{a^2 \left(a^2+b^2\right)}+\frac{2 (a C-b B) \log (\tan (c+d x))}{a^2}+\frac{i (B+i C) \log (-\tan (c+d x)+i)}{a+i b}-\frac{(C+i B) \log (\tan (c+d x)+i)}{a-i b}-\frac{2 B \cot (c+d x)}{a}}{2 d}","\frac{b^2 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)}-\frac{x (a B+b C)}{a^2+b^2}-\frac{(b B-a C) \log (\sin (c+d x))}{a^2 d}-\frac{B \cot (c+d x)}{a d}",1,"((-2*B*Cot[c + d*x])/a + (I*(B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b) + (2*(-(b*B) + a*C)*Log[Tan[c + d*x]])/a^2 - ((I*B + C)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b^2*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)))/(2*d)","C",1
31,1,163,137,1.4222994,"\int \frac{\cot ^4(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{a+b \tan (c+d x)} \, dx","Integrate[(Cot[c + d*x]^4*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x]),x]","\frac{\frac{2 (b B-a C) \cot (c+d x)}{a^2}-\frac{2 \left(a^2 B+a b C-b^2 B\right) \log (\tan (c+d x))}{a^3}+\frac{2 b^3 (a C-b B) \log (a+b \tan (c+d x))}{a^3 \left(a^2+b^2\right)}+\frac{(B+i C) \log (-\tan (c+d x)+i)}{a+i b}+\frac{(B-i C) \log (\tan (c+d x)+i)}{a-i b}-\frac{B \cot ^2(c+d x)}{a}}{2 d}","\frac{x (b B-a C)}{a^2+b^2}+\frac{(b B-a C) \cot (c+d x)}{a^2 d}-\frac{\left(a^2 B+a b C-b^2 B\right) \log (\sin (c+d x))}{a^3 d}-\frac{b^3 (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)}-\frac{B \cot ^2(c+d x)}{2 a d}",1,"((2*(b*B - a*C)*Cot[c + d*x])/a^2 - (B*Cot[c + d*x]^2)/a + ((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*(a^2*B - b^2*B + a*b*C)*Log[Tan[c + d*x]])/a^3 + ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b^3*(-(b*B) + a*C)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)))/(2*d)","C",1
32,1,444,208,4.2650118,"\int \frac{\tan ^2(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^2,x]","\frac{2 b^2 C \left(a^2+b^2\right)^2 \tan ^2(c+d x)+2 i a^2 \left(2 a^3 C-a^2 b B+4 a b^2 C-3 b^3 B\right) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left(2 \left(a^2+b^2\right)^2 (2 a C-b B) \log (\cos (c+d x))+2 (a+i b)^2 (c+d x) \left(-2 i a^3 C+i a^2 b (B+4 i C)+2 a b^2 (B+i C)+b^3 C\right)+a^2 \left(-2 a^3 C+a^2 b B-4 a b^2 C+3 b^3 B\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)\right)+b \tan (c+d x) \left(2 \left(a^2+b^2\right)^2 (2 a C-b B) \log (\cos (c+d x))+a^2 \left(-2 a^3 C+a^2 b B-4 a b^2 C+3 b^3 B\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 \left(-2 i a^5 C (c+d x+i)+i a^4 b B (c+d x+i)+a^3 b^2 C (-4 i c-4 i d x+3)+a^2 b^3 (C (c+d x)+i B (3 c+3 d x+i))+a b^4 (C-2 B (c+d x))-b^5 C (c+d x)\right)\right)}{2 b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","\frac{a (b B-a C) \tan ^2(c+d x)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(-2 a^2 C+a b B-b^2 C\right) \tan (c+d x)}{b^2 d \left(a^2+b^2\right)}+\frac{\left(a^2 B+2 a b C-b^2 B\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{x \left(a^2 (-C)+2 a b B+b^2 C\right)}{\left(a^2+b^2\right)^2}+\frac{a^2 \left(-2 a^3 C+a^2 b B-4 a b^2 C+3 b^3 B\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^2}",1,"(a*(2*(a + I*b)^2*(2*a*b^2*(B + I*C) + I*a^2*b*(B + (4*I)*C) - (2*I)*a^3*C + b^3*C)*(c + d*x) + 2*(a^2 + b^2)^2*(-(b*B) + 2*a*C)*Log[Cos[c + d*x]] + a^2*(a^2*b*B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*(2*(a^3*b^2*C*(3 - (4*I)*c - (4*I)*d*x) - b^5*C*(c + d*x) + I*a^4*b*B*(I + c + d*x) - (2*I)*a^5*C*(I + c + d*x) + a*b^4*(C - 2*B*(c + d*x)) + a^2*b^3*(C*(c + d*x) + I*B*(I + 3*c + 3*d*x))) + 2*(a^2 + b^2)^2*(-(b*B) + 2*a*C)*Log[Cos[c + d*x]] + a^2*(a^2*b*B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] + 2*b^2*(a^2 + b^2)^2*C*Tan[c + d*x]^2 + (2*I)*a^2*(-(a^2*b*B) - 3*b^3*B + 2*a^3*C + 4*a*b^2*C)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(2*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
33,1,324,157,2.1776202,"\int \frac{\tan (c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Tan[c + d*x]*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^2,x]","\frac{-2 i a \left(a^3 C+3 a b^2 C-2 b^3 B\right) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left(a \left(a^3 C+3 a b^2 C-2 b^3 B\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+2 (a+i b)^2 (c+d x) \left(i a^2 C+2 a b C-b^2 B\right)-2 C \left(a^2+b^2\right)^2 \log (\cos (c+d x))\right)+b \tan (c+d x) \left(a \left(a^3 C+3 a b^2 C-2 b^3 B\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)-2 C \left(a^2+b^2\right)^2 \log (\cos (c+d x))+2 (a+i b) \left(i a^3 C (c+d x+i)+a^2 b (B+C (c+d x+i))-a b^2 (B (c+d x+i)-2 i C (c+d x))-i b^3 B (c+d x)\right)\right)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}","-\frac{a^2 (b B-a C)}{b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2 (-C)+2 a b B+b^2 C\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^2}-\frac{x \left(a^2 B+2 a b C-b^2 B\right)}{\left(a^2+b^2\right)^2}-\frac{a \left(a^3 (-C)-3 a b^2 C+2 b^3 B\right) \log (a+b \tan (c+d x))}{b^2 d \left(a^2+b^2\right)^2}",1,"(a*(2*(a + I*b)^2*(-(b^2*B) + I*a^2*C + 2*a*b*C)*(c + d*x) - 2*(a^2 + b^2)^2*C*Log[Cos[c + d*x]] + a*(-2*b^3*B + a^3*C + 3*a*b^2*C)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*(2*(a + I*b)*((-I)*b^3*B*(c + d*x) + I*a^3*C*(I + c + d*x) - a*b^2*((-2*I)*C*(c + d*x) + B*(I + c + d*x)) + a^2*b*(B + C*(I + c + d*x))) - 2*(a^2 + b^2)^2*C*Log[Cos[c + d*x]] + a*(-2*b^3*B + a^3*C + 3*a*b^2*C)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] - (2*I)*a*(-2*b^3*B + a^3*C + 3*a*b^2*C)*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
34,1,140,115,2.1903529,"\int \frac{B \tan (c+d x)+C \tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 \left(\left(a^2 (-B)-2 a b C+b^2 B\right) \log (a+b \tan (c+d x))-\frac{a \left(a^2+b^2\right) (a C-b B)}{b (a+b \tan (c+d x))}\right)}{\left(a^2+b^2\right)^2}+\frac{(B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{(B-i C) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","\frac{a (b B-a C)}{b d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{\left(a^2 B+2 a b C-b^2 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2 (-C)+2 a b B+b^2 C\right)}{\left(a^2+b^2\right)^2}",1,"(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*((-(a^2*B) + b^2*B - 2*a*b*C)*Log[a + b*Tan[c + d*x]] - (a*(a^2 + b^2)*(-(b*B) + a*C))/(b*(a + b*Tan[c + d*x]))))/(a^2 + b^2)^2)/(2*d)","C",1
35,1,190,111,2.2235351,"\int \frac{\cot (c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{C ((-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)))}{a^2+b^2}-(b B-a C) \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (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}\right)}{2 b d}","-\frac{b B-a C}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{\left(a^2 (-C)+2 a b B+b^2 C\right) \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 a b C-b^2 B\right)}{\left(a^2+b^2\right)^2}",1,"((C*(((-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]]))/(a^2 + b^2) - (b*B - a*C)*((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*b*d)","C",1
36,1,159,137,2.4113228,"\int \frac{\cot ^2(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^2,x]","-\frac{\frac{2 b (a C-b B)}{a \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{2 B \log (\tan (c+d x))}{a^2}+\frac{2 b \left(-2 a^3 C+3 a^2 b B+b^3 B\right) \log (a+b \tan (c+d x))}{a^2 \left(a^2+b^2\right)^2}+\frac{(B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac{(B-i C) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","\frac{b (b B-a C)}{a d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{x \left(a^2 (-C)+2 a b B+b^2 C\right)}{\left(a^2+b^2\right)^2}+\frac{B \log (\sin (c+d x))}{a^2 d}-\frac{b \left(-2 a^3 C+3 a^2 b B+b^3 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left(a^2+b^2\right)^2}",1,"-1/2*(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (2*B*Log[Tan[c + d*x]])/a^2 + ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*(3*a^2*b*B + b^3*B - 2*a^3*C)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) + (2*b*(-(b*B) + a*C))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/d","C",1
37,1,193,192,3.5671161,"\int \frac{\cot ^3(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^2} \, dx","Integrate[(Cot[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^2,x]","\frac{\frac{2 (a C-2 b B) \log (\tan (c+d x))}{a^3}+\frac{2 b^2 (a C-b B)}{a^2 \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{2 B \cot (c+d x)}{a^2}-\frac{2 b^2 \left(3 a^3 C-4 a^2 b B+a b^2 C-2 b^3 B\right) \log (a+b \tan (c+d x))}{a^3 \left(a^2+b^2\right)^2}+\frac{i (B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{(C+i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d}","-\frac{(2 b B-a C) \log (\sin (c+d x))}{a^3 d}-\frac{b \left(a^2 B-a b C+2 b^2 B\right)}{a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))}-\frac{x \left(a^2 B+2 a b C-b^2 B\right)}{\left(a^2+b^2\right)^2}+\frac{b^2 \left(-3 a^3 C+4 a^2 b B-a b^2 C+2 b^3 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)^2}-\frac{B \cot (c+d x)}{a d (a+b \tan (c+d x))}",1,"((-2*B*Cot[c + d*x])/a^2 + (I*(B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (2*(-2*b*B + a*C)*Log[Tan[c + d*x]])/a^3 - ((I*B + C)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b^2*(-4*a^2*b*B - 2*b^3*B + 3*a^3*C + a*b^2*C)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^2) + (2*b^2*(-(b*B) + a*C))/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(2*d)","C",1
38,1,1146,331,6.8563042,"\int \frac{\tan ^3(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]","\frac{(a C-b B) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) (B+C \tan (c+d x)) a^4}{2 (a-i b)^2 (a+i b)^2 b^2 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \left(2 C \sin (c+d x) a^5-b B \sin (c+d x) a^4+5 b^2 C \sin (c+d x) a^3-4 b^3 B \sin (c+d x) a^2\right) (B+C \tan (c+d x))}{(a-i b)^2 (a+i b)^2 b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{C \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \tan (c+d x) (B+C \tan (c+d x))}{b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left(B a^3+3 b C a^2-3 b^2 B a-b^3 C\right) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^3 (a+i b)^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left(6 a^2 B b^{13}+6 i a^3 B b^{12}-10 a^3 C b^{12}+15 a^4 B b^{11}-10 i a^4 C b^{11}+15 i a^5 B b^{10}-29 a^5 C b^{10}+13 a^6 B b^9-29 i a^6 C b^9+13 i a^7 B b^8-31 a^7 C b^8+5 a^8 B b^7-31 i a^8 C b^7+5 i a^9 B b^6-15 a^9 C b^6+a^{10} B b^5-15 i a^{10} C b^5+i a^{11} B b^4-3 a^{11} C b^4-3 i a^{12} C b^3\right) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^6 (a+i b)^5 b^7 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}-\frac{i \left(-3 C a^7+b B a^6-9 b^2 C a^5+3 b^3 B a^4-10 b^4 C a^3+6 b^5 B a^2\right) \tan ^{-1}(\tan (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 \left(a^2+b^2\right)^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{(3 a C-b B) \log (\cos (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac{\left(-3 C a^7+b B a^6-9 b^2 C a^5+3 b^3 B a^4-10 b^4 C a^3+6 b^5 B a^2\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{2 b^4 \left(a^2+b^2\right)^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}","\frac{a (b B-a C) \tan ^3(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a \left(-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right) \tan ^2(c+d x)}{2 b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right)}{\left(a^2+b^2\right)^3}-\frac{\left(-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right) \tan (c+d x)}{b^3 d \left(a^2+b^2\right)^2}+\frac{a^2 \left(-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right) \log (a+b \tan (c+d x))}{b^4 d \left(a^2+b^2\right)^3}",1,"(a^4*(-(b*B) + a*C)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(B + C*Tan[c + d*x]))/(2*(a - I*b)^2*(a + I*b)^2*b^2*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*(c + d*x)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(B + C*Tan[c + d*x]))/((a - I*b)^3*(a + I*b)^3*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((I*a^11*b^4*B + a^10*b^5*B + (5*I)*a^9*b^6*B + 5*a^8*b^7*B + (13*I)*a^7*b^8*B + 13*a^6*b^9*B + (15*I)*a^5*b^10*B + 15*a^4*b^11*B + (6*I)*a^3*b^12*B + 6*a^2*b^13*B - (3*I)*a^12*b^3*C - 3*a^11*b^4*C - (15*I)*a^10*b^5*C - 15*a^9*b^6*C - (31*I)*a^8*b^7*C - 31*a^7*b^8*C - (29*I)*a^6*b^9*C - 29*a^5*b^10*C - (10*I)*a^4*b^11*C - 10*a^3*b^12*C)*(c + d*x)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(B + C*Tan[c + d*x]))/((a - I*b)^6*(a + I*b)^5*b^7*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) - (I*(a^6*b*B + 3*a^4*b^3*B + 6*a^2*b^5*B - 3*a^7*C - 9*a^5*b^2*C - 10*a^3*b^4*C)*ArcTan[Tan[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(B + C*Tan[c + d*x]))/(b^4*(a^2 + b^2)^3*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((-(b*B) + 3*a*C)*Log[Cos[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(B + C*Tan[c + d*x]))/(b^4*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + ((a^6*b*B + 3*a^4*b^3*B + 6*a^2*b^5*B - 3*a^7*C - 9*a^5*b^2*C - 10*a^3*b^4*C)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(B + C*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)^3*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*(-(a^4*b*B*Sin[c + d*x]) - 4*a^2*b^3*B*Sin[c + d*x] + 2*a^5*C*Sin[c + d*x] + 5*a^3*b^2*C*Sin[c + d*x])*(B + C*Tan[c + d*x]))/((a - I*b)^2*(a + I*b)^2*b^3*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3) + (C*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*Tan[c + d*x]*(B + C*Tan[c + d*x]))/(b^3*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3)","C",1
39,1,462,250,4.9182105,"\int \frac{\tan ^2(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (B+C \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x)) \left(-2 C \left(a^2+b^2\right)^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a^3 b^2 \left(a^2+b^2\right) (b B-a C)-2 a b \left(a^2+b^2\right) \left(a^3 C+4 a b^2 C-3 b^3 B\right) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+2 b^3 (c+d x) \left(a^3 C-3 a^2 b B-3 a b^2 C+b^3 B\right) (a \cos (c+d x)+b \sin (c+d x))^2+2 i a (c+d x) \left(a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right) (a \cos (c+d x)+b \sin (c+d x))^2+a \left(a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\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 \left(a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\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 (B \cos (c+d x)+C \sin (c+d x))}","\frac{a (b B-a C) \tan ^2(c+d x)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 \left(a^3 (-C)-3 a b^2 C+2 b^3 B\right)}{b^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right) \log (\cos (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{x \left(a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right)}{\left(a^2+b^2\right)^3}+\frac{a \left(a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right) \log (a+b \tan (c+d x))}{b^3 d \left(a^2+b^2\right)^3}",1,"(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(a^3*b^2*(a^2 + b^2)*(b*B - a*C) - 2*a*b*(a^2 + b^2)*(-3*b^3*B + a^3*C + 4*a*b^2*C)*Sin[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + 2*b^3*(-3*a^2*b*B + b^3*B + a^3*C - 3*a*b^2*C)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (2*I)*a*(a^2*b^3*B - 3*b^5*B + a^5*C + 3*a^3*b^2*C + 6*a*b^4*C)*(c + d*x)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - (2*I)*a*(a^2*b^3*B - 3*b^5*B + a^5*C + 3*a^3*b^2*C + 6*a*b^4*C)*ArcTan[Tan[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 2*(a^2 + b^2)^3*C*Log[Cos[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + a*(a^2*b^3*B - 3*b^5*B + a^5*C + 3*a^3*b^2*C + 6*a*b^4*C)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)*(B + C*Tan[c + d*x]))/(2*b^3*(a^2 + b^2)^3*d*(B*Cos[c + d*x] + C*Sin[c + d*x])*(a + b*Tan[c + d*x])^3)","C",1
40,1,288,189,5.4529346,"\int \frac{\tan (c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Tan[c + d*x]*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]","\frac{(b B-a C) \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)+C \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (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}\right)-\frac{a C+b B}{b (a+b \tan (c+d x))^2}-\frac{2 C \tan (c+d x)}{(a+b \tan (c+d x))^2}}{2 b d}","-\frac{a^2 (b B-a C)}{2 b^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a \left(a^3 (-C)-3 a b^2 C+2 b^3 B\right)}{b^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}-\frac{x \left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right)}{\left(a^2+b^2\right)^3}",1,"(-((b*B + a*C)/(b*(a + b*Tan[c + d*x])^2)) - (2*C*Tan[c + d*x])/(a + b*Tan[c + d*x])^2 + C*((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) + (b*B - a*C)*((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
41,1,188,179,4.0661015,"\int \frac{B \tan (c+d x)+C \tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{a (b B-a C)}{b \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{2 \left(a^2 B+2 a b C-b^2 B\right)}{\left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{2 \left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^3}+\frac{(B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^3}+\frac{(B-i C) \log (\tan (c+d x)+i)}{(a-i b)^3}}{2 d}","\frac{a (b B-a C)}{2 b d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{a^2 B+2 a b C-b^2 B}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right)}{\left(a^2+b^2\right)^3}",1,"(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^3 + ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b)^3 - (2*(a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 + (a*(b*B - a*C))/(b*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (2*(a^2*B - b^2*B + 2*a*b*C))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*d)","C",1
42,1,243,175,4.7713721,"\int \frac{\cot (c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]","-\frac{(b B-a C) \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)+C \left(\frac{2 b \left(\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (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}\right)}{2 b d}","-\frac{b B-a C}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{a^2 (-C)+2 a b B+b^2 C}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{\left(a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right)}{\left(a^2+b^2\right)^3}",1,"-1/2*(C*((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) + (b*B - a*C)*((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))/(b*d)","C",1
43,1,223,215,3.1511866,"\int \frac{\cot ^2(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]","\frac{\frac{2 B \log (\tan (c+d x))}{a^3}+\frac{b (b B-a C)}{a \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{2 b \left(-2 a^3 C+3 a^2 b B+b^3 B\right)}{a^2 \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{2 b \left(-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right) \log (a+b \tan (c+d x))}{a^3 \left(a^2+b^2\right)^3}-\frac{(B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{(B-i C) \log (\tan (c+d x)+i)}{(a-i b)^3}}{2 d}","\frac{B \log (\sin (c+d x))}{a^3 d}+\frac{b (b B-a C)}{2 a d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(-2 a^3 C+3 a^2 b B+b^3 B\right)}{a^2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{x \left(a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right)}{\left(a^2+b^2\right)^3}-\frac{b \left(-3 a^5 C+6 a^4 b B+a^3 b^2 C+3 a^2 b^3 B+b^5 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left(a^2+b^2\right)^3}",1,"(-(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^3) + (2*B*Log[Tan[c + d*x]])/a^3 - ((B - I*C)*Log[I + Tan[c + d*x]])/(a - I*b)^3 - (2*b*(6*a^4*b*B + 3*a^2*b^3*B + b^5*B - 3*a^5*C + a^3*b^2*C)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^3) + (b*(b*B - a*C))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (2*b*(3*a^2*b*B + b^3*B - 2*a^3*C))/(a^2*(a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*d)","C",1
44,1,288,287,6.4248385,"\int \frac{\cot ^3(c+d x) \left(B \tan (c+d x)+C \tan ^2(c+d x)\right)}{(a+b \tan (c+d x))^3} \, dx","Integrate[(Cot[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]","-\frac{(3 b B-a C) \log (\tan (c+d x))}{a^4 d}-\frac{B \cot (c+d x)}{a^3 d}-\frac{b^2 (b B-a C)}{2 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{b^2 \left(-3 a^3 C+4 a^2 b B-a b^2 C+2 b^3 B\right)}{a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b^2 \left(-6 a^5 C+10 a^4 b B-3 a^3 b^2 C+9 a^2 b^3 B-a b^4 C+3 b^5 B\right) \log (a+b \tan (c+d x))}{a^4 d \left(a^2+b^2\right)^3}+\frac{(B+i C) \log (-\tan (c+d x)+i)}{2 d (-b+i a)^3}-\frac{(C+i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^3}","-\frac{(3 b B-a C) \log (\sin (c+d x))}{a^4 d}-\frac{b \left(2 a^2 B-a b C+3 b^2 B\right)}{2 a^2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}-\frac{x \left(a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right)}{\left(a^2+b^2\right)^3}-\frac{b \left(a^4 B-3 a^3 b C+6 a^2 b^2 B-a b^3 C+3 b^4 B\right)}{a^3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}+\frac{b^2 \left(-6 a^5 C+10 a^4 b B-3 a^3 b^2 C+9 a^2 b^3 B-a b^4 C+3 b^5 B\right) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left(a^2+b^2\right)^3}-\frac{B \cot (c+d x)}{a d (a+b \tan (c+d x))^2}",1,"-((B*Cot[c + d*x])/(a^3*d)) + ((B + I*C)*Log[I - Tan[c + d*x]])/(2*(I*a - b)^3*d) - ((3*b*B - a*C)*Log[Tan[c + d*x]])/(a^4*d) - ((I*B + C)*Log[I + Tan[c + d*x]])/(2*(a - I*b)^3*d) + (b^2*(10*a^4*b*B + 9*a^2*b^3*B + 3*b^5*B - 6*a^5*C - 3*a^3*b^2*C - a*b^4*C)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b^2*(b*B - a*C))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (b^2*(4*a^2*b*B + 2*b^3*B - 3*a^3*C - a*b^2*C))/(a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","C",1
45,1,110,132,0.4081514,"\int \tan ^2(c+d x) (b \tan (c+d x))^n \left(A+B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Tan[c + d*x]^2*(b*Tan[c + d*x])^n*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{\tan ^3(c+d x) (b \tan (c+d x))^n \left((n+4) (A-C) \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\tan ^2(c+d x)\right)+B (n+3) \tan (c+d x) \, _2F_1\left(1,\frac{n+4}{2};\frac{n+6}{2};-\tan ^2(c+d x)\right)+C (n+4)\right)}{d (n+3) (n+4)}","\frac{(A-C) (b \tan (c+d x))^{n+3} \, _2F_1\left(1,\frac{n+3}{2};\frac{n+5}{2};-\tan ^2(c+d x)\right)}{b^3 d (n+3)}+\frac{B (b \tan (c+d x))^{n+4} \, _2F_1\left(1,\frac{n+4}{2};\frac{n+6}{2};-\tan ^2(c+d x)\right)}{b^4 d (n+4)}+\frac{C (b \tan (c+d x))^{n+3}}{b^3 d (n+3)}",1,"(Tan[c + d*x]^3*(b*Tan[c + d*x])^n*(C*(4 + n) + (A - C)*(4 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Tan[c + d*x]^2] + B*(3 + n)*Hypergeometric2F1[1, (4 + n)/2, (6 + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]))/(d*(3 + n)*(4 + n))","A",1
46,1,115,154,0.3934925,"\int \tan ^m(c+d x) (b \tan (c+d x))^n \left(A+B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Tan[c + d*x]^m*(b*Tan[c + d*x])^n*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{\tan ^{m+1}(c+d x) (b \tan (c+d x))^n \left(\frac{(A-C) \, _2F_1\left(1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);-\tan ^2(c+d x)\right)}{m+n+1}+\frac{B \tan (c+d x) \, _2F_1\left(1,\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);-\tan ^2(c+d x)\right)}{m+n+2}+\frac{C}{m+n+1}\right)}{d}","\frac{(A-C) \tan ^{m+1}(c+d x) (b \tan (c+d x))^n \, _2F_1\left(1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);-\tan ^2(c+d x)\right)}{d (m+n+1)}+\frac{B \tan ^{m+2}(c+d x) (b \tan (c+d x))^n \, _2F_1\left(1,\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);-\tan ^2(c+d x)\right)}{d (m+n+2)}+\frac{C \tan ^{m+1}(c+d x) (b \tan (c+d x))^n}{d (m+n+1)}",1,"(Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n*(C/(1 + m + n) + ((A - C)*Hypergeometric2F1[1, (1 + m + n)/2, (3 + m + n)/2, -Tan[c + d*x]^2])/(1 + m + n) + (B*Hypergeometric2F1[1, (2 + m + n)/2, (4 + m + n)/2, -Tan[c + d*x]^2]*Tan[c + d*x])/(2 + m + n)))/d","A",1
47,1,133,170,0.5334515,"\int \tan ^m(c+d x) \sqrt{b \tan (c+d x)} \left(A+B \tan (c+d x)+C \tan ^2(c+d x)\right) \, dx","Integrate[Tan[c + d*x]^m*Sqrt[b*Tan[c + d*x]]*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]","\frac{2 \sqrt{b \tan (c+d x)} \tan ^{m+1}(c+d x) \left((2 m+5) (A-C) \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(c+d x)\right)+B (2 m+3) \tan (c+d x) \, _2F_1\left(1,\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);-\tan ^2(c+d x)\right)+C (2 m+5)\right)}{d (2 m+3) (2 m+5)}","\frac{2 (A-C) \sqrt{b \tan (c+d x)} \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(c+d x)\right)}{d (2 m+3)}+\frac{2 B \sqrt{b \tan (c+d x)} \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{1}{4} (2 m+5);\frac{1}{4} (2 m+9);-\tan ^2(c+d x)\right)}{d (2 m+5)}+\frac{2 C \sqrt{b \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)}",1,"(2*Tan[c + d*x]^(1 + m)*Sqrt[b*Tan[c + d*x]]*(C*(5 + 2*m) + (A - C)*(5 + 2*m)*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[c + d*x]^2] + B*(3 + 2*m)*Hypergeometric2F1[1, (5 + 2*m)/4, (9 + 2*m)/4, -Tan[c + d*x]^2]*Tan[c + d*x]))/(d*(3 + 2*m)*(5 + 2*m))","A",1
48,1,133,170,0.5366743,"\int \frac{\tan ^m(c+d x) \left(A+B \tan (c+d x)+C \tan ^2(c+d x)\right)}{\sqrt{b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2))/Sqrt[b*Tan[c + d*x]],x]","\frac{2 \tan ^{m+1}(c+d x) \left((2 m+3) (A-C) \, _2F_1\left(1,\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);-\tan ^2(c+d x)\right)+B (2 m+1) \tan (c+d x) \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(c+d x)\right)+C (2 m+3)\right)}{d (2 m+1) (2 m+3) \sqrt{b \tan (c+d x)}}","\frac{2 (A-C) \tan ^{m+1}(c+d x) \, _2F_1\left(1,\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);-\tan ^2(c+d x)\right)}{d (2 m+1) \sqrt{b \tan (c+d x)}}+\frac{2 B \tan ^{m+2}(c+d x) \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(c+d x)\right)}{d (2 m+3) \sqrt{b \tan (c+d x)}}+\frac{2 C \tan ^{m+1}(c+d x)}{d (2 m+1) \sqrt{b \tan (c+d x)}}",1,"(2*Tan[c + d*x]^(1 + m)*(C*(3 + 2*m) + (A - C)*(3 + 2*m)*Hypergeometric2F1[1, (1 + 2*m)/4, (5 + 2*m)/4, -Tan[c + d*x]^2] + B*(1 + 2*m)*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[c + d*x]^2]*Tan[c + d*x]))/(d*(1 + 2*m)*(3 + 2*m)*Sqrt[b*Tan[c + d*x]])","A",1
49,0,0,328,28.5590985,"\int \frac{\tan ^m(c+d x) \left(A+B \tan (c+d x)+C \tan ^2(c+d x)\right)}{\sqrt{a+b \tan (c+d x)}} \, dx","Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2))/Sqrt[a + b*Tan[c + d*x]],x]","\int \frac{\tan ^m(c+d x) \left(A+B \tan (c+d x)+C \tan ^2(c+d x)\right)}{\sqrt{a+b \tan (c+d x)}} \, dx","-\frac{\left(\sqrt{-b^2} (A-C)+b B\right) \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} \left(-\frac{b \tan (c+d x)}{a}\right)^{-m} F_1\left(\frac{1}{2};1,-m;\frac{3}{2};\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}},\frac{b \tan (c+d x)}{a}+1\right)}{b d \left(a-\sqrt{-b^2}\right)}-\frac{\left(b B-\sqrt{-b^2} (A-C)\right) \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} \left(-\frac{b \tan (c+d x)}{a}\right)^{-m} F_1\left(\frac{1}{2};1,-m;\frac{3}{2};\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}},\frac{b \tan (c+d x)}{a}+1\right)}{b d \left(a+\sqrt{-b^2}\right)}+\frac{2 C \tan ^m(c+d x) \sqrt{a+b \tan (c+d x)} \left(-\frac{b \tan (c+d x)}{a}\right)^{-m} \, _2F_1\left(\frac{1}{2},-m;\frac{3}{2};\frac{b \tan (c+d x)}{a}+1\right)}{b d}",1,"Integrate[(Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2))/Sqrt[a + b*Tan[c + d*x]], x]","F",-1
50,1,300,353,6.3816419,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}-\frac{\frac{(a C d-5 b (B d+c C)) (a+b \tan (e+f x))^4}{4 b f}-\frac{5 \left(3 (-a A d-a B c+a C d+A b c-b B d-b c C) \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)-(d (A-C)+B c) \left(-6 b^2 \left(6 a^2-b^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)\right)}{6 f}}{5 b}","\frac{b \tan (e+f x) \left(a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right)}{f}-\frac{\log (\cos (e+f x)) \left(a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right)}{f}+x \left(a^3 (A c-B d-c C)-3 a^2 b (d (A-C)+B c)-3 a b^2 (A c-B d-c C)+b^3 (d (A-C)+B c)\right)+\frac{(d (A-C)+B c) (a+b \tan (e+f x))^3}{3 f}+\frac{(a+b \tan (e+f x))^2 (a A d+a B c-a C d+A b c-b B d-b c C)}{2 f}-\frac{(a C d-5 b (B d+c C)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac{C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}",1,"(C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^4)/(5*b*f) - (((a*C*d - 5*b*(c*C + B*d))*(a + b*Tan[e + f*x])^4)/(4*b*f) - (5*(3*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*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) - (B*c + (A - 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)))/(6*f))/(5*b)","C",1
51,1,243,248,3.434373,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{-6 (-a A d-a B c+a C d+A b c-b B d-b c C) \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 (d (A-C)+B c) \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)+\frac{(4 b (B d+c C)-a C d) (a+b \tan (e+f x))^3}{b}+3 C d \tan (e+f x) (a+b \tan (e+f x))^3}{12 b f}","-\frac{\log (\cos (e+f x)) \left(a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right)}{f}+x \left(a^2 (A c-B d-c C)-2 a b (d (A-C)+B c)-b^2 (A c-B d-c C)\right)+\frac{(d (A-C)+B c) (a+b \tan (e+f x))^2}{2 f}+\frac{b \tan (e+f x) (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac{(a C d-4 b (B d+c C)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac{C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f}",1,"(((-(a*C*d) + 4*b*(c*C + B*d))*(a + b*Tan[e + f*x])^3)/b + 3*C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^3 - 6*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)*(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*(B*c + (A - 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))/(12*b*f)","C",1
52,1,161,161,1.6168445,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{3 (a+i b) (d-i c) (A+i B-C) \log (-\tan (e+f x)+i)+3 (a-i b) (d+i c) (A-i B-C) \log (\tan (e+f x)+i)+6 d \tan (e+f x) (a B+A b-b C)+\frac{(3 a C d+3 b B d-b c C) (c+d \tan (e+f x))^2}{d^2}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^2}{d}}{6 f}","-\frac{\log (\cos (e+f x)) (a A d+a B c-a C d+A b c-b B d-b c C)}{f}+x (a (A c-B d-c C)-b (d (A-C)+B c))+\frac{d \tan (e+f x) (a B+A b-b C)}{f}-\frac{(-3 a C d-3 b B d+b c C) (c+d \tan (e+f x))^2}{6 d^2 f}+\frac{b C \tan (e+f x) (c+d \tan (e+f x))^2}{3 d f}",1,"(3*(a + I*b)*(A + I*B - C)*((-I)*c + d)*Log[I - Tan[e + f*x]] + 3*(a - I*b)*(A - I*B - C)*(I*c + d)*Log[I + Tan[e + f*x]] + 6*(A*b + a*B - b*C)*d*Tan[e + f*x] + ((-(b*c*C) + 3*b*B*d + 3*a*C*d)*(c + d*Tan[e + f*x])^2)/d^2 + (2*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^2)/d)/(6*f)","C",1
53,1,76,73,0.4695165,"\int (c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{-2 (d (A-C)+B c) \log (\cos (e+f x))+2 A c f x-2 (B d+c C) \tan ^{-1}(\tan (e+f x))+2 (B d+c C) \tan (e+f x)+C d \tan ^2(e+f x)}{2 f}","-\frac{(d (A-C)+B c) \log (\cos (e+f x))}{f}+x (A c-B d-c C)+\frac{B d \tan (e+f x)}{f}+\frac{C (c+d \tan (e+f x))^2}{2 d f}",1,"(2*A*c*f*x - 2*(c*C + B*d)*ArcTan[Tan[e + f*x]] - 2*(B*c + (A - C)*d)*Log[Cos[e + f*x]] + 2*(c*C + B*d)*Tan[e + f*x] + C*d*Tan[e + f*x]^2)/(2*f)","A",1
54,1,148,156,1.1934013,"\int \frac{(c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{a+b \tan (e+f x)} \, dx","Integrate[((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]","\frac{\frac{2 (b c-a d) \left(a (a C-b B)+A b^2\right) \log (a+b \tan (e+f x))}{b^2 \left(a^2+b^2\right)}+\frac{(d-i c) (A+i B-C) \log (-\tan (e+f x)+i)}{a+i b}+\frac{(d+i c) (A-i B-C) \log (\tan (e+f x)+i)}{a-i b}+\frac{2 C d \tan (e+f x)}{b}}{2 f}","\frac{(b c-a d) \left(A b^2-a (b B-a C)\right) \log (a+b \tan (e+f x))}{b^2 f \left(a^2+b^2\right)}+\frac{\log (\cos (e+f x)) (-a A d-a B c+a C d+A b c-b B d-b c C)}{f \left(a^2+b^2\right)}+\frac{x (a (A c-B d-c C)+b (d (A-C)+B c))}{a^2+b^2}+\frac{C d \tan (e+f x)}{b f}",1,"(((A + I*B - C)*((-I)*c + d)*Log[I - Tan[e + f*x]])/(a + I*b) + ((A - I*B - C)*(I*c + d)*Log[I + Tan[e + f*x]])/(a - I*b) + (2*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)) + (2*C*d*Tan[e + f*x])/b)/(2*f)","C",1
55,1,589,265,6.86395,"\int \frac{(c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^2} \, dx","Integrate[((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]","\frac{-2 i a \tan ^{-1}(\tan (e+f x)) (a+b \tan (e+f x)) \left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right)+a^2 \left(2 (a+i b)^2 (e+f x) \left(i a^2 C d+2 a b C d+A b^2 (c-i d)+b^2 (-i B c-B d-c C)\right)-2 C d \left(a^2+b^2\right)^2 \log (\cos (e+f x))+\left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right) \log \left((a \cos (e+f x)+b \sin (e+f x))^2\right)\right)+b \tan (e+f x) \left(-2 a C d \left(a^2+b^2\right)^2 \log (\cos (e+f x))+a \left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right) \log \left((a \cos (e+f x)+b \sin (e+f x))^2\right)+2 (a+i b) \left(i a^4 C d (e+f x+i)+a^3 b (c C+d (B+C (e+f x+i)))-i a^2 b^2 (i A c (e+f x)+A d (e+f x-i)+B c (e+f x-i)-i B d (e+f x+i)-i c C (e+f x+i)-2 C d (e+f x))+a b^3 (A c (i e+i f x+1)+A d (e+f x+i)+B c (e+f x+i)-i B d (e+f x)-i c C (e+f x))-i A b^4 c\right)\right)}{2 a b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}","-\frac{(b c-a d) \left(A b^2-a (b B-a C)\right)}{b^2 f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{\log (\cos (e+f x)) \left(-\left(a^2 (d (A-C)+B c)\right)+2 a b (A c-B d-c C)+b^2 (d (A-C)+B c)\right)}{f \left(a^2+b^2\right)^2}+\frac{x \left(a^2 (A c-B d-c C)+2 a b (d (A-C)+B c)-b^2 (A c-B d-c C)\right)}{\left(a^2+b^2\right)^2}+\frac{\left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right) \log (a+b \tan (e+f x))}{b^2 f \left(a^2+b^2\right)^2}",1,"(a^2*(2*(a + I*b)^2*(A*b^2*(c - I*d) + I*a^2*C*d + 2*a*b*C*d + b^2*((-I)*B*c - c*C - B*d))*(e + f*x) - 2*(a^2 + b^2)^2*C*d*Log[Cos[e + f*x]] + (a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]) + b*(2*(a + I*b)*((-I)*A*b^4*c + I*a^4*C*d*(I + e + f*x) + a*b^3*(A*c*(1 + I*e + I*f*x) - I*c*C*(e + f*x) - I*B*d*(e + f*x) + B*c*(I + e + f*x) + A*d*(I + e + f*x)) - I*a^2*b^2*(I*A*c*(e + f*x) - 2*C*d*(e + f*x) + B*c*(-I + e + f*x) + A*d*(-I + e + f*x) - I*c*C*(I + e + f*x) - I*B*d*(I + e + f*x)) + a^3*b*(c*C + d*(B + C*(I + e + f*x)))) - 2*a*(a^2 + b^2)^2*C*d*Log[Cos[e + f*x]] + a*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2])*Tan[e + f*x] - (2*I)*a*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*ArcTan[Tan[e + f*x]]*(a + b*Tan[e + f*x]))/(2*a*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))","C",1
56,1,379,320,6.3822351,"\int \frac{(c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^3} \, dx","Integrate[((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]","-\frac{C (c+d \tan (e+f x))}{b f (a+b \tan (e+f x))^2}-\frac{-\frac{-a C d-b B d+b c C}{2 b f (a+b \tan (e+f x))^2}+\frac{\frac{\left(2 a b^2 (d (A-C)+B c)-2 b^3 (A c-B d-c C)\right) \left(-\frac{2 a b}{\left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{b}{2 \left(a^2+b^2\right) (a+b \tan (e+f x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right)^3}-\frac{\log (-\tan (e+f x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (e+f x)+i)}{2 (b+i a)^3}\right)}{b}-2 b (d (A-C)+B c) \left(-\frac{b}{\left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{2 a b \log (a+b \tan (e+f x))}{\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)}{2 b f}}{b}","-\frac{(b c-a d) \left(A b^2-a (b B-a C)\right)}{2 b^2 f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)}{b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}+\frac{\left(-\left(a^3 (d (A-C)+B c)\right)+3 a^2 b (A c-B d-c C)+3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c 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 (A c-B d-c C)+3 a^2 b (d (A-C)+B c)-3 a b^2 (A c-B d-c C)-b^3 (d (A-C)+B c)\right)}{\left(a^2+b^2\right)^3}",1,"-((C*(c + d*Tan[e + f*x]))/(b*f*(a + b*Tan[e + f*x])^2)) - (-1/2*(b*c*C - b*B*d - a*C*d)/(b*f*(a + b*Tan[e + f*x])^2) + (((-2*b^3*(A*c - c*C - B*d) + 2*a*b^2*(B*c + (A - C)*d))*(-1/2*Log[I - Tan[e + f*x]]/(I*a - b)^3 + Log[I + Tan[e + f*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[e + f*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b^2)*(a + b*Tan[e + f*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[e + f*x]))))/b - 2*b*(B*c + (A - C)*d)*(((-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 + (2*a*b*Log[a + b*Tan[e + f*x]])/(a^2 + b^2)^2 - b/((a^2 + b^2)*(a + b*Tan[e + f*x]))))/(2*b*f))/b","C",1
57,1,573,661,6.6459157,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}+\frac{-\frac{3 (-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}+\frac{\frac{3 b \tan (e+f x) (c+d \tan (e+f x))^3 \left(5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right)}{2 d f}-\frac{\frac{(c+d \tan (e+f x))^3 \left(b \left(6 c \left(5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right)-120 d^3 \left(a^2 B+2 a b (A-C)-b^2 B\right)\right)-24 a d \left(5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right)\right)}{3 d f}-\frac{60 \left(d^2 \left(a^3 (B c-d (A-C))+3 a^2 b (A c+B d-c C)-3 a b^2 (B c-d (A-C))-b^3 (A c+B d-c C)\right) \left(-i (c-i d)^2 \log (\tan (e+f x)+i)+i (c+i d)^2 \log (-\tan (e+f x)+i)-2 d^2 \tan (e+f x)\right)+d^2 \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-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)}{f}}{4 d}}{5 d}}{6 d}","\frac{\log (\cos (e+f x)) \left(-\left(a^3 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)+3 a^2 b \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+3 a b^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-b^3 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)\right)}{f}-x \left(a^3 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+3 a^2 b \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-3 a b^2 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-b^3 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)+\frac{(c+d \tan (e+f x))^3 \left(4 a^3 C d^3-3 a^2 b d^2 (3 c C-16 B d)+3 a b^2 d \left(20 d^2 (A-C)-5 B c d+2 c^2 C\right)-\left(b^3 \left(5 c d^2 (A-C)-2 B c^2 d+20 B d^3+c^3 C\right)\right)\right)}{60 d^4 f}+\frac{\left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right) (c+d \tan (e+f x))^2}{2 f}+\frac{d \tan (e+f x) \left(a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right)}{f}+\frac{b \tan (e+f x) (c+d \tan (e+f x))^3 \left(5 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right)}{20 d^3 f}-\frac{(-a C d-2 b B d+b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{10 d^2 f}+\frac{C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^3}{6 d f}",1,"(C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(6*d*f) + ((-3*(b*c*C - 2*b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(5*d*f) + ((3*b*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(2*d*f) - (((-24*a*d*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d)) + b*(-120*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3 + 6*c*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))))*(c + d*Tan[e + f*x])^3)/(3*d*f) - (60*(d^2*(3*a^2*b*(A*c - c*C + B*d) - b^3*(A*c - c*C + B*d) + a^3*(B*c - (A - C)*d) - 3*a*b^2*(B*c - (A - C)*d))*(I*(c + I*d)^2*Log[I - Tan[e + f*x]] - I*(c - I*d)^2*Log[I + Tan[e + f*x]] - 2*d^2*Tan[e + f*x]) + (a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^2*((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)))/f)/(4*d))/(5*d))/(6*d)","C",1
58,1,383,443,6.4977339,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}+\frac{\frac{b \tan (e+f x) (2 a C d+5 b B d-2 b c C) (c+d \tan (e+f x))^3}{4 d f}-\frac{\frac{(c+d \tan (e+f x))^3 \left(-8 a^2 C d^2+10 a b d (c C-4 B d)-\left(b^2 \left(20 d^2 (A-C)-5 B c d+2 c^2 C\right)\right)\right)}{3 d f}-\frac{10 \left(d \left(a^2 (B c-d (A-C))+2 a b (A c+B d-c C)-b^2 (B c-d (A-C))\right) \left(-i (c-i d)^2 \log (\tan (e+f x)+i)+i (c+i d)^2 \log (-\tan (e+f x)+i)-2 d^2 \tan (e+f x)\right)+d \left(a^2 B+2 a b (A-C)-b^2 B\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)}{f}}{4 d}}{5 d}","\frac{\log (\cos (e+f x)) \left(-\left(a^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)+2 a b \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+b^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)}{f}-x \left(a^2 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+2 a b \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-b^2 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)\right)+\frac{(c+d \tan (e+f x))^3 \left(8 a^2 C d^2-10 a b d (c C-4 B d)+b^2 \left(20 d^2 (A-C)-5 B c d+2 c^2 C\right)\right)}{60 d^3 f}+\frac{\left(a^2 B+2 a b (A-C)-b^2 B\right) (c+d \tan (e+f x))^2}{2 f}+\frac{d \tan (e+f x) \left(a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right)}{f}-\frac{b \tan (e+f x) (-2 a C d-5 b B d+2 b c C) (c+d \tan (e+f x))^3}{20 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3}{5 d f}",1,"(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(5*d*f) + ((b*(-2*b*c*C + 5*b*B*d + 2*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(4*d*f) - (((-8*a^2*C*d^2 + 10*a*b*d*(c*C - 4*B*d) - b^2*(2*c^2*C - 5*B*c*d + 20*(A - C)*d^2))*(c + d*Tan[e + f*x])^3)/(3*d*f) - (10*(d*(2*a*b*(A*c - c*C + B*d) + a^2*(B*c - (A - C)*d) - b^2*(B*c - (A - C)*d))*(I*(c + I*d)^2*Log[I - Tan[e + f*x]] - I*(c - I*d)^2*Log[I + Tan[e + f*x]] - 2*d^2*Tan[e + f*x]) + (a^2*B - b^2*B + 2*a*b*(A - C))*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)))/f)/(4*d))/(5*d)","C",1
59,1,241,266,2.8872054,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{6 (-a A d+a B c+a C d+A b c+b B d-b c C) \left(-2 d^2 \tan (e+f x)+i \left((c+i d)^2 \log (-\tan (e+f x)+i)-(c-i d)^2 \log (\tan (e+f x)+i)\right)\right)+6 (a B+A b-b C) \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)+\frac{(4 a C d+4 b B d-b c C) (c+d \tan (e+f x))^3}{d}+3 b C \tan (e+f x) (c+d \tan (e+f x))^3}{12 d f}","-\frac{\log (\cos (e+f x)) \left(A \left(2 a c d+b \left(c^2-d^2\right)\right)+a \left(B c^2-B d^2-2 c C d\right)-b \left(2 B c d+c^2 C-C d^2\right)\right)}{f}-x \left(a \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+b \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)+\frac{(a B+A b-b C) (c+d \tan (e+f x))^2}{2 f}+\frac{d \tan (e+f x) (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac{(-4 a C d-4 b B d+b c C) (c+d \tan (e+f x))^3}{12 d^2 f}+\frac{b C \tan (e+f x) (c+d \tan (e+f x))^3}{4 d f}",1,"(((-(b*c*C) + 4*b*B*d + 4*a*C*d)*(c + d*Tan[e + f*x])^3)/d + 3*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^3 + 6*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)*(I*((c + I*d)^2*Log[I - Tan[e + f*x]] - (c - I*d)^2*Log[I + Tan[e + f*x]]) - 2*d^2*Tan[e + f*x]) + 6*(A*b + a*B - b*C)*((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))/(12*d*f)","C",1
60,1,176,131,1.1825795,"\int (c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{3 (d (C-A)+B c) \left(-2 d^2 \tan (e+f x)+i \left((c+i d)^2 \log (-\tan (e+f x)+i)-(c-i d)^2 \log (\tan (e+f x)+i)\right)\right)+3 B \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)+2 C (c+d \tan (e+f x))^3}{6 d f}","-\frac{\left(2 c d (A-C)+B \left(c^2-d^2\right)\right) \log (\cos (e+f x))}{f}-x \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+\frac{d \tan (e+f x) (d (A-C)+B c)}{f}+\frac{B (c+d \tan (e+f x))^2}{2 f}+\frac{C (c+d \tan (e+f x))^3}{3 d f}",1,"(2*C*(c + d*Tan[e + f*x])^3 + 3*(B*c + (-A + C)*d)*(I*((c + I*d)^2*Log[I - Tan[e + f*x]] - (c - I*d)^2*Log[I + Tan[e + f*x]]) - 2*d^2*Tan[e + f*x]) + 3*B*((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))/(6*d*f)","C",1
61,1,190,254,3.0299073,"\int \frac{(c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{a+b \tan (e+f x)} \, dx","Integrate[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]","\frac{\frac{2 (b c-a d)^2 \left(a (a C-b B)+A b^2\right) \log (a+b \tan (e+f x))}{b^2 \left(a^2+b^2\right)}+\frac{b (c-i d)^2 (i A+B-i C) \log (\tan (e+f x)+i)}{a-i b}+\frac{b (c+i d)^2 (-i A+B+i C) \log (-\tan (e+f x)+i)}{a+i b}+\frac{2 d \tan (e+f x) (-a C d+b B d+b c C)}{b}+C (c+d \tan (e+f x))^2}{2 b f}","-\frac{\log (\cos (e+f x)) \left(A \left(2 a c d-b \left(c^2-d^2\right)\right)+a \left(B c^2-B d^2-2 c C d\right)+b \left(2 B c d+c^2 C-C d^2\right)\right)}{f \left(a^2+b^2\right)}-\frac{x \left(a \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-b \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)}{a^2+b^2}+\frac{(b c-a d)^2 \left(A b^2-a (b B-a C)\right) \log (a+b \tan (e+f x))}{b^3 f \left(a^2+b^2\right)}+\frac{d \tan (e+f x) (-a C d+b B d+b c C)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f}",1,"((b*((-I)*A + B + I*C)*(c + I*d)^2*Log[I - Tan[e + f*x]])/(a + I*b) + (b*(I*A + B - I*C)*(c - I*d)^2*Log[I + Tan[e + f*x]])/(a - I*b) + (2*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)^2*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)) + (2*d*(b*c*C + b*B*d - a*C*d)*Tan[e + f*x])/b + C*(c + d*Tan[e + f*x])^2)/(2*b*f)","C",1
62,1,2640,415,8.0201899,"\int \frac{(c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^2} \, dx","Integrate[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","-\frac{\log (\cos (e+f x)) \left(a^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)+2 a b \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-b^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)}{f \left(a^2+b^2\right)^2}-\frac{x \left(a^2 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-2 a b \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-b^2 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)\right)}{\left(a^2+b^2\right)^2}-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^2}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{d^2 \tan (e+f x) \left(2 a^2 C-a b B+A b^2+b^2 C\right)}{b^2 f \left(a^2+b^2\right)}-\frac{(b c-a d) \left(-2 a^4 C d+a^3 b B d+a^2 b^2 (B c-4 C d)-a b^3 (2 A c-3 B d-2 c C)-b^4 (2 A d+B c)\right) \log (a+b \tan (e+f x))}{b^3 f \left(a^2+b^2\right)^2}",1,"((-I)*(-2*a^6*A*b^6*c^2 + (2*I)*a^5*A*b^7*c^2 - 2*a^4*A*b^8*c^2 + (2*I)*a^3*A*b^9*c^2 + a^7*b^5*B*c^2 - I*a^6*b^6*B*c^2 - a^3*b^9*B*c^2 + I*a^2*b^10*B*c^2 + 2*a^6*b^6*c^2*C - (2*I)*a^5*b^7*c^2*C + 2*a^4*b^8*c^2*C - (2*I)*a^3*b^9*c^2*C + 2*a^7*A*b^5*c*d - (2*I)*a^6*A*b^6*c*d - 2*a^3*A*b^9*c*d + (2*I)*a^2*A*b^10*c*d + 4*a^6*b^6*B*c*d - (4*I)*a^5*b^7*B*c*d + 4*a^4*b^8*B*c*d - (4*I)*a^3*b^9*B*c*d - 2*a^9*b^3*c*C*d + (2*I)*a^8*b^4*c*C*d - 8*a^7*b^5*c*C*d + (8*I)*a^6*b^6*c*C*d - 6*a^5*b^7*c*C*d + (6*I)*a^4*b^8*c*C*d + 2*a^6*A*b^6*d^2 - (2*I)*a^5*A*b^7*d^2 + 2*a^4*A*b^8*d^2 - (2*I)*a^3*A*b^9*d^2 - a^9*b^3*B*d^2 + I*a^8*b^4*B*d^2 - 4*a^7*b^5*B*d^2 + (4*I)*a^6*b^6*B*d^2 - 3*a^5*b^7*B*d^2 + (3*I)*a^4*b^8*B*d^2 + 2*a^10*b^2*C*d^2 - (2*I)*a^9*b^3*C*d^2 + 6*a^8*b^4*C*d^2 - (6*I)*a^7*b^5*C*d^2 + 4*a^6*b^6*C*d^2 - (4*I)*a^5*b^7*C*d^2)*(e + f*x)*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)/(a^2*(a - I*b)^4*(a + I*b)^3*b^5*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2) - (I*(2*a*A*b^4*c^2 - a^2*b^3*B*c^2 + b^5*B*c^2 - 2*a*b^4*c^2*C - 2*a^2*A*b^3*c*d + 2*A*b^5*c*d - 4*a*b^4*B*c*d + 2*a^4*b*c*C*d + 6*a^2*b^3*c*C*d - 2*a*A*b^4*d^2 + a^4*b*B*d^2 + 3*a^2*b^3*B*d^2 - 2*a^5*C*d^2 - 4*a^3*b^2*C*d^2)*ArcTan[Tan[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)/(b^3*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2) + ((-2*b*c*C*d - b*B*d^2 + 2*a*C*d^2)*Log[Cos[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)/(b^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2) + ((2*a*A*b^4*c^2 - a^2*b^3*B*c^2 + b^5*B*c^2 - 2*a*b^4*c^2*C - 2*a^2*A*b^3*c*d + 2*A*b^5*c*d - 4*a*b^4*B*c*d + 2*a^4*b*c*C*d + 6*a^2*b^3*c*C*d - 2*a*A*b^4*d^2 + a^4*b*B*d^2 + 3*a^2*b^3*B*d^2 - 2*a^5*C*d^2 - 4*a^3*b^2*C*d^2)*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)/(2*b^3*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2) + (Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(a^5*b*C*d^2 + 2*a^3*b^3*C*d^2 + a*b^5*C*d^2 + a^4*A*b^2*c^2*(e + f*x) - a^2*A*b^4*c^2*(e + f*x) + 2*a^3*b^3*B*c^2*(e + f*x) - a^4*b^2*c^2*C*(e + f*x) + a^2*b^4*c^2*C*(e + f*x) + 4*a^3*A*b^3*c*d*(e + f*x) - 2*a^4*b^2*B*c*d*(e + f*x) + 2*a^2*b^4*B*c*d*(e + f*x) - 4*a^3*b^3*c*C*d*(e + f*x) - a^4*A*b^2*d^2*(e + f*x) + a^2*A*b^4*d^2*(e + f*x) - 2*a^3*b^3*B*d^2*(e + f*x) + a^4*b^2*C*d^2*(e + f*x) - a^2*b^4*C*d^2*(e + f*x) - a^5*b*C*d^2*Cos[2*(e + f*x)] - 2*a^3*b^3*C*d^2*Cos[2*(e + f*x)] - a*b^5*C*d^2*Cos[2*(e + f*x)] + a^4*A*b^2*c^2*(e + f*x)*Cos[2*(e + f*x)] - a^2*A*b^4*c^2*(e + f*x)*Cos[2*(e + f*x)] + 2*a^3*b^3*B*c^2*(e + f*x)*Cos[2*(e + f*x)] - a^4*b^2*c^2*C*(e + f*x)*Cos[2*(e + f*x)] + a^2*b^4*c^2*C*(e + f*x)*Cos[2*(e + f*x)] + 4*a^3*A*b^3*c*d*(e + f*x)*Cos[2*(e + f*x)] - 2*a^4*b^2*B*c*d*(e + f*x)*Cos[2*(e + f*x)] + 2*a^2*b^4*B*c*d*(e + f*x)*Cos[2*(e + f*x)] - 4*a^3*b^3*c*C*d*(e + f*x)*Cos[2*(e + f*x)] - a^4*A*b^2*d^2*(e + f*x)*Cos[2*(e + f*x)] + a^2*A*b^4*d^2*(e + f*x)*Cos[2*(e + f*x)] - 2*a^3*b^3*B*d^2*(e + f*x)*Cos[2*(e + f*x)] + a^4*b^2*C*d^2*(e + f*x)*Cos[2*(e + f*x)] - a^2*b^4*C*d^2*(e + f*x)*Cos[2*(e + f*x)] + a^2*A*b^4*c^2*Sin[2*(e + f*x)] + A*b^6*c^2*Sin[2*(e + f*x)] - a^3*b^3*B*c^2*Sin[2*(e + f*x)] - a*b^5*B*c^2*Sin[2*(e + f*x)] + a^4*b^2*c^2*C*Sin[2*(e + f*x)] + a^2*b^4*c^2*C*Sin[2*(e + f*x)] - 2*a^3*A*b^3*c*d*Sin[2*(e + f*x)] - 2*a*A*b^5*c*d*Sin[2*(e + f*x)] + 2*a^4*b^2*B*c*d*Sin[2*(e + f*x)] + 2*a^2*b^4*B*c*d*Sin[2*(e + f*x)] - 2*a^5*b*c*C*d*Sin[2*(e + f*x)] - 2*a^3*b^3*c*C*d*Sin[2*(e + f*x)] + a^4*A*b^2*d^2*Sin[2*(e + f*x)] + a^2*A*b^4*d^2*Sin[2*(e + f*x)] - a^5*b*B*d^2*Sin[2*(e + f*x)] - a^3*b^3*B*d^2*Sin[2*(e + f*x)] + 2*a^6*C*d^2*Sin[2*(e + f*x)] + 3*a^4*b^2*C*d^2*Sin[2*(e + f*x)] + a^2*b^4*C*d^2*Sin[2*(e + f*x)] + a^3*A*b^3*c^2*(e + f*x)*Sin[2*(e + f*x)] - a*A*b^5*c^2*(e + f*x)*Sin[2*(e + f*x)] + 2*a^2*b^4*B*c^2*(e + f*x)*Sin[2*(e + f*x)] - a^3*b^3*c^2*C*(e + f*x)*Sin[2*(e + f*x)] + a*b^5*c^2*C*(e + f*x)*Sin[2*(e + f*x)] + 4*a^2*A*b^4*c*d*(e + f*x)*Sin[2*(e + f*x)] - 2*a^3*b^3*B*c*d*(e + f*x)*Sin[2*(e + f*x)] + 2*a*b^5*B*c*d*(e + f*x)*Sin[2*(e + f*x)] - 4*a^2*b^4*c*C*d*(e + f*x)*Sin[2*(e + f*x)] - a^3*A*b^3*d^2*(e + f*x)*Sin[2*(e + f*x)] + a*A*b^5*d^2*(e + f*x)*Sin[2*(e + f*x)] - 2*a^2*b^4*B*d^2*(e + f*x)*Sin[2*(e + f*x)] + a^3*b^3*C*d^2*(e + f*x)*Sin[2*(e + f*x)] - a*b^5*C*d^2*(e + f*x)*Sin[2*(e + f*x)])*(c + d*Tan[e + f*x])^2)/(2*a*(a - I*b)^2*(a + I*b)^2*b^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2)","C",1
63,1,2499,597,8.1614598,"\int \frac{(c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^3} \, dx","Integrate[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^2}{2 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{(b c-a d) \left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right)}{b^3 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{\log (\cos (e+f x)) \left(a^3 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)+3 a^2 b \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-3 a b^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-b^3 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)\right)}{f \left(a^2+b^2\right)^3}-\frac{x \left(a^3 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-3 a^2 b \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-3 a b^2 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+b^3 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)}{\left(a^2+b^2\right)^3}+\frac{\left(a^6 C d^2+3 a^4 b^2 C d^2-a^3 b^3 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-3 a^2 b^4 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-2 C d^2\right)+3 a b^5 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)+b^6 \left(c (2 B d+c C)-A \left(c^2-d^2\right)\right)\right) \log (a+b \tan (e+f x))}{b^3 f \left(a^2+b^2\right)^3}",1,"((-(A*b^4*c^2) + a*b^3*B*c^2 - a^2*b^2*c^2*C + 2*a*A*b^3*c*d - 2*a^2*b^2*B*c*d + 2*a^3*b*c*C*d - a^2*A*b^2*d^2 + a^3*b*B*d^2 - a^4*C*d^2)*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(c + d*Tan[e + f*x])^2)/(2*(a - I*b)^2*(a + I*b)^2*b*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + ((a^3*A*c^2 - 3*a*A*b^2*c^2 + 3*a^2*b*B*c^2 - b^3*B*c^2 - a^3*c^2*C + 3*a*b^2*c^2*C + 6*a^2*A*b*c*d - 2*A*b^3*c*d - 2*a^3*B*c*d + 6*a*b^2*B*c*d - 6*a^2*b*c*C*d + 2*b^3*c*C*d - a^3*A*d^2 + 3*a*A*b^2*d^2 - 3*a^2*b*B*d^2 + b^3*B*d^2 + a^3*C*d^2 - 3*a*b^2*C*d^2)*(e + f*x)*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/((a - I*b)^3*(a + I*b)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + (((3*I)*a^9*A*b^6*c^2 + 3*a^8*A*b^7*c^2 + (5*I)*a^7*A*b^8*c^2 + 5*a^6*A*b^9*c^2 + I*a^5*A*b^10*c^2 + a^4*A*b^11*c^2 - I*a^3*A*b^12*c^2 - a^2*A*b^13*c^2 - I*a^10*b^5*B*c^2 - a^9*b^6*B*c^2 + I*a^8*b^7*B*c^2 + a^7*b^8*B*c^2 + (5*I)*a^6*b^9*B*c^2 + 5*a^5*b^10*B*c^2 + (3*I)*a^4*b^11*B*c^2 + 3*a^3*b^12*B*c^2 - (3*I)*a^9*b^6*c^2*C - 3*a^8*b^7*c^2*C - (5*I)*a^7*b^8*c^2*C - 5*a^6*b^9*c^2*C - I*a^5*b^10*c^2*C - a^4*b^11*c^2*C + I*a^3*b^12*c^2*C + a^2*b^13*c^2*C - (2*I)*a^10*A*b^5*c*d - 2*a^9*A*b^6*c*d + (2*I)*a^8*A*b^7*c*d + 2*a^7*A*b^8*c*d + (10*I)*a^6*A*b^9*c*d + 10*a^5*A*b^10*c*d + (6*I)*a^4*A*b^11*c*d + 6*a^3*A*b^12*c*d - (6*I)*a^9*b^6*B*c*d - 6*a^8*b^7*B*c*d - (10*I)*a^7*b^8*B*c*d - 10*a^6*b^9*B*c*d - (2*I)*a^5*b^10*B*c*d - 2*a^4*b^11*B*c*d + (2*I)*a^3*b^12*B*c*d + 2*a^2*b^13*B*c*d + (2*I)*a^10*b^5*c*C*d + 2*a^9*b^6*c*C*d - (2*I)*a^8*b^7*c*C*d - 2*a^7*b^8*c*C*d - (10*I)*a^6*b^9*c*C*d - 10*a^5*b^10*c*C*d - (6*I)*a^4*b^11*c*C*d - 6*a^3*b^12*c*C*d - (3*I)*a^9*A*b^6*d^2 - 3*a^8*A*b^7*d^2 - (5*I)*a^7*A*b^8*d^2 - 5*a^6*A*b^9*d^2 - I*a^5*A*b^10*d^2 - a^4*A*b^11*d^2 + I*a^3*A*b^12*d^2 + a^2*A*b^13*d^2 + I*a^10*b^5*B*d^2 + a^9*b^6*B*d^2 - I*a^8*b^7*B*d^2 - a^7*b^8*B*d^2 - (5*I)*a^6*b^9*B*d^2 - 5*a^5*b^10*B*d^2 - (3*I)*a^4*b^11*B*d^2 - 3*a^3*b^12*B*d^2 + I*a^13*b^2*C*d^2 + a^12*b^3*C*d^2 + (5*I)*a^11*b^4*C*d^2 + 5*a^10*b^5*C*d^2 + (13*I)*a^9*b^6*C*d^2 + 13*a^8*b^7*C*d^2 + (15*I)*a^7*b^8*C*d^2 + 15*a^6*b^9*C*d^2 + (6*I)*a^5*b^10*C*d^2 + 6*a^4*b^11*C*d^2)*(e + f*x)*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/(a^2*(a - I*b)^6*(a + I*b)^5*b^5*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) - (I*(3*a^2*A*b^4*c^2 - A*b^6*c^2 - a^3*b^3*B*c^2 + 3*a*b^5*B*c^2 - 3*a^2*b^4*c^2*C + b^6*c^2*C - 2*a^3*A*b^3*c*d + 6*a*A*b^5*c*d - 6*a^2*b^4*B*c*d + 2*b^6*B*c*d + 2*a^3*b^3*c*C*d - 6*a*b^5*c*C*d - 3*a^2*A*b^4*d^2 + A*b^6*d^2 + a^3*b^3*B*d^2 - 3*a*b^5*B*d^2 + a^6*C*d^2 + 3*a^4*b^2*C*d^2 + 6*a^2*b^4*C*d^2)*ArcTan[Tan[e + f*x]]*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/(b^3*(a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) - (C*d^2*Log[Cos[e + f*x]]*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/(b^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + ((3*a^2*A*b^4*c^2 - A*b^6*c^2 - a^3*b^3*B*c^2 + 3*a*b^5*B*c^2 - 3*a^2*b^4*c^2*C + b^6*c^2*C - 2*a^3*A*b^3*c*d + 6*a*A*b^5*c*d - 6*a^2*b^4*B*c*d + 2*b^6*B*c*d + 2*a^3*b^3*c*C*d - 6*a*b^5*c*C*d - 3*a^2*A*b^4*d^2 + A*b^6*d^2 + a^3*b^3*B*d^2 - 3*a*b^5*B*d^2 + a^6*C*d^2 + 3*a^4*b^2*C*d^2 + 6*a^2*b^4*C*d^2)*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)/(2*b^3*(a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3) + (Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(3*a*A*b^4*c^2*Sin[e + f*x] - 2*a^2*b^3*B*c^2*Sin[e + f*x] + b^5*B*c^2*Sin[e + f*x] + a^3*b^2*c^2*C*Sin[e + f*x] - 2*a*b^4*c^2*C*Sin[e + f*x] - 4*a^2*A*b^3*c*d*Sin[e + f*x] + 2*A*b^5*c*d*Sin[e + f*x] + 2*a^3*b^2*B*c*d*Sin[e + f*x] - 4*a*b^4*B*c*d*Sin[e + f*x] + 6*a^2*b^3*c*C*d*Sin[e + f*x] + a^3*A*b^2*d^2*Sin[e + f*x] - 2*a*A*b^4*d^2*Sin[e + f*x] + 3*a^2*b^3*B*d^2*Sin[e + f*x] - a^5*C*d^2*Sin[e + f*x] - 4*a^3*b^2*C*d^2*Sin[e + f*x])*(c + d*Tan[e + f*x])^2)/(a*(a - I*b)^2*(a + I*b)^2*b^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3)","C",1
64,1,419,603,6.5741238,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}+\frac{-\frac{2 b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{5 d f}-\frac{-\frac{(c+d \tan (e+f x))^4 \left(5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left(15 d^2 (A-C)-3 B c d+c^2 C\right)\right)}{2 d f}+\frac{5 \left(d \left(a^2 B+2 a b (A-C)-b^2 B\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 d \left(a^2 (B c-d (A-C))+2 a b (A c+B d-c C)-b^2 (B c-d (A-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)}{f}}{5 d}}{6 d}","-\frac{d \tan (e+f x) \left(-\left(a^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)+2 a b \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+b^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)}{f}+\frac{(c+d \tan (e+f x))^4 \left(5 a^2 C d^2-6 a b d (c C-5 B d)+b^2 \left(15 d^2 (A-C)-3 B c d+c^2 C\right)\right)}{60 d^3 f}+\frac{\log (\cos (e+f x)) \left(-\left(a^2 \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right)+2 a b \left(-A \left(c^3-3 c d^2\right)+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)+b^2 \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right)}{f}+x \left(a^2 \left(A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right)-2 a b \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)+b^2 \left(-A \left(c^3-3 c d^2\right)+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)\right)+\frac{\left(a^2 B+2 a b (A-C)-b^2 B\right) (c+d \tan (e+f x))^3}{3 f}+\frac{(c+d \tan (e+f x))^2 \left(a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right)}{2 f}-\frac{b \tan (e+f x) (-a C d-3 b B d+b c C) (c+d \tan (e+f x))^4}{15 d^2 f}+\frac{C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^4}{6 d f}",1,"(C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f) + ((-2*b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(5*d*f) - (-1/2*((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15*(A - C)*d^2))*(c + d*Tan[e + f*x])^4)/(d*f) + (5*(3*d*(2*a*b*(A*c - c*C + B*d) + a^2*(B*c - (A - C)*d) - b^2*(B*c - (A - C)*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) + (a^2*B - b^2*B + 2*a*b*(A - C))*d*((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)))/f)/(5*d))/(6*d)","C",1
65,1,297,389,6.3321413,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac{\frac{(-5 a C d-5 b B d+b c C) (c+d \tan (e+f x))^4}{4 d f}+\frac{5 \left((a B+A b-b C) \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 (-a A d+a B c+a C d+A b c+b B d-b c C) \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{d \tan (e+f x) \left(A \left(2 a c d+b \left(c^2-d^2\right)\right)+a \left(B c^2-B d^2-2 c C d\right)-b \left(2 B c d+c^2 C-C d^2\right)\right)}{f}-\frac{\log (\cos (e+f x)) \left(A \left(3 a c^2 d-a d^3+b c^3-3 b c d^2\right)+a \left(B c^3-3 B c d^2-3 c^2 C d+C d^3\right)-b \left(3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)\right)}{f}+x \left(a \left(A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right)-b \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right)+\frac{(a B+A b-b C) (c+d \tan (e+f x))^3}{3 f}+\frac{(c+d \tan (e+f x))^2 (a A d+a B c-a C d+A b c-b B d-b c C)}{2 f}-\frac{(-5 a C d-5 b B d+b c C) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac{b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}",1,"(b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(5*d*f) - (((b*c*C - 5*b*B*d - 5*a*C*d)*(c + d*Tan[e + f*x])^4)/(4*d*f) + (5*(3*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*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) + (A*b + a*B - b*C)*((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
66,1,212,191,2.4361397,"\int (c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{-6 (d (C-A)+B c) \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)+2 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 C (c+d \tan (e+f x))^4}{12 d f}","\frac{d \tan (e+f x) \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)}{f}-\frac{\left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right) \log (\cos (e+f x))}{f}-x \left(-A \left(c^3-3 c d^2\right)+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)+\frac{(d (A-C)+B c) (c+d \tan (e+f x))^2}{2 f}+\frac{B (c+d \tan (e+f x))^3}{3 f}+\frac{C (c+d \tan (e+f x))^4}{4 d f}",1,"(3*C*(c + d*Tan[e + f*x])^4 - 6*(B*c + (-A + C)*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) + 2*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
67,1,255,363,4.8295962,"\int \frac{(c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{a+b \tan (e+f x)} \, dx","Integrate[((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]","\frac{\frac{6 (b c-a d)^3 \left(a (a C-b B)+A b^2\right) \log (a+b \tan (e+f x))}{b^2 \left(a^2+b^2\right)}+\frac{3 b^2 (c+i d)^3 (-i A+B+i C) \log (-\tan (e+f x)+i)}{a+i b}-\frac{3 b^2 (d+i c)^3 (A-i B-C) \log (\tan (e+f x)+i)}{a-i b}+3 (-a C d+b B d+b c C) (c+d \tan (e+f x))^2+\frac{6 d (b c-a d) \tan (e+f x) (-a C d+b B d+b c C)}{b}+6 b d^2 \tan (e+f x) (d (A-C)+B c)+2 b C (c+d \tan (e+f x))^3}{6 b^2 f}","-\frac{\log (\cos (e+f x)) \left(A \left(a d \left(3 c^2-d^2\right)-b \left(c^3-3 c d^2\right)\right)+a \left(B c^3-3 B c d^2-3 c^2 C d+C d^3\right)+b \left(3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)\right)}{f \left(a^2+b^2\right)}-\frac{x \left(a \left(-A \left(c^3-3 c d^2\right)+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)-b \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right)}{a^2+b^2}+\frac{(b c-a d)^3 \left(A b^2-a (b B-a C)\right) \log (a+b \tan (e+f x))}{b^4 f \left(a^2+b^2\right)}+\frac{d \tan (e+f x) \left((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right)}{b^3 f}+\frac{(-a C d+b B d+b c C) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f}",1,"((3*b^2*((-I)*A + B + I*C)*(c + I*d)^3*Log[I - Tan[e + f*x]])/(a + I*b) - (3*b^2*(A - I*B - C)*(I*c + d)^3*Log[I + Tan[e + f*x]])/(a - I*b) + (6*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)^3*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)) + 6*b*d^2*(B*c + (A - C)*d)*Tan[e + f*x] + (6*d*(b*c - a*d)*(b*c*C + b*B*d - a*C*d)*Tan[e + f*x])/b + 3*(b*c*C + b*B*d - a*C*d)*(c + d*Tan[e + f*x])^2 + 2*b*C*(c + d*Tan[e + f*x])^3)/(6*b^2*f)","C",1
68,1,2467,574,8.5771981,"\int \frac{(c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^2} \, dx","Integrate[((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\log (\cos (e+f x)) \left(-\left(a^2 \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right)+2 a b \left(A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right)+b^2 \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right)}{f \left(a^2+b^2\right)^2}-\frac{x \left(a^2 \left(-A \left(c^3-3 c d^2\right)+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right)-2 a b \left(d (A-C) \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)+b^2 \left(A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right)\right)}{\left(a^2+b^2\right)^2}-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^3}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{d \left(3 a^2 C-2 a b B+2 A b^2+b^2 C\right) (c+d \tan (e+f x))^2}{2 b^2 f \left(a^2+b^2\right)}-\frac{d^2 \tan (e+f x) \left(3 a^3 C d-a^2 b (2 B d+3 c C)-A b^2 (b c-a d)+a b^2 (B c+2 C d)-b^3 (B d+2 c C)\right)}{b^3 f \left(a^2+b^2\right)}-\frac{(b c-a d)^2 \left(-3 a^4 C d+2 a^3 b B d+a^2 b^2 (B c-d (A+5 C))-2 a b^3 (A c-2 B d-c C)-b^4 (3 A d+B c)\right) \log (a+b \tan (e+f x))}{b^4 f \left(a^2+b^2\right)^2}",1,"((a^2*A*c^3 - A*b^2*c^3 + 2*a*b*B*c^3 - a^2*c^3*C + b^2*c^3*C + 6*a*A*b*c^2*d - 3*a^2*B*c^2*d + 3*b^2*B*c^2*d - 6*a*b*c^2*C*d - 3*a^2*A*c*d^2 + 3*A*b^2*c*d^2 - 6*a*b*B*c*d^2 + 3*a^2*c*C*d^2 - 3*b^2*c*C*d^2 - 2*a*A*b*d^3 + a^2*B*d^3 - b^2*B*d^3 + 2*a*b*C*d^3)*(e + f*x)*Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/((a - I*b)^2*(a + I*b)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) - (I*(-2*a^6*A*b^8*c^3 + (2*I)*a^5*A*b^9*c^3 - 2*a^4*A*b^10*c^3 + (2*I)*a^3*A*b^11*c^3 + a^7*b^7*B*c^3 - I*a^6*b^8*B*c^3 - a^3*b^11*B*c^3 + I*a^2*b^12*B*c^3 + 2*a^6*b^8*c^3*C - (2*I)*a^5*b^9*c^3*C + 2*a^4*b^10*c^3*C - (2*I)*a^3*b^11*c^3*C + 3*a^7*A*b^7*c^2*d - (3*I)*a^6*A*b^8*c^2*d - 3*a^3*A*b^11*c^2*d + (3*I)*a^2*A*b^12*c^2*d + 6*a^6*b^8*B*c^2*d - (6*I)*a^5*b^9*B*c^2*d + 6*a^4*b^10*B*c^2*d - (6*I)*a^3*b^11*B*c^2*d - 3*a^9*b^5*c^2*C*d + (3*I)*a^8*b^6*c^2*C*d - 12*a^7*b^7*c^2*C*d + (12*I)*a^6*b^8*c^2*C*d - 9*a^5*b^9*c^2*C*d + (9*I)*a^4*b^10*c^2*C*d + 6*a^6*A*b^8*c*d^2 - (6*I)*a^5*A*b^9*c*d^2 + 6*a^4*A*b^10*c*d^2 - (6*I)*a^3*A*b^11*c*d^2 - 3*a^9*b^5*B*c*d^2 + (3*I)*a^8*b^6*B*c*d^2 - 12*a^7*b^7*B*c*d^2 + (12*I)*a^6*b^8*B*c*d^2 - 9*a^5*b^9*B*c*d^2 + (9*I)*a^4*b^10*B*c*d^2 + 6*a^10*b^4*c*C*d^2 - (6*I)*a^9*b^5*c*C*d^2 + 18*a^8*b^6*c*C*d^2 - (18*I)*a^7*b^7*c*C*d^2 + 12*a^6*b^8*c*C*d^2 - (12*I)*a^5*b^9*c*C*d^2 - a^9*A*b^5*d^3 + I*a^8*A*b^6*d^3 - 4*a^7*A*b^7*d^3 + (4*I)*a^6*A*b^8*d^3 - 3*a^5*A*b^9*d^3 + (3*I)*a^4*A*b^10*d^3 + 2*a^10*b^4*B*d^3 - (2*I)*a^9*b^5*B*d^3 + 6*a^8*b^6*B*d^3 - (6*I)*a^7*b^7*B*d^3 + 4*a^6*b^8*B*d^3 - (4*I)*a^5*b^9*B*d^3 - 3*a^11*b^3*C*d^3 + (3*I)*a^10*b^4*C*d^3 - 8*a^9*b^5*C*d^3 + (8*I)*a^8*b^6*C*d^3 - 5*a^7*b^7*C*d^3 + (5*I)*a^6*b^8*C*d^3)*(e + f*x)*Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(a^2*(a - I*b)^4*(a + I*b)^3*b^7*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) - (I*(2*a*A*b^5*c^3 - a^2*b^4*B*c^3 + b^6*B*c^3 - 2*a*b^5*c^3*C - 3*a^2*A*b^4*c^2*d + 3*A*b^6*c^2*d - 6*a*b^5*B*c^2*d + 3*a^4*b^2*c^2*C*d + 9*a^2*b^4*c^2*C*d - 6*a*A*b^5*c*d^2 + 3*a^4*b^2*B*c*d^2 + 9*a^2*b^4*B*c*d^2 - 6*a^5*b*c*C*d^2 - 12*a^3*b^3*c*C*d^2 + a^4*A*b^2*d^3 + 3*a^2*A*b^4*d^3 - 2*a^5*b*B*d^3 - 4*a^3*b^3*B*d^3 + 3*a^6*C*d^3 + 5*a^4*b^2*C*d^3)*ArcTan[Tan[e + f*x]]*Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(b^4*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + ((-3*b^2*c^2*C*d - 3*b^2*B*c*d^2 + 6*a*b*c*C*d^2 - A*b^2*d^3 + 2*a*b*B*d^3 - 3*a^2*C*d^3 + b^2*C*d^3)*Cos[e + f*x]*Log[Cos[e + f*x]]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(b^4*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + ((2*a*A*b^5*c^3 - a^2*b^4*B*c^3 + b^6*B*c^3 - 2*a*b^5*c^3*C - 3*a^2*A*b^4*c^2*d + 3*A*b^6*c^2*d - 6*a*b^5*B*c^2*d + 3*a^4*b^2*c^2*C*d + 9*a^2*b^4*c^2*C*d - 6*a*A*b^5*c*d^2 + 3*a^4*b^2*B*c*d^2 + 9*a^2*b^4*B*c*d^2 - 6*a^5*b*c*C*d^2 - 12*a^3*b^3*c*C*d^2 + a^4*A*b^2*d^3 + 3*a^2*A*b^4*d^3 - 2*a^5*b*B*d^3 - 4*a^3*b^3*B*d^3 + 3*a^6*C*d^3 + 5*a^4*b^2*C*d^3)*Cos[e + f*x]*Log[(a*Cos[e + f*x] + b*Sin[e + f*x])^2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(2*b^4*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + (C*d^3*Sec[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(2*b^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + ((a*Cos[e + f*x] + b*Sin[e + f*x])^2*(3*b*c*C*d^2*Sin[e + f*x] + b*B*d^3*Sin[e + f*x] - 2*a*C*d^3*Sin[e + f*x])*(c + d*Tan[e + f*x])^3)/(b^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2) + (Cos[e + f*x]*(a*Cos[e + f*x] + b*Sin[e + f*x])*(A*b^5*c^3*Sin[e + f*x] - a*b^4*B*c^3*Sin[e + f*x] + a^2*b^3*c^3*C*Sin[e + f*x] - 3*a*A*b^4*c^2*d*Sin[e + f*x] + 3*a^2*b^3*B*c^2*d*Sin[e + f*x] - 3*a^3*b^2*c^2*C*d*Sin[e + f*x] + 3*a^2*A*b^3*c*d^2*Sin[e + f*x] - 3*a^3*b^2*B*c*d^2*Sin[e + f*x] + 3*a^4*b*c*C*d^2*Sin[e + f*x] - a^3*A*b^2*d^3*Sin[e + f*x] + a^4*b*B*d^3*Sin[e + f*x] - a^5*C*d^3*Sin[e + f*x])*(c + d*Tan[e + f*x])^3)/(a*(a - I*b)*(a + I*b)*b^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)","C",1
69,1,1451,798,15.8924783,"\int \frac{(c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^3} \, dx","Integrate[((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]","-\frac{2 (b c-a d)^2 (a \cos (e+f x)+b \sin (e+f x))^3 \left(2 C d \tan \left(\frac{1}{2} (e+f x)\right) a^6-b \left(B d \tan \left(\frac{1}{2} (e+f x)\right)+C \left(d-c \tan \left(\frac{1}{2} (e+f x)\right)\right)\right) a^5+b^2 \left(c \left(C-2 B \tan \left(\frac{1}{2} (e+f x)\right)\right)+d \left(B+4 C \tan \left(\frac{1}{2} (e+f x)\right)\right)\right) a^4-b^3 \left(A d+C d-3 A c \tan \left(\frac{1}{2} (e+f x)\right)+c C \tan \left(\frac{1}{2} (e+f x)\right)+B \left(c+3 d \tan \left(\frac{1}{2} (e+f x)\right)\right)\right) a^3+b^4 \left(c C+B d+A \left(c+2 d \tan \left(\frac{1}{2} (e+f x)\right)\right)\right) a^2-b^5 \left(B c+A \left(d-c \tan \left(\frac{1}{2} (e+f x)\right)\right)\right) a+A b^6 c\right) (c+d \tan (e+f x))^3}{a^3 b^3 \left(a^2+b^2\right)^2 f (c \cos (e+f x)+d \sin (e+f x))^3 \left(a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)-2 b \tan \left(\frac{1}{2} (e+f x)\right)\right) (a+b \tan (e+f x))^3}-\frac{2 C d^3 (a \cos (e+f x)+b \sin (e+f x))^3 \tan \left(\frac{1}{2} (e+f x)\right) (c+d \tan (e+f x))^3}{b^3 f (c \cos (e+f x)+d \sin (e+f x))^3 \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) (a+b \tan (e+f x))^3}+\frac{2 \left(A b^2+a (a C-b B)\right) (a d-b c)^3 (a \cos (e+f x)+b \sin (e+f x))^3 \left(a+2 b \tan \left(\frac{1}{2} (e+f x)\right)\right) (c+d \tan (e+f x))^3}{a^3 b^2 \left(a^2+b^2\right) f (c \cos (e+f x)+d \sin (e+f x))^3 \left(-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 b \tan \left(\frac{1}{2} (e+f x)\right)+a\right)^2 (a+b \tan (e+f x))^3}+\frac{\left(\left(-C c^3-3 B d c^2+3 C d^2 c+B d^3+A \left(c^3-3 c d^2\right)\right) a^3+3 b \left(B \left(c^3-3 c d^2\right)-(A-C) d \left(d^2-3 c^2\right)\right) a^2+3 b^2 \left(-A c^3+C c^3+3 B d c^2+3 A d^2 c-3 C d^2 c-B d^3\right) a+b^3 \left((A-C) d \left(d^2-3 c^2\right)-B \left(c^3-3 c d^2\right)\right)\right) (e+f x) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{\left(a^2+b^2\right)^3 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}-\frac{d^2 (3 b c C-3 a d C+b B d) \log \left(1-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{b^4 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}+\frac{\left(\left(B \left(c^3-3 c d^2\right)-(A-C) d \left(d^2-3 c^2\right)\right) a^3+3 b \left(-A c^3+C c^3+3 B d c^2+3 A d^2 c-3 C d^2 c-B d^3\right) a^2-3 b^2 \left(B \left(c^3-3 c d^2\right)-(A-C) d \left(d^2-3 c^2\right)\right) a+b^3 \left(-C c^3-3 B d c^2+3 C d^2 c+B d^3+A \left(c^3-3 c d^2\right)\right)\right) \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{\left(a^2+b^2\right)^3 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}-\frac{(b c-a d) \left(-3 C d^2 a^6+b B d^2 a^5+b^2 d (B c-9 C d) a^4+b^3 B \left(c^2+3 d^2\right) a^3+b^4 \left(3 C c^2+6 B d c-10 C d^2+A \left(d^2-3 c^2\right)\right) a^2+b^5 \left(8 c (C-A) d-3 B \left(c^2-2 d^2\right)\right) a+b^6 \left(A \left(c^2-3 d^2\right)-c (c C+3 B d)\right)\right) \log \left(a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)-2 b \tan \left(\frac{1}{2} (e+f x)\right)\right) (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}{b^4 \left(a^2+b^2\right)^3 f (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}","-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^3}{2 b \left(a^2+b^2\right) f (a+b \tan (e+f x))^2}+\frac{\left(-3 C d a^4+b B d a^3+b^2 (2 B c+(A-7 C) d) a^2-b^3 (4 A c-4 C c-5 B d) a-b^4 (2 B c+3 A d)\right) (c+d \tan (e+f x))^2}{2 b^2 \left(a^2+b^2\right)^2 f (a+b \tan (e+f x))}-\frac{\left(\left(C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left(c^3-3 c d^2\right)\right) a^3-3 b \left((A-C) d \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right) a^2+3 b^2 \left(A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right) a+b^3 \left((A-C) d \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right)\right) x}{\left(a^2+b^2\right)^3}-\frac{\left(\left((A-C) d \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right) a^3+3 b \left(C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left(c^3-3 c d^2\right)\right) a^2-3 b^2 \left((A-C) d \left(3 c^2-d^2\right)+B \left(c^3-3 c d^2\right)\right) a+b^3 \left(A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right)\right) \log (\cos (e+f x))}{\left(a^2+b^2\right)^3 f}-\frac{(b c-a d) \left(-3 C d^2 a^6+b B d^2 a^5+b^2 d (B c-9 C d) a^4+b^3 B \left(c^2+3 d^2\right) a^3+b^4 \left(3 C c^2+6 B d c-10 C d^2-A \left(3 c^2-d^2\right)\right) a^2-b^5 \left(8 c (A-C) d+3 B \left(c^2-2 d^2\right)\right) a-b^6 \left(c (c C+3 B d)-A \left(c^2-3 d^2\right)\right)\right) \log (a+b \tan (e+f x))}{b^4 \left(a^2+b^2\right)^3 f}-\frac{d^2 \left(-3 C d a^4+b B d a^3+b^2 (B c-6 C d) a^2-b^3 (2 A c-2 C c-3 B d) a-b^4 (B c+(2 A+C) d)\right) \tan (e+f x)}{b^3 \left(a^2+b^2\right)^2 f}",1,"((3*a*b^2*(-(A*c^3) + c^3*C + 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 - B*d^3) + a^3*(-(c^3*C) - 3*B*c^2*d + 3*c*C*d^2 + B*d^3 + A*(c^3 - 3*c*d^2)) + b^3*((A - C)*d*(-3*c^2 + d^2) - B*(c^3 - 3*c*d^2)) + 3*a^2*b*(-((A - C)*d*(-3*c^2 + d^2)) + B*(c^3 - 3*c*d^2)))*(e + f*x)*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/((a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3) - (d^2*(3*b*c*C + b*B*d - 3*a*C*d)*Log[1 - Tan[(e + f*x)/2]^2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(b^4*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3) + ((3*a^2*b*(-(A*c^3) + c^3*C + 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 - B*d^3) + b^3*(-(c^3*C) - 3*B*c^2*d + 3*c*C*d^2 + B*d^3 + A*(c^3 - 3*c*d^2)) + a^3*(-((A - C)*d*(-3*c^2 + d^2)) + B*(c^3 - 3*c*d^2)) - 3*a*b^2*(-((A - C)*d*(-3*c^2 + d^2)) + B*(c^3 - 3*c*d^2)))*Log[1 + Tan[(e + f*x)/2]^2]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/((a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3) - ((b*c - a*d)*(a^5*b*B*d^2 - 3*a^6*C*d^2 + a^4*b^2*d*(B*c - 9*C*d) + a^3*b^3*B*(c^2 + 3*d^2) + b^6*(-(c*(c*C + 3*B*d)) + A*(c^2 - 3*d^2)) + a*b^5*(8*c*(-A + C)*d - 3*B*(c^2 - 2*d^2)) + a^2*b^4*(3*c^2*C + 6*B*c*d - 10*C*d^2 + A*(-3*c^2 + d^2)))*Log[-2*b*Tan[(e + f*x)/2] + a*(-1 + Tan[(e + f*x)/2]^2)]*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(b^4*(a^2 + b^2)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3) - (2*C*d^3*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*Tan[(e + f*x)/2]*(c + d*Tan[e + f*x])^3)/(b^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(-1 + Tan[(e + f*x)/2]^2)*(a + b*Tan[e + f*x])^3) + (2*(A*b^2 + a*(-(b*B) + a*C))*(-(b*c) + a*d)^3*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(a + 2*b*Tan[(e + f*x)/2])*(c + d*Tan[e + f*x])^3)/(a^3*b^2*(a^2 + b^2)*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + 2*b*Tan[(e + f*x)/2] - a*Tan[(e + f*x)/2]^2)^2*(a + b*Tan[e + f*x])^3) - (2*(b*c - a*d)^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(A*b^6*c + 2*a^6*C*d*Tan[(e + f*x)/2] - a*b^5*(B*c + A*(d - c*Tan[(e + f*x)/2])) - a^5*b*(B*d*Tan[(e + f*x)/2] + C*(d - c*Tan[(e + f*x)/2])) + a^4*b^2*(c*(C - 2*B*Tan[(e + f*x)/2]) + d*(B + 4*C*Tan[(e + f*x)/2])) + a^2*b^4*(c*C + B*d + A*(c + 2*d*Tan[(e + f*x)/2])) - a^3*b^3*(A*d + C*d - 3*A*c*Tan[(e + f*x)/2] + c*C*Tan[(e + f*x)/2] + B*(c + 3*d*Tan[(e + f*x)/2])))*(c + d*Tan[e + f*x])^3)/(a^3*b^3*(a^2 + b^2)^2*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(-2*b*Tan[(e + f*x)/2] + a*(-1 + Tan[(e + f*x)/2]^2))*(a + b*Tan[e + f*x])^3)","A",0
70,1,258,337,4.5153937,"\int \frac{(a+b \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{c+d \tan (e+f x)} \, dx","Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]),x]","\frac{6 b^2 d \tan (e+f x) (a B+A b-b C)+\frac{6 (a d-b c)^3 \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d^2 \left(c^2+d^2\right)}+\frac{3 d^2 (a-i b)^3 (i A+B-i C) \log (\tan (e+f x)+i)}{c-i d}+\frac{3 d^2 (a+i b)^3 (-i A+B+i C) \log (-\tan (e+f x)+i)}{c+i d}-3 (-a C d-b B d+b c C) (a+b \tan (e+f x))^2-\frac{6 b (b c-a d) \tan (e+f x) (a C d+b B d-b c C)}{d}+2 C d (a+b \tan (e+f x))^3}{6 d^2 f}","-\frac{\log (\cos (e+f x)) \left(a^3 (B c-d (A-C))+3 a^2 b (A c+B d-c C)-3 a b^2 (B c-d (A-C))-b^3 (A c+B d-c C)\right)}{f \left(c^2+d^2\right)}+\frac{x \left(a^3 (A c+B d-c C)-3 a^2 b (B c-d (A-C))-3 a b^2 (A c+B d-c C)+b^3 (B c-d (A-C))\right)}{c^2+d^2}-\frac{(b c-a d)^3 \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d^4 f \left(c^2+d^2\right)}+\frac{b \tan (e+f x) \left(b d^2 (a B+A b-b C)+(b c-a d) (-a C d-b B d+b c C)\right)}{d^3 f}-\frac{(-a C d-b B d+b c C) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac{C (a+b \tan (e+f x))^3}{3 d f}",1,"((3*(a + I*b)^3*((-I)*A + B + I*C)*d^2*Log[I - Tan[e + f*x]])/(c + I*d) + (3*(a - I*b)^3*(I*A + B - I*C)*d^2*Log[I + Tan[e + f*x]])/(c - I*d) + (6*(-(b*c) + a*d)^3*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)) + 6*b^2*(A*b + a*B - b*C)*d*Tan[e + f*x] - (6*b*(b*c - a*d)*(-(b*c*C) + b*B*d + a*C*d)*Tan[e + f*x])/d - 3*(b*c*C - b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2 + 2*C*d*(a + b*Tan[e + f*x])^3)/(6*d^2*f)","C",1
71,1,190,236,2.9896649,"\int \frac{(a+b \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{c+d \tan (e+f x)} \, dx","Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]),x]","\frac{\frac{2 (b c-a d)^2 \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d^2 \left(c^2+d^2\right)}+\frac{d (a-i b)^2 (i A+B-i C) \log (\tan (e+f x)+i)}{c-i d}+\frac{d (a+i b)^2 (-i A+B+i C) \log (-\tan (e+f x)+i)}{c+i d}+\frac{2 b \tan (e+f x) (a C d+b B d-b c C)}{d}+C (a+b \tan (e+f x))^2}{2 d f}","-\frac{\log (\cos (e+f x)) \left(a^2 (B c-d (A-C))+2 a b (A c+B d-c C)-b^2 (B c-d (A-C))\right)}{f \left(c^2+d^2\right)}+\frac{x \left(a^2 (A c+B d-c C)-2 a b (B c-d (A-C))-b^2 (A c+B d-c C)\right)}{c^2+d^2}+\frac{(b c-a d)^2 \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d^3 f \left(c^2+d^2\right)}-\frac{b \tan (e+f x) (-a C d-b B d+b c C)}{d^2 f}+\frac{C (a+b \tan (e+f x))^2}{2 d f}",1,"(((a + I*b)^2*((-I)*A + B + I*C)*d*Log[I - Tan[e + f*x]])/(c + I*d) + ((a - I*b)^2*(I*A + B - I*C)*d*Log[I + Tan[e + f*x]])/(c - I*d) + (2*(b*c - a*d)^2*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)) + (2*b*(-(b*c*C) + b*B*d + a*C*d)*Tan[e + f*x])/d + C*(a + b*Tan[e + f*x])^2)/(2*d*f)","C",1
72,1,148,156,1.0856482,"\int \frac{(a+b \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{c+d \tan (e+f x)} \, dx","Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]),x]","\frac{\frac{2 (a d-b c) \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d^2 \left(c^2+d^2\right)}+\frac{(b-i a) (A+i B-C) \log (-\tan (e+f x)+i)}{c+i d}+\frac{(b+i a) (A-i B-C) \log (\tan (e+f x)+i)}{c-i d}+\frac{2 b C \tan (e+f x)}{d}}{2 f}","-\frac{(b c-a d) \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d^2 f \left(c^2+d^2\right)}-\frac{\log (\cos (e+f x)) (-a A d+a B c+a C d+A b c+b B d-b c C)}{f \left(c^2+d^2\right)}+\frac{x (a (A c+B d-c C)-b (B c-d (A-C)))}{c^2+d^2}+\frac{b C \tan (e+f x)}{d f}",1,"((((-I)*a + b)*(A + I*B - C)*Log[I - Tan[e + f*x]])/(c + I*d) + ((I*a + b)*(A - I*B - C)*Log[I + Tan[e + f*x]])/(c - I*d) + (2*(-(b*c) + a*d)*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)) + (2*b*C*Tan[e + f*x])/d)/(2*f)","C",1
73,1,117,99,0.1970656,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x]),x]","\frac{\frac{2 \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d \left(c^2+d^2\right)}+\frac{(-i A+B+i C) \log (-\tan (e+f x)+i)}{c+i d}+\frac{(i A+B-i C) \log (\tan (e+f x)+i)}{c-i d}}{2 f}","\frac{\left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{d f \left(c^2+d^2\right)}-\frac{(B c-d (A-C)) \log (\cos (e+f x))}{f \left(c^2+d^2\right)}+\frac{x (A c+B d-c C)}{c^2+d^2}",1,"((((-I)*A + B + I*C)*Log[I - Tan[e + f*x]])/(c + I*d) + ((I*A + B - I*C)*Log[I + Tan[e + f*x]])/(c - I*d) + (2*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d*(c^2 + d^2)))/(2*f)","C",1
74,1,313,165,1.5048192,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])),x]","-\frac{\frac{\log \left(\sqrt{-b^2}-b \tan (e+f x)\right) \left(\frac{\sqrt{-b^2} (a (A c+B d-c C)+b d (C-A)+b B c)}{b}+a A d-a B c-a C d+A b c+b B d-b c C\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{\log \left(\sqrt{-b^2}+b \tan (e+f x)\right) \left(\frac{b (a (A c+B d-c C)+b d (C-A)+b B c)}{\sqrt{-b^2}}+a A d-a B c-a C d+A b c+b B d-b c C\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{2 \left(a (a C-b B)+A b^2\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (a d-b c)}+\frac{2 \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)}}{2 f}","\frac{x (a (A c+B d-c C)+b (B c-d (A-C)))}{\left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{\left(A b^2-a (b B-a C)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right) (b c-a d)}-\frac{\left(A d^2-B c d+c^2 C\right) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left(c^2+d^2\right) (b c-a d)}",1,"-1/2*(((A*b*c - a*B*c - b*c*C + a*A*d + b*B*d - a*C*d + (Sqrt[-b^2]*(b*B*c + b*(-A + C)*d + a*(A*c - c*C + B*d)))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) + (2*(A*b^2 + a*(-(b*B) + a*C))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(-(b*c) + a*d)) + ((A*b*c - a*B*c - b*c*C + a*A*d + b*B*d - a*C*d + (b*(b*B*c + b*(-A + C)*d + a*(A*c - c*C + B*d)))/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) + (2*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)))/f","A",1
75,1,543,281,6.9565041,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])),x]","\frac{-\frac{(b c-a d) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right) \left(\frac{\sqrt{-b^2} \left(a^2 (A c+B d-c C)+2 a b (d (C-A)+B c)-b^2 (A c+B d-c C)\right)}{b}+a^2 A d+a^2 (-B) c-a^2 C d+2 a A b c+2 a b B d-2 a b c C-A b^2 d+b^2 B c+b^2 C d\right)}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{(b c-a d) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right) \left(\frac{\sqrt{-b^2} \left(-\left(a^2 (A c+B d-c C)\right)-2 a b (d (C-A)+B c)+b^2 (A c+B d-c C)\right)}{b}+a^2 A d+a^2 (-B) c-a^2 C d+2 a A b c+2 a b B d-2 a b c C-A b^2 d+b^2 B c+b^2 C d\right)}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{d \left(a^2+b^2\right) \left(A d^2-B c d+c^2 C\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)}+\frac{\left(a^4 C d-2 a^3 b B d+a^2 b^2 (3 A d+B c-C d)+2 a b^3 c (C-A)+b^4 (A d-B c)\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (a d-b c)}-\frac{A b^2}{a+b \tan (e+f x)}+\frac{a (b B-a C)}{a+b \tan (e+f x)}}{f \left(a^2+b^2\right) (b c-a d)}","\frac{x \left(a^2 (A c+B d-c C)+2 a b (B c-d (A-C))-b^2 (A c+B d-c C)\right)}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)}-\frac{A b^2-a (b B-a C)}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))}+\frac{\left(a^4 (-C) d+2 a^3 b B d-a^2 b^2 (3 A d+B c-C d)+2 a b^3 c (A-C)+b^4 (B c-A 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 \left(A d^2-B c d+c^2 C\right) \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*((b*c - a*d)*(2*a*A*b*c - a^2*B*c + b^2*B*c - 2*a*b*c*C + a^2*A*d - A*b^2*d + 2*a*b*B*d - a^2*C*d + b^2*C*d + (Sqrt[-b^2]*(a^2*(A*c - c*C + B*d) - b^2*(A*c - c*C + B*d) + 2*a*b*(B*c + (-A + C)*d)))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) + ((2*a*b^3*c*(-A + C) - 2*a^3*b*B*d + a^4*C*d + b^4*(-(B*c) + A*d) + a^2*b^2*(B*c + 3*A*d - C*d))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(-(b*c) + a*d)) - ((b*c - a*d)*(2*a*A*b*c - a^2*B*c + b^2*B*c - 2*a*b*c*C + a^2*A*d - A*b^2*d + 2*a*b*B*d - a^2*C*d + b^2*C*d + (Sqrt[-b^2]*(-(a^2*(A*c - c*C + B*d)) + b^2*(A*c - c*C + B*d) - 2*a*b*(B*c + (-A + C)*d)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) + ((a^2 + b^2)*d*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)) - (A*b^2)/(a + b*Tan[e + f*x]) + (a*(b*B - a*C))/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f)","A",1
76,1,898,477,9.027695,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])),x]","-\frac{A b^2-a (b B-a C)}{2 \left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x))^2}-\frac{-\frac{-2 \left(-A d a^2+b c (A-C) a+b^2 (B c-A d)\right) b^2-a \left(2 b (A b-C b-a B) (b c-a d)-2 a \left(A b^2-a (b B-a C)\right) d\right)}{\left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x))}-\frac{-\frac{2 b \left(a^2+b^2\right)^2 \left(C c^2-B d c+A d^2\right) \log (c+d \tan (e+f x)) d^2}{(b c-a d) \left(c^2+d^2\right)}-\frac{b (b c-a d)^2 \left(-B c a^3+A d a^3-C d a^3+3 A b c a^2-3 b c C a^2+3 b B d a^2+3 b^2 B c a-3 A b^2 d a+3 b^2 C d a-A b^3 c+b^3 c C-b^3 B d+\frac{\sqrt{-b^2} \left((A c-C c+B d) a^3+3 b (B c-(A-C) d) a^2-3 b^2 (A c-C c+B d) a-b^3 (B c-(A-C) d)\right)}{b}\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 \left(C d^2 a^6-3 b B d^2 a^5+3 b^2 d (B c+2 A d-C d) a^4-b^3 \left(8 c (A-C) d+B \left(c^2-d^2\right)\right) a^3-3 b^4 \left(c (c C+2 B d)-A \left(c^2+d^2\right)\right) a^2+3 b^5 B c^2 a+b^6 \left(c (c C-B d)-A \left(c^2-d^2\right)\right)\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (b c-a d)}-\frac{b (b c-a d)^2 \left(-B c a^3+A d a^3-C d a^3+3 A b c a^2-3 b c C a^2+3 b B d a^2+3 b^2 B c a-3 A b^2 d a+3 b^2 C d a-A b^3 c+b^3 c C-b^3 B d-\frac{\sqrt{-b^2} \left((A c-C c+B d) a^3+3 b (B c-(A-C) d) a^2-3 b^2 (A c-C c+B d) a-b^3 (B c-(A-C) d)\right)}{b}\right) \log \left(b \tan (e+f x)+\sqrt{-b^2}\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)}}{b \left(a^2+b^2\right) (b c-a d) f}}{2 \left(a^2+b^2\right) (b c-a d)}","-\frac{A b^2-a (b B-a C)}{2 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^2}+\frac{x \left(a^3 (A c+B d-c C)+3 a^2 b (B c-d (A-C))-3 a b^2 (A c+B d-c C)-b^3 (B c-d (A-C))\right)}{\left(a^2+b^2\right)^3 \left(c^2+d^2\right)}-\frac{a^4 (-C) d+2 a^3 b B d-a^2 b^2 (3 A d+B c-C d)+2 a b^3 c (A-C)+b^4 (B c-A d)}{f \left(a^2+b^2\right)^2 (b c-a d)^2 (a+b \tan (e+f x))}+\frac{\left(a^6 C d^2-3 a^5 b B d^2+3 a^4 b^2 d (2 A d+B c-C d)-a^3 b^3 \left(8 c d (A-C)+B \left(c^2-d^2\right)\right)-3 a^2 b^4 \left(c (2 B d+c C)-A \left(c^2+d^2\right)\right)+3 a b^5 B c^2+b^6 \left(c (c C-B d)-A \left(c^2-d^2\right)\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^2 \left(A d^2-B c d+c^2 C\right) \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*(A*b^2 - a*(b*B - a*C))/((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*A*b*c - A*b^3*c - a^3*B*c + 3*a*b^2*B*c - 3*a^2*b*c*C + b^3*c*C + a^3*A*d - 3*a*A*b^2*d + 3*a^2*b*B*d - b^3*B*d - a^3*C*d + 3*a*b^2*C*d + (Sqrt[-b^2]*(a^3*(A*c - c*C + B*d) - 3*a*b^2*(A*c - c*C + B*d) + 3*a^2*b*(B*c - (A - C)*d) - b^3*(B*c - (A - C)*d)))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2))) + (2*b*(3*a*b^5*B*c^2 - 3*a^5*b*B*d^2 + a^6*C*d^2 + 3*a^4*b^2*d*(B*c + 2*A*d - C*d) + b^6*(c*(c*C - B*d) - A*(c^2 - d^2)) - a^3*b^3*(8*c*(A - C)*d + B*(c^2 - d^2)) - 3*a^2*b^4*(c*(c*C + 2*B*d) - A*(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*A*b*c - A*b^3*c - a^3*B*c + 3*a*b^2*B*c - 3*a^2*b*c*C + b^3*c*C + a^3*A*d - 3*a*A*b^2*d + 3*a^2*b*B*d - b^3*B*d - a^3*C*d + 3*a*b^2*C*d - (Sqrt[-b^2]*(a^3*(A*c - c*C + B*d) - 3*a*b^2*(A*c - c*C + B*d) + 3*a^2*b*(B*c - (A - C)*d) - b^3*(B*c - (A - C)*d)))/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^2*(c^2*C - B*c*d + A*d^2)*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)) - (-(a*(-2*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B - b*C)*(b*c - a*d))) - 2*b^2*(a*b*c*(A - C) - a^2*A*d + 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
77,1,2463,579,8.5483061,"\int \frac{(a+b \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^2} \, dx","Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\log (\cos (e+f x)) \left(a^3 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+3 a^2 b \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)-3 a b^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)-b^3 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)\right)}{f \left(c^2+d^2\right)^2}-\frac{x \left(a^3 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)-3 a^2 b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)-3 a b^2 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)+b^3 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{\left(c^2+d^2\right)^2}+\frac{b^2 \tan (e+f x) \left(a d \left(d^2 (A+2 C)-B c d+3 c^2 C\right)-b \left(c d^2 (A+2 C)-2 B c^2 d-B d^3+3 c^3 C\right)\right)}{d^3 f \left(c^2+d^2\right)}-\frac{\left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^3}{d f \left(c^2+d^2\right) (c+d \tan (e+f x))}+\frac{b \left(d^2 (2 A+C)-2 B c d+3 c^2 C\right) (a+b \tan (e+f x))^2}{2 d^2 f \left(c^2+d^2\right)}+\frac{(b c-a d)^2 \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(c^2 d^2 (A+5 C)+3 A d^4-2 B c^3 d-4 B c d^3+3 c^4 C\right)\right) \log (c+d \tan (e+f x))}{d^4 f \left(c^2+d^2\right)^2}",1,"((a^3*A*c^2 - 3*a*A*b^2*c^2 - 3*a^2*b*B*c^2 + b^3*B*c^2 - a^3*c^2*C + 3*a*b^2*c^2*C + 6*a^2*A*b*c*d - 2*A*b^3*c*d + 2*a^3*B*c*d - 6*a*b^2*B*c*d - 6*a^2*b*c*C*d + 2*b^3*c*C*d - a^3*A*d^2 + 3*a*A*b^2*d^2 + 3*a^2*b*B*d^2 - b^3*B*d^2 + a^3*C*d^2 - 3*a*b^2*C*d^2)*(e + f*x)*Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3)/((c - I*d)^2*(c + I*d)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) + (((3*I)*b^3*c^11*C*d^3 - (2*I)*b^3*B*c^10*d^4 - (6*I)*a*b^2*c^10*C*d^4 + 3*b^3*c^10*C*d^4 + I*A*b^3*c^9*d^5 + (3*I)*a*b^2*B*c^9*d^5 - 2*b^3*B*c^9*d^5 + (3*I)*a^2*b*c^9*C*d^5 - 6*a*b^2*c^9*C*d^5 + (8*I)*b^3*c^9*C*d^5 + A*b^3*c^8*d^6 + 3*a*b^2*B*c^8*d^6 - (6*I)*b^3*B*c^8*d^6 + 3*a^2*b*c^8*C*d^6 - (18*I)*a*b^2*c^8*C*d^6 + 8*b^3*c^8*C*d^6 - (3*I)*a^2*A*b*c^7*d^7 + (4*I)*A*b^3*c^7*d^7 - I*a^3*B*c^7*d^7 + (12*I)*a*b^2*B*c^7*d^7 - 6*b^3*B*c^7*d^7 + (12*I)*a^2*b*c^7*C*d^7 - 18*a*b^2*c^7*C*d^7 + (5*I)*b^3*c^7*C*d^7 + (2*I)*a^3*A*c^6*d^8 - 3*a^2*A*b*c^6*d^8 - (6*I)*a*A*b^2*c^6*d^8 + 4*A*b^3*c^6*d^8 - a^3*B*c^6*d^8 - (6*I)*a^2*b*B*c^6*d^8 + 12*a*b^2*B*c^6*d^8 - (4*I)*b^3*B*c^6*d^8 - (2*I)*a^3*c^6*C*d^8 + 12*a^2*b*c^6*C*d^8 - (12*I)*a*b^2*c^6*C*d^8 + 5*b^3*c^6*C*d^8 + 2*a^3*A*c^5*d^9 - 6*a*A*b^2*c^5*d^9 + (3*I)*A*b^3*c^5*d^9 - 6*a^2*b*B*c^5*d^9 + (9*I)*a*b^2*B*c^5*d^9 - 4*b^3*B*c^5*d^9 - 2*a^3*c^5*C*d^9 + (9*I)*a^2*b*c^5*C*d^9 - 12*a*b^2*c^5*C*d^9 + (2*I)*a^3*A*c^4*d^10 - (6*I)*a*A*b^2*c^4*d^10 + 3*A*b^3*c^4*d^10 - (6*I)*a^2*b*B*c^4*d^10 + 9*a*b^2*B*c^4*d^10 - (2*I)*a^3*c^4*C*d^10 + 9*a^2*b*c^4*C*d^10 + 2*a^3*A*c^3*d^11 + (3*I)*a^2*A*b*c^3*d^11 - 6*a*A*b^2*c^3*d^11 + I*a^3*B*c^3*d^11 - 6*a^2*b*B*c^3*d^11 - 2*a^3*c^3*C*d^11 + 3*a^2*A*b*c^2*d^12 + a^3*B*c^2*d^12)*(e + f*x)*Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3)/(c^2*(c - I*d)^4*(c + I*d)^3*d^7*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) - (I*(3*b^3*c^6*C - 2*b^3*B*c^5*d - 6*a*b^2*c^5*C*d + A*b^3*c^4*d^2 + 3*a*b^2*B*c^4*d^2 + 3*a^2*b*c^4*C*d^2 + 5*b^3*c^4*C*d^2 - 4*b^3*B*c^3*d^3 - 12*a*b^2*c^3*C*d^3 - 3*a^2*A*b*c^2*d^4 + 3*A*b^3*c^2*d^4 - a^3*B*c^2*d^4 + 9*a*b^2*B*c^2*d^4 + 9*a^2*b*c^2*C*d^4 + 2*a^3*A*c*d^5 - 6*a*A*b^2*c*d^5 - 6*a^2*b*B*c*d^5 - 2*a^3*c*C*d^5 + 3*a^2*A*b*d^6 + a^3*B*d^6)*ArcTan[Tan[e + f*x]]*Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3)/(d^4*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) + ((-3*b^3*c^2*C + 2*b^3*B*c*d + 6*a*b^2*c*C*d - A*b^3*d^2 - 3*a*b^2*B*d^2 - 3*a^2*b*C*d^2 + b^3*C*d^2)*Cos[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])^3)/(d^4*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) + ((3*b^3*c^6*C - 2*b^3*B*c^5*d - 6*a*b^2*c^5*C*d + A*b^3*c^4*d^2 + 3*a*b^2*B*c^4*d^2 + 3*a^2*b*c^4*C*d^2 + 5*b^3*c^4*C*d^2 - 4*b^3*B*c^3*d^3 - 12*a*b^2*c^3*C*d^3 - 3*a^2*A*b*c^2*d^4 + 3*A*b^3*c^2*d^4 - a^3*B*c^2*d^4 + 9*a*b^2*B*c^2*d^4 + 9*a^2*b*c^2*C*d^4 + 2*a^3*A*c*d^5 - 6*a*A*b^2*c*d^5 - 6*a^2*b*B*c*d^5 - 2*a^3*c*C*d^5 + 3*a^2*A*b*d^6 + a^3*B*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])^2*(a + b*Tan[e + f*x])^3)/(2*d^4*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) + (b^3*C*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^3)/(2*d^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) + ((c*Cos[e + f*x] + d*Sin[e + f*x])^2*(-2*b^3*c*C*Sin[e + f*x] + b^3*B*d*Sin[e + f*x] + 3*a*b^2*C*d*Sin[e + f*x])*(a + b*Tan[e + f*x])^3)/(d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2) + (Cos[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])*(-(b^3*c^5*C*Sin[e + f*x]) + b^3*B*c^4*d*Sin[e + f*x] + 3*a*b^2*c^4*C*d*Sin[e + f*x] - A*b^3*c^3*d^2*Sin[e + f*x] - 3*a*b^2*B*c^3*d^2*Sin[e + f*x] - 3*a^2*b*c^3*C*d^2*Sin[e + f*x] + 3*a*A*b^2*c^2*d^3*Sin[e + f*x] + 3*a^2*b*B*c^2*d^3*Sin[e + f*x] + a^3*c^2*C*d^3*Sin[e + f*x] - 3*a^2*A*b*c*d^4*Sin[e + f*x] - a^3*B*c*d^4*Sin[e + f*x] + a^3*A*d^5*Sin[e + f*x])*(a + b*Tan[e + f*x])^3)/(c*(c - I*d)*(c + I*d)*d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^2)","C",1
78,1,2636,417,7.9431297,"\int \frac{(a+b \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^2} \, dx","Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2,x]","\text{Result too large to show}","\frac{\log (\cos (e+f x)) \left(a^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+2 a b \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)-b^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{f \left(c^2+d^2\right)^2}-\frac{x \left(a^2 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)-2 a b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)-b^2 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)\right)}{\left(c^2+d^2\right)^2}-\frac{\left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^2}{d f \left(c^2+d^2\right) (c+d \tan (e+f x))}-\frac{(b c-a d) \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right)\right) \log (c+d \tan (e+f x))}{d^3 f \left(c^2+d^2\right)^2}+\frac{b^2 \tan (e+f x) \left(d^2 (A+C)-B c d+2 c^2 C\right)}{d^2 f \left(c^2+d^2\right)}",1,"(((-2*I)*b^2*c^10*C*d^2 + I*b^2*B*c^9*d^3 + (2*I)*a*b*c^9*C*d^3 - 2*b^2*c^9*C*d^3 + b^2*B*c^8*d^4 + 2*a*b*c^8*C*d^4 - (6*I)*b^2*c^8*C*d^4 - (2*I)*a*A*b*c^7*d^5 - I*a^2*B*c^7*d^5 + (4*I)*b^2*B*c^7*d^5 + (8*I)*a*b*c^7*C*d^5 - 6*b^2*c^7*C*d^5 + (2*I)*a^2*A*c^6*d^6 - 2*a*A*b*c^6*d^6 - (2*I)*A*b^2*c^6*d^6 - a^2*B*c^6*d^6 - (4*I)*a*b*B*c^6*d^6 + 4*b^2*B*c^6*d^6 - (2*I)*a^2*c^6*C*d^6 + 8*a*b*c^6*C*d^6 - (4*I)*b^2*c^6*C*d^6 + 2*a^2*A*c^5*d^7 - 2*A*b^2*c^5*d^7 - 4*a*b*B*c^5*d^7 + (3*I)*b^2*B*c^5*d^7 - 2*a^2*c^5*C*d^7 + (6*I)*a*b*c^5*C*d^7 - 4*b^2*c^5*C*d^7 + (2*I)*a^2*A*c^4*d^8 - (2*I)*A*b^2*c^4*d^8 - (4*I)*a*b*B*c^4*d^8 + 3*b^2*B*c^4*d^8 - (2*I)*a^2*c^4*C*d^8 + 6*a*b*c^4*C*d^8 + 2*a^2*A*c^3*d^9 + (2*I)*a*A*b*c^3*d^9 - 2*A*b^2*c^3*d^9 + I*a^2*B*c^3*d^9 - 4*a*b*B*c^3*d^9 - 2*a^2*c^3*C*d^9 + 2*a*A*b*c^2*d^10 + a^2*B*c^2*d^10)*(e + f*x)*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2)/(c^2*(c - I*d)^4*(c + I*d)^3*d^5*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2) - (I*(-2*b^2*c^5*C + b^2*B*c^4*d + 2*a*b*c^4*C*d - 4*b^2*c^3*C*d^2 - 2*a*A*b*c^2*d^3 - a^2*B*c^2*d^3 + 3*b^2*B*c^2*d^3 + 6*a*b*c^2*C*d^3 + 2*a^2*A*c*d^4 - 2*A*b^2*c*d^4 - 4*a*b*B*c*d^4 - 2*a^2*c*C*d^4 + 2*a*A*b*d^5 + a^2*B*d^5)*ArcTan[Tan[e + f*x]]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2)/(d^3*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2) + ((2*b^2*c*C - b^2*B*d - 2*a*b*C*d)*Log[Cos[e + f*x]]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + b*Tan[e + f*x])^2)/(d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2) + ((-2*b^2*c^5*C + b^2*B*c^4*d + 2*a*b*c^4*C*d - 4*b^2*c^3*C*d^2 - 2*a*A*b*c^2*d^3 - a^2*B*c^2*d^3 + 3*b^2*B*c^2*d^3 + 6*a*b*c^2*C*d^3 + 2*a^2*A*c*d^4 - 2*A*b^2*c*d^4 - 4*a*b*B*c*d^4 - 2*a^2*c*C*d^4 + 2*a*A*b*d^5 + a^2*B*d^5)*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])^2)/(2*d^3*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2) + (Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])*(b^2*c^5*C*d + 2*b^2*c^3*C*d^3 + b^2*c*C*d^5 + a^2*A*c^4*d^2*(e + f*x) - A*b^2*c^4*d^2*(e + f*x) - 2*a*b*B*c^4*d^2*(e + f*x) - a^2*c^4*C*d^2*(e + f*x) + b^2*c^4*C*d^2*(e + f*x) + 4*a*A*b*c^3*d^3*(e + f*x) + 2*a^2*B*c^3*d^3*(e + f*x) - 2*b^2*B*c^3*d^3*(e + f*x) - 4*a*b*c^3*C*d^3*(e + f*x) - a^2*A*c^2*d^4*(e + f*x) + A*b^2*c^2*d^4*(e + f*x) + 2*a*b*B*c^2*d^4*(e + f*x) + a^2*c^2*C*d^4*(e + f*x) - b^2*c^2*C*d^4*(e + f*x) - b^2*c^5*C*d*Cos[2*(e + f*x)] - 2*b^2*c^3*C*d^3*Cos[2*(e + f*x)] - b^2*c*C*d^5*Cos[2*(e + f*x)] + a^2*A*c^4*d^2*(e + f*x)*Cos[2*(e + f*x)] - A*b^2*c^4*d^2*(e + f*x)*Cos[2*(e + f*x)] - 2*a*b*B*c^4*d^2*(e + f*x)*Cos[2*(e + f*x)] - a^2*c^4*C*d^2*(e + f*x)*Cos[2*(e + f*x)] + b^2*c^4*C*d^2*(e + f*x)*Cos[2*(e + f*x)] + 4*a*A*b*c^3*d^3*(e + f*x)*Cos[2*(e + f*x)] + 2*a^2*B*c^3*d^3*(e + f*x)*Cos[2*(e + f*x)] - 2*b^2*B*c^3*d^3*(e + f*x)*Cos[2*(e + f*x)] - 4*a*b*c^3*C*d^3*(e + f*x)*Cos[2*(e + f*x)] - a^2*A*c^2*d^4*(e + f*x)*Cos[2*(e + f*x)] + A*b^2*c^2*d^4*(e + f*x)*Cos[2*(e + f*x)] + 2*a*b*B*c^2*d^4*(e + f*x)*Cos[2*(e + f*x)] + a^2*c^2*C*d^4*(e + f*x)*Cos[2*(e + f*x)] - b^2*c^2*C*d^4*(e + f*x)*Cos[2*(e + f*x)] + 2*b^2*c^6*C*Sin[2*(e + f*x)] - b^2*B*c^5*d*Sin[2*(e + f*x)] - 2*a*b*c^5*C*d*Sin[2*(e + f*x)] + A*b^2*c^4*d^2*Sin[2*(e + f*x)] + 2*a*b*B*c^4*d^2*Sin[2*(e + f*x)] + a^2*c^4*C*d^2*Sin[2*(e + f*x)] + 3*b^2*c^4*C*d^2*Sin[2*(e + f*x)] - 2*a*A*b*c^3*d^3*Sin[2*(e + f*x)] - a^2*B*c^3*d^3*Sin[2*(e + f*x)] - b^2*B*c^3*d^3*Sin[2*(e + f*x)] - 2*a*b*c^3*C*d^3*Sin[2*(e + f*x)] + a^2*A*c^2*d^4*Sin[2*(e + f*x)] + A*b^2*c^2*d^4*Sin[2*(e + f*x)] + 2*a*b*B*c^2*d^4*Sin[2*(e + f*x)] + a^2*c^2*C*d^4*Sin[2*(e + f*x)] + b^2*c^2*C*d^4*Sin[2*(e + f*x)] - 2*a*A*b*c*d^5*Sin[2*(e + f*x)] - a^2*B*c*d^5*Sin[2*(e + f*x)] + a^2*A*d^6*Sin[2*(e + f*x)] + a^2*A*c^3*d^3*(e + f*x)*Sin[2*(e + f*x)] - A*b^2*c^3*d^3*(e + f*x)*Sin[2*(e + f*x)] - 2*a*b*B*c^3*d^3*(e + f*x)*Sin[2*(e + f*x)] - a^2*c^3*C*d^3*(e + f*x)*Sin[2*(e + f*x)] + b^2*c^3*C*d^3*(e + f*x)*Sin[2*(e + f*x)] + 4*a*A*b*c^2*d^4*(e + f*x)*Sin[2*(e + f*x)] + 2*a^2*B*c^2*d^4*(e + f*x)*Sin[2*(e + f*x)] - 2*b^2*B*c^2*d^4*(e + f*x)*Sin[2*(e + f*x)] - 4*a*b*c^2*C*d^4*(e + f*x)*Sin[2*(e + f*x)] - a^2*A*c*d^5*(e + f*x)*Sin[2*(e + f*x)] + A*b^2*c*d^5*(e + f*x)*Sin[2*(e + f*x)] + 2*a*b*B*c*d^5*(e + f*x)*Sin[2*(e + f*x)] + a^2*c*C*d^5*(e + f*x)*Sin[2*(e + f*x)] - b^2*c*C*d^5*(e + f*x)*Sin[2*(e + f*x)])*(a + b*Tan[e + f*x])^2)/(2*c*(c - I*d)^2*(c + I*d)^2*d^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^2)","C",1
79,1,606,292,6.7829527,"\int \frac{(a+b \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^2} \, dx","Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2,x]","\frac{-2 i c \tan ^{-1}(\tan (e+f x)) (c+d \tan (e+f x)) \left(a d^2 \left(2 c d (A-C)+B \left(d^2-c^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right)+c^2 \left(2 (c+i d)^2 (e+f x) \left(a d^2 (A-i B-C)+b \left(d^2 (-B-i A)+i c^2 C+2 c C d\right)\right)+\left(a d^2 \left(2 c d (A-C)+B \left(d^2-c^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)-2 b C \left(c^2+d^2\right)^2 \log (\cos (e+f x))\right)+d \tan (e+f x) \left(c \left(a d^2 \left(2 c d (A-C)+B \left(d^2-c^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right) \log \left((c \cos (e+f x)+d \sin (e+f x))^2\right)+2 (c+i d) \left(a d \left(-c^2 d (-A (e+f x)+B (i e+i f x+1)+C (e+f x+i))+c d^2 (A (i e+i f x+1)+B (e+f x+i)-i C (e+f x))-i A d^3+c^3 C\right)+b c \left(-i c d^2 (A (e+f x-i)-i B (e+f x+i)-2 C (e+f x))+d^3 (A (e+f x+i)-i B (e+f x))+c^2 d (B+C (e+f x+i))+i c^3 C (e+f x+i)\right)\right)-2 b c C \left(c^2+d^2\right)^2 \log (\cos (e+f x))\right)}{2 c d^2 f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}","\frac{(b c-a d) \left(A d^2-B c d+c^2 C\right)}{d^2 f \left(c^2+d^2\right) (c+d \tan (e+f x))}-\frac{\log (\cos (e+f x)) \left(-A \left(2 a c d-b \left(c^2-d^2\right)\right)+a \left(B c^2-B d^2+2 c C d\right)-b \left(-2 B c d+c^2 C-C d^2\right)\right)}{f \left(c^2+d^2\right)^2}-\frac{x \left(a \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)-b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{\left(c^2+d^2\right)^2}+\frac{\left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right) \log (c+d \tan (e+f x))}{d^2 f \left(c^2+d^2\right)^2}",1,"(c^2*(2*(c + I*d)^2*(a*(A - I*B - C)*d^2 + b*(I*c^2*C + 2*c*C*d + ((-I)*A - B)*d^2))*(e + f*x) - 2*b*C*(c^2 + d^2)^2*Log[Cos[e + f*x]] + (b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d + B*(-c^2 + d^2)))*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) + d*(2*(c + I*d)*(b*c*(I*c^3*C*(I + e + f*x) + d^3*((-I)*B*(e + f*x) + A*(I + e + f*x)) - I*c*d^2*(-2*C*(e + f*x) + A*(-I + e + f*x) - I*B*(I + e + f*x)) + c^2*d*(B + C*(I + e + f*x))) + a*d*(c^3*C - I*A*d^3 + c*d^2*(A*(1 + I*e + I*f*x) - I*C*(e + f*x) + B*(I + e + f*x)) - c^2*d*(B*(1 + I*e + I*f*x) - A*(e + f*x) + C*(I + e + f*x)))) - 2*b*c*C*(c^2 + d^2)^2*Log[Cos[e + f*x]] + c*(b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d + B*(-c^2 + d^2)))*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2])*Tan[e + f*x] - (2*I)*c*(b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d + B*(-c^2 + d^2)))*ArcTan[Tan[e + f*x]]*(c + d*Tan[e + f*x]))/(2*c*d^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))","C",1
80,1,207,140,2.5500506,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^2,x]","\frac{(d (C-A)+B c) \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}-\frac{2 C}{c+d \tan (e+f x)}}{2 d f}","-\frac{A d^2-B c d+c^2 C}{d f \left(c^2+d^2\right) (c+d \tan (e+f x))}+\frac{\left(2 c d (A-C)-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+c^2 C-C d^2\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) - (2*C)/(c + d*Tan[e + f*x]) + (B*c + (-A + C)*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
81,1,592,293,7.5036967,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2),x]","-\frac{\frac{b^2 \left(c^2+d^2\right) \left(A b^2-a (b B-a C)\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (b c-a d)}-\frac{b (b c-a d) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right) \left(-\frac{\sqrt{-b^2} \left(a \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)+b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{b}+2 a A c d-a B c^2+a B d^2-2 a c C d+A b c^2-A b d^2+2 b B c d-b c^2 C+b C d^2\right)}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{b (b c-a d) \log \left(\sqrt{-b^2}+b \tan (e+f x)\right) \left(\frac{\sqrt{-b^2} \left(a \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)+b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{b}+2 a A c d-a B c^2+a B d^2-2 a c C d+A b c^2-A b d^2+2 b B c d-b c^2 C+b C d^2\right)}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{b \left(b \left(c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right)-a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)}}{b f \left(c^2+d^2\right) (a d-b c)}-\frac{A d^2-c (B d-c C)}{f \left(c^2+d^2\right) (a d-b c) (c+d \tan (e+f x))}","-\frac{x \left(a \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)+b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)^2}+\frac{b \left(A b^2-a (b B-a C)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right) (b c-a d)^2}+\frac{A d^2-B c d+c^2 C}{f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))}-\frac{\left(b \left(c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right)-a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\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,"-((-1/2*(b*(b*c - a*d)*(A*b*c^2 - a*B*c^2 - b*c^2*C + 2*a*A*c*d + 2*b*B*c*d - 2*a*c*C*d - A*b*d^2 + a*B*d^2 + b*C*d^2 - (Sqrt[-b^2]*(a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d - B*(c^2 - d^2))))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) + (b^2*(A*b^2 - a*(b*B - a*C))*(c^2 + d^2)*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b*c - a*d)*(A*b*c^2 - a*B*c^2 - b*c^2*C + 2*a*A*c*d + 2*b*B*c*d - 2*a*c*C*d - A*b*d^2 + a*B*d^2 + b*C*d^2 + (Sqrt[-b^2]*(a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d - B*(c^2 - d^2))))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (b*(b*(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(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)) - (A*d^2 - c*(-(c*C) + B*d))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))","B",1
82,1,984,509,8.908137,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2),x]","-\frac{A b^2-a (b B-a C)}{\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{d^2 \left(2 A d b^2-a A (b c-a d)-(b B-a C) (b c+a d)\right)-c \left((A b-C b-a B) d (b c-a d)-2 c \left(A b^2-a (b B-a C)\right) d\right)}{(a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))}-\frac{-\frac{\left(c^2+d^2\right) \left(-2 C d a^4+3 b B d a^3-b^2 (B c+4 A d) a^2+b^3 (2 A c-2 C c+B d) a+b^4 (B c-2 A d)\right) \log (a+b \tan (e+f x)) b^2}{\left(a^2+b^2\right) (b c-a d)}+\frac{(b c-a d)^2 \left(-B c^2 a^2+B d^2 a^2+2 A c d a^2-2 c C d a^2+2 A b c^2 a-2 A b d^2 a+2 b C d^2 a-2 b c^2 C a+4 b B c d a+b^2 B c^2-b^2 B d^2-2 A b^2 c d+2 b^2 c C d-\frac{\sqrt{-b^2} \left(\left(C c^2-2 B d c-C d^2-A \left(c^2-d^2\right)\right) a^2+2 b \left(2 c (A-C) d-B \left(c^2-d^2\right)\right) a-b^2 \left(C c^2-2 B d c-C d^2-A \left(c^2-d^2\right)\right)\right)}{b}\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right) b}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{(b c-a d)^2 \left(-B c^2 a^2+B d^2 a^2+2 A c d a^2-2 c C d a^2+2 A b c^2 a-2 A b d^2 a+2 b C d^2 a-2 b c^2 C a+4 b B c d a+b^2 B c^2-b^2 B d^2-2 A b^2 c d+2 b^2 c C d+\frac{\sqrt{-b^2} \left(\left(C c^2-2 B d c-C d^2-A \left(c^2-d^2\right)\right) a^2+2 b \left(2 c (A-C) d-B \left(c^2-d^2\right)\right) a-b^2 \left(C c^2-2 B d c-C d^2-A \left(c^2-d^2\right)\right)\right)}{b}\right) \log \left(b \tan (e+f x)+\sqrt{-b^2}\right) b}{2 \left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{\left(a^2+b^2\right) d \left(b \left(2 C c^4-3 B d c^3+4 A d^2 c^2-B d^3 c+2 A d^4\right)-a d^2 \left(2 c (A-C) d-B \left(c^2-d^2\right)\right)\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)}","-\frac{d \left(A \left(a^2 d^2+b^2 \left(c^2+2 d^2\right)\right)+a^2 \left(-B c d+2 c^2 C+C d^2\right)-a b B \left(c^2+d^2\right)+b^2 c (c C-B d)\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 \left(a^2 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)+2 a b \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)-b^2 \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)\right)}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)^2}-\frac{A b^2-a (b B-a C)}{f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac{b \left(-2 a^4 C d+3 a^3 b B d-a^2 b^2 (4 A d+B c)+a b^3 (2 A c+B d-2 c C)+b^4 (B c-2 A 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{d \left(b \left(4 A c^2 d^2+2 A d^4-3 B c^3 d-B c d^3+2 c^4 C\right)-a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\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,"-((A*b^2 - a*(b*B - a*C))/((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*A*b*c^2 - a^2*B*c^2 + b^2*B*c^2 - 2*a*b*c^2*C + 2*a^2*A*c*d - 2*A*b^2*c*d + 4*a*b*B*c*d - 2*a^2*c*C*d + 2*b^2*c*C*d - 2*a*A*b*d^2 + a^2*B*d^2 - b^2*B*d^2 + 2*a*b*C*d^2 - (Sqrt[-b^2]*(a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d - B*(c^2 - d^2))))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (b^2*(c^2 + d^2)*(3*a^3*b*B*d - 2*a^4*C*d + b^4*(B*c - 2*A*d) - a^2*b^2*(B*c + 4*A*d) + a*b^3*(2*A*c - 2*c*C + B*d))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + (b*(b*c - a*d)^2*(2*a*A*b*c^2 - a^2*B*c^2 + b^2*B*c^2 - 2*a*b*c^2*C + 2*a^2*A*c*d - 2*A*b^2*c*d + 4*a*b*B*c*d - 2*a^2*c*C*d + 2*b^2*c*C*d - 2*a*A*b*d^2 + a^2*B*d^2 - b^2*B*d^2 + 2*a*b*C*d^2 + (Sqrt[-b^2]*(a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d - B*(c^2 - d^2))))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (b*(a^2 + b^2)*d*(b*(2*c^4*C - 3*B*c^3*d + 4*A*c^2*d^2 - B*c*d^3 + 2*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(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)) - (-(c*(-2*c*(A*b^2 - a*(b*B - a*C))*d + (A*b - a*B - b*C)*d*(b*c - a*d))) + d^2*(2*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(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
83,1,1758,841,8.6218361,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]","-\frac{A b^2-a (b B-a C)}{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(3 A d b^2-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right)-a \left(2 b (A b-C b-a B) (b c-a d)-3 a \left(A b^2-a (b B-a C)\right) 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{d^2 \left((-b c-a d) \left(2 b (A b-C b-a B) (b c-a d)-3 a \left(A b^2-a (b B-a C)\right) d\right)+\left(2 b^2 d-a (b c-a d)\right) \left(3 A d b^2-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right)\right)-c \left(d (b c-a d) \left(-3 b \left(A b^2-a (b B-a C)\right) d-2 a (A b-C b-a B) (b c-a d)+b \left(3 A d b^2-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right)\right)-2 c d \left(b^2 \left(3 A d b^2-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right)-a \left(2 b (A b-C b-a B) (b c-a d)-3 a \left(A b^2-a (b B-a C)\right) d\right)\right)\right)}{(a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))}-\frac{-\frac{\left(\sqrt{-b^2} \left(-B c^2 b^4+B d^2 b^4+2 A c d b^4-2 c C d b^4-3 a A c^2 b^3+3 a A d^2 b^3-3 a C d^2 b^3+3 a c^2 C b^3-6 a B c d b^3+3 a^2 B c^2 b^2-3 a^2 B d^2 b^2-6 a^2 A c d b^2+6 a^2 c C d b^2+a^3 A c^2 b-a^3 A d^2 b+a^3 C d^2 b-a^3 c^2 C b+2 a^3 B c d b\right)-b^2 \left(B c^2 a^3-B d^2 a^3-2 A c d a^3+2 c C d a^3-3 A b c^2 a^2+3 A b d^2 a^2-3 b C d^2 a^2+3 b c^2 C a^2-6 b B c d a^2-3 b^2 B c^2 a+3 b^2 B d^2 a+6 A b^2 c d a-6 b^2 c C d a+A b^3 c^2-A b^3 d^2+b^3 C d^2-b^3 c^2 C+2 b^3 B c d\right)\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right) (b c-a d)^3}{b \left(a^2+b^2\right) \left(c^2+d^2\right)}+\frac{\left(\left(B c^2 a^3-B d^2 a^3-2 A c d a^3+2 c C d a^3-3 A b c^2 a^2+3 A b d^2 a^2-3 b C d^2 a^2+3 b c^2 C a^2-6 b B c d a^2-3 b^2 B c^2 a+3 b^2 B d^2 a+6 A b^2 c d a-6 b^2 c C d a+A b^3 c^2-A b^3 d^2+b^3 C d^2-b^3 c^2 C+2 b^3 B c d\right) b^2+\sqrt{-b^2} \left(-B c^2 b^4+B d^2 b^4+2 A c d b^4-2 c C d b^4-3 a A c^2 b^3+3 a A d^2 b^3-3 a C d^2 b^3+3 a c^2 C b^3-6 a B c d b^3+3 a^2 B c^2 b^2-3 a^2 B d^2 b^2-6 a^2 A c d b^2+6 a^2 c C d b^2+a^3 A c^2 b-a^3 A d^2 b+a^3 C d^2 b-a^3 c^2 C b+2 a^3 B c d b\right)\right) \log \left(b \tan (e+f x)+\sqrt{-b^2}\right) (b c-a d)^3}{b \left(a^2+b^2\right) \left(c^2+d^2\right)}-\frac{2 b^2 \left(c^2+d^2\right) \left(-3 C d^2 a^6+6 b B d^2 a^5-b^2 d (4 B c+(10 A-C) d) a^4+b^3 \left(10 c (A-C) d+B \left(c^2+3 d^2\right)\right) a^3+3 b^4 \left(c (c C+2 B d)-A \left(c^2+3 d^2\right)\right) a^2+b^5 \left(2 c (A-C) d-B \left(3 c^2-d^2\right)\right) a-b^6 \left(c (c C-2 B d)-A \left(c^2-3 d^2\right)\right)\right) \log (a+b \tan (e+f x))}{\left(a^2+b^2\right) (b c-a d)}-\frac{2 b \left(a^2+b^2\right)^2 d^2 \left(b \left(3 C c^4-4 B d c^3+(5 A+C) d^2 c^2-2 B d^3 c+3 A d^4\right)-a d^2 \left(2 c (A-C) d-B \left(c^2-d^2\right)\right)\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right) (b c-a d)}}{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{\left(b \left(3 C c^4-4 B d c^3+(5 A+C) d^2 c^2-2 B d^3 c+3 A d^4\right)-a d^2 \left(2 c (A-C) d-B \left(c^2-d^2\right)\right)\right) \log (c \cos (e+f x)+d \sin (e+f x)) d^2}{(b c-a d)^4 \left(c^2+d^2\right)^2 f}-\frac{\left(-d \left(3 C c^2-B d c+(A+2 C) d^2\right) a^4+3 b B d \left(c^2+d^2\right) a^3-b^2 \left(B c^3+4 A d c^2+2 C d c^2-B d^2 c+6 A d^3\right) a^2+b^3 (2 A c-2 C c+B d) \left(c^2+d^2\right) a-b^4 \left(d \left(2 A c^2+C c^2+3 A d^2\right)-B \left(c^3+2 d^2 c\right)\right)\right) d}{\left(a^2+b^2\right)^2 (b c-a d)^3 \left(c^2+d^2\right) f (c+d \tan (e+f x))}-\frac{\left(\left(C c^2-2 B d c-C d^2-A \left(c^2-d^2\right)\right) a^3+3 b \left(2 c (A-C) d-B \left(c^2-d^2\right)\right) a^2-3 b^2 \left(C c^2-2 B d c-C d^2-A \left(c^2-d^2\right)\right) a-b^3 \left(2 c (A-C) d-B \left(c^2-d^2\right)\right)\right) x}{\left(a^2+b^2\right)^3 \left(c^2+d^2\right)^2}-\frac{b \left(-3 C d^2 a^6+6 b B d^2 a^5-b^2 d (4 B c+(10 A-C) d) a^4+b^3 \left(10 c (A-C) d+B \left(c^2+3 d^2\right)\right) a^3+3 b^4 \left(c (c C+2 B d)-A \left(c^2+3 d^2\right)\right) a^2+b^5 \left(2 c (A-C) d-B \left(3 c^2-d^2\right)\right) a-b^6 \left(c (c C-2 B d)-A \left(c^2-3 d^2\right)\right)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{\left(a^2+b^2\right)^3 (b c-a d)^4 f}-\frac{-3 C d a^4+5 b B d a^3-b^2 (2 B c+(7 A-C) d) a^2+b^3 (4 A c-4 C c+B d) a+b^4 (2 B c-3 A d)}{2 \left(a^2+b^2\right)^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac{A b^2-a (b B-a C)}{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))}",1,"-1/2*(A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (-((-(a*(-3*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B - b*C)*(b*c - a*d))) + b^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(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*c - a*d)^3*(-(b^2*(-3*a^2*A*b*c^2 + A*b^3*c^2 + a^3*B*c^2 - 3*a*b^2*B*c^2 + 3*a^2*b*c^2*C - b^3*c^2*C - 2*a^3*A*c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d + 2*b^3*B*c*d + 2*a^3*c*C*d - 6*a*b^2*c*C*d + 3*a^2*A*b*d^2 - A*b^3*d^2 - a^3*B*d^2 + 3*a*b^2*B*d^2 - 3*a^2*b*C*d^2 + b^3*C*d^2)) + Sqrt[-b^2]*(a^3*A*b*c^2 - 3*a*A*b^3*c^2 + 3*a^2*b^2*B*c^2 - b^4*B*c^2 - a^3*b*c^2*C + 3*a*b^3*c^2*C - 6*a^2*A*b^2*c*d + 2*A*b^4*c*d + 2*a^3*b*B*c*d - 6*a*b^3*B*c*d + 6*a^2*b^2*c*C*d - 2*b^4*c*C*d - a^3*A*b*d^2 + 3*a*A*b^3*d^2 - 3*a^2*b^2*B*d^2 + b^4*B*d^2 + a^3*b*C*d^2 - 3*a*b^3*C*d^2))*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/(b*(a^2 + b^2)*(c^2 + d^2))) - (2*b^2*(c^2 + d^2)*(6*a^5*b*B*d^2 - 3*a^6*C*d^2 - a^4*b^2*d*(4*B*c + (10*A - C)*d) - b^6*(c*(c*C - 2*B*d) - A*(c^2 - 3*d^2)) + a*b^5*(2*c*(A - C)*d - B*(3*c^2 - d^2)) + 3*a^2*b^4*(c*(c*C + 2*B*d) - A*(c^2 + 3*d^2)) + a^3*b^3*(10*c*(A - C)*d + B*(c^2 + 3*d^2)))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + ((b*c - a*d)^3*(b^2*(-3*a^2*A*b*c^2 + A*b^3*c^2 + a^3*B*c^2 - 3*a*b^2*B*c^2 + 3*a^2*b*c^2*C - b^3*c^2*C - 2*a^3*A*c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d + 2*b^3*B*c*d + 2*a^3*c*C*d - 6*a*b^2*c*C*d + 3*a^2*A*b*d^2 - A*b^3*d^2 - a^3*B*d^2 + 3*a*b^2*B*d^2 - 3*a^2*b*C*d^2 + b^3*C*d^2) + Sqrt[-b^2]*(a^3*A*b*c^2 - 3*a*A*b^3*c^2 + 3*a^2*b^2*B*c^2 - b^4*B*c^2 - a^3*b*c^2*C + 3*a*b^3*c^2*C - 6*a^2*A*b^2*c*d + 2*A*b^4*c*d + 2*a^3*b*B*c*d - 6*a*b^3*B*c*d + 6*a^2*b^2*c*C*d - 2*b^4*c*C*d - a^3*A*b*d^2 + 3*a*A*b^3*d^2 - 3*a^2*b^2*B*d^2 + b^4*B*d^2 + a^3*b*C*d^2 - 3*a*b^3*C*d^2))*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(b*(a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)^2*d^2*(b*(3*c^4*C - 4*B*c^3*d + c^2*(5*A + C)*d^2 - 2*B*c*d^3 + 3*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(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*((-(b*c) - a*d)*(-3*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B - b*C)*(b*c - a*d)) + (2*b^2*d - a*(b*c - a*d))*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d))) - c*(d*(b*c - a*d)*(-3*b*(A*b^2 - a*(b*B - a*C))*d - 2*a*(A*b - a*B - b*C)*(b*c - a*d) + b*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d))) - 2*c*d*(-(a*(-3*a*(A*b^2 - a*(b*B - a*C))*d + 2*b*(A*b - a*B - b*C)*(b*c - a*d))) + b^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*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)))/(2*(a^2 + b^2)*(b*c - a*d))","B",0
84,1,1445,804,15.597668,"\int \frac{(a+b \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^3} \, dx","Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3,x]","\frac{\left(\left(-C c^3+3 B d c^2+3 C d^2 c-B d^3+A \left(c^3-3 c d^2\right)\right) a^3-3 b \left((A-C) d \left(d^2-3 c^2\right)+B \left(c^3-3 c d^2\right)\right) a^2+3 b^2 \left(-A c^3+C c^3-3 B d c^2+3 A d^2 c-3 C d^2 c+B d^3\right) a+b^3 \left((A-C) d \left(d^2-3 c^2\right)+B \left(c^3-3 c d^2\right)\right)\right) (e+f x) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{\left(c^2+d^2\right)^3 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}-\frac{b^2 (-3 b c C+3 a d C+b B d) \log \left(1-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{d^4 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}+\frac{\left(\left((A-C) d \left(d^2-3 c^2\right)+B \left(c^3-3 c d^2\right)\right) a^3-3 b \left(-A c^3+C c^3-3 B d c^2+3 A d^2 c-3 C d^2 c+B d^3\right) a^2-3 b^2 \left((A-C) d \left(d^2-3 c^2\right)+B \left(c^3-3 c d^2\right)\right) a+b^3 \left(-A c^3+C c^3-3 B d c^2+3 A d^2 c-3 C d^2 c+B d^3\right)\right) \log \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{\left(c^2+d^2\right)^3 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}+\frac{(a d-b c) \left(a^2 \left(-\left((A-C) d \left(d^2-3 c^2\right)\right)-B \left(c^3-3 c d^2\right)\right) d^3-a b \left(8 c (C-A) d^3+B \left(c^4+6 d^2 c^2-3 d^4\right)\right) d^2+b^2 \left(3 C c^6-B d c^5+9 C d^2 c^4-3 B d^3 c^3-(A-10 C) d^4 c^2-6 B d^5 c+3 A d^6\right)\right) \log \left(c \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)-2 d \tan \left(\frac{1}{2} (e+f x)\right)\right) (c \cos (e+f x)+d \sin (e+f x))^3 (a+b \tan (e+f x))^3}{d^4 \left(c^2+d^2\right)^3 f (a \cos (e+f x)+b \sin (e+f x))^3 (c+d \tan (e+f x))^3}-\frac{2 (b c-a d)^2 (c \cos (e+f x)+d \sin (e+f x))^3 \left(a d \left(C \tan \left(\frac{1}{2} (e+f x)\right) c^5+d \left(C-2 B \tan \left(\frac{1}{2} (e+f x)\right)\right) c^4-d^2 \left(B-3 A \tan \left(\frac{1}{2} (e+f x)\right)+C \tan \left(\frac{1}{2} (e+f x)\right)\right) c^3+(A+C) d^3 c^2+d^4 \left(A \tan \left(\frac{1}{2} (e+f x)\right)-B\right) c+A d^5\right)+b c \left(2 C \tan \left(\frac{1}{2} (e+f x)\right) c^5-d \left(C+B \tan \left(\frac{1}{2} (e+f x)\right)\right) c^4+d^2 \left(B+4 C \tan \left(\frac{1}{2} (e+f x)\right)\right) c^3-d^3 \left(A+C+3 B \tan \left(\frac{1}{2} (e+f x)\right)\right) c^2+d^4 \left(B+2 A \tan \left(\frac{1}{2} (e+f x)\right)\right) c-A d^5\right)\right) (a+b \tan (e+f x))^3}{c^3 d^3 \left(c^2+d^2\right)^2 f (a \cos (e+f x)+b \sin (e+f x))^3 \left(c \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)-2 d \tan \left(\frac{1}{2} (e+f x)\right)\right) (c+d \tan (e+f x))^3}-\frac{2 b^3 C (c \cos (e+f x)+d \sin (e+f x))^3 \tan \left(\frac{1}{2} (e+f x)\right) (a+b \tan (e+f x))^3}{d^3 f (a \cos (e+f x)+b \sin (e+f x))^3 \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) (c+d \tan (e+f x))^3}+\frac{2 (b c-a d)^3 \left(C c^2-B d c+A d^2\right) (c \cos (e+f x)+d \sin (e+f x))^3 \left(c+2 d \tan \left(\frac{1}{2} (e+f x)\right)\right) (a+b \tan (e+f x))^3}{c^3 d^2 \left(c^2+d^2\right) f (a \cos (e+f x)+b \sin (e+f x))^3 \left(-c \tan ^2\left(\frac{1}{2} (e+f x)\right)+2 d \tan \left(\frac{1}{2} (e+f x)\right)+c\right)^2 (c+d \tan (e+f x))^3}","-\frac{\left(C c^2-B d c+A d^2\right) (a+b \tan (e+f x))^3}{2 d \left(c^2+d^2\right) f (c+d \tan (e+f x))^2}-\frac{\left(2 a \left(2 c (A-C) d-B \left(c^2-d^2\right)\right) d^2+b \left(3 C c^4-B d c^3-(A-7 C) d^2 c^2-5 B d^3 c+3 A d^4\right)\right) (a+b \tan (e+f x))^2}{2 d^2 \left(c^2+d^2\right)^2 f (c+d \tan (e+f x))}-\frac{\left(\left(C c^3-3 B d c^2-3 C d^2 c+B d^3-A \left(c^3-3 c d^2\right)\right) a^3-3 b \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) a^2+3 b^2 \left(A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right) a+b^3 \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)\right) x}{\left(c^2+d^2\right)^3}-\frac{\left(-\left(\left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) a^3\right)+3 b \left(A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right) a^2+3 b^2 \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) a-b^3 \left(A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right)\right) \log (\cos (e+f x))}{\left(c^2+d^2\right)^3 f}-\frac{(b c-a d) \left(a^2 \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) d^3+a b \left(8 c (A-C) d^3-B \left(c^4+6 d^2 c^2-3 d^4\right)\right) d^2+b^2 \left(3 C c^6-B d c^5+9 C d^2 c^4-3 B d^3 c^3-(A-10 C) d^4 c^2-6 B d^5 c+3 A d^6\right)\right) \log (c+d \tan (e+f x))}{d^4 \left(c^2+d^2\right)^3 f}+\frac{b^2 \left(a \left(2 c (A-C) d-B \left(c^2-d^2\right)\right) d^2+b \left(3 C c^4-B d c^3+6 C d^2 c^2-3 B d^3 c+(2 A+C) d^4\right)\right) \tan (e+f x)}{d^3 \left(c^2+d^2\right)^2 f}",1,"((3*a*b^2*(-(A*c^3) + c^3*C - 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 + B*d^3) + a^3*(-(c^3*C) + 3*B*c^2*d + 3*c*C*d^2 - B*d^3 + A*(c^3 - 3*c*d^2)) - 3*a^2*b*((A - C)*d*(-3*c^2 + d^2) + B*(c^3 - 3*c*d^2)) + b^3*((A - C)*d*(-3*c^2 + d^2) + B*(c^3 - 3*c*d^2)))*(e + f*x)*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3)/((c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3) - (b^2*(-3*b*c*C + b*B*d + 3*a*C*d)*Log[1 - Tan[(e + f*x)/2]^2]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3)/(d^4*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3) + ((-3*a^2*b*(-(A*c^3) + c^3*C - 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 + B*d^3) + b^3*(-(A*c^3) + c^3*C - 3*B*c^2*d + 3*A*c*d^2 - 3*c*C*d^2 + B*d^3) + a^3*((A - C)*d*(-3*c^2 + d^2) + B*(c^3 - 3*c*d^2)) - 3*a*b^2*((A - C)*d*(-3*c^2 + d^2) + B*(c^3 - 3*c*d^2)))*Log[1 + Tan[(e + f*x)/2]^2]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3)/((c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3) + ((-(b*c) + a*d)*(b^2*(3*c^6*C - B*c^5*d + 9*c^4*C*d^2 - 3*B*c^3*d^3 - c^2*(A - 10*C)*d^4 - 6*B*c*d^5 + 3*A*d^6) + a^2*d^3*(-((A - C)*d*(-3*c^2 + d^2)) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c*(-A + C)*d^3 + B*(c^4 + 6*c^2*d^2 - 3*d^4)))*Log[-2*d*Tan[(e + f*x)/2] + c*(-1 + Tan[(e + f*x)/2]^2)]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^3)/(d^4*(c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + d*Tan[e + f*x])^3) - (2*b^3*C*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*Tan[(e + f*x)/2]*(a + b*Tan[e + f*x])^3)/(d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(-1 + Tan[(e + f*x)/2]^2)*(c + d*Tan[e + f*x])^3) + (2*(b*c - a*d)^3*(c^2*C - B*c*d + A*d^2)*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(c + 2*d*Tan[(e + f*x)/2])*(a + b*Tan[e + f*x])^3)/(c^3*d^2*(c^2 + d^2)*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(c + 2*d*Tan[(e + f*x)/2] - c*Tan[(e + f*x)/2]^2)^2*(c + d*Tan[e + f*x])^3) - (2*(b*c - a*d)^2*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a*d*(c^2*(A + C)*d^3 + A*d^5 + c^5*C*Tan[(e + f*x)/2] + c*d^4*(-B + A*Tan[(e + f*x)/2]) + c^4*d*(C - 2*B*Tan[(e + f*x)/2]) - c^3*d^2*(B - 3*A*Tan[(e + f*x)/2] + C*Tan[(e + f*x)/2])) + b*c*(-(A*d^5) + 2*c^5*C*Tan[(e + f*x)/2] + c*d^4*(B + 2*A*Tan[(e + f*x)/2]) - c^4*d*(C + B*Tan[(e + f*x)/2]) - c^2*d^3*(A + C + 3*B*Tan[(e + f*x)/2]) + c^3*d^2*(B + 4*C*Tan[(e + f*x)/2])))*(a + b*Tan[e + f*x])^3)/(c^3*d^3*(c^2 + d^2)^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^3*(-2*d*Tan[(e + f*x)/2] + c*(-1 + Tan[(e + f*x)/2]^2))*(c + d*Tan[e + f*x])^3)","A",0
85,1,2499,597,8.126963,"\int \frac{(a+b \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^3} \, dx","Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{\log (\cos (e+f x)) \left(-\left(a^2 \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)\right)+2 a b \left(A c^3-3 A c d^2+3 B c^2 d-B d^3-c^3 C+3 c C d^2\right)+b^2 \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)\right)}{f \left(c^2+d^2\right)^3}-\frac{x \left(a^2 \left(-A \left(c^3-3 c d^2\right)-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right)-2 a b \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)+b^2 \left(A c^3-3 A c d^2+3 B c^2 d-B d^3-c^3 C+3 c C d^2\right)\right)}{\left(c^2+d^2\right)^3}-\frac{\left(-a^2 d^3 \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)+2 a b d^3 \left(A c^3-3 A c d^2+3 B c^2 d-B d^3-c^3 C+3 c C d^2\right)-b^2 \left(-3 c^2 d^4 (A-2 C)+A d^6+B c^3 d^3-3 B c d^5+c^6 C+3 c^4 C d^2\right)\right) \log (c+d \tan (e+f x))}{d^3 f \left(c^2+d^2\right)^3}-\frac{\left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^2}{2 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}+\frac{(b c-a d) \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right)}{d^3 f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}",1,"((-(b^2*c^4*C) + b^2*B*c^3*d + 2*a*b*c^3*C*d - A*b^2*c^2*d^2 - 2*a*b*B*c^2*d^2 - a^2*c^2*C*d^2 + 2*a*A*b*c*d^3 + a^2*B*c*d^3 - a^2*A*d^4)*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])*(a + b*Tan[e + f*x])^2)/(2*(c - I*d)^2*(c + I*d)^2*d*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3) + ((a^2*A*c^3 - A*b^2*c^3 - 2*a*b*B*c^3 - a^2*c^3*C + b^2*c^3*C + 6*a*A*b*c^2*d + 3*a^2*B*c^2*d - 3*b^2*B*c^2*d - 6*a*b*c^2*C*d - 3*a^2*A*c*d^2 + 3*A*b^2*c*d^2 + 6*a*b*B*c*d^2 + 3*a^2*c*C*d^2 - 3*b^2*c*C*d^2 - 2*a*A*b*d^3 - a^2*B*d^3 + b^2*B*d^3 + 2*a*b*C*d^3)*(e + f*x)*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)/((c - I*d)^3*(c + I*d)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3) + ((I*b^2*c^13*C*d^2 + b^2*c^12*C*d^3 + (5*I)*b^2*c^11*C*d^4 - (2*I)*a*A*b*c^10*d^5 - I*a^2*B*c^10*d^5 + I*b^2*B*c^10*d^5 + (2*I)*a*b*c^10*C*d^5 + 5*b^2*c^10*C*d^5 + (3*I)*a^2*A*c^9*d^6 - 2*a*A*b*c^9*d^6 - (3*I)*A*b^2*c^9*d^6 - a^2*B*c^9*d^6 - (6*I)*a*b*B*c^9*d^6 + b^2*B*c^9*d^6 - (3*I)*a^2*c^9*C*d^6 + 2*a*b*c^9*C*d^6 + (13*I)*b^2*c^9*C*d^6 + 3*a^2*A*c^8*d^7 + (2*I)*a*A*b*c^8*d^7 - 3*A*b^2*c^8*d^7 + I*a^2*B*c^8*d^7 - 6*a*b*B*c^8*d^7 - I*b^2*B*c^8*d^7 - 3*a^2*c^8*C*d^7 - (2*I)*a*b*c^8*C*d^7 + 13*b^2*c^8*C*d^7 + (5*I)*a^2*A*c^7*d^8 + 2*a*A*b*c^7*d^8 - (5*I)*A*b^2*c^7*d^8 + a^2*B*c^7*d^8 - (10*I)*a*b*B*c^7*d^8 - b^2*B*c^7*d^8 - (5*I)*a^2*c^7*C*d^8 - 2*a*b*c^7*C*d^8 + (15*I)*b^2*c^7*C*d^8 + 5*a^2*A*c^6*d^9 + (10*I)*a*A*b*c^6*d^9 - 5*A*b^2*c^6*d^9 + (5*I)*a^2*B*c^6*d^9 - 10*a*b*B*c^6*d^9 - (5*I)*b^2*B*c^6*d^9 - 5*a^2*c^6*C*d^9 - (10*I)*a*b*c^6*C*d^9 + 15*b^2*c^6*C*d^9 + I*a^2*A*c^5*d^10 + 10*a*A*b*c^5*d^10 - I*A*b^2*c^5*d^10 + 5*a^2*B*c^5*d^10 - (2*I)*a*b*B*c^5*d^10 - 5*b^2*B*c^5*d^10 - I*a^2*c^5*C*d^10 - 10*a*b*c^5*C*d^10 + (6*I)*b^2*c^5*C*d^10 + a^2*A*c^4*d^11 + (6*I)*a*A*b*c^4*d^11 - A*b^2*c^4*d^11 + (3*I)*a^2*B*c^4*d^11 - 2*a*b*B*c^4*d^11 - (3*I)*b^2*B*c^4*d^11 - a^2*c^4*C*d^11 - (6*I)*a*b*c^4*C*d^11 + 6*b^2*c^4*C*d^11 - I*a^2*A*c^3*d^12 + 6*a*A*b*c^3*d^12 + I*A*b^2*c^3*d^12 + 3*a^2*B*c^3*d^12 + (2*I)*a*b*B*c^3*d^12 - 3*b^2*B*c^3*d^12 + I*a^2*c^3*C*d^12 - 6*a*b*c^3*C*d^12 - a^2*A*c^2*d^13 + A*b^2*c^2*d^13 + 2*a*b*B*c^2*d^13 + a^2*c^2*C*d^13)*(e + f*x)*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)/(c^2*(c - I*d)^6*(c + I*d)^5*d^5*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3) - (I*(b^2*c^6*C + 3*b^2*c^4*C*d^2 - 2*a*A*b*c^3*d^3 - a^2*B*c^3*d^3 + b^2*B*c^3*d^3 + 2*a*b*c^3*C*d^3 + 3*a^2*A*c^2*d^4 - 3*A*b^2*c^2*d^4 - 6*a*b*B*c^2*d^4 - 3*a^2*c^2*C*d^4 + 6*b^2*c^2*C*d^4 + 6*a*A*b*c*d^5 + 3*a^2*B*c*d^5 - 3*b^2*B*c*d^5 - 6*a*b*c*C*d^5 - a^2*A*d^6 + A*b^2*d^6 + 2*a*b*B*d^6 + a^2*C*d^6)*ArcTan[Tan[e + f*x]]*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)/(d^3*(c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3) - (b^2*C*Log[Cos[e + f*x]]*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)/(d^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3) + ((b^2*c^6*C + 3*b^2*c^4*C*d^2 - 2*a*A*b*c^3*d^3 - a^2*B*c^3*d^3 + b^2*B*c^3*d^3 + 2*a*b*c^3*C*d^3 + 3*a^2*A*c^2*d^4 - 3*A*b^2*c^2*d^4 - 6*a*b*B*c^2*d^4 - 3*a^2*c^2*C*d^4 + 6*b^2*c^2*C*d^4 + 6*a*A*b*c*d^5 + 3*a^2*B*c*d^5 - 3*b^2*B*c*d^5 - 6*a*b*c*C*d^5 - a^2*A*d^6 + A*b^2*d^6 + 2*a*b*B*d^6 + a^2*C*d^6)*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^3*(a + b*Tan[e + f*x])^2)/(2*d^3*(c^2 + d^2)^3*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3) + (Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(-(b^2*c^5*C*Sin[e + f*x]) + A*b^2*c^3*d^2*Sin[e + f*x] + 2*a*b*B*c^3*d^2*Sin[e + f*x] + a^2*c^3*C*d^2*Sin[e + f*x] - 4*b^2*c^3*C*d^2*Sin[e + f*x] - 4*a*A*b*c^2*d^3*Sin[e + f*x] - 2*a^2*B*c^2*d^3*Sin[e + f*x] + 3*b^2*B*c^2*d^3*Sin[e + f*x] + 6*a*b*c^2*C*d^3*Sin[e + f*x] + 3*a^2*A*c*d^4*Sin[e + f*x] - 2*A*b^2*c*d^4*Sin[e + f*x] - 4*a*b*B*c*d^4*Sin[e + f*x] - 2*a^2*c*C*d^4*Sin[e + f*x] + 2*a*A*b*d^5*Sin[e + f*x] + a^2*B*d^5*Sin[e + f*x])*(a + b*Tan[e + f*x])^2)/(c*(c - I*d)^2*(c + I*d)^2*d^2*f*(a*Cos[e + f*x] + b*Sin[e + f*x])^2*(c + d*Tan[e + f*x])^3)","C",1
86,1,378,352,6.3441559,"\int \frac{(a+b \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^3} \, dx","Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3,x]","-\frac{C (a+b \tan (e+f x))}{d f (c+d \tan (e+f x))^2}-\frac{\frac{-a C d+b B d+b c C}{2 d f (c+d \tan (e+f x))^2}+\frac{\frac{\left(2 c d^2 (a B+A b-b C)+2 d^3 (b B-a (A-C))\right) \left(-\frac{2 c d}{\left(c^2+d^2\right)^2 (c+d \tan (e+f x))}-\frac{d}{2 \left(c^2+d^2\right) (c+d \tan (e+f x))^2}+\frac{d \left(3 c^2-d^2\right) \log (c+d \tan (e+f x))}{\left(c^2+d^2\right)^3}-\frac{\log (-\tan (e+f x)+i)}{2 (-d+i c)^3}+\frac{\log (\tan (e+f x)+i)}{2 (d+i c)^3}\right)}{d}-2 d (a B+A b-b C) \left(-\frac{d}{\left(c^2+d^2\right) (c+d \tan (e+f x))}+\frac{2 c d \log (c+d \tan (e+f x))}{\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)}{2 d f}}{d}","\frac{(b c-a d) \left(A d^2-B c d+c^2 C\right)}{2 d^2 f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}-\frac{a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)}{d^2 f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}+\frac{\left(A \left(a d \left(3 c^2-d^2\right)-b \left(c^3-3 c d^2\right)\right)-a \left(B c^3-3 B c d^2+3 c^2 C d-C d^3\right)+b \left(-3 B c^2 d+B d^3+c^3 C-3 c 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 \left(-A \left(c^3-3 c d^2\right)-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right)-b \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)\right)}{\left(c^2+d^2\right)^3}",1,"-((C*(a + b*Tan[e + f*x]))/(d*f*(c + d*Tan[e + f*x])^2)) - ((b*c*C + b*B*d - a*C*d)/(2*d*f*(c + d*Tan[e + f*x])^2) + (((2*c*(A*b + a*B - b*C)*d^2 + 2*(b*B - a*(A - C))*d^3)*(-1/2*Log[I - Tan[e + f*x]]/(I*c - d)^3 + Log[I + Tan[e + f*x]]/(2*(I*c + d)^3) + (d*(3*c^2 - d^2)*Log[c + d*Tan[e + f*x]])/(c^2 + d^2)^3 - d/(2*(c^2 + d^2)*(c + d*Tan[e + f*x])^2) - (2*c*d)/((c^2 + d^2)^2*(c + d*Tan[e + f*x]))))/d - 2*(A*b + a*B - b*C)*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 + (2*c*d*Log[c + d*Tan[e + f*x]])/(c^2 + d^2)^2 - d/((c^2 + d^2)*(c + d*Tan[e + f*x]))))/(2*d*f))/d","C",1
87,1,261,209,5.2794341,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^3,x]","-\frac{-(d (C-A)+B c) \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)+\frac{C}{(c+d \tan (e+f x))^2}}{2 d f}","-\frac{A d^2-B c d+c^2 C}{2 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^2}-\frac{2 c d (A-C)-B \left(c^2-d^2\right)}{f \left(c^2+d^2\right)^2 (c+d \tan (e+f x))}+\frac{\left(d (A-C) \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 \left(c^3-3 c d^2\right)-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right)}{\left(c^2+d^2\right)^3}",1,"-1/2*(C/(c + d*Tan[e + f*x])^2 + 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 + C)*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))/(d*f)","C",1
88,1,912,487,9.2444331,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3),x]","-\frac{A d^2-c (B d-c C)}{2 (a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))^2}-\frac{-\frac{-2 \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(2 d (b c-a d) (B c-(A-C) d)-2 b c \left(C c^2-B d c+A 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(A b^2-a (b B-a C)\right) \left(c^2+d^2\right)^2 \log (a+b \tan (e+f x)) b^3}{\left(a^2+b^2\right) (b c-a d)}-\frac{(b c-a d)^2 \left(A b c^3-a B c^3-b C c^3+3 a A d c^2+3 b B d c^2-3 a C d c^2-3 A b d^2 c+3 a B d^2 c+3 b C d^2 c-a A d^3-b B d^3+a C d^3-\frac{\sqrt{-b^2} \left(a \left(C c^3-3 B d c^2-3 C d^2 c+B d^3-A \left(c^3-3 c d^2\right)\right)+b \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)\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)^2 \left(A b c^3-a B c^3-b C c^3+3 a A d c^2+3 b B d c^2-3 a C d c^2-3 A b d^2 c+3 a B d^2 c+3 b C d^2 c-a A d^3-b B d^3+a C d^3+\frac{\sqrt{-b^2} \left(b (A-C) d \left(3 c^2-d^2\right)-b B \left(c^3-3 c d^2\right)-a \left(A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\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 \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) d^3-a b \left(8 c^3 (A-C) d-B \left(3 c^4-6 d^2 c^2-d^4\right)\right) d^2+b^2 \left(C c^6-3 B d c^5+3 (2 A-C) d^2 c^4+B d^3 c^3+3 A d^4 c^2+A d^6\right)\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)}","-\frac{x \left(a \left(-A \left(c^3-3 c d^2\right)-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right)+b \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right)^3}-\frac{\left(a^2 d^3 \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)-a b d^2 \left(8 c^3 d (A-C)-B \left(3 c^4-6 c^2 d^2-d^4\right)\right)+b^2 \left(3 c^4 d^2 (2 A-C)+3 A c^2 d^4+A d^6-3 B c^5 d+B c^3 d^3+c^6 C\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^2 \left(A b^2-a (b B-a C)\right) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left(a^2+b^2\right) (b c-a d)^3}+\frac{A d^2-B c d+c^2 C}{2 f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))^2}+\frac{b \left(c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right)-a d^2 \left(2 c d (A-C)-B \left(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))}",1,"-1/2*(A*d^2 - c*(-(c*C) + B*d))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-((-((b*(b*c - a*d)^2*(A*b*c^3 - a*B*c^3 - b*c^3*C + 3*a*A*c^2*d + 3*b*B*c^2*d - 3*a*c^2*C*d - 3*A*b*c*d^2 + 3*a*B*c*d^2 + 3*b*c*C*d^2 - a*A*d^3 - b*B*d^3 + a*C*d^3 - (Sqrt[-b^2]*(a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) + b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2))))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2))) + (2*b^3*(A*b^2 - a*(b*B - a*C))*(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)^2*(A*b*c^3 - a*B*c^3 - b*c^3*C + 3*a*A*c^2*d + 3*b*B*c^2*d - 3*a*c^2*C*d - 3*A*b*c*d^2 + 3*a*B*c*d^2 + 3*b*c*C*d^2 - a*A*d^3 - b*B*d^3 + a*C*d^3 + (Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) - b*B*(c^3 - 3*c*d^2) - a*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) - (2*b*(b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^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*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)) - c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - 2*b*c*(c^2*C - B*c*d + A*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",1
89,1,1732,861,8.7396133,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3),x]","-\frac{A b^2-a (b B-a C)}{\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(3 A d b^2-a A (b c-a d)-(b B-a C) (b c+2 a d)\right)-c \left((A b-C b-a B) d (b c-a d)-3 c \left(A b^2-a (b B-a C)\right) d\right)}{2 (a d-b c) \left(c^2+d^2\right) f (c+d \tan (e+f x))^2}-\frac{-\frac{d^2 \left(\left(2 b d^2-2 c (a d-b c)\right) \left(3 A d b^2-a A (b c-a d)-(b B-a C) (b c+2 a d)\right)-2 a d \left((A b-C b-a B) d (b c-a d)-3 c \left(A b^2-a (b B-a C)\right) d\right)\right)-c \left(2 d (a d-b c) \left(-3 \left(A b^2-a (b B-a C)\right) d^2+\left(3 A d b^2-a A (b c-a d)-(b B-a C) (b c+2 a d)\right) d-c (A b-C b-a B) (b c-a d)\right)-2 b c \left(d^2 \left(3 A d b^2-a A (b c-a d)-(b B-a C) (b c+2 a d)\right)-c \left((A b-C b-a B) d (b c-a d)-3 c \left(A b^2-a (b B-a C)\right) d\right)\right)\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)^2 \left(-3 C d a^4+4 b B d a^3-b^2 (B c+(5 A+C) d) a^2+2 b^3 (A c-C c+B d) a+b^4 (B c-3 A d)\right) \log (a+b \tan (e+f x)) b^3}{\left(a^2+b^2\right) (b c-a d)}-\frac{2 \left(a^2+b^2\right) d \left(a^2 \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) d^3-2 a b \left(c (A-C) d \left(5 c^2+d^2\right)-B \left(2 c^4-3 d^2 c^2-d^4\right)\right) d^2+b^2 \left(3 C c^6-6 B d c^5+(10 A-C) d^2 c^4-3 B d^3 c^3+9 A d^4 c^2-B d^5 c+3 A d^6\right)\right) \log (c+d \tan (e+f x)) b}{(b c-a d) \left(c^2+d^2\right)}-\frac{(b c-a d)^3 \left(\sqrt{-b^2} \left(A c^3 b^3-B d^3 b^3-3 A c d^2 b^3+3 c C d^2 b^3-c^3 C b^3+3 B c^2 d b^3-2 a B c^3 b^2-2 a A d^3 b^2+2 a C d^3 b^2+6 a B c d^2 b^2+6 a A c^2 d b^2-6 a c^2 C d b^2-a^2 A c^3 b+a^2 B d^3 b+3 a^2 A c d^2 b-3 a^2 c C d^2 b+a^2 c^3 C b-3 a^2 B c^2 d b\right)-b^2 \left(2 a A b c^3-a^2 B c^3+b^2 B c^3-2 a b C c^3-3 A b^2 d c^2+3 a^2 A d c^2+6 a b B d c^2-3 a^2 C d c^2+3 b^2 C d c^2-6 a A b d^2 c+3 a^2 B d^2 c-3 b^2 B d^2 c+6 a b C d^2 c+A b^2 d^3-a^2 A d^3-2 a b B d^3+a^2 C d^3-b^2 C d^3\right)\right) \log \left(\sqrt{-b^2}-b \tan (e+f x)\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right) b}+\frac{(b c-a d)^3 \left(\left(2 a A b c^3-a^2 B c^3+b^2 B c^3-2 a b C c^3-3 A b^2 d c^2+3 a^2 A d c^2+6 a b B d c^2-3 a^2 C d c^2+3 b^2 C d c^2-6 a A b d^2 c+3 a^2 B d^2 c-3 b^2 B d^2 c+6 a b C d^2 c+A b^2 d^3-a^2 A d^3-2 a b B d^3+a^2 C d^3-b^2 C d^3\right) b^2+\sqrt{-b^2} \left(A c^3 b^3-B d^3 b^3-3 A c d^2 b^3+3 c C d^2 b^3-c^3 C b^3+3 B c^2 d b^3-2 a B c^3 b^2-2 a A d^3 b^2+2 a C d^3 b^2+6 a B c d^2 b^2+6 a A c^2 d b^2-6 a c^2 C d b^2-a^2 A c^3 b+a^2 B d^3 b+3 a^2 A c d^2 b-3 a^2 c C d^2 b+a^2 c^3 C b-3 a^2 B c^2 d b\right)\right) \log \left(b \tan (e+f x)+\sqrt{-b^2}\right)}{\left(a^2+b^2\right) \left(c^2+d^2\right) b}}{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{\left(-3 C d a^4+4 b B d a^3-b^2 (B c+(5 A+C) d) a^2+2 b^3 (A c-C c+B d) a+b^4 (B c-3 A d)\right) \log (a \cos (e+f x)+b \sin (e+f x)) b^2}{\left(a^2+b^2\right)^2 (b c-a d)^4 f}-\frac{\left(\left(C c^3-3 B d c^2-3 C d^2 c+B d^3-A \left(c^3-3 c d^2\right)\right) a^2+2 b \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) a+b^2 \left(A c^3-C c^3+3 B d c^2-3 A d^2 c+3 C d^2 c-B d^3\right)\right) x}{\left(a^2+b^2\right)^2 \left(c^2+d^2\right)^3}+\frac{d \left(a^2 \left((A-C) d \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right) d^3-2 a b \left(c (A-C) d \left(5 c^2+d^2\right)-B \left(2 c^4-3 d^2 c^2-d^4\right)\right) d^2+b^2 \left(3 C c^6-6 B d c^5+(10 A-C) d^2 c^4-3 B d^3 c^3+9 A d^4 c^2-B d^5 c+3 A d^6\right)\right) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^4 \left(c^2+d^2\right)^3 f}-\frac{d \left(d^2 \left(2 c C d+B \left(c^2-d^2\right)\right) a^3+b \left(3 C c^4-3 B d c^3+2 C d^2 c^2-B d^3 c+C d^4\right) a^2+b^2 \left(2 c C d^3-B \left(c^4+d^2 c^2+2 d^4\right)\right) a+b^3 c \left(2 C c^3-3 B d c^2-B d^3\right)-A \left(-\left(\left(c^4+6 d^2 c^2+3 d^4\right) b^3\right)+2 a c d^3 b^2-2 a^2 d^2 \left(2 c^2+d^2\right) b+2 a^3 c d^3\right)\right)}{\left(a^2+b^2\right) (b c-a d)^3 \left(c^2+d^2\right)^2 f (c+d \tan (e+f x))}-\frac{d \left(\left(3 C c^2-B d c+2 C d^2\right) a^2-2 b B \left(c^2+d^2\right) a+b^2 c (c C-B d)+A \left(\left(2 c^2+3 d^2\right) b^2+a^2 d^2\right)\right)}{2 \left(a^2+b^2\right) (b c-a d)^2 \left(c^2+d^2\right) f (c+d \tan (e+f x))^2}-\frac{A b^2-a (b B-a C)}{\left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))^2}",1,"-((A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2)) - (-1/2*(-(c*(-3*c*(A*b^2 - a*(b*B - a*C))*d + (A*b - a*B - b*C)*d*(b*c - a*d))) + d^2*(3*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 2*a*d)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-((-(((b*c - a*d)^3*(-(b^2*(2*a*A*b*c^3 - a^2*B*c^3 + b^2*B*c^3 - 2*a*b*c^3*C + 3*a^2*A*c^2*d - 3*A*b^2*c^2*d + 6*a*b*B*c^2*d - 3*a^2*c^2*C*d + 3*b^2*c^2*C*d - 6*a*A*b*c*d^2 + 3*a^2*B*c*d^2 - 3*b^2*B*c*d^2 + 6*a*b*c*C*d^2 - a^2*A*d^3 + A*b^2*d^3 - 2*a*b*B*d^3 + a^2*C*d^3 - b^2*C*d^3)) + Sqrt[-b^2]*(-(a^2*A*b*c^3) + A*b^3*c^3 - 2*a*b^2*B*c^3 + a^2*b*c^3*C - b^3*c^3*C + 6*a*A*b^2*c^2*d - 3*a^2*b*B*c^2*d + 3*b^3*B*c^2*d - 6*a*b^2*c^2*C*d + 3*a^2*A*b*c*d^2 - 3*A*b^3*c*d^2 + 6*a*b^2*B*c*d^2 - 3*a^2*b*c*C*d^2 + 3*b^3*c*C*d^2 - 2*a*A*b^2*d^3 + a^2*b*B*d^3 - b^3*B*d^3 + 2*a*b^2*C*d^3))*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/(b*(a^2 + b^2)*(c^2 + d^2))) - (2*b^3*(c^2 + d^2)^2*(4*a^3*b*B*d - 3*a^4*C*d + b^4*(B*c - 3*A*d) + 2*a*b^3*(A*c - c*C + B*d) - a^2*b^2*(B*c + (5*A + C)*d))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) + ((b*c - a*d)^3*(b^2*(2*a*A*b*c^3 - a^2*B*c^3 + b^2*B*c^3 - 2*a*b*c^3*C + 3*a^2*A*c^2*d - 3*A*b^2*c^2*d + 6*a*b*B*c^2*d - 3*a^2*c^2*C*d + 3*b^2*c^2*C*d - 6*a*A*b*c*d^2 + 3*a^2*B*c*d^2 - 3*b^2*B*c*d^2 + 6*a*b*c*C*d^2 - a^2*A*d^3 + A*b^2*d^3 - 2*a*b*B*d^3 + a^2*C*d^3 - b^2*C*d^3) + Sqrt[-b^2]*(-(a^2*A*b*c^3) + A*b^3*c^3 - 2*a*b^2*B*c^3 + a^2*b*c^3*C - b^3*c^3*C + 6*a*A*b^2*c^2*d - 3*a^2*b*B*c^2*d + 3*b^3*B*c^2*d - 6*a*b^2*c^2*C*d + 3*a^2*A*b*c*d^2 - 3*A*b^3*c*d^2 + 6*a*b^2*B*c*d^2 - 3*a^2*b*c*C*d^2 + 3*b^3*c*C*d^2 - 2*a*A*b^2*d^3 + a^2*b*B*d^3 - b^3*B*d^3 + 2*a*b^2*C*d^3))*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(b*(a^2 + b^2)*(c^2 + d^2)) - (2*b*(a^2 + b^2)*d*(b^2*(3*c^6*C - 6*B*c^5*d + c^4*(10*A - C)*d^2 - 3*B*c^3*d^3 + 9*A*c^2*d^4 - B*c*d^5 + 3*A*d^6) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - 2*a*b*d^2*(c*(A - C)*d*(5*c^2 + d^2) - B*(2*c^4 - 3*c^2*d^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)) - (d^2*(-2*a*d*(-3*c*(A*b^2 - a*(b*B - a*C))*d + (A*b - a*B - b*C)*d*(b*c - a*d)) + (2*b*d^2 - 2*c*(-(b*c) + a*d))*(3*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 2*a*d))) - c*(2*d*(-(b*c) + a*d)*(-3*(A*b^2 - a*(b*B - a*C))*d^2 - c*(A*b - a*B - b*C)*(b*c - a*d) + d*(3*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 2*a*d))) - 2*b*c*(-(c*(-3*c*(A*b^2 - a*(b*B - a*C))*d + (A*b - a*B - b*C)*d*(b*c - a*d))) + d^2*(3*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 2*a*d)))))/((-(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))","B",0
90,1,1232,464,6.3920251,"\int (a+b \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{2 C (c+d \tan (e+f x))^{3/2} (a+b \tan (e+f x))^3}{9 d f}+\frac{2 \left(\frac{2 \left(\frac{3 b \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right) \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{10 d f}-\frac{2 \left(\frac{2 \left(b \left(\frac{3}{4} c \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)-\frac{315}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)-\frac{15}{8} a d \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)\right) (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac{i \left(-\frac{15}{8} a d \left(4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right)+\frac{3}{4} b c \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)+\frac{15}{8} a d \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)+\frac{5}{2} i d \left(\frac{63}{4} a \left(B a^2+2 b (A-C) a-b^2 B\right) d^2+\frac{3}{4} b \left(4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right)-\frac{3}{4} b \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)\right)-b \left(\frac{3}{4} c \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)-\frac{315}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)\right) \left(\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right) (c-i d)^{3/2}}{i d-c}+2 \sqrt{c+d \tan (e+f x)}\right)}{2 f}-\frac{i \left(-\frac{15}{8} a d \left(4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right)+\frac{3}{4} b c \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)+\frac{15}{8} a d \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)-\frac{5}{2} i d \left(\frac{63}{4} a \left(B a^2+2 b (A-C) a-b^2 B\right) d^2+\frac{3}{4} b \left(4 c (2 c C-3 B d) b^2-a d (16 c C+9 B d) b+a^2 (21 A-13 C) d^2\right)-\frac{3}{4} b \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)\right)-b \left(\frac{3}{4} c \left(21 b (A b-C b+a B) d^2+4 (b c-a d) (2 b c C-2 a d C-3 b B d)\right)-\frac{315}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)\right) \left(\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right) (c+i d)^{3/2}}{-c-i d}+2 \sqrt{c+d \tan (e+f x)}\right)}{2 f}\right)}{5 d}\right)}{7 d}-\frac{3 (2 b c C-2 a d C-3 b B d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}\right)}{9 d}","\frac{2 (c+d \tan (e+f x))^{3/2} \left(40 a^3 C d^3-6 a^2 b d^2 (16 c C-45 B d)+9 a b^2 d \left(35 d^2 (A-C)-14 B c d+8 c^2 C\right)-\left(b^3 \left(42 c d^2 (A-C)-24 B c^2 d+105 B d^3+16 c^3 C\right)\right)\right)}{315 d^4 f}+\frac{2 \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 b \tan (e+f x) (c+d \tan (e+f x))^{3/2} \left(21 b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-3 b B d+2 b c C)\right)}{105 d^3 f}-\frac{(a-i b)^3 \sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{(a+i b)^3 \sqrt{c+i d} (i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 (-2 a C d-3 b B d+2 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{21 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}}{9 d f}",1,"(2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2))/(9*d*f) + (2*((-3*(2*b*c*C - 3*b*B*d - 2*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2))/(7*d*f) + (2*((3*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(10*d*f) - (2*((2*((-15*a*d*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 + b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*(c + d*Tan[e + f*x])^(3/2))/(3*d*f) + ((I/2)*((-15*a*d*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/8 + (3*b*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4 + (15*a*d*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 + ((5*I)/2)*d*((63*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (3*b*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/4 - (3*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4) - b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*((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)*((-15*a*d*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/8 + (3*b*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4 + (15*a*d*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/8 - ((5*I)/2)*d*((63*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (3*b*(a^2*(21*A - 13*C)*d^2 + 4*b^2*c*(2*c*C - 3*B*d) - a*b*d*(16*c*C + 9*B*d)))/4 - (3*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4) - b*((-315*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (3*c*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d)))/4))*((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))/(5*d)))/(7*d)))/(9*d)","B",1
91,1,314,325,4.8181727,"\int (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{2 \left((c+d \tan (e+f x))^{3/2} \left(20 a^2 C d^2+14 a b d (5 B d-2 c C)+b^2 \left(35 d^2 (A-C)-14 B c d+8 c^2 C\right)\right)+\frac{105}{2} d^3 (a-i b)^2 (i A+B-i C) \left(\sqrt{c+d \tan (e+f x)}-\sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)+\frac{105}{2} d^3 (a+i b)^2 (-i A+B+i C) \left(\sqrt{c+d \tan (e+f x)}-\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)+3 b d \tan (e+f x) (4 a C d+7 b B d-4 b c C) (c+d \tan (e+f x))^{3/2}+15 C d^2 (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}\right)}{105 d^3 f}","\frac{2 (c+d \tan (e+f x))^{3/2} \left(20 a^2 C d^2-14 a b d (2 c C-5 B d)+b^2 \left(35 d^2 (A-C)-14 B c d+8 c^2 C\right)\right)}{105 d^3 f}+\frac{2 \left(a^2 B+2 a b (A-C)-b^2 B\right) \sqrt{c+d \tan (e+f x)}}{f}-\frac{(a-i b)^2 \sqrt{c-i d} (B+i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}-\frac{(a+i b)^2 \sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 b \tan (e+f x) (-4 a C d-7 b B d+4 b c C) (c+d \tan (e+f x))^{3/2}}{35 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}}{7 d f}",1,"(2*((20*a^2*C*d^2 + 14*a*b*d*(-2*c*C + 5*B*d) + b^2*(8*c^2*C - 14*B*c*d + 35*(A - C)*d^2))*(c + d*Tan[e + f*x])^(3/2) + 3*b*d*(-4*b*c*C + 7*b*B*d + 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2) + 15*C*d^2*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2) + (105*(a - I*b)^2*(I*A + B - I*C)*d^3*(-(Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]) + Sqrt[c + d*Tan[e + f*x]]))/2 + (105*(a + I*b)^2*((-I)*A + B + I*C)*d^3*(-(Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]) + Sqrt[c + d*Tan[e + f*x]]))/2))/(105*d^3*f)","A",1
92,1,220,224,1.9910216,"\int (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{15 d (b+i a) (A-i B-C) \left(\sqrt{c+d \tan (e+f x)}-\sqrt{c-i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)+15 d (b-i a) (A+i B-C) \left(\sqrt{c+d \tan (e+f x)}-\sqrt{c+i d} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)\right)+\frac{2 (5 a C d+5 b B d-2 b c C) (c+d \tan (e+f x))^{3/2}}{d}+6 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{15 d f}","\frac{2 (a B+A b-b C) \sqrt{c+d \tan (e+f x)}}{f}-\frac{(b+i a) \sqrt{c-i d} (A-i B-C) \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} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 (-5 a C d-5 b B d+2 b c C) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}",1,"((2*(-2*b*c*C + 5*b*B*d + 5*a*C*d)*(c + d*Tan[e + f*x])^(3/2))/d + 6*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2) + 15*(I*a + b)*(A - I*B - C)*d*(-(Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]) + Sqrt[c + d*Tan[e + f*x]]) + 15*((-I)*a + b)*(A + I*B - C)*d*(-(Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]) + Sqrt[c + d*Tan[e + f*x]]))/(15*d*f)","A",1
93,1,150,155,0.5576804,"\int \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{-3 i d \sqrt{c-i d} (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+3 i d \sqrt{c+i d} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)+2 \sqrt{c+d \tan (e+f x)} (3 B d+c C+C d \tan (e+f x))}{3 d f}","-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}-\frac{\sqrt{c+i d} (B-i (A-C)) \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}+\frac{2 C (c+d \tan (e+f x))^{3/2}}{3 d f}",1,"((-3*I)*(A - I*B - C)*Sqrt[c - I*d]*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (3*I)*(A + I*B - C)*Sqrt[c + I*d]*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + 2*Sqrt[c + d*Tan[e + f*x]]*(c*C + 3*B*d + C*d*Tan[e + f*x]))/(3*d*f)","A",1
94,1,233,234,0.6888273,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{a+b \tan (e+f x)} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]","\frac{2 \sqrt{b} C \left(a^2+b^2\right) \sqrt{c+d \tan (e+f x)}+b^{3/2} (b-i a) \sqrt{c-i d} (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+b^{3/2} (b+i a) \sqrt{c+i d} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)-2 \sqrt{b c-a d} \left(a (a C-b B)+A b^2\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)}","-\frac{2 \sqrt{b c-a d} \left(A b^2-a (b B-a C)\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)}-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)}+\frac{\sqrt{c+i d} (i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)}+\frac{2 C \sqrt{c+d \tan (e+f x)}}{b f}",1,"(b^(3/2)*((-I)*a + b)*(A - I*B - C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + b^(3/2)*(I*a + b)*(A + I*B - C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + 2*Sqrt[b]*(a^2 + b^2)*C*Sqrt[c + d*Tan[e + f*x]])/(b^(3/2)*(a^2 + b^2)*f)","A",1
95,1,764,317,6.3863765,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^2} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]","-\frac{2 C \sqrt{c+d \tan (e+f x)}}{b f (a+b \tan (e+f x))}-\frac{2 \left(-\frac{\sqrt{c+d \tan (e+f x)} \left(\frac{1}{2} b^2 (-a C d-A b c+2 b c C)-a \left(-\frac{1}{2} a (-a C d-b B d+b c C)-\frac{1}{2} b^2 (d (A-C)+B c)\right)\right)}{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}{4} a^2 d (b c-a d) \left(a^2 (-C)-a b B+A b^2-2 b^2 C\right)+\frac{1}{4} b^2 (b c-a d) \left(a^2 C d+a b (2 A c-B d-2 c C)+b^2 (A d+2 B c)\right)+\frac{1}{2} a b^2 (b c-a d) (-a A d-a B c+a C d+A b c-b B d-b c 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} b (b c-a d) \left(a^2 (A c-B d-c C)+2 a b (d (A-C)+B c)-b^2 (A c-B d-c C)\right)+\frac{1}{2} i b (b c-a d) \left(-\left(a^2 (d (A-C)+B c)\right)+2 a b (A c-B d-c C)+b^2 (d (A-C)+B c)\right)\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 (b c-a d) \left(a^2 (A c-B d-c C)+2 a b (d (A-C)+B c)-b^2 (A c-B d-c C)\right)-\frac{1}{2} i b (b c-a d) \left(-\left(a^2 (d (A-C)+B c)\right)+2 a b (A c-B d-c C)+b^2 (d (A-C)+B 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}","-\frac{\left(A b^2-a (b B-a C)\right) \sqrt{c+d \tan (e+f x)}}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}-\frac{\left(a^4 C d+a^3 b B d-a^2 b^2 (3 A d+2 B c-5 C d)+a b^3 (4 A c-3 B d-4 c C)+b^4 (A d+2 B c)\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 \sqrt{b c-a d}}-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2}-\frac{\sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2}",1,"(-2*C*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x])) - (2*(-((((I*Sqrt[c - I*d]*((b*(b*c - a*d)*(a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) + 2*a*b*(B*c + (A - C)*d)))/2 + (I/2)*b*(b*c - a*d)*(2*a*b*(A*c - c*C - B*d) - a^2*(B*c + (A - C)*d) + b^2*(B*c + (A - C)*d)))*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)*(a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) + 2*a*b*(B*c + (A - C)*d)))/2 - (I/2)*b*(b*c - a*d)*(2*a*b*(A*c - c*C - B*d) - a^2*(B*c + (A - C)*d) + b^2*(B*c + (A - C)*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]*(-1/4*(a^2*(A*b^2 - a*b*B - a^2*C - 2*b^2*C)*d*(b*c - a*d)) + (a*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))/2 + (b^2*(b*c - a*d)*(a^2*C*d + b^2*(2*B*c + A*d) + a*b*(2*A*c - 2*c*C - B*d)))/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))/((a^2 + b^2)*(b*c - a*d))) - (((b^2*(-(A*b*c) + 2*b*c*C - a*C*d))/2 - a*(-1/2*(b^2*(B*c + (A - C)*d)) - (a*(b*c*C - b*B*d - a*C*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","B",1
96,1,2819,543,6.381501,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^3} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{\left(A b^2-a (b B-a C)\right) \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{c+d \tan (e+f x)} \left(a^4 C d+3 a^3 b B d-a^2 b^2 (7 A d+4 B c-9 C d)+a b^3 (8 A c-5 B d-8 c C)+b^4 (A d+4 B c)\right)}{4 b f \left(a^2+b^2\right)^2 (b c-a d) (a+b \tan (e+f x))}+\frac{\left(a^6 C d^2+3 a^5 b B d^2-3 a^4 b^2 d (5 A d+4 B c-6 C d)+2 a^3 b^3 \left(20 c d (A-C)+B \left(4 c^2-13 d^2\right)\right)-3 a^2 b^4 \left(8 A c^2-6 A d^2-16 B c d-8 c^2 C+5 C d^2\right)-3 a b^5 \left(8 c d (A-C)+B \left(8 c^2-d^2\right)\right)-b^6 \left(4 c (B d+2 c C)-A \left(8 c^2+d^2\right)\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 (b c-a d)^{3/2}}-\frac{\sqrt{c-i d} (A-i B-C) \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} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)^3}",1,"(-2*C*Sqrt[c + d*Tan[e + f*x]])/(3*b*f*(a + b*Tan[e + f*x])^2) - (2*(-1/2*(((b^2*(-3*A*b*c + 4*b*c*C - a*C*d))/2 - a*((-3*b^2*(B*c + (A - C)*d))/2 - (a*(b*c*C - 3*b*B*d - a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*b*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4) + a*((3*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)) - (d*((3*b^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))/2) - I*(a*(b*c - a*d)*((3*b*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4) - b*((3*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)) - (d*((3*b^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))/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*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4) + a*((3*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)) - (d*((3*b^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))/2) + I*(a*(b*c - a*d)*((3*b*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4) - b*((3*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)) - (d*((3*b^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))/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*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 + 3*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + (3*b*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4)) + (a^2*d*((3*b^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))/2 + b^2*((3*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 + (-(b*c) + (a*d)/2)*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))))*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^2*(b*c - a*d)*(a^2*C*d + b^2*(4*B*c + A*d) + a*b*(4*A*c - 4*c*C - B*d)))/4 - a*((3*a*(3*A*b^2 - 3*a*b*B - a^2*C - 4*b^2*C)*d*(b*c - a*d))/4 - 3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])))/(2*(a^2 + b^2)*(b*c - a*d))))/(3*b)","B",1
97,1,1290,550,6.4137338,"\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{2 C (c+d \tan (e+f x))^{5/2} (a+b \tan (e+f x))^3}{11 d f}+\frac{2 \left(\frac{(-6 b c C+6 a d C+11 b B d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}+\frac{2 \left(\frac{b \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right) \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{14 d f}-\frac{2 \left(\frac{2 \left(b \left(\frac{1}{4} c \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)-\frac{693}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)-\frac{7}{8} a d \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)\right) (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac{i \left(-\frac{7}{8} a d \left(4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right)+\frac{1}{4} b c \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)+\frac{7}{8} a d \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)+\frac{7}{2} i d \left(\frac{99}{4} a \left(B a^2+2 b (A-C) a-b^2 B\right) d^2+\frac{1}{4} b \left(4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right)-\frac{1}{4} b \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)\right)-b \left(\frac{1}{4} c \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)-\frac{693}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)\right) \left(\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left(\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right) (c-i d)^{3/2}}{i d-c}+2 \sqrt{c+d \tan (e+f x)}\right)\right)}{2 f}-\frac{i \left(-\frac{7}{8} a d \left(4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right)+\frac{1}{4} b c \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)+\frac{7}{8} a d \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)-\frac{7}{2} i d \left(\frac{99}{4} a \left(B a^2+2 b (A-C) a-b^2 B\right) d^2+\frac{1}{4} b \left(4 c (6 c C-11 B d) b^2-a d (48 c C+55 B d) b+3 a^2 (33 A-25 C) d^2\right)-\frac{1}{4} b \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)\right)-b \left(\frac{1}{4} c \left(99 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-11 b B d)\right)-\frac{693}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)\right) \left(\frac{2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left(\frac{2 \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right) (c+i d)^{3/2}}{-c-i d}+2 \sqrt{c+d \tan (e+f x)}\right)\right)}{2 f}\right)}{7 d}\right)}{9 d}\right)}{11 d}","\frac{2 (c+d \tan (e+f x))^{5/2} \left(168 a^3 C d^3-2 a^2 b d^2 (192 c C-847 B d)+33 a b^2 d \left(63 d^2 (A-C)-18 B c d+8 c^2 C\right)-\left(b^3 \left(198 c d^2 (A-C)-88 B c^2 d+693 B d^3+48 c^3 C\right)\right)\right)}{3465 d^4 f}+\frac{2 \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \sqrt{c+d \tan (e+f x)} \left(a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right)}{f}+\frac{2 b \tan (e+f x) (c+d \tan (e+f x))^{5/2} \left(99 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-11 b B d+6 b c C)\right)}{693 d^3 f}+\frac{(a+i b)^3 (c+i d)^{3/2} (i A-B-i C) \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} (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}-\frac{2 (-6 a C d-11 b B d+6 b c C) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2}}{11 d f}",1,"(2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2))/(11*d*f) + (2*(((-6*b*c*C + 11*b*B*d + 6*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2))/(9*d*f) + (2*((b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(14*d*f) - (2*((2*((-7*a*d*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 + b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4))*(c + d*Tan[e + f*x])^(5/2))/(5*d*f) + ((I/2)*((-7*a*d*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/8 + (b*c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4 + (7*a*d*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 + ((7*I)/2)*d*((99*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (b*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/4 - (b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4) - b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4))*((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)*((-7*a*d*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/8 + (b*c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4 + (7*a*d*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/8 - ((7*I)/2)*d*((99*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 + (b*(3*a^2*(33*A - 25*C)*d^2 + 4*b^2*c*(6*c*C - 11*B*d) - a*b*d*(48*c*C + 55*B*d)))/4 - (b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4) - b*((-693*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d)))/4))*((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))/(7*d)))/(9*d)))/(11*d)","B",1
98,1,350,396,6.198491,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{2 \left((c+d \tan (e+f x))^{5/2} \left(28 a^2 C d^2+18 a b d (7 B d-2 c C)+b^2 \left(63 d^2 (A-C)-18 B c d+8 c^2 C\right)\right)+\frac{105}{2} d^3 (a-i b)^2 (i A+B-i C) \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{105}{2} d^3 (a+i b)^2 (-i A+B+i C) \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 b d \tan (e+f x) (4 a C d+9 b B d-4 b c C) (c+d \tan (e+f x))^{5/2}+35 C d^2 (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}\right)}{315 d^3 f}","\frac{2 (c+d \tan (e+f x))^{5/2} \left(28 a^2 C d^2-18 a b d (2 c C-7 B d)+b^2 \left(63 d^2 (A-C)-18 B c d+8 c^2 C\right)\right)}{315 d^3 f}+\frac{2 \left(a^2 B+2 a b (A-C)-b^2 B\right) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \sqrt{c+d \tan (e+f x)} \left(a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right)}{f}-\frac{(a-i b)^2 (c-i d)^{3/2} (B+i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{(a+i b)^2 (c+i d)^{3/2} (i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 b \tan (e+f x) (-4 a C d-9 b B d+4 b c C) (c+d \tan (e+f x))^{5/2}}{63 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}}{9 d f}",1,"(2*((28*a^2*C*d^2 + 18*a*b*d*(-2*c*C + 7*B*d) + b^2*(8*c^2*C - 18*B*c*d + 63*(A - C)*d^2))*(c + d*Tan[e + f*x])^(5/2) + 5*b*d*(-4*b*c*C + 9*b*B*d + 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2) + 35*C*d^2*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2) + (105*(a - I*b)^2*(I*A + B - I*C)*d^3*(-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])))/2 + (105*(a + I*b)^2*((-I)*A + B + I*C)*d^3*(-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])))/2))/(315*d^3*f)","A",1
99,1,260,273,4.4413272,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{\frac{35}{3} d (b+i a) (A-i B-C) \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) (A+i B-C) \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{2 (7 a C d+7 b B d-2 b c C) (c+d \tan (e+f x))^{5/2}}{d}+10 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{35 d f}","\frac{2 (a B+A b-b C) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac{2 \sqrt{c+d \tan (e+f x)} (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac{(b+i a) (c-i d)^{3/2} (A-i B-C) \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} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 (-7 a C d-7 b B d+2 b c C) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}",1,"((2*(-2*b*c*C + 7*b*B*d + 7*a*C*d)*(c + d*Tan[e + f*x])^(5/2))/d + 10*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2) + (35*(I*a + b)*(A - I*B - C)*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)*(A + I*B - C)*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
100,1,202,187,1.2352665,"\int (c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{5 (i A+B-i C) \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+B+i C) \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 C (c+d \tan (e+f x))^{5/2}}{d}}{15 f}","\frac{2 (d (A-C)+B c) \sqrt{c+d \tan (e+f x)}}{f}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}-\frac{(c+i d)^{3/2} (B-i (A-C)) \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}+\frac{2 C (c+d \tan (e+f x))^{5/2}}{5 d f}",1,"((6*C*(c + d*Tan[e + f*x])^(5/2))/d + 5*(I*A + B - I*C)*(-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 + B + I*C)*(-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
101,1,266,271,2.5522114,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{a+b \tan (e+f x)} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]","\frac{\frac{3 i b \left((a-i b) (c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)-(a+i b) (c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)\right)}{a^2+b^2}-\frac{6 (b c-a d)^{3/2} \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{3/2} \left(a^2+b^2\right)}+\frac{6 (-a C d+b B d+b c C) \sqrt{c+d \tan (e+f x)}}{b}+2 C (c+d \tan (e+f x))^{3/2}}{3 b f}","-\frac{2 (b c-a d)^{3/2} \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{5/2} f \left(a^2+b^2\right)}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)}-\frac{(c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)}+\frac{2 (-a C d+b B d+b c C) \sqrt{c+d \tan (e+f x)}}{b^2 f}+\frac{2 C (c+d \tan (e+f x))^{3/2}}{3 b f}",1,"(((3*I)*b*(-((a + I*b)*(A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]) + (a - I*b)*(A + I*B - C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]))/(a^2 + b^2) - (6*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)) + (6*(b*c*C + b*B*d - a*C*d)*Sqrt[c + d*Tan[e + f*x]])/b + 2*C*(c + d*Tan[e + f*x])^(3/2))/(3*b*f)","A",1
102,1,1732,372,6.3134481,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^2} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]","\frac{-A c \sqrt{c+d \tan (e+f x)} b^{11/2}+2 C d \tan (e+f x) \sqrt{c+d \tan (e+f x)} b^{11/2}-2 B c \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^5-3 A d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^5+a B c \sqrt{c+d \tan (e+f x)} b^{9/2}+a A d \sqrt{c+d \tan (e+f x)} b^{9/2}+2 a C d \sqrt{c+d \tan (e+f x)} b^{9/2}-2 a B c \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^4-3 a A d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^4-4 a A c \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^4+4 a c C \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^4+5 a B d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^4-a^2 A c \sqrt{c+d \tan (e+f x)} b^{7/2}-a^2 c C \sqrt{c+d \tan (e+f x)} b^{7/2}-a^2 B d \sqrt{c+d \tan (e+f x)} b^{7/2}+4 a^2 C d \tan (e+f x) \sqrt{c+d \tan (e+f x)} b^{7/2}-4 a^2 A c \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^3+4 a^2 c C \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^3+5 a^2 B d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^3+2 a^2 B c \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^3+a^2 A d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^3-7 a^2 C d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^3+(b-i a)^2 (i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right) (a+b \tan (e+f x)) b^{5/2}+(i a+b)^2 (-i A+B+i C) (c+i d)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right) (a+b \tan (e+f x)) b^{5/2}+a^3 B c \sqrt{c+d \tan (e+f x)} b^{5/2}+a^3 A d \sqrt{c+d \tan (e+f x)} b^{5/2}+5 a^3 C d \sqrt{c+d \tan (e+f x)} b^{5/2}+2 a^3 B c \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^2+a^3 A d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^2-7 a^3 C d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b^2+a^3 B d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b^2-a^4 c C \sqrt{c+d \tan (e+f x)} b^{3/2}-a^4 B d \sqrt{c+d \tan (e+f x)} b^{3/2}+2 a^4 C d \tan (e+f x) \sqrt{c+d \tan (e+f x)} b^{3/2}+a^4 B d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) b-3 a^4 C d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right) \tan (e+f x) b+3 a^5 C d \sqrt{c+d \tan (e+f x)} \sqrt{b}-3 a^5 C d \sqrt{b c-a d} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{5/2} \left(a^2+b^2\right)^2 f (a+b \tan (e+f x))}","-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{3/2}}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{d \left(3 a^2 C-a b B+A b^2+2 b^2 C\right) \sqrt{c+d \tan (e+f x)}}{b^2 f \left(a^2+b^2\right)}+\frac{\sqrt{b c-a d} \left(-3 a^4 C d+a^3 b B d+a^2 b^2 (d (A-7 C)+2 B c)-a b^3 (4 A c-5 B d-4 c C)-b^4 (3 A d+2 B c)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{5/2} f \left(a^2+b^2\right)^2}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2}-\frac{(c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2}",1,"(-4*a^2*A*b^3*c*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + 2*a^3*b^2*B*c*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] - 2*a*b^4*B*c*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + 4*a^2*b^3*c*C*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + a^3*A*b^2*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] - 3*a*A*b^4*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + a^4*b*B*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] + 5*a^2*b^3*B*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] - 3*a^5*C*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] - 7*a^3*b^2*C*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]] - 4*a*A*b^4*c*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] + 2*a^2*b^3*B*c*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] - 2*b^5*B*c*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] + 4*a*b^4*c*C*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] + a^2*A*b^3*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] - 3*A*b^5*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] + a^3*b^2*B*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] + 5*a*b^4*B*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] - 3*a^4*b*C*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] - 7*a^2*b^3*C*d*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]*Tan[e + f*x] + b^(5/2)*((-I)*a + b)^2*(I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]*(a + b*Tan[e + f*x]) + b^(5/2)*(I*a + b)^2*((-I)*A + B + I*C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]*(a + b*Tan[e + f*x]) - a^2*A*b^(7/2)*c*Sqrt[c + d*Tan[e + f*x]] - A*b^(11/2)*c*Sqrt[c + d*Tan[e + f*x]] + a^3*b^(5/2)*B*c*Sqrt[c + d*Tan[e + f*x]] + a*b^(9/2)*B*c*Sqrt[c + d*Tan[e + f*x]] - a^4*b^(3/2)*c*C*Sqrt[c + d*Tan[e + f*x]] - a^2*b^(7/2)*c*C*Sqrt[c + d*Tan[e + f*x]] + a^3*A*b^(5/2)*d*Sqrt[c + d*Tan[e + f*x]] + a*A*b^(9/2)*d*Sqrt[c + d*Tan[e + f*x]] - a^4*b^(3/2)*B*d*Sqrt[c + d*Tan[e + f*x]] - a^2*b^(7/2)*B*d*Sqrt[c + d*Tan[e + f*x]] + 3*a^5*Sqrt[b]*C*d*Sqrt[c + d*Tan[e + f*x]] + 5*a^3*b^(5/2)*C*d*Sqrt[c + d*Tan[e + f*x]] + 2*a*b^(9/2)*C*d*Sqrt[c + d*Tan[e + f*x]] + 2*a^4*b^(3/2)*C*d*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]] + 4*a^2*b^(7/2)*C*d*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]] + 2*b^(11/2)*C*d*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(b^(5/2)*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))","B",1
103,1,7678,532,6.5641563,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^3} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{3/2}}{2 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}-\frac{\sqrt{c+d \tan (e+f x)} \left(3 a^4 C d+a^3 b B d-a^2 b^2 (5 A d+4 B c-11 C d)+a b^3 (8 A c-7 B d-8 c C)+b^4 (3 A d+4 B c)\right)}{4 b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{\left(3 a^6 C d^2+a^5 b B d^2+a^4 b^2 d (3 d (A+2 C)+4 B c)-2 a^3 b^3 \left(12 c d (A-C)+B \left(4 c^2-9 d^2\right)\right)+a^2 b^4 \left(24 A c^2-26 A d^2-48 B c d-24 c^2 C+35 C d^2\right)+a b^5 \left(40 c d (A-C)+3 B \left(8 c^2-5 d^2\right)\right)-b^6 \left(8 A c^2-3 A d^2-12 B c d-8 c^2 C\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{4 b^{5/2} f \left(a^2+b^2\right)^3 \sqrt{b c-a d}}-\frac{(c-i d)^{3/2} (A-i B-C) \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} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)^3}",1,"Result too large to show","B",1
104,1,564,503,6.4502084,"\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}+\frac{2 \left(\frac{b \tan (e+f x) (4 a C d+11 b B d-4 b c C) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac{2 \left(\frac{(c+d \tan (e+f x))^{7/2} \left(-36 a^2 C d^2+22 a b d (2 c C-9 B d)-\left(b^2 \left(99 d^2 (A-C)-22 B c d+8 c^2 C\right)\right)\right)}{14 d f}+\frac{i \left(\frac{99}{4} d^2 \left(-\left(a^2 (A-C)\right)+2 a b B+b^2 (A-C)\right)+\frac{99}{4} i d^2 \left(a^2 B+2 a b (A-C)-b^2 B\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{99}{4} d^2 \left(-\left(a^2 (A-C)\right)+2 a b B+b^2 (A-C)\right)-\frac{99}{4} i d^2 \left(a^2 B+2 a b (A-C)-b^2 B\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}\right)}{9 d}\right)}{11 d}","-\frac{2 \sqrt{c+d \tan (e+f x)} \left(-\left(a^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)+2 a b \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+b^2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right)}{f}+\frac{2 (c+d \tan (e+f x))^{7/2} \left(36 a^2 C d^2-22 a b d (2 c C-9 B d)+b^2 \left(99 d^2 (A-C)-22 B c d+8 c^2 C\right)\right)}{693 d^3 f}+\frac{2 \left(a^2 B+2 a b (A-C)-b^2 B\right) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 (c+d \tan (e+f x))^{3/2} \left(a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right)}{3 f}-\frac{(a-i b)^2 (c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}+\frac{(a+i b)^2 (c+i d)^{5/2} (i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 b \tan (e+f x) (-4 a C d-11 b B d+4 b c C) (c+d \tan (e+f x))^{7/2}}{99 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{7/2}}{11 d f}",1,"(2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*d*f) + (2*((b*(-4*b*c*C + 11*b*B*d + 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(9*d*f) - (2*(((-36*a^2*C*d^2 + 22*a*b*d*(2*c*C - 9*B*d) - b^2*(8*c^2*C - 22*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(14*d*f) + ((I/2)*(((99*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2 + (99*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/4)*((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)*(((-99*I)/4)*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2 + (99*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^2)/4)*((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)))/(11*d)","A",1
105,1,324,353,5.3588206,"\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{\frac{63}{2} i d (a-i b) (A-i B-C) \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{63}{2} i d (a+i b) (A+i B-C) \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{2 (9 a C d+9 b B d-2 b c C) (c+d \tan (e+f x))^{7/2}}{d}+14 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{63 d f}","\frac{2 \sqrt{c+d \tan (e+f x)} \left(A \left(2 a c d+b \left(c^2-d^2\right)\right)+a \left(B c^2-B d^2-2 c C d\right)-b \left(2 B c d+c^2 C-C d^2\right)\right)}{f}+\frac{2 (a B+A b-b C) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac{2 (c+d \tan (e+f x))^{3/2} (a A d+a B c-a C d+A b c-b B d-b c C)}{3 f}-\frac{(b+i a) (c-i d)^{5/2} (A-i B-C) \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} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f}-\frac{2 (-9 a C d-9 b B d+2 b c C) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac{2 b C \tan (e+f x) (c+d \tan (e+f x))^{7/2}}{9 d f}",1,"((2*(-2*b*c*C + 9*b*B*d + 9*a*C*d)*(c + d*Tan[e + f*x])^(7/2))/d + 14*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2) + ((63*I)/2)*(a - I*b)*(A - I*B - C)*d*((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) - ((63*I)/2)*(a + I*b)*(A + I*B - C)*d*((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))/(63*d*f)","A",1
106,1,262,229,2.0663353,"\int (c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{7 i (A-i B-C) \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-C) \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 C (c+d \tan (e+f x))^{7/2}}{d}}{14 f}","\frac{2 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right) \sqrt{c+d \tan (e+f x)}}{f}+\frac{2 (d (A-C)+B c) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f}-\frac{(c+i d)^{5/2} (B-i (A-C)) \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}+\frac{2 C (c+d \tan (e+f x))^{7/2}}{7 d f}",1,"((4*C*(c + d*Tan[e + f*x])^(7/2))/d + (7*I)*(A - I*B - C)*((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 - C)*((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
107,1,322,336,5.3055674,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{a+b \tan (e+f x)} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]","\frac{\frac{15 \left(b^{7/2} (b-i a) (c-i d)^{5/2} (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)+b^{7/2} (b+i a) (c+i d)^{5/2} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)-2 (b c-a d)^{5/2} \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)\right)}{b^{5/2} \left(a^2+b^2\right)}+\frac{30 \sqrt{c+d \tan (e+f x)} \left((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right)}{b^2}+\frac{10 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{b}+6 C (c+d \tan (e+f x))^{5/2}}{15 b f}","-\frac{2 (b c-a d)^{5/2} \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{7/2} f \left(a^2+b^2\right)}+\frac{2 \sqrt{c+d \tan (e+f x)} \left((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right)}{b^3 f}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)}+\frac{(c+i d)^{5/2} (i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)}+\frac{2 (-a C d+b B d+b c C) (c+d \tan (e+f x))^{3/2}}{3 b^2 f}+\frac{2 C (c+d \tan (e+f x))^{5/2}}{5 b f}",1,"((15*(b^(7/2)*((-I)*a + b)*(A - I*B - C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + b^(7/2)*(I*a + b)*(A + I*B - C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]))/(b^(5/2)*(a^2 + b^2)) + (30*(b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(b*c*C + b*B*d - a*C*d))*Sqrt[c + d*Tan[e + f*x]])/b^2 + (10*(b*c*C + b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/b + 6*C*(c + d*Tan[e + f*x])^(5/2))/(15*b*f)","A",1
108,1,6112,473,6.5322786,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^2} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]","\text{Result too large to show}","-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{5/2}}{b f \left(a^2+b^2\right) (a+b \tan (e+f x))}+\frac{d \left(5 a^2 C-3 a b B+3 A b^2+2 b^2 C\right) (c+d \tan (e+f x))^{3/2}}{3 b^2 f \left(a^2+b^2\right)}-\frac{d \sqrt{c+d \tan (e+f x)} \left(5 a^3 C d-a^2 b (3 B d+5 c C)-A b^2 (b c-a d)+a b^2 (B c+4 C d)-2 b^3 (B d+2 c C)\right)}{b^3 f \left(a^2+b^2\right)}+\frac{(b c-a d)^{3/2} \left(-5 a^4 C d+3 a^3 b B d+a^2 b^2 (2 B c-d (A+9 C))-a b^3 (4 A c-7 B d-4 c C)-b^4 (5 A d+2 B c)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{b^{7/2} f \left(a^2+b^2\right)^2}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b)^2}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b)^2}",1,"Result too large to show","B",1
109,1,18214,643,6.8914917,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^3} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]","\text{Result too large to show}","-\frac{\left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{5/2}}{2 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^2}+\frac{(c+d \tan (e+f x))^{3/2} \left(-5 a^4 C d+a^3 b B d+a^2 b^2 (3 A d+4 B c-13 C d)-a b^3 (8 A c-9 B d-8 c C)-b^4 (5 A d+4 B c)\right)}{4 b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))}-\frac{d \sqrt{c+d \tan (e+f x)} \left(-15 a^4 C d+3 a^3 b B d+a^2 b^2 (d (A-31 C)+4 B c)-a b^3 (8 A c-11 B d-8 c C)-b^4 (7 A d+4 B c+8 C d)\right)}{4 b^3 f \left(a^2+b^2\right)^2}+\frac{\sqrt{b c-a d} \left(-15 a^6 C d^2+3 a^5 b B d^2+a^4 b^2 d (d (A-46 C)+4 B c)+2 a^3 b^3 \left(4 c d (A-C)+B \left(4 c^2+3 d^2\right)\right)-3 a^2 b^4 \left(8 A c^2-6 A d^2-16 B c d-8 c^2 C+21 C d^2\right)-a b^5 \left(56 c d (A-C)+B \left(24 c^2-35 d^2\right)\right)-b^6 \left(4 c (5 B d+2 c C)-A \left(8 c^2-15 d^2\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{4 b^{7/2} f \left(a^2+b^2\right)^3}-\frac{(c-i d)^{5/2} (A-i B-C) \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} (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (-b+i a)^3}",1,"Result too large to show","B",1
110,1,1200,407,6.4323135,"\int \frac{(a+b \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]","\frac{2 C \sqrt{c+d \tan (e+f x)} (a+b \tan (e+f x))^3}{7 d f}+\frac{2 \left(\frac{(-6 b c C+6 a d C+7 b B d) \sqrt{c+d \tan (e+f x)} (a+b \tan (e+f x))^2}{5 d f}+\frac{2 \left(\frac{b \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right) \tan (e+f x) \sqrt{c+d \tan (e+f x)}}{6 d f}-\frac{2 \left(\frac{2 \sqrt{c+d \tan (e+f x)} \left(b \left(\frac{1}{4} c \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)-\frac{105}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)-\frac{3}{8} a d \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)\right)}{d f}+\frac{i \sqrt{c-i d} \left(\frac{1}{4} b c \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)+\frac{3}{8} a d \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)-\frac{3}{8} a d ((4 b c+a d) (6 b c C-6 a d C-7 b B d)-5 a d (6 b c C-a (7 A-C) d))-b \left(\frac{1}{4} c \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)-\frac{105}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)+\frac{3}{2} i d \left(\frac{35}{4} a \left(B a^2+2 b (A-C) a-b^2 B\right) d^2-\frac{1}{4} b \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)+\frac{1}{4} b ((4 b c+a d) (6 b c C-6 a d C-7 b B d)-5 a d (6 b c C-a (7 A-C) d))\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{(i d-c) f}-\frac{i \sqrt{c+i d} \left(\frac{1}{4} b c \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)+\frac{3}{8} a d \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)-\frac{3}{8} a d ((4 b c+a d) (6 b c C-6 a d C-7 b B d)-5 a d (6 b c C-a (7 A-C) d))-b \left(\frac{1}{4} c \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)-\frac{105}{8} \left(B a^2+2 b (A-C) a-b^2 B\right) d^3\right)-\frac{3}{2} i d \left(\frac{35}{4} a \left(B a^2+2 b (A-C) a-b^2 B\right) d^2-\frac{1}{4} b \left(35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right)+\frac{1}{4} b ((4 b c+a d) (6 b c C-6 a d C-7 b B d)-5 a d (6 b c C-a (7 A-C) d))\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{(-c-i d) f}\right)}{3 d}\right)}{5 d}\right)}{7 d}","\frac{2 \sqrt{c+d \tan (e+f x)} \left(72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left(15 d^2 (A-C)-10 B c d+8 c^2 C\right)-\left(b^3 \left(70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right)\right)\right)}{105 d^4 f}+\frac{2 b \tan (e+f x) \sqrt{c+d \tan (e+f x)} \left(35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right)}{105 d^3 f}-\frac{(-b+i a)^3 (A+i B-C) \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 (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}-\frac{2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}{35 d^2 f}+\frac{2 C (a+b \tan (e+f x))^3 \sqrt{c+d \tan (e+f x)}}{7 d f}",1,"(2*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]])/(7*d*f) + (2*(((-6*b*c*C + 7*b*B*d + 6*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*d*f) + (2*((b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(6*d*f) - (2*((I*Sqrt[c - I*d]*((b*c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4 + (3*a*d*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - (3*a*d*(-5*a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - b*((-105*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4) + ((3*I)/2)*d*((35*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 - (b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4 + (b*(-5*a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*((b*c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4 + (3*a*d*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - (3*a*d*(-5*a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 - b*((-105*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4) - ((3*I)/2)*d*((35*a*(a^2*B - b^2*B + 2*a*b*(A - C))*d^2)/4 - (b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4 + (b*(-5*a*d*(6*b*c*C - a*(7*A - C)*d) + (4*b*c + a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((-c - I*d)*f) + (2*((-3*a*d*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/8 + b*((-105*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/8 + (c*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d)))/4))*Sqrt[c + d*Tan[e + f*x]])/(d*f)))/(3*d)))/(5*d)))/(7*d)","B",1
111,1,275,287,6.031759,"\int \frac{(a+b \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{2 \sqrt{c+d \tan (e+f x)} \left(12 a^2 C d^2+10 a b d (3 B d-2 c C)+b^2 \left(15 d^2 (A-C)-10 B c d+8 c^2 C\right)\right)}{d^2}-\frac{15 d (a-i b)^2 (i A+B-i C) \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)^2 (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}+\frac{2 b \tan (e+f x) (4 a C d+5 b B d-4 b c C) \sqrt{c+d \tan (e+f x)}}{d}+6 C (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}{15 d f}","\frac{2 \sqrt{c+d \tan (e+f x)} \left(12 a^2 C d^2-10 a b d (2 c C-3 B d)+b^2 \left(15 d^2 (A-C)-10 B c d+8 c^2 C\right)\right)}{15 d^3 f}-\frac{(a-i b)^2 (B+i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f \sqrt{c-i d}}+\frac{(a+i b)^2 (i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}-\frac{2 b \tan (e+f x) (-4 a C d-5 b B d+4 b c C) \sqrt{c+d \tan (e+f x)}}{15 d^2 f}+\frac{2 C (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}{5 d f}",1,"((-15*(a - I*b)^2*(I*A + B - I*C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((15*I)*(a + I*b)^2*(A + I*B - C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*(12*a^2*C*d^2 + 10*a*b*d*(-2*c*C + 3*B*d) + b^2*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2))*Sqrt[c + d*Tan[e + f*x]])/d^2 + (2*b*(-4*b*c*C + 5*b*B*d + 4*a*C*d)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/d + 6*C*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(15*d*f)","A",1
112,1,192,194,1.4568023,"\int \frac{(a+b \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]","\frac{2 \left(-\frac{3 i d (a-i b) (A-i B-C) \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) (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{2 \sqrt{c+i d}}+\frac{(3 a C d+3 b B d-2 b c C) \sqrt{c+d \tan (e+f x)}}{d}+b C \tan (e+f x) \sqrt{c+d \tan (e+f x)}\right)}{3 d f}","-\frac{(b+i a) (A-i B-C) \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) (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}-\frac{2 (-3 a C d-3 b B d+2 b c C) \sqrt{c+d \tan (e+f x)}}{3 d^2 f}+\frac{2 b C \tan (e+f x) \sqrt{c+d \tan (e+f x)}}{3 d f}",1,"(2*((((-3*I)/2)*(a - I*b)*(A - I*B - C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (((3*I)/2)*(a + I*b)*(A + I*B - C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + ((-2*b*c*C + 3*b*B*d + 3*a*C*d)*Sqrt[c + d*Tan[e + f*x]])/d + b*C*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]]))/(3*d*f)","A",1
113,1,129,133,0.2155241,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/Sqrt[c + d*Tan[e + f*x]],x]","\frac{-\frac{i (A-i B-C) \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-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{\sqrt{c+i d}}+\frac{2 C \sqrt{c+d \tan (e+f x)}}{d}}{f}","-\frac{(i A+B-i C) \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-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f \sqrt{c+i d}}+\frac{2 C \sqrt{c+d \tan (e+f x)}}{d f}",1,"(((-I)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*C*Sqrt[c + d*Tan[e + f*x]])/d)/f","A",1
114,1,194,210,0.4175762,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{-\frac{2 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} \sqrt{b c-a d}}+\frac{(b-i a) (A-i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{\sqrt{c-i d}}+\frac{(b+i a) (A+i B-C) \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 \left(A b^2-a (b B-a 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) \sqrt{b c-a d}}-\frac{(i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (a-i b) \sqrt{c-i d}}-\frac{(A+i B-C) \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)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((I*a + b)*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/((a^2 + b^2)*f)","A",1
115,1,521,327,6.2165915,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]),x]","-\frac{\left(A b^2-a (b B-a C)\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}{2} a^2 d \left(A b^2-a (b B-a C)\right)+\frac{1}{2} b^2 \left(-2 a A (b c-a d)-2 (b B-a C) \left(b c-\frac{a d}{2}\right)+A b^2 d\right)-a b (b c-a d) (-a B+A b-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(-(b c-a d) \left(a^2 (A-C)+2 a b B-b^2 (A-C)\right)+i (b c-a d) \left(a^2 B-2 a b (A-C)-b^2 B\right)\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 c-a d) \left(a^2 (A-C)+2 a b B-b^2 (A-C)\right)-i (b c-a d) \left(a^2 B-2 a b (A-C)-b^2 B\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)}","-\frac{\left(A b^2-a (b B-a C)\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{\left(a^4 (-C) d+3 a^3 b B d-a^2 b^2 (5 A d+2 B c-3 C d)+a b^3 (4 A c-B d-4 c C)+b^4 (2 B c-A 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 (b c-a d)^{3/2}}-\frac{(i A+B-i C) \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{(B-i (A-C)) \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*Sqrt[c - I*d]*(I*(a^2*B - b^2*B - 2*a*b*(A - C))*(b*c - a*d) - (2*a*b*B + a^2*(A - C) - b^2*(A - C))*(b*c - a*d))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*((-I)*(a^2*B - b^2*B - 2*a*b*(A - C))*(b*c - a*d) - (2*a*b*B + a^2*(A - C) - b^2*(A - C))*(b*c - a*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*(A*b^2 - a*(b*B - a*C))*d)/2 - a*b*(A*b - a*B - b*C)*(b*c - a*d) + (b^2*(A*b^2*d - 2*a*A*(b*c - a*d) - 2*(b*B - a*C)*(b*c - (a*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))) - ((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))","A",1
116,1,920,511,6.7874025,"\int \frac{(a+b \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 C (a+b \tan (e+f x))^3}{5 d f \sqrt{c+d \tan (e+f x)}}+\frac{2 \left(\frac{(-6 b c C+6 a d C+5 b B d) (a+b \tan (e+f x))^2}{3 d f \sqrt{c+d \tan (e+f x)}}+\frac{2 \left(\frac{\left(15 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-5 b B d)\right) (a+b \tan (e+f x))}{2 d f \sqrt{c+d \tan (e+f x)}}+\frac{\frac{2 \left(\frac{1}{2} \left(-15 A b^3 d^3+45 a^2 A b d^3+15 a^3 B d^3-45 a b^2 B d^3+15 b^3 C d^3-45 a^2 b C d^3\right) \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)+\frac{\left(d^2 \left(\frac{1}{2} \left(-30 A c d^2 b^3+30 c C d^2 b^3-48 c^3 C b^3+40 B c^2 d b^3+15 a A d^3 b^2-15 a C d^3 b^2-110 a B c d^2 b^2+144 a c^2 C d b^2+40 a^2 B d^3 b-144 a^2 c C d^2 b+15 a^3 A d^3+33 a^3 C d^3\right)+\frac{1}{2} \left(15 B d^3 b^3+30 A c d^2 b^3-30 c C d^2 b^3+48 c^3 C b^3-40 B c^2 d b^3-60 a A d^3 b^2+60 a C d^3 b^2+110 a B c d^2 b^2-144 a c^2 C d b^2-85 a^2 B d^3 b+144 a^2 c C d^2 b-48 a^3 C d^3\right)\right)-\frac{1}{2} c d \left(-15 A b^3 d^3+45 a^2 A b d^3+15 a^3 B d^3-45 a b^2 B d^3+15 b^3 C d^3-45 a^2 b C d^3\right)\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{(i c-d) \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)}{(i c+d) \sqrt{c+d \tan (e+f x)}}\right)}{d}\right)}{d}-\frac{2 \left(-15 B d^3 b^3-30 A c d^2 b^3+30 c C d^2 b^3-48 c^3 C b^3+40 B c^2 d b^3+60 a A d^3 b^2-60 a C d^3 b^2-110 a B c d^2 b^2+144 a c^2 C d b^2+85 a^2 B d^3 b-144 a^2 c C d^2 b+48 a^3 C d^3\right)}{d \sqrt{c+d \tan (e+f x)}}}{4 d f}\right)}{3 d}\right)}{5 d}","\frac{2 b \sqrt{c+d \tan (e+f x)} \left(6 a^2 d^2 \left(d^2 (5 A+7 C)-5 B c d+12 c^2 C\right)-15 a b d \left(c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right)+b^2 \left(6 c^2 d^2 (5 A+3 C)+15 d^4 (A-C)-40 B c^3 d-25 B c d^3+48 c^4 C\right)\right)}{15 d^4 f \left(c^2+d^2\right)}-\frac{2 b^2 \tan (e+f x) \sqrt{c+d \tan (e+f x)} \left(4 (b c-a d) \left(d^2 (5 A+C)-5 B c d+6 c^2 C\right)-5 d^2 ((A-C) (b c-a d)+B (a c+b d))\right)}{15 d^3 f \left(c^2+d^2\right)}+\frac{2 b \left(d^2 (5 A+C)-5 B c d+6 c^2 C\right) (a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}}{5 d^2 f \left(c^2+d^2\right)}-\frac{2 \left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^3}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{(-b+i a)^3 (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}-\frac{(a-i b)^3 (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}",1,"(2*C*(a + b*Tan[e + f*x])^3)/(5*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*(((-6*b*c*C + 5*b*B*d + 6*a*C*d)*(a + b*Tan[e + f*x])^2)/(3*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*(((15*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 5*b*B*d - 6*a*C*d))*(a + b*Tan[e + f*x]))/(2*d*f*Sqrt[c + d*Tan[e + f*x]]) + ((-2*(-48*b^3*c^3*C + 40*b^3*B*c^2*d + 144*a*b^2*c^2*C*d - 30*A*b^3*c*d^2 - 110*a*b^2*B*c*d^2 - 144*a^2*b*c*C*d^2 + 30*b^3*c*C*d^2 + 60*a*A*b^2*d^3 + 85*a^2*b*B*d^3 - 15*b^3*B*d^3 + 48*a^3*C*d^3 - 60*a*b^2*C*d^3))/(d*Sqrt[c + d*Tan[e + f*x]]) + (2*(((45*a^2*A*b*d^3 - 15*A*b^3*d^3 + 15*a^3*B*d^3 - 45*a*b^2*B*d^3 - 45*a^2*b*C*d^3 + 15*b^3*C*d^3)*(((-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]))/2 + ((-1/2*(c*d*(45*a^2*A*b*d^3 - 15*A*b^3*d^3 + 15*a^3*B*d^3 - 45*a*b^2*B*d^3 - 45*a^2*b*C*d^3 + 15*b^3*C*d^3)) + d^2*((-48*b^3*c^3*C + 40*b^3*B*c^2*d + 144*a*b^2*c^2*C*d - 30*A*b^3*c*d^2 - 110*a*b^2*B*c*d^2 - 144*a^2*b*c*C*d^2 + 30*b^3*c*C*d^2 + 15*a^3*A*d^3 + 15*a*A*b^2*d^3 + 40*a^2*b*B*d^3 + 33*a^3*C*d^3 - 15*a*b^2*C*d^3)/2 + (48*b^3*c^3*C - 40*b^3*B*c^2*d - 144*a*b^2*c^2*C*d + 30*A*b^3*c*d^2 + 110*a*b^2*B*c*d^2 + 144*a^2*b*c*C*d^2 - 30*b^3*c*C*d^2 - 60*a*A*b^2*d^3 - 85*a^2*b*B*d^3 + 15*b^3*B*d^3 - 48*a^3*C*d^3 + 60*a*b^2*C*d^3)/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)/(4*d*f)))/(3*d)))/(5*d)","C",1
117,1,476,343,6.509814,"\int \frac{(a+b \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]","\frac{2 C (a+b \tan (e+f x))^2}{3 d f \sqrt{c+d \tan (e+f x)}}+\frac{2 \left(\frac{(4 a C d+3 b B d-4 b c C) (a+b \tan (e+f x))}{d f \sqrt{c+d \tan (e+f x)}}+\frac{-\frac{2 \left(8 a^2 C d^2+9 a b B d^2-16 a b c C d+3 A b^2 d^2-6 b^2 B c d+8 b^2 c^2 C-3 b^2 C d^2\right)}{d \sqrt{c+d \tan (e+f x)}}+\frac{2 \left(\frac{3}{2} d^2 \left(a^2 B+2 a b (A-C)-b^2 B\right) \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)+\frac{\left(-\frac{3}{2} c d^3 \left(a^2 B+2 a b (A-C)-b^2 B\right)-\frac{3}{2} d^4 \left(-\left(a^2 (A-C)\right)+2 a b B+b^2 (A-C)\right)\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}\right)}{d}}{2 d f}\right)}{3 d}","-\frac{2 \left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^2}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}+\frac{2 b \sqrt{c+d \tan (e+f x)} \left(6 a d \left(d^2 (A+C)-B c d+2 c^2 C\right)-b \left(c d^2 (3 A+5 C)-6 B c^2 d-3 B d^3+8 c^3 C\right)\right)}{3 d^3 f \left(c^2+d^2\right)}-\frac{(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{3/2}}-\frac{(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}+\frac{2 b^2 \tan (e+f x) \left(d^2 (3 A+C)-3 B c d+4 c^2 C\right) \sqrt{c+d \tan (e+f x)}}{3 d^2 f \left(c^2+d^2\right)}",1,"(2*C*(a + b*Tan[e + f*x])^2)/(3*d*f*Sqrt[c + d*Tan[e + f*x]]) + (2*(((-4*b*c*C + 3*b*B*d + 4*a*C*d)*(a + b*Tan[e + f*x]))/(d*f*Sqrt[c + d*Tan[e + f*x]]) + ((-2*(8*b^2*c^2*C - 6*b^2*B*c*d - 16*a*b*c*C*d + 3*A*b^2*d^2 + 9*a*b*B*d^2 + 8*a^2*C*d^2 - 3*b^2*C*d^2))/(d*Sqrt[c + d*Tan[e + f*x]]) + (2*((3*(a^2*B - b^2*B + 2*a*b*(A - C))*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]))/2 + (((-3*c*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/2 - (3*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*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
118,1,290,201,2.5905296,"\int \frac{(a+b \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]","\frac{\frac{(-a A d+a B c+a C d+A b c+b B d-b c C) \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)}}+(a B+A b-b C) \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)-\frac{2 (2 a C d+b B d-2 b c C)}{d \sqrt{c+d \tan (e+f x)}}+\frac{2 C (a+b \tan (e+f x))}{\sqrt{c+d \tan (e+f x)}}}{d f}","\frac{2 (b c-a d) \left(A d^2-B c d+c^2 C\right)}{d^2 f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{(b+i a) (A-i B-C) \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) (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{3/2}}+\frac{2 b C \sqrt{c+d \tan (e+f x)}}{d^2 f}",1,"((A*b + a*B - b*C)*(((-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]) - (2*(-2*b*c*C + b*B*d + 2*a*C*d))/(d*Sqrt[c + d*Tan[e + f*x]]) + ((A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*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*C*(a + b*Tan[e + f*x]))/Sqrt[c + d*Tan[e + f*x]])/(d*f)","C",1
119,1,218,157,1.0106614,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(3/2),x]","\frac{\frac{(d (C-A)+B c) \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)}}-i B \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{2 C}{\sqrt{c+d \tan (e+f x)}}}{d f}","-\frac{2 \left(A d^2-B c d+c^2 C\right)}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{(i A+B-i C) \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-C)) \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*(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]) - (2*C)/Sqrt[c + d*Tan[e + f*x]] + ((B*c + (-A + C)*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]]))/(d*f)","C",1
120,1,296,262,4.9502791,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]","\frac{\frac{2 \sqrt{b} \left(c^2+d^2\right) \left(a (a C-b B)+A b^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{i \left(\frac{(a+i b) (c+i d) (A-i B-C) (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) (A+i B-C) (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 \left(A d^2-B c d+c^2 C\right)}{\sqrt{c+d \tan (e+f x)}}}{f \left(c^2+d^2\right) (a d-b c)}","-\frac{2 \sqrt{b} \left(A b^2-a (b B-a C)\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) (b c-a d)^{3/2}}+\frac{2 \left(A d^2-B c d+c^2 C\right)}{f \left(c^2+d^2\right) (b c-a d) \sqrt{c+d \tan (e+f x)}}+\frac{(A-i B-C) \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{(i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b) (c+i d)^{3/2}}",1,"(((-I)*(((a + I*b)*(A - I*B - C)*(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)*(A + I*B - C)*(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*Sqrt[b]*(A*b^2 + a*(-(b*B) + a*C))*(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*(c^2*C - B*c*d + A*d^2))/Sqrt[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*(c^2 + d^2)*f)","A",1
121,1,2078,447,6.2859404,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)),x]","\text{Result too large to show}","-\frac{d \left(A \left(2 a^2 d^2+b^2 \left(c^2+3 d^2\right)\right)+a^2 \left(-2 B c d+3 c^2 C+C d^2\right)-a b B \left(c^2+d^2\right)+2 b^2 c (c C-B d)\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{A b^2-a (b B-a C)}{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{\sqrt{b} \left(-3 a^4 C d+5 a^3 b B d-a^2 b^2 (d (7 A-C)+2 B c)+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A 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 A+B-i C) \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{(B-i (A-C)) \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,"-((A*b^2 - a*(b*B - a*C))/((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]*((b*(-(b*c) + a*d)*((-3*(A*b^2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 + a*(-1/2*(a*d*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2 - (b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2) - I*((a*(-(b*c) + a*d)*((-3*(A*b^2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 - b*(-1/2*(a*d*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2 - (b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/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*(A*b^2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 + a*(-1/2*(a*d*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2 - (b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2) + I*((a*(-(b*c) + a*d)*((-3*(A*b^2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 - b*(-1/2*(a*d*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2 - (b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/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]*(-1/2*(a*b*(-(b*c) + a*d)*((-3*(A*b^2 - a*(b*B - a*C))*d^2)/2 - c*(A*b - a*B - b*C)*(b*c - a*d) + (d*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2)) + (a^2*b*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/2 + b^2*(-1/2*(a*d*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/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*(-(c*((-3*c*(A*b^2 - a*(b*B - a*C))*d)/2 + (A*b - a*B - b*C)*d*(b*c - a*d))) + (d^2*(3*A*b^2*d - 2*a*A*(b*c - a*d) - (b*B - a*C)*(2*b*c + a*d)))/2))/((-(b*c) + a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]))/((a^2 + b^2)*(b*c - a*d))","B",1
122,1,670,585,6.9064156,"\int \frac{(a+b \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]","\frac{2 C (a+b \tan (e+f x))^3}{3 d f (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(-\frac{3 (-2 a C d-b B d+2 b c C) (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(-\frac{3 (a+b \tan (e+f x)) \left(b d^2 (a B+A b-b C)+4 (b c-a d) (-2 a C d-b B d+2 b c C)\right)}{2 d f (c+d \tan (e+f x))^{3/2}}-\frac{3 \left(-\frac{2 \left(16 a^3 C d^3+9 a^2 b B d^3-48 a^2 b c C d^2-18 a b^2 B c d^2+48 a b^2 c^2 C d-2 A b^3 c d^2+8 b^3 B c^2 d+b^3 B d^3-16 b^3 c^3 C+2 b^3 c C d^2\right)}{3 d (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(\frac{\left(\frac{3}{2} c d^4 \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right)+\frac{3}{2} d^5 \left(-\left(a^3 (A-C)\right)+3 a^2 b B+3 a b^2 (A-C)-b^3 B\right)\right) \left(\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{3 (-d+i c) (c+d \tan (e+f x))^{3/2}}-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{3 (d+i c) (c+d \tan (e+f x))^{3/2}}\right)}{d}-\frac{3}{2} d^3 \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\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)\right)}{3 d}\right)}{4 d f}\right)}{d}\right)}{3 d}","\frac{2 b \sqrt{c+d \tan (e+f x)} \left(6 a^2 d^3 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+3 a b d \left(-c^2 d^2 (A-17 C)+d^4 (5 A+3 C)-2 B c^3 d-8 B c d^3+8 c^4 C\right)-b^2 \left(2 c^3 d^2 (A+15 C)+8 c d^4 (A+C)-8 B c^4 d-17 B c^2 d^3-3 B d^5+16 c^5 C\right)\right)}{3 d^4 f \left(c^2+d^2\right)^2}+\frac{2 b^2 \tan (e+f x) \sqrt{c+d \tan (e+f x)} \left(3 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(c^2 d^2 (A+15 C)+d^4 (7 A+C)-4 B c^3 d-10 B c d^3+8 c^4 C\right)\right)}{3 d^3 f \left(c^2+d^2\right)^2}-\frac{2 \left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^3}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{2 (a+b \tan (e+f x))^2 \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(2 A d^4-B c^3 d-3 B c d^3+2 c^4 C+4 c^2 C d^2\right)\right)}{d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(-b+i a)^3 (A+i B-C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{5/2}}-\frac{(a-i b)^3 (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}",1,"(2*C*(a + b*Tan[e + f*x])^3)/(3*d*f*(c + d*Tan[e + f*x])^(3/2)) + (2*((-3*(2*b*c*C - b*B*d - 2*a*C*d)*(a + b*Tan[e + f*x])^2)/(d*f*(c + d*Tan[e + f*x])^(3/2)) + (2*((-3*(b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - b*B*d - 2*a*C*d))*(a + b*Tan[e + f*x]))/(2*d*f*(c + d*Tan[e + f*x])^(3/2)) - (3*((-2*(-16*b^3*c^3*C + 8*b^3*B*c^2*d + 48*a*b^2*c^2*C*d - 2*A*b^3*c*d^2 - 18*a*b^2*B*c*d^2 - 48*a^2*b*c*C*d^2 + 2*b^3*c*C*d^2 + 9*a^2*b*B*d^3 + b^3*B*d^3 + 16*a^3*C*d^3))/(3*d*(c + d*Tan[e + f*x])^(3/2)) + (2*((((3*c*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^4)/2 + (3*(3*a^2*b*B - b^3*B - a^3*(A - C) + 3*a*b^2*(A - C))*d^5)/2)*(-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*(c + d*Tan[e + f*x])^(3/2)) + Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]/(3*(I*c - d)*(c + d*Tan[e + f*x])^(3/2))))/d - (3*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*d^3*(-(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]])))/2))/(3*d)))/(4*d*f)))/d))/(3*d)","C",1
123,1,502,358,6.5622472,"\int \frac{(a+b \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]","\frac{2 C (a+b \tan (e+f x))^2}{d f (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(-\frac{(4 a C d+b B d-4 b c C) (a+b \tan (e+f x))}{d f (c+d \tan (e+f x))^{3/2}}-\frac{-\frac{2 \left(8 a^2 C d^2+a b B d^2-16 a b c C d-A b^2 d^2-2 b^2 B c d+8 b^2 c^2 C+b^2 C d^2\right)}{3 d (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(\frac{\left(\frac{3}{2} c d^3 \left(a^2 B+2 a b (A-C)-b^2 B\right)+\frac{3}{2} d^4 \left(-\left(a^2 (A-C)\right)+2 a b B+b^2 (A-C)\right)\right) \left(\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c+i d}\right)}{3 (-d+i c) (c+d \tan (e+f x))^{3/2}}-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{c+d \tan (e+f x)}{c-i d}\right)}{3 (d+i c) (c+d \tan (e+f x))^{3/2}}\right)}{d}-\frac{3}{2} d^2 \left(a^2 B+2 a b (A-C)-b^2 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)\right)}{3 d}}{2 d f}\right)}{d}","-\frac{2 \left(A d^2-B c d+c^2 C\right) (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{2 (b c-a d) \left(3 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-2 c^2 d^2 (A-5 C)+4 A d^4-B c^3 d-7 B c d^3+4 c^4 C\right)\right)}{3 d^3 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b)^2 (i A+B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{f (c-i d)^{5/2}}-\frac{(a+i b)^2 (B-i (A-C)) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (c+i d)^{5/2}}+\frac{2 b^2 \left(d^2 (A+3 C)-B c d+4 c^2 C\right) \sqrt{c+d \tan (e+f x)}}{3 d^3 f \left(c^2+d^2\right)}",1,"(2*C*(a + b*Tan[e + f*x])^2)/(d*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(-(((-4*b*c*C + b*B*d + 4*a*C*d)*(a + b*Tan[e + f*x]))/(d*f*(c + d*Tan[e + f*x])^(3/2))) - ((-2*(8*b^2*c^2*C - 2*b^2*B*c*d - 16*a*b*c*C*d - A*b^2*d^2 + a*b*B*d^2 + 8*a^2*C*d^2 + b^2*C*d^2))/(3*d*(c + d*Tan[e + f*x])^(3/2)) + (2*((((3*c*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3)/2 + (3*(2*a*b*B - a^2*(A - C) + b^2*(A - C))*d^4)/2)*(-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)]/((I*c + d)*(c + d*Tan[e + f*x])^(3/2)) + Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]/(3*(I*c - d)*(c + d*Tan[e + f*x])^(3/2))))/d - (3*(a^2*B - b^2*B + 2*a*b*(A - C))*d^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]])))/2))/(3*d))/(2*d*f)))/d","C",1
124,1,300,273,3.0197568,"\int \frac{(a+b \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{d (-a A d+a B c+a C d+A b c+b B d-b c C) \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)-3 d (a B+A b-b C) (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)+2 (c-i d) (c+i d) (-2 a C d+b B d+2 b c C)+6 C d (c-i d) (c+i d) (a+b \tan (e+f x))}{3 d^2 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}","\frac{2 (b c-a d) \left(A d^2-B c d+c^2 C\right)}{3 d^2 f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right)}{d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b) (i A+B-i C) \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) (A+i B-C) \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*(c - I*d)*(c + I*d)*(2*b*c*C + b*B*d - 2*a*C*d) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*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*C*(c - I*d)*(c + I*d)*d*(a + b*Tan[e + f*x]) - 3*(A*b + a*B - b*C)*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
125,1,223,209,0.9381117,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{(d (C-A)+B c) \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)-3 B (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)+2 C \left(c^2+d^2\right)}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}","-\frac{2 \left(A d^2-B c d+c^2 C\right)}{3 d f \left(c^2+d^2\right) (c+d \tan (e+f x))^{3/2}}-\frac{2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)}{f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(i A+B-i C) \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-C)) \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*C*(c^2 + d^2) + (B*c + (-A + C)*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)]) - 3*B*(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*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2))","C",1
126,1,1948,365,6.2756508,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)),x]","-\frac{2 \left(A d^2-c (B d-c C)\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{2 \left(-\frac{3}{2} \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right)}{(a d-b c) \left(c^2+d^2\right) f \sqrt{c+d \tan (e+f x)}}-\frac{2 \left(\frac{\frac{i \sqrt{c-i d} \left(\frac{1}{2} b (a d-b c) \left(-\frac{3}{2} c (b c-a d) (B c-(A-C) d)-\frac{3}{2} b d \left(C c^2-B d c+A d^2\right)-\frac{3}{2} d \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)\right)+a \left(-\frac{3}{2} \left(\frac{b d^2}{2}-\frac{1}{2} c (a d-b c)\right) \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)-\frac{1}{2} a d \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)-\frac{1}{2} b \left(-\frac{3}{2} \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right)\right)-i \left(\frac{1}{2} a (a d-b c) \left(-\frac{3}{2} c (b c-a d) (B c-(A-C) d)-\frac{3}{2} b d \left(C c^2-B d c+A d^2\right)-\frac{3}{2} d \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)\right)-b \left(-\frac{3}{2} \left(\frac{b d^2}{2}-\frac{1}{2} c (a d-b c)\right) \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)-\frac{1}{2} a d \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)-\frac{1}{2} b \left(-\frac{3}{2} \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right)\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right)}{(i d-c) f}-\frac{i \sqrt{c+i d} \left(\frac{1}{2} b (a d-b c) \left(-\frac{3}{2} c (b c-a d) (B c-(A-C) d)-\frac{3}{2} b d \left(C c^2-B d c+A d^2\right)-\frac{3}{2} d \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)\right)+a \left(-\frac{3}{2} \left(\frac{b d^2}{2}-\frac{1}{2} c (a d-b c)\right) \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)-\frac{1}{2} a d \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)-\frac{1}{2} b \left(-\frac{3}{2} \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right)\right)+i \left(\frac{1}{2} a (a d-b c) \left(-\frac{3}{2} c (b c-a d) (B c-(A-C) d)-\frac{3}{2} b d \left(C c^2-B d c+A d^2\right)-\frac{3}{2} d \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)\right)-b \left(-\frac{3}{2} \left(\frac{b d^2}{2}-\frac{1}{2} c (a d-b c)\right) \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)-\frac{1}{2} a d \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)-\frac{1}{2} b \left(-\frac{3}{2} \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right)\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{(-c-i d) f}}{a^2+b^2}+\frac{2 \sqrt{b c-a d} \left(\frac{1}{2} b \left(-\frac{3}{2} \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right) d^2-c \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right) a^2-\frac{1}{2} b (a d-b c) \left(-\frac{3}{2} c (b c-a d) (B c-(A-C) d)-\frac{3}{2} b d \left(C c^2-B d c+A d^2\right)-\frac{3}{2} d \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)\right) a+b^2 \left(-\frac{3}{2} \left(\frac{b d^2}{2}-\frac{1}{2} c (a d-b c)\right) \left(a A c d-a (c C-B d) d-A b \left(c^2+d^2\right)\right)-\frac{1}{2} a d \left(\frac{3}{2} d (b c-a d) (B c-(A-C) d)-\frac{3}{2} b c \left(C c^2-B d c+A d^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right)}{\sqrt{b} \left(a^2+b^2\right) (a d-b c) f}\right)}{(a d-b c) \left(c^2+d^2\right)}\right)}{3 (a d-b c) \left(c^2+d^2\right)}","-\frac{2 b^{3/2} \left(A b^2-a (b B-a C)\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) (b c-a d)^{5/2}}+\frac{2 \left(A d^2-B c d+c^2 C\right)}{3 f \left(c^2+d^2\right) (b c-a d) (c+d \tan (e+f x))^{3/2}}+\frac{2 \left(b \left(c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right)-a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{f \left(c^2+d^2\right)^2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}+\frac{(A-i B-C) \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{(i A-B-i C) \tanh ^{-1}\left(\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right)}{f (a+i b) (c+i d)^{5/2}}",1,"(-2*(A*d^2 - c*(-(c*C) + B*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)*(B*c - (A - C)*d))/2 - (3*b*d*(c^2*C - B*c*d + A*d^2))/2 - (3*d*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2))/2 + a*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - (a*d*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2))/2 - (b*((-3*d^2*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - c*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2)))/2) - I*((a*(-(b*c) + a*d)*((-3*c*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*d*(c^2*C - B*c*d + A*d^2))/2 - (3*d*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2))/2 - b*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - (a*d*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2))/2 - (b*((-3*d^2*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - c*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*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*c*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*d*(c^2*C - B*c*d + A*d^2))/2 - (3*d*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2))/2 + a*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - (a*d*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2))/2 - (b*((-3*d^2*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - c*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2)))/2) + I*((a*(-(b*c) + a*d)*((-3*c*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*d*(c^2*C - B*c*d + A*d^2))/2 - (3*d*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2))/2 - b*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - (a*d*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2))/2 - (b*((-3*d^2*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - c*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*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]*(-1/2*(a*b*(-(b*c) + a*d)*((-3*c*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*d*(c^2*C - B*c*d + A*d^2))/2 - (3*d*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2)) + (a^2*b*((-3*d^2*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - c*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2)))/2 + b^2*((-3*((b*d^2)/2 - (c*(-(b*c) + a*d))/2)*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - (a*d*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/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)))/((-(b*c) + a*d)*(c^2 + d^2)) - (2*((-3*d^2*(a*A*c*d - a*d*(c*C - B*d) - A*b*(c^2 + d^2)))/2 - c*((3*d*(b*c - a*d)*(B*c - (A - C)*d))/2 - (3*b*c*(c^2*C - B*c*d + A*d^2))/2)))/((-(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))","B",1
127,1,6052,679,6.4265191,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((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(A \left(2 a^2 d^2+b^2 \left(3 c^2+5 d^2\right)\right)+a^2 \left(-2 B c d+5 c^2 C+3 C d^2\right)-3 a b B \left(c^2+d^2\right)+2 b^2 c (c C-B d)\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{A b^2-a (b B-a C)}{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{d \left(2 a^3 d^2 \left(B c^2-B d^2+2 c C d\right)+a^2 b \left(-6 B c^3 d-2 B c d^3+5 c^4 C+2 c^2 C d^2+C d^4\right)-A \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)-a b^2 \left(B c^4+3 B d^4-4 c C d^3\right)+2 b^3 c \left(-3 B c^2 d-B d^3+2 c^3 C\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{b^{3/2} \left(-5 a^4 C d+7 a^3 b B d-a^2 b^2 (d (9 A+C)+2 B c)+a b^3 (4 A c+3 B d-4 c C)+b^4 (2 B c-5 A 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{(i A+B-i C) \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{(B-i (A-C)) \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,"Result too large to show","B",1
128,1,1202,679,9.9505941,"\int (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (c+d \tan (e+f x))^{3/2} (a+b \tan (e+f x))^{5/2}}{4 d f}+\frac{\frac{(-5 b c C+5 a d C+8 b B d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{6 d f}+\frac{\frac{3 \left(16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 d f}+\frac{\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(24 b \left(B a^2+2 b (A-C) a-b^2 B\right) d^3-\frac{3}{8} (b c-a d) \left(16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right)\right)}{b f}+\frac{\frac{24 b \left(b \left((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right)+\sqrt{-b^2} \left((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) 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) d^3}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{24 b \left(b \left((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right)-\sqrt{-b^2} \left((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right) d^3}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+\frac{\sqrt{-b^2} d}{b}}}-\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{1}{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}} \left(\left(5 C c^4-8 B d c^3+16 (A-C) d^2 c^2+64 B d^3 c+128 (A-C) d^4\right) b^4-20 a d \left(C c^3-2 B d c^2+8 (A-C) d^2 c-16 B d^3\right) b^3+30 a^2 d^2 \left(C c^2-4 B d c-8 (A-C) d^2\right) b^2-20 a^3 d^3 (c C+2 B d) b+5 a^4 C d^4\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}}}\right) \sqrt{\frac{c+d \tan (e+f x)}{c-\frac{a d}{b}}}}{8 \sqrt{c+d \tan (e+f x)} \sqrt{d}}}{b^2 f}}{2 d}}{3 d}}{4 d}","\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(64 b d^3 \left(a^2 B+2 a b (A-C)-b^2 B\right)-(b c-a d) \left(16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right)\right)}{64 b d^3 f}-\frac{\left(5 a^4 C d^4-20 a^3 b d^3 (2 B d+c C)+30 a^2 b^2 d^2 \left(-8 d^2 (A-C)-4 B c d+c^2 C\right)-20 a b^3 d \left(8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right)+b^4 \left(16 c^2 d^2 (A-C)+128 d^4 (A-C)-8 B c^3 d+64 B c d^3+5 c^4 C\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)}{64 b^{3/2} d^{7/2} f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left(16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right)}{32 d^3 f}-\frac{(a-i b)^{5/2} \sqrt{c-i d} (i A+B-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)}{f}-\frac{(a+i b)^{5/2} \sqrt{c+i d} (B-i (A-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)}{f}-\frac{(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}",1,"(C*(a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2))/(4*d*f) + (((-5*b*c*C + 8*b*B*d + 5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(6*d*f) + ((3*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*d*f) + (((24*b*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3 - (3*(b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((24*b*d^3*(b*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d)) + Sqrt[-b^2]*(a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) - 3*a^2*b*(B*c + (A - C)*d) + b^3*(B*c + (A - C)*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]) - (24*b*d^3*(b*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d)) - Sqrt[-b^2]*(a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) - 3*a^2*b*(B*c + (A - C)*d) + b^3*(B*c + (A - C)*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*Sqrt[b]*Sqrt[c - (a*d)/b]*Sqrt[(c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b)))^(-1)]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b))]*(5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 20*a*b^3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 64*B*c*d^3 + 128*(A - C)*d^4))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b))])]*Sqrt[(c + d*Tan[e + f*x])/(c - (a*d)/b)])/(8*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*d))/(3*d))/(4*d)","A",0
129,1,835,505,8.8756698,"\int (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac{\frac{\frac{3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(8 b (A b-C b+a B) d^2+(b c-a d) (b c C-a d C-2 b B d)\right)}{4 b f}+\frac{\frac{6 b \left(\sqrt{-b^2} \left((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right)+b \left((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) 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) d^2}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}+\frac{6 b \left(\sqrt{-b^2} \left((A c-C c-B d) a^2-2 b (B c+(A-C) d) a-b^2 (A c-C c-B d)\right)-b \left((B c+(A-C) d) a^2+2 b (A c-C c-B d) a-b^2 (B c+(A-C) d)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right) d^2}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+\frac{\sqrt{-b^2} d}{b}}}-\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \left(-\left(\left(C c^3-2 B d c^2+8 (A-C) d^2 c-16 B d^3\right) b^3\right)+3 a d \left(C c^2-4 B d c-8 (A-C) d^2\right) b^2-3 a^2 d^2 (c C+2 B d) b+a^3 C d^3\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right) \sqrt{\frac{b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt{c+d \tan (e+f x)} \sqrt{d}}}{b^2 f}}{2 d}-\frac{3 (b c C-a d C-2 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d f}}{3 d}","-\frac{\left(a^3 C d^3-3 a^2 b d^2 (2 B d+c C)+3 a b^2 d \left(-8 d^2 (A-C)-4 B c d+c^2 C\right)-\left(b^3 \left(8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\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)}{8 b^{3/2} d^{5/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(8 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right)}{8 b d^2 f}-\frac{(a-i b)^{3/2} \sqrt{c-i d} (i A+B-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)}{f}+\frac{(a+i b)^{3/2} \sqrt{c+i d} (i A-B-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)}{f}-\frac{(-a C d-2 b B d+b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}",1,"(C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*d*f) + ((-3*(b*c*C - 2*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(4*d*f) + ((3*(8*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*f) + ((6*b*d^2*(Sqrt[-b^2]*(a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d)) + b*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*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]) + (6*b*d^2*(Sqrt[-b^2]*(a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d)) - b*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*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*Sqrt[b]*Sqrt[c - (a*d)/b]*(a^3*C*d^3 - 3*a^2*b*d^2*(c*C + 2*B*d) + 3*a*b^2*d*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3))*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[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*d))/(3*d)","A",1
130,1,619,381,7.7867101,"\int \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{\frac{-\frac{\sqrt{b} \sqrt{c-\frac{a d}{b}} \left(a^2 C d^2-2 a b d (2 B d+c C)+b^2 \left(-8 d^2 (A-C)-4 B c d+c^2 C\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)}{2 \sqrt{d} \sqrt{c+d \tan (e+f x)}}+\frac{2 b d \left(b (a A d+a B c-a C d+A b c-b B d-b c C)-\sqrt{-b^2} (-a (A c-B d-c C)+b d (A-C)+b B c)\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{2 b d \left(\sqrt{-b^2} (-a (A c-B d-c C)+b d (A-C)+b B c)+b (a A d+a B c-a C d+A b c-b B d-b c C)\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}}}{b^2 f}+\frac{(a C d+4 b B d-b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{2 b f}}{2 d}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}","-\frac{\left(a^2 C d^2-2 a b d (2 B d+c C)+b^2 \left(-8 d^2 (A-C)-4 B c d+c^2 C\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} d^{3/2} f}-\frac{\sqrt{a-i b} \sqrt{c-i d} (i A+B-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)}{f}-\frac{\sqrt{a+i b} \sqrt{c+i d} (B-i (A-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)}{f}-\frac{(-a C d-4 b B d+b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}",1,"(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*d*f) + (((-(b*c*C) + 4*b*B*d + a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(2*b*f) + ((2*b*d*(b*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) - Sqrt[-b^2]*(b*B*c + b*(A - C)*d - a*(A*c - c*C - B*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]) - (2*b*d*(b*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) + Sqrt[-b^2]*(b*B*c + b*(A - C)*d - a*(A*c - c*C - B*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]) - (Sqrt[b]*Sqrt[c - (a*d)/b]*(a^2*C*d^2 - 2*a*b*d*(c*C + 2*B*d) + b^2*(c^2*C - 4*B*c*d - 8*(A - 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)])/(2*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*d)","A",1
131,1,441,287,4.2441639,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{a+b \tan (e+f x)}} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[a + b*Tan[e + f*x]],x]","\frac{\frac{b \left(\sqrt{-b^2} (A c-B d-c C)+b d (A-C)+b B c\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} (A c-B d-c C)-b (d (A-C)+B c)\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} \sqrt{c-\frac{a d}{b}} (-a C d+2 b B d+b c C) \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 C \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 f}","-\frac{\sqrt{c-i d} (i A+B-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)}{f \sqrt{a-i b}}-\frac{\sqrt{c+i d} (B-i (A-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)}{f \sqrt{a+i b}}+\frac{(-a C d+2 b B d+b c C) \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} \sqrt{d} f}+\frac{C \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b f}",1,"((b*(b*B*c + b*(A - C)*d + Sqrt[-b^2]*(A*c - c*C - B*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*(Sqrt[-b^2]*(A*c - c*C - B*d) - b*(B*c + (A - C)*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*C*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + (Sqrt[b]*Sqrt[c - (a*d)/b]*(b*c*C + 2*b*B*d - a*C*d)*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^2*f)","A",1
132,1,621058,300,35.7309624,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{3/2}} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{c+d \tan (e+f x)}}{b f \left(a^2+b^2\right) \sqrt{a+b \tan (e+f x)}}-\frac{\sqrt{c-i d} (i A+B-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)}{f (a-i b)^{3/2}}-\frac{\sqrt{c+i d} (B-i (A-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)}{f (a+i b)^{3/2}}+\frac{2 C \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)}{b^{3/2} f}",1,"Result too large to show","C",0
133,1,600,370,7.0136235,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{5/2}} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2),x]","-\frac{C \sqrt{c+d \tan (e+f x)}}{b f (a+b \tan (e+f x))^{3/2}}-\frac{-\frac{2 \sqrt{c+d \tan (e+f x)} \left(\frac{1}{2} b^2 (-a C d-2 A b c+3 b c C)-a \left(-\frac{1}{2} a (-a C d-2 b B d+b c C)-\left(b^2 (d (A-C)+B c)\right)\right)\right)}{3 f \left(a^2+b^2\right) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{2 \sqrt{c+d \tan (e+f x)} \left(\frac{1}{2} b^2 (b c-a d) \left(a^2 C d+a b (3 A c-B d-3 c C)+b^2 (A d+3 B c)\right)-a \left(\frac{1}{2} a d (b c-a d) \left(a^2 (-C)-2 a b B+2 A b^2-3 b^2 C\right)-\frac{3}{2} b^2 (b c-a d) (-a A d-a B c+a C d+A b c-b B d-b c C)\right)\right)}{f \left(a^2+b^2\right) (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{3 b (b c-a d) \left(\frac{(a-i b)^2 \sqrt{c+i d} (B-i (A-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}}+\frac{(a+i b)^2 \sqrt{-c+i d} (i A+B-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}}\right)}{2 f \left(a^2+b^2\right)}\right)}{3 \left(a^2+b^2\right) (b c-a d)}}{b}","-\frac{2 \left(A b^2-a (b B-a C)\right) \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{2 \sqrt{c+d \tan (e+f x)} \left(a^4 C d+2 a^3 b B d-a^2 b^2 (5 A d+3 B c-7 C d)+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right)}{3 b f \left(a^2+b^2\right)^2 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{\sqrt{c-i d} (i A+B-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)}{f (a-i b)^{5/2}}-\frac{\sqrt{c+i d} (B-i (A-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)}{f (a+i b)^{5/2}}",1,"-((C*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x])^(3/2))) - ((-2*((b^2*(-2*A*b*c + 3*b*c*C - a*C*d))/2 - a*(-(b^2*(B*c + (A - C)*d)) - (a*(b*c*C - 2*b*B*d - a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*((-3*b*(b*c - a*d)*(((a + I*b)^2*(I*A + B - I*C)*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] + ((a - I*b)^2*(B - I*(A - C))*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*(a^2 + b^2)*f) - (2*((b^2*(b*c - a*d)*(a^2*C*d + b^2*(3*B*c + A*d) + a*b*(3*A*c - 3*c*C - B*d)))/2 - a*((a*(2*A*b^2 - 2*a*b*B - a^2*C - 3*b^2*C)*d*(b*c - a*d))/2 - (3*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])))/(3*(a^2 + b^2)*(b*c - a*d)))/b","A",1
134,1,1109,597,7.3483016,"\int \frac{\sqrt{c+d \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{7/2}} \, dx","Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]","-\frac{\sqrt{c+d \tan (e+f x)} C}{2 b f (a+b \tan (e+f x))^{5/2}}-\frac{-\frac{2 \sqrt{c+d \tan (e+f x)} \left(\frac{1}{2} b^2 (-4 A b c+5 b C c-a C d)-a \left(-2 (B c+(A-C) d) b^2-\frac{1}{2} a (b c C-a d C-4 b B d)\right)\right)}{5 \left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left(-\frac{2 \sqrt{c+d \tan (e+f x)} \left(b^2 (b c-a d) \left(C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right)-a \left(a \left(-C a^2-4 b B a+4 A b^2-5 b^2 C\right) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b C c-a A d-b B d+a C d)\right)\right)}{3 \left(a^2+b^2\right) (b c-a d) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left(-\frac{15 b \left(\frac{(i a-b)^3 (A-i B-C) \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}}-\frac{(i a+b)^3 (A+i B-C) \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) (b c-a d)^2}{2 \left(a^2+b^2\right) f}-\frac{2 \left(b^2 \left((b c-a d) \left(b^2 d-\frac{3}{2} a (b c-a d)\right) \left(C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right)+\left(\frac{a d}{2}-\frac{3 b c}{2}\right) \left(a \left(-C a^2-4 b B a+4 A b^2-5 b^2 C\right) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b C c-a A d-b B d+a C d)\right)\right)-a \left(\frac{3}{2} b (b c-a d) \left(b \left(-C a^2-4 b B a+4 A b^2-5 b^2 C\right) d (b c-a d)+5 a b (A b c-a B c-b C c-a A d-b B d+a C d) (b c-a d)+b \left(C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right) (b c-a d)\right)-a d \left(b^2 (b c-a d) \left(C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right)-a \left(a \left(-C a^2-4 b B a+4 A b^2-5 b^2 C\right) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b C c-a A d-b B d+a C d)\right)\right)\right)\right) \sqrt{c+d \tan (e+f x)}}{\left(a^2+b^2\right) f \sqrt{a+b \tan (e+f x)} (b c-a d)}\right)}{3 \left(a^2+b^2\right) (b c-a d)}\right)}{5 \left(a^2+b^2\right) (b c-a d)}}{2 b}","-\frac{2 \left(A b^2-a (b B-a C)\right) \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 \sqrt{c+d \tan (e+f x)} \left(a^4 C d+4 a^3 b B d-a^2 b^2 (9 A d+5 B c-11 C d)+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right)}{15 b f \left(a^2+b^2\right)^2 (b c-a d) (a+b \tan (e+f x))^{3/2}}+\frac{2 \sqrt{c+d \tan (e+f x)} \left(2 a^6 C d^2+8 a^5 b B d^2-a^4 b^2 d (33 A d+25 B c-39 C d)+a^3 b^3 \left(80 c d (A-C)+B \left(15 c^2-49 d^2\right)\right)-a^2 b^4 \left(45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right)-a b^5 \left(40 c d (A-C)+B \left(45 c^2-3 d^2\right)\right)-b^6 \left(5 c (B d+3 c C)-A \left(15 c^2+2 d^2\right)\right)\right)}{15 b f \left(a^2+b^2\right)^3 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}-\frac{\sqrt{c-i d} (i A+B-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)}{f (a-i b)^{7/2}}-\frac{\sqrt{c+i d} (B-i (A-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)}{f (a+i b)^{7/2}}",1,"-1/2*(C*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x])^(5/2)) - ((-2*((b^2*(-4*A*b*c + 5*b*c*C - a*C*d))/2 - a*(-2*b^2*(B*c + (A - C)*d) - (a*(b*c*C - 4*b*B*d - a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*((-2*(b^2*(b*c - a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) - a*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*((-15*b*(b*c - a*d)^2*(((I*a - b)^3*(A - I*B - C)*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] - ((I*a + b)^3*(A + I*B - C)*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*(a^2 + b^2)*f) - (2*(b^2*((b*c - a*d)*(b^2*d - (3*a*(b*c - a*d))/2)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) + ((-3*b*c)/2 + (a*d)/2)*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))) - a*((3*b*(b*c - a*d)*(b*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) + 5*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + b*(b*c - a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d))))/2 - a*d*(b^2*(b*c - a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) - a*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)))))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])))/(3*(a^2 + b^2)*(b*c - a*d))))/(5*(a^2 + b^2)*(b*c - a*d)))/(2*b)","A",1
135,1,1304,682,9.1402643,"\int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\frac{(-3 b c C+3 a d C+8 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{6 d f}+\frac{\frac{\left(48 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-8 b B d)\right) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 b f}+\frac{\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(24 b \left(B a^2+2 b (A-C) a-b^2 B\right) d^3-\frac{3}{8} (a d-b c) \left(48 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-8 b B d)\right)\right)}{b f}+\frac{-\frac{24 b^2 \left(\sqrt{-b^2} \left(\left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right) a^2+2 b \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) a-b^2 \left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right)\right)-b \left(-\left(\left(2 c (A-C) d+B \left(c^2-d^2\right)\right) a^2\right)+2 b \left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right) a+b^2 \left(2 c (A-C) d+B \left(c^2-d^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right) d^2}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+\frac{\sqrt{-b^2} d}{b}}}+\frac{24 \left(-d^2 \left(-\left(\left(2 c (A-C) d+B \left(c^2-d^2\right)\right) a^2\right)+2 b \left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right) a+b^2 \left(2 c (A-C) d+B \left(c^2-d^2\right)\right)\right) b^5-\sqrt{-b^2} d^2 \left(\left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right) a^2+2 b \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) a-b^2 \left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right)\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)}{b^2 \sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}+\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{1}{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}} \left(\left(3 C c^4-8 B d c^3+48 (A-C) d^2 c^2-192 B d^3 c-128 (A-C) d^4\right) b^4-12 a d \left(C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right) b^3+6 a^2 d^2 \left(3 C c^2+12 B d c+8 (A-C) d^2\right) b^2-4 a^3 d^3 (3 c C+2 B d) b+3 a^4 C d^4\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}}}\right) \sqrt{\frac{c+d \tan (e+f x)}{c-\frac{a d}{b}}}}{8 \sqrt{c+d \tan (e+f x)} \sqrt{d}}}{b^2 f}}{2 b}}{3 d}}{4 d}","\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(64 b d^3 \left(a^2 B+2 a b (A-C)-b^2 B\right)+(b c-a d) \left(48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right)\right)}{64 b^2 d^2 f}+\frac{\left(3 a^4 C d^4-4 a^3 b d^3 (2 B d+3 c C)+6 a^2 b^2 d^2 \left(8 d^2 (A-C)+12 B c d+3 c^2 C\right)-12 a b^3 d \left(-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right)+b^4 \left(48 c^2 d^2 (A-C)-128 d^4 (A-C)-8 B c^3 d-192 B c d^3+3 c^4 C\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)}{64 b^{5/2} d^{5/2} f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left(48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right)}{96 b d^2 f}-\frac{(a-i b)^{3/2} (c-i d)^{3/2} (B+i (A-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)}{f}-\frac{(a+i b)^{3/2} (c+i d)^{3/2} (B-i (A-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)}{f}-\frac{(-3 a C d-8 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}",1,"(C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2))/(4*d*f) + (((-3*b*c*C + 8*b*B*d + 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(6*d*f) + (((48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*b*f) + (((24*b*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3 - (3*(-(b*c) + a*d)*(48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d)))/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((24*(-(b^4*Sqrt[-b^2]*d^2*(a^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d + B*(c^2 - d^2)))) - b^5*d^2*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(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]])])/(b^2*Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (24*b^2*d^2*(Sqrt[-b^2]*(a^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d + B*(c^2 - d^2))) - b*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(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]) + (3*Sqrt[b]*Sqrt[c - (a*d)/b]*Sqrt[(c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b)))^(-1)]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b))]*(3*a^4*C*d^4 - 4*a^3*b*d^3*(3*c*C + 2*B*d) + 6*a^2*b^2*d^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - 12*a*b^3*d*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2 + 16*B*d^3) + b^4*(3*c^4*C - 8*B*c^3*d + 48*c^2*(A - C)*d^2 - 192*B*c*d^3 - 128*(A - C)*d^4))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b))])]*Sqrt[(c + d*Tan[e + f*x])/(c - (a*d)/b)])/(8*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*b))/(3*d))/(4*d)","A",0
136,1,867,508,9.0020861,"\int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac{\frac{(-b c C+a d C+6 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 b f}+\frac{\frac{3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(8 b (A b-C b+a B) d^2-(b c-a d) (b c C-a d C-6 b B d)\right)}{4 b f}+\frac{\frac{6 d \left(b \left(2 a A c d-2 a c C d+A b \left(c^2-d^2\right)+a B \left(c^2-d^2\right)-b \left(C c^2+2 B d c-C d^2\right)\right)-\sqrt{-b^2} \left(a \left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right)+b \left(2 c (A-C) d+B \left(c^2-d^2\right)\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) b^2}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{6 d \left(b \left(2 a A c d-2 a c C d+A b \left(c^2-d^2\right)+a B \left(c^2-d^2\right)-b \left(C c^2+2 B d c-C d^2\right)\right)+\sqrt{-b^2} \left(a \left(C c^2+2 B d c-C d^2-A \left(c^2-d^2\right)\right)+b \left(2 c (A-C) d+B \left(c^2-d^2\right)\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right) b^2}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+\frac{\sqrt{-b^2} d}{b}}}+\frac{3 \sqrt{c-\frac{a d}{b}} \left(-\left(\left(C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right) b^3\right)+a d \left(3 C c^2+12 B d c+8 (A-C) d^2\right) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right) \sqrt{\frac{b c+b d \tan (e+f x)}{b c-a d}} \sqrt{b}}{4 \sqrt{d} \sqrt{c+d \tan (e+f x)}}}{b^2 f}}{2 b}}{3 d}","\frac{\left(a^3 C d^3-a^2 b d^2 (2 B d+3 c C)+a b^2 d \left(8 d^2 (A-C)+12 B c d+3 c^2 C\right)-\left(b^3 \left(-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\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)}{8 b^{5/2} d^{3/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right)}{8 b^2 d f}-\frac{\sqrt{a-i b} (c-i d)^{3/2} (i A+B-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)}{f}-\frac{\sqrt{a+i b} (c+i d)^{3/2} (B-i (A-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)}{f}-\frac{(-a C d-6 b B d+b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}",1,"(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*d*f) + (((-(b*c*C) + 6*b*B*d + a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(4*b*f) + ((3*(8*b*(A*b + a*B - b*C)*d^2 - (b*c - a*d)*(b*c*C - 6*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*f) + ((6*b^2*d*(b*(2*a*A*c*d - 2*a*c*C*d + A*b*(c^2 - d^2) + a*B*(c^2 - d^2) - b*(c^2*C + 2*B*c*d - C*d^2)) - Sqrt[-b^2]*(a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d + B*(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]) - (6*b^2*d*(b*(2*a*A*c*d - 2*a*c*C*d + A*b*(c^2 - d^2) + a*B*(c^2 - d^2) - b*(c^2*C + 2*B*c*d - C*d^2)) + Sqrt[-b^2]*(a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d + B*(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]) + (3*Sqrt[b]*Sqrt[c - (a*d)/b]*(a^3*C*d^3 - a^2*b*d^2*(3*c*C + 2*B*d) + a*b^2*d*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - b^3*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2 + 16*B*d^3))*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[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*b))/(3*d)","A",1
137,1,613,384,7.6906617,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{a+b \tan (e+f x)}} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[a + b*Tan[e + f*x]],x]","\frac{\frac{\frac{\sqrt{b} \sqrt{c-\frac{a d}{b}} \left(3 a^2 C d^2-2 a b d (2 B d+3 c C)+b^2 \left(8 d^2 (A-C)+12 B c d+3 c^2 C\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)}{2 \sqrt{d} \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2 \left(\sqrt{-b^2} \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)-b \left(2 c d (A-C)+B \left(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{2 b^2 \left(\sqrt{-b^2} \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-C d^2\right)+b \left(2 c d (A-C)+B \left(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{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}}}{b^2 f}+\frac{(-3 a C d+4 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{2 b f}}{2 b}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}","\frac{\left(3 a^2 C d^2-2 a b d (2 B d+3 c C)+b^2 \left(8 d^2 (A-C)+12 B c d+3 c^2 C\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^{5/2} \sqrt{d} f}-\frac{(c-i d)^{3/2} (i A+B-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)}{f \sqrt{a-i b}}+\frac{(c+i d)^{3/2} (i A-B-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)}{f \sqrt{a+i b}}+\frac{(-3 a C d+4 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^2 f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}",1,"(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*b*f) + (((3*b*c*C + 4*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(2*b*f) + ((-2*b^2*(Sqrt[-b^2]*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b*(2*c*(A - C)*d + B*(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]) - (2*b^2*(Sqrt[-b^2]*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d + B*(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]) + (Sqrt[b]*Sqrt[c - (a*d)/b]*(3*a^2*C*d^2 - 2*a*b*d*(3*c*C + 2*B*d) + b^2*(3*c^2*C + 12*B*c*d + 8*(A - 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)])/(2*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*b)","A",1
138,1,1073629,382,39.7092302,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{3/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{3/2}}{b f \left(a^2+b^2\right) \sqrt{a+b \tan (e+f x)}}+\frac{d \left(3 a^2 C-2 a b B+2 A b^2+b^2 C\right) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 f \left(a^2+b^2\right)}-\frac{(c-i d)^{3/2} (i A+B-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)}{f (a-i b)^{3/2}}-\frac{(c+i d)^{3/2} (B-i (A-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)}{f (a+i b)^{3/2}}+\frac{\sqrt{d} (-3 a C d+2 b B d+3 b c C) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{b^{5/2} f}",1,"Result too large to show","C",0
139,1,1347065,402,41.0503341,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{5/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{3/2}}{3 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^{3/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right)}{b^2 f \left(a^2+b^2\right)^2 \sqrt{a+b \tan (e+f x)}}-\frac{(c-i d)^{3/2} (i A+B-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)}{f (a-i b)^{5/2}}-\frac{(c+i d)^{3/2} (B-i (A-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)}{f (a+i b)^{5/2}}+\frac{2 C 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)}{b^{5/2} f}",1,"Result too large to show","C",0
140,1,3134,586,9.0622272,"\int \frac{(c+d \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{7/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{3/2}}{5 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^{5/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left(3 a^4 C d+2 a^3 b B d-a^2 b^2 (7 A d+5 B c-13 C d)+2 a b^3 (5 A c-4 B d-5 c C)+b^4 (3 A d+5 B c)\right)}{15 b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))^{3/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left(3 a^6 C d^2+2 a^5 b B d^2+a^4 b^2 d (d (8 A+C)+10 B c)-a^3 b^3 \left(50 c d (A-C)+B \left(15 c^2-39 d^2\right)\right)+a^2 b^4 \left(45 A c^2-49 A d^2-90 B c d-45 c^2 C+58 C d^2\right)+a b^5 \left(70 c d (A-C)+B \left(45 c^2-23 d^2\right)\right)+b^6 \left(5 c (4 B d+3 c C)-3 A \left(5 c^2-d^2\right)\right)\right)}{15 b^2 f \left(a^2+b^2\right)^3 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{(c-i d)^{3/2} (i A+B-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)}{f (a-i b)^{7/2}}-\frac{(c+i d)^{3/2} (B-i (A-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)}{f (a+i b)^{7/2}}",1,"-((C*(c + d*Tan[e + f*x])^(3/2))/(b*f*(a + b*Tan[e + f*x])^(5/2))) - (-1/4*((3*b*c*C - 2*b*B*d - 3*a*C*d)*Sqrt[c + d*Tan[e + f*x]])/(b*f*(a + b*Tan[e + f*x])^(5/2)) - ((-2*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*((-2*(b^2*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))) - a*((5*b*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))/2 - 2*a*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*((-15*b^2*(b*c - a*d)^2*(((3*a^2*A*b*c^2 - A*b^3*c^2 - a^3*B*c^2 + 3*a*b^2*B*c^2 - 3*a^2*b*c^2*C + b^3*c^2*C - 2*a^3*A*c*d + 6*a*A*b^2*c*d - 6*a^2*b*B*c*d + 2*b^3*B*c*d + 2*a^3*c*C*d - 6*a*b^2*c*C*d - 3*a^2*A*b*d^2 + A*b^3*d^2 + a^3*B*d^2 - 3*a*b^2*B*d^2 + 3*a^2*b*C*d^2 - b^3*C*d^2 + I*(-(a^3*A*c^2) + 3*a*A*b^2*c^2 - 3*a^2*b*B*c^2 + b^3*B*c^2 + a^3*c^2*C - 3*a*b^2*c^2*C - 6*a^2*A*b*c*d + 2*A*b^3*c*d + 2*a^3*B*c*d - 6*a*b^2*B*c*d + 6*a^2*b*c*C*d - 2*b^3*c*C*d + a^3*A*d^2 - 3*a*A*b^2*d^2 + 3*a^2*b*B*d^2 - b^3*B*d^2 - a^3*C*d^2 + 3*a*b^2*C*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]) + ((-3*a^2*A*b*c^2 + A*b^3*c^2 + a^3*B*c^2 - 3*a*b^2*B*c^2 + 3*a^2*b*c^2*C - b^3*c^2*C + 2*a^3*A*c*d - 6*a*A*b^2*c*d + 6*a^2*b*B*c*d - 2*b^3*B*c*d - 2*a^3*c*C*d + 6*a*b^2*c*C*d + 3*a^2*A*b*d^2 - A*b^3*d^2 - a^3*B*d^2 + 3*a*b^2*B*d^2 - 3*a^2*b*C*d^2 + b^3*C*d^2 + I*(-(a^3*A*c^2) + 3*a*A*b^2*c^2 - 3*a^2*b*B*c^2 + b^3*B*c^2 + a^3*c^2*C - 3*a*b^2*c^2*C - 6*a^2*A*b*c*d + 2*A*b^3*c*d + 2*a^3*B*c*d - 6*a*b^2*B*c*d + 6*a^2*b*c*C*d - 2*b^3*c*C*d + a^3*A*d^2 - 3*a*A*b^2*d^2 + 3*a^2*b*B*d^2 - b^3*B*d^2 - a^3*C*d^2 + 3*a*b^2*C*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*(a^2 + b^2)*f) - (2*(b^2*((b^2*d - (3*a*(b*c - a*d))/2)*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))) + ((-3*b*c)/2 + (a*d)/2)*((5*b*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))/2 - 2*a*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))))) - a*((3*b*(b*c - a*d)*((-5*a*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))/2 - 2*b*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))) + b*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))))/2 - a*d*(b^2*(((2*b^2*d - (5*a*(b*c - a*d))/2)*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 + ((-5*b*c)/2 + (a*d)/2)*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))) - a*((5*b*(b*c - a*d)*((b*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - (b*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d)))/4 - 2*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))/2 - 2*a*d*((b^2*(8*A*b^2*c^2 + 3*a^2*C*d^2 - 2*a*b*d*(3*c*C - B*d) - 5*b^2*c*(c*C + 2*B*d)))/4 - a*(-1/4*(a*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(3*b*c*C - 2*b*B*d - 3*a*C*d))) + 2*b^3*(2*c*(A - C)*d + B*(c^2 - d^2))))))))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])))/(3*(a^2 + b^2)*(b*c - a*d))))/(5*(a^2 + b^2)*(b*c - a*d)))/(2*b))/b","B",1
141,1,1261,697,9.7305399,"\int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\frac{(-b c C+a d C+8 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{6 b f}+\frac{\frac{\left(48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 b f}+\frac{\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(24 b^2 d^2 (A b c+a B c-b C c+a A d-b B d-a C d)-\frac{3}{8} (a d-b c) \left(48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right)\right)}{b f}+\frac{-\frac{24 d \left(\sqrt{-b^2} \left(b (A-C) d \left(3 c^2-d^2\right)+b B \left(c^3-3 c d^2\right)-a \left(A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right)\right)-b \left(A \left(b c^3+3 a d c^2-3 b d^2 c-a d^3\right)-b \left(C c^3+3 B d c^2-3 C d^2 c-B d^3\right)+a \left(B c^3-3 C d c^2-3 B d^2 c+C d^3\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) b^3}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{24 d \left(\sqrt{-b^2} \left(b (A-C) d \left(3 c^2-d^2\right)+b B \left(c^3-3 c d^2\right)-a \left(A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right)\right)+b \left(A \left(b c^3+3 a d c^2-3 b d^2 c-a d^3\right)-b \left(C c^3+3 B d c^2-3 C d^2 c-B d^3\right)+a \left(B c^3-3 C d c^2-3 B d^2 c+C d^3\right)\right)\right) \tanh ^{-1}\left(\frac{\sqrt{c+\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right) b^3}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+\frac{\sqrt{-b^2} d}{b}}}-\frac{3 \sqrt{c-\frac{a d}{b}} \sqrt{\frac{1}{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}} \left(\left(5 C c^4-40 B d c^3-240 (A-C) d^2 c^2+320 B d^3 c+128 (A-C) d^4\right) b^4-4 a d \left(5 C c^3+30 B d c^2+40 (A-C) d^2 c-16 B d^3\right) b^3+2 a^2 d^2 \left(15 C c^2+20 B d c+8 (A-C) d^2\right) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4\right) \sinh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left(c-\frac{a d}{b}\right)}}}\right) \sqrt{\frac{c+d \tan (e+f x)}{c-\frac{a d}{b}}} \sqrt{b}}{8 \sqrt{d} \sqrt{c+d \tan (e+f x)}}}{b^2 f}}{2 b}}{3 b}}{4 d}","-\frac{\left(5 a^4 C d^4-4 a^3 b d^3 (2 B d+5 c C)+2 a^2 b^2 d^2 \left(8 d^2 (A-C)+20 B c d+15 c^2 C\right)-4 a b^3 d \left(40 c d^2 (A-C)+30 B c^2 d-16 B d^3+5 c^3 C\right)+b^4 \left(-240 c^2 d^2 (A-C)+128 d^4 (A-C)-40 B c^3 d+320 B c d^3+5 c^4 C\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)}{64 b^{7/2} d^{3/2} f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left(48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right)}{96 b^2 d f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left(48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right)\right)}{64 b^3 d f}-\frac{\sqrt{a-i b} (c-i d)^{5/2} (i A+B-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)}{f}+\frac{\sqrt{a+i b} (c+i d)^{5/2} (i A-B-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)}{f}-\frac{(-a C d-8 b B d+b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}",1,"(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f) + (((-(b*c*C) + 8*b*B*d + a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(6*b*f) + (((48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*b*f) + (((24*b^2*d^2*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) - (3*(-(b*c) + a*d)*(48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d)))/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((-24*b^3*d*(Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - a*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3)) - b*(A*(b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*d^2 + C*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]) - (24*b^3*d*(Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - a*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3)) + b*(A*(b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*d^2 + C*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]*Sqrt[c - (a*d)/b]*Sqrt[(c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b)))^(-1)]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b))]*(5*a^4*C*d^4 - 4*a^3*b*d^3*(5*c*C + 2*B*d) + 2*a^2*b^2*d^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - 4*a*b^3*d*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 40*B*c^3*d - 240*c^2*(A - C)*d^2 + 320*B*c*d^3 + 128*(A - C)*d^4))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b))])]*Sqrt[(c + d*Tan[e + f*x])/(c - (a*d)/b)])/(8*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*b))/(3*b))/(4*d)","A",0
142,1,780,505,8.9682414,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{a+b \tan (e+f x)}} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[a + b*Tan[e + f*x]],x]","\frac{\frac{\frac{-\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \left(5 a^3 C d^3-3 a^2 b d^2 (2 B d+5 c C)+a b^2 d \left(8 d^2 (A-C)+20 B c d+15 c^2 C\right)-\left(b^3 \left(40 c d^2 (A-C)+30 B c^2 d-16 B d^3+5 c^3 C\right)\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{d} \sqrt{c+d \tan (e+f x)}}+\frac{6 b^3 \left(\sqrt{-b^2} \left(A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right)+b d (A-C) \left(3 c^2-d^2\right)+b B \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{6 b^3 \left(-\sqrt{-b^2} \left(A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right)+b d (A-C) \left(3 c^2-d^2\right)+b B \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}}}{b^2 f}+\frac{3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left((b c-a d) (-5 a C d+6 b B d+5 b c C)+8 b^2 d (d (A-C)+B c)\right)}{4 b f}}{2 b}+\frac{(-5 a C d+6 b B d+5 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 b f}}{3 b}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}","-\frac{\left(5 a^3 C d^3-3 a^2 b d^2 (2 B d+5 c C)+a b^2 d \left(8 d^2 (A-C)+20 B c d+15 c^2 C\right)-\left(b^3 \left(40 c d^2 (A-C)+30 B c^2 d-16 B d^3+5 c^3 C\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)}{8 b^{7/2} \sqrt{d} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left((b c-a d) (-5 a C d+6 b B d+5 b c C)+8 b^2 d (d (A-C)+B c)\right)}{8 b^3 f}-\frac{(c-i d)^{5/2} (i A+B-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)}{f \sqrt{a-i b}}-\frac{(c+i d)^{5/2} (B-i (A-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)}{f \sqrt{a+i b}}+\frac{(-5 a C d+6 b B d+5 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b^2 f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}",1,"(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*b*f) + (((5*b*c*C + 6*b*B*d - 5*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(4*b*f) + ((3*(8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(5*b*c*C + 6*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*f) + ((6*b^3*(b*(A - C)*d*(3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) + Sqrt[-b^2]*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*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^3*(b*(A - C)*d*(3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - Sqrt[-b^2]*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*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]*Sqrt[c - (a*d)/b]*(5*a^3*C*d^3 - 3*a^2*b*d^2*(5*c*C + 2*B*d) + a*b^2*d*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - b^3*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2 - 16*B*d^3))*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[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*b))/(3*b)","A",1
143,1,1654245,535,44.3840626,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{3/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{5/2}}{b f \left(a^2+b^2\right) \sqrt{a+b \tan (e+f x)}}+\frac{d \left(5 a^2 C-4 a b B+4 A b^2+b^2 C\right) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 f \left(a^2+b^2\right)}+\frac{\sqrt{d} \left(15 a^2 C d^2-6 a b d (2 B d+5 c C)+b^2 \left(8 d^2 (A-C)+20 B c d+15 c^2 C\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^{7/2} f}-\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right)}{4 b^3 f \left(a^2+b^2\right)}-\frac{(c-i d)^{5/2} (i A+B-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)}{f (a-i b)^{3/2}}-\frac{(c+i d)^{5/2} (B-i (A-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)}{f (a+i b)^{3/2}}",1,"Result too large to show","C",0
144,1,2018669,545,47.1024438,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{5/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{5/2}}{3 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^{3/2}}+\frac{2 (c+d \tan (e+f x))^{3/2} \left(-5 a^4 C d+2 a^3 b B d+a^2 b^2 (d (A-11 C)+3 B c)-2 a b^3 (3 A c-4 B d-3 c C)-b^4 (5 A d+3 B c)\right)}{3 b^2 f \left(a^2+b^2\right)^2 \sqrt{a+b \tan (e+f x)}}-\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(-5 a^4 C d+2 a^3 b B d+2 a^2 b^2 (B c-5 C d)-2 a b^3 (2 A c-3 B d-2 c C)-b^4 (d (4 A+C)+2 B c)\right)}{b^3 f \left(a^2+b^2\right)^2}-\frac{(c-i d)^{5/2} (i A+B-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)}{f (a-i b)^{5/2}}-\frac{(c+i d)^{5/2} (B-i (A-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)}{f (a+i b)^{5/2}}+\frac{d^{3/2} (-5 a C d+2 b B d+5 b c C) \tanh ^{-1}\left(\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right)}{b^{7/2} f}",1,"Result too large to show","C",0
145,1,2345519,590,49.1310784,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{7/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{5/2}}{5 b f \left(a^2+b^2\right) (a+b \tan (e+f x))^{5/2}}-\frac{2 (c+d \tan (e+f x))^{3/2} \left(a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right)}{3 b^2 f \left(a^2+b^2\right)^2 (a+b \tan (e+f x))^{3/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left(a^6 C d^2+3 a^4 b^2 C d^2-a^3 b^3 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-3 a^2 b^4 \left(-A \left(c^2-d^2\right)+2 B c d+c^2 C-2 C d^2\right)+3 a b^5 \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)+b^6 \left(c (2 B d+c C)-A \left(c^2-d^2\right)\right)\right)}{b^3 f \left(a^2+b^2\right)^3 \sqrt{a+b \tan (e+f x)}}-\frac{(c-i d)^{5/2} (i A+B-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)}{f (a-i b)^{7/2}}-\frac{(c+i d)^{5/2} (B-i (A-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)}{f (a+i b)^{7/2}}+\frac{2 C 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^{7/2} f}",1,"Result too large to show","C",0
146,1,2719441,946,53.637837,"\int \frac{(c+d \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(a+b \tan (e+f x))^{9/2}} \, dx","Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(9/2),x]","\text{Result too large to show}","-\frac{(i A+B-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)^{5/2}}{(a-i b)^{9/2} f}-\frac{2 \left(A b^2-a (b B-a C)\right) (c+d \tan (e+f x))^{5/2}}{7 b \left(a^2+b^2\right) f (a+b \tan (e+f x))^{7/2}}-\frac{2 \left(5 C d a^4+2 b B d a^3-b^2 (7 B c+9 A d-19 C d) a^2+2 b^3 (7 A c-7 C c-6 B d) a+b^4 (7 B c+5 A d)\right) (c+d \tan (e+f x))^{3/2}}{35 b^2 \left(a^2+b^2\right)^2 f (a+b \tan (e+f x))^{5/2}}-\frac{(B-i (A-C)) (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)}{(a+i b)^{9/2} f}-\frac{2 \left(15 C d^3 a^8+6 b B d^3 a^7+2 b^2 d^2 (7 B c+4 A d+26 C d) a^6+2 b^3 d \left(56 c (A-C) d+B \left(35 c^2-12 d^2\right)\right) a^5-b^4 \left(105 B c^3+525 A d c^2-525 C d c^2-749 B d^2 c-311 A d^3+221 C d^3\right) a^4-2 b^5 \left(210 C c^3+700 B d c^2-798 C d^2 c-317 B d^3-42 A \left(5 c^3-19 c d^2\right)\right) a^3+2 b^6 \left(315 B c^3+875 A d c^2-875 C d c^2-812 B d^2 c-261 A d^3+291 C d^3\right) a^2-2 b^7 \left(210 A c^3-210 C c^3-525 B d c^2-406 A d^2 c+406 C d^2 c+88 B d^3\right) a-b^8 \left(5 d \left(49 A c^2-49 C c^2-3 A d^2\right)+7 B \left(15 c^3-23 c d^2\right)\right)\right) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left(a^2+b^2\right)^4 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left(15 C d^2 a^6+6 b B d^2 a^5+b^2 d (14 B c+8 A d+37 C d) a^4-b^3 \left(98 c (A-C) d+B \left(35 c^2-75 d^2\right)\right) a^3+3 b^4 \left(35 A c^2-35 C c^2-70 B d c-39 A d^2+54 C d^2\right) a^2+b^5 \left(182 c (A-C) d+B \left(105 c^2-71 d^2\right)\right) a+b^6 \left(7 c (5 c C+8 B d)-5 A \left(7 c^2-3 d^2\right)\right)\right) \sqrt{c+d \tan (e+f x)}}{105 b^3 \left(a^2+b^2\right)^3 f (a+b \tan (e+f x))^{3/2}}",1,"Result too large to show","C",0
147,1,785,505,8.4984089,"\int \frac{(a+b \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{\frac{\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \left(5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left(8 d^2 (A-C)-4 B c d+3 c^2 C\right)-\left(b^3 \left(8 c d^2 (A-C)-6 B c^2 d+16 B d^3+5 c^3 C\right)\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{d} \sqrt{c+d \tan (e+f x)}}-\frac{6 d^3 \left(\sqrt{-b^2} \left(-\left(a^3 (A-C)\right)+3 a^2 b B+3 a b^2 (A-C)-b^3 B\right)-b \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\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 d^3 \left(\sqrt{-b^2} \left(-\left(a^3 (A-C)\right)+3 a^2 b B+3 a b^2 (A-C)-b^3 B\right)+b \left(a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\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}}}{b d f}+\frac{3 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(8 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-6 b B d+5 b c C)\right)}{4 d f}}{2 d}+\frac{(5 a C d+6 b B d-5 b c C) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{4 d f}}{3 d}+\frac{C (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 d f}","\frac{\left(5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left(8 d^2 (A-C)-4 B c d+3 c^2 C\right)-\left(b^3 \left(8 c d^2 (A-C)-6 B c^2 d+16 B d^3+5 c^3 C\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)}{8 \sqrt{b} d^{7/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(8 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-6 b B d+5 b c C)\right)}{8 d^3 f}-\frac{(a-i b)^{5/2} (i A+B-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)}{f \sqrt{c-i d}}-\frac{(a+i b)^{5/2} (B-i (A-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)}{f \sqrt{c+i d}}-\frac{(-5 a C d-6 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{12 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}}{3 d f}",1,"(C*(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]])/(3*d*f) + (((-5*b*c*C + 6*b*B*d + 5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(4*d*f) + ((3*(8*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 6*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d*f) + ((-6*(Sqrt[-b^2]*(3*a^2*b*B - b^3*B - a^3*(A - C) + 3*a*b^2*(A - C)) - b*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C)))*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*(Sqrt[-b^2]*(3*a^2*b*B - b^3*B - a^3*(A - C) + 3*a*b^2*(A - C)) + b*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C)))*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]*Sqrt[c - (a*d)/b]*(5*a^3*C*d^3 - 15*a^2*b*d^2*(c*C - 2*B*d) + 5*a*b^2*d*(3*c^2*C - 4*B*c*d + 8*(A - C)*d^2) - b^3*(5*c^3*C - 6*B*c^2*d + 8*c*(A - C)*d^2 + 16*B*d^3))*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[d]*Sqrt[c + d*Tan[e + f*x]]))/(b*d*f))/(2*d))/(3*d)","A",1
148,1,582,383,7.0771757,"\int \frac{(a+b \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]","\frac{\frac{\sqrt{c-\frac{a d}{b}} \left(3 a^2 C d^2+6 a b d (2 B d-c C)+b^2 \left(8 d^2 (A-C)-4 B c d+3 c^2 C\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{d} \sqrt{c+d \tan (e+f x)}}+\frac{4 d^2 \left(b \left(a^2 B+2 a b (A-C)-b^2 B\right)-\sqrt{-b^2} \left(a^2 (C-A)+2 a b B+b^2 (A-C)\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)}{b \sqrt{\sqrt{-b^2}-a} \sqrt{\frac{\sqrt{-b^2} d}{b}-c}}-\frac{4 d^2 \left(b \left(a^2 B+2 a b (A-C)-b^2 B\right)+\sqrt{-b^2} \left(a^2 (C-A)+2 a b B+b^2 (A-C)\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)}{b \sqrt{a+\sqrt{-b^2}} \sqrt{\frac{\sqrt{-b^2} d}{b}+c}}+(3 a C d+4 b B d-3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}+2 C d (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}","\frac{\left(3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left(8 d^2 (A-C)-4 B c d+3 c^2 C\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} d^{5/2} f}-\frac{(a-i b)^{3/2} (i A+B-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)}{f \sqrt{c-i d}}+\frac{(a+i b)^{3/2} (i A-B-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)}{f \sqrt{c+i d}}-\frac{(-3 a C d-4 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}",1,"((4*(b*(a^2*B - b^2*B + 2*a*b*(A - C)) - Sqrt[-b^2]*(2*a*b*B + b^2*(A - C) + a^2*(-A + 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]])])/(b*Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (4*(b*(a^2*B - b^2*B + 2*a*b*(A - C)) + Sqrt[-b^2]*(2*a*b*B + b^2*(A - C) + a^2*(-A + 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]])])/(b*Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + (-3*b*c*C + 4*b*B*d + 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + 2*C*d*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]] + (Sqrt[c - (a*d)/b]*(3*a^2*C*d^2 + 6*a*b*d*(-(c*C) + 2*B*d) + b^2*(3*c^2*C - 4*B*c*d + 8*(A - C)*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[d]*Sqrt[c + d*Tan[e + f*x]]))/(4*d^2*f)","A",1
149,1,450,290,6.6307209,"\int \frac{\sqrt{a+b \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{\sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(Sqrt[a + b*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]],x]","\frac{C \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{d f}-\frac{\frac{d \left(\sqrt{-b^2} (a (C-A)+b B)-b (a B+A b-b C)\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(\sqrt{-b^2} (a (C-A)+b B)+b (a B+A b-b C)\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} \sqrt{c-\frac{a d}{b}} (-a C d-2 b B d+b c C) \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 d f}","-\frac{\sqrt{a-i b} (i A+B-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)}{f \sqrt{c-i d}}+\frac{\sqrt{a+i b} (i A-B-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)}{f \sqrt{c+i d}}-\frac{(-a C d-2 b B d+b c 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} d^{3/2} f}+\frac{C \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{d f}",1,"(C*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d*f) - (((-(b*(A*b + a*B - b*C)) + Sqrt[-b^2]*(b*B + a*(-A + C)))*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*(A*b + a*B - b*C) + Sqrt[-b^2]*(b*B + a*(-A + C)))*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]) + (Sqrt[b]*Sqrt[c - (a*d)/b]*(b*c*C - 2*b*B*d - a*C*d)*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
150,1,362,239,2.300882,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\frac{\left(\sqrt{-b^2} (A-C)+b B\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{\left(\sqrt{-b^2} (C-A)+b B\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{2 \sqrt{b} C \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 f}","-\frac{(B+i (A-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)}{f \sqrt{a-i b} \sqrt{c-i d}}+\frac{(i A-B-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)}{f \sqrt{a+i b} \sqrt{c+i d}}+\frac{2 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} \sqrt{d} f}",1,"(((b*B + Sqrt[-b^2]*(A - C))*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*B + Sqrt[-b^2]*(-A + C))*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]) + (2*Sqrt[b]*C*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*f)","A",1
151,1,264,251,2.5820081,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\frac{2 \left(a (a C-b B)+A b^2\right) \sqrt{c+d \tan (e+f x)}}{(a d-b c) \sqrt{a+b \tan (e+f x)}}+\frac{(a+i b) (i A+B-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{(b+i a) (A+i B-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}}}{f \left(a^2+b^2\right)}","-\frac{2 \left(A b^2-a (b B-a C)\right) \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 A+B-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)}{f (a-i b)^{3/2} \sqrt{c-i d}}-\frac{(B-i (A-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)}{f (a+i b)^{3/2} \sqrt{c+i d}}",1,"(((a + I*b)*(I*A + B - I*C)*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)*(A + I*B - C)*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*(A*b^2 + a*(-(b*B) + a*C))*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
152,1,388,375,6.2790739,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]),x]","\frac{\frac{2 \left(a^2+b^2\right) \left(a (a C-b B)+A b^2\right) \sqrt{c+d \tan (e+f x)}}{(a d-b c) (a+b \tan (e+f x))^{3/2}}+\frac{2 \sqrt{c+d \tan (e+f x)} \left(2 a^4 C d-5 a^3 b B d+a^2 b^2 (8 A d+3 B c-4 C d)+a b^3 (-6 A c+B d+6 c C)+b^4 (2 A d-3 B c)\right)}{(b c-a d)^2 \sqrt{a+b \tan (e+f x)}}+\frac{3 i (a-i b)^2 (A+i B-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{3 (a+i b)^2 (i A+B-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}}}{3 f \left(a^2+b^2\right)^2}","-\frac{2 \left(A b^2-a (b B-a C)\right) \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{2 \sqrt{c+d \tan (e+f x)} \left(-2 a^4 C d+5 a^3 b B d-a^2 b^2 (8 A d+3 B c-4 C d)+a b^3 (6 A c-B d-6 c C)+b^4 (3 B c-2 A d)\right)}{3 f \left(a^2+b^2\right)^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}-\frac{(i A+B-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)}{f (a-i b)^{5/2} \sqrt{c-i d}}-\frac{(B-i (A-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)}{f (a+i b)^{5/2} \sqrt{c+i d}}",1,"((3*(a + I*b)^2*(I*A + B - I*C)*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]) + ((3*I)*(a - I*b)^2*(A + I*B - C)*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*(a^2 + b^2)*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*Tan[e + f*x]])/((-(b*c) + a*d)*(a + b*Tan[e + f*x])^(3/2)) + (2*(-5*a^3*b*B*d + 2*a^4*C*d + b^4*(-3*B*c + 2*A*d) + a*b^3*(-6*A*c + 6*c*C + B*d) + a^2*b^2*(3*B*c + 8*A*d - 4*C*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
153,1,1653959,528,44.5309755,"\int \frac{(a+b \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{\sqrt{b} \left(15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left(8 d^2 (A-C)-12 B c d+15 c^2 C\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^{7/2} f}+\frac{b \left(d^2 (4 A+C)-4 B c d+5 c^2 C\right) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 f \left(c^2+d^2\right)}-\frac{2 \left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^{5/2}}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{b \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(3 (b c-a d) \left(d^2 (4 A+C)-4 B c d+5 c^2 C\right)-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right)}{4 d^3 f \left(c^2+d^2\right)}-\frac{(a-i b)^{5/2} (i A+B-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)}{f (c-i d)^{3/2}}-\frac{(a+i b)^{5/2} (B-i (A-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)}{f (c+i d)^{3/2}}",1,"Result too large to show","C",0
154,1,1073499,380,39.8561933,"\int \frac{(a+b \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{b \left(d^2 (2 A+C)-2 B c d+3 c^2 C\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 \left(A d^2-B c d+c^2 C\right) (a+b \tan (e+f x))^{3/2}}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b)^{3/2} (i A+B-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)}{f (c-i d)^{3/2}}-\frac{(a+i b)^{3/2} (B-i (A-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)}{f (c+i d)^{3/2}}-\frac{\sqrt{b} (-3 a C d-2 b B d+3 b c C) \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}",1,"Result too large to show","C",0
155,1,621084,299,35.5870582,"\int \frac{\sqrt{a+b \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(Sqrt[a + b*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(A d^2-B c d+c^2 C\right) \sqrt{a+b \tan (e+f x)}}{d f \left(c^2+d^2\right) \sqrt{c+d \tan (e+f x)}}-\frac{\sqrt{a-i b} (i A+B-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)}{f (c-i d)^{3/2}}-\frac{\sqrt{a+i b} (B-i (A-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)}{f (c+i d)^{3/2}}+\frac{2 \sqrt{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)}{d^{3/2} f}",1,"Result too large to show","C",0
156,1,275,251,3.2090348,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{\frac{2 \left(A d^2-B c d+c^2 C\right) \sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}+(b c-a d) \left(\frac{(c+i d) (i A+B-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{(d+i c) (A+i B-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 \left(A d^2-B c d+c^2 C\right) \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{(B+i (A-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)}{f \sqrt{a-i b} (c-i d)^{3/2}}+\frac{(i A-B-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)}{f \sqrt{a+i b} (c+i d)^{3/2}}",1,"-(((b*c - a*d)*(((I*A + B - I*C)*(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]) + ((A + I*B - C)*(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*(c^2*C - B*c*d + A*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
157,1,484,383,6.7203789,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{2 \left(A b^2-a (b B-a C)\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 \sqrt{a+b \tan (e+f x)} \left(\frac{1}{2} d^2 \left(-a A (b c-a d)-(b B-a C) (a d+b c)+2 A b^2 d\right)-c \left(\frac{1}{2} d (b c-a d) (-a B+A b-b C)-c d \left(A b^2-a (b B-a C)\right)\right)\right)}{f \left(c^2+d^2\right) (a d-b c) \sqrt{c+d \tan (e+f x)}}+\frac{(b c-a d)^2 \left(\frac{(a+i b) (c+i d) (i A+B-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{(b+i a) (c-i d) (A+i B-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)}{2 f \left(c^2+d^2\right) (a d-b c)}\right)}{\left(a^2+b^2\right) (b c-a d)}","-\frac{2 d \sqrt{a+b \tan (e+f x)} \left(A \left(a^2 d^2+b^2 \left(c^2+2 d^2\right)\right)+a^2 \left(-B c d+2 c^2 C+C d^2\right)-a b B \left(c^2+d^2\right)+b^2 c (c C-B d)\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{2 \left(A b^2-a (b B-a C)\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{(i A+B-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)}{f (a-i b)^{3/2} (c-i d)^{3/2}}-\frac{(B-i (A-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)}{f (a+i b)^{3/2} (c+i d)^{3/2}}",1,"(-2*(A*b^2 - a*(b*B - a*C)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - (2*(((b*c - a*d)^2*(((a + I*b)*(I*A + B - I*C)*(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)*(A + I*B - C)*(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*(-(b*c) + a*d)*(c^2 + d^2)*f) - (2*(-(c*(-(c*(A*b^2 - a*(b*B - a*C))*d) + ((A*b - a*B - b*C)*d*(b*c - a*d))/2)) + (d^2*(2*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + a*d)))/2)*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))","A",1
158,1,902,598,6.9038427,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)),x]","-\frac{2 \left(A b^2-a (b B-a C)\right)}{3 \left(a^2+b^2\right) (b c-a d) f (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 A d b^2-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right)-a \left(\frac{3}{2} b (A b-C b-a B) (b c-a d)-2 a \left(A b^2-a (b B-a C)\right) d\right)\right)}{\left(a^2+b^2\right) (b c-a d) f \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{2 \left(-\frac{3 \left(\frac{(A+i B-C) (i c+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)^2}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(a+i b)^2 (i A+B-i C) (c+i d) \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)^3}{4 (a d-b c) \left(c^2+d^2\right) f}-\frac{2 \left(d^2 \left(\left(-\frac{b c}{2}-\frac{a d}{2}\right) \left(\frac{3}{2} b (A b-C b-a B) (b c-a d)-2 a \left(A b^2-a (b B-a C)\right) d\right)+\frac{1}{2} \left(b^2 d-\frac{1}{2} a (b c-a d)\right) \left(4 A d b^2-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right)\right)-c \left(\frac{1}{2} d (b c-a d) \left(-2 b \left(A b^2-a (b B-a C)\right) d-\frac{3}{2} a (A b-C b-a B) (b c-a d)+\frac{1}{2} b \left(4 A d b^2-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right)\right)-c d \left(\frac{1}{2} b^2 \left(4 A d b^2-3 a A (b c-a d)-(b B-a C) (3 b c+a d)\right)-a \left(\frac{3}{2} b (A b-C b-a B) (b c-a d)-2 a \left(A b^2-a (b B-a C)\right) d\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)}{\left(a^2+b^2\right) (b c-a d)}\right)}{3 \left(a^2+b^2\right) (b c-a d)}","-\frac{2 \left(A b^2-a (b B-a C)\right)}{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 \sqrt{a+b \tan (e+f x)} \left(a^4 (-d) \left(d^2 (3 A+5 C)-3 B c d+8 c^2 C\right)+8 a^3 b B d \left(c^2+d^2\right)-a^2 b^2 \left(11 A c^2 d+17 A d^3+3 B c^3-3 B c d^2+5 c^2 C d-C d^3\right)+2 a b^3 \left(c^2+d^2\right) (3 A c+B d-3 c C)-b^4 \left(d \left(5 A c^2+8 A d^2+3 c^2 C\right)-3 B \left(c^3+2 c d^2\right)\right)\right)}{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{2 \left(-4 a^4 C d+7 a^3 b B d-a^2 b^2 (2 d (5 A-C)+3 B c)+a b^3 (6 A c+B d-6 c C)+b^4 (3 B c-4 A 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{(i A+B-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)}{f (a-i b)^{5/2} (c-i d)^{3/2}}-\frac{(B-i (A-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)}{f (a+i b)^{5/2} (c+i d)^{3/2}}",1,"(-2*(A*b^2 - a*(b*B - a*C)))/(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*(-(a*(-2*a*(A*b^2 - a*(b*B - a*C))*d + (3*b*(A*b - a*B - b*C)*(b*c - a*d))/2)) + (b^2*(4*A*b^2*d - 3*a*A*(b*c - a*d) - (b*B - a*C)*(3*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*A + B - I*C)*(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]) + ((a - I*b)^2*(A + I*B - C)*(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*(d^2*((-1/2*(b*c) - (a*d)/2)*(-2*a*(A*b^2 - a*(b*B - a*C))*d + (3*b*(A*b - a*B - b*C)*(b*c - a*d))/2) + ((b^2*d - (a*(b*c - a*d))/2)*(4*A*b^2*d - 3*a*A*(b*c - a*d) - (b*B - a*C)*(3*b*c + a*d)))/2) - c*((d*(b*c - a*d)*(-2*b*(A*b^2 - a*(b*B - a*C))*d - (3*a*(A*b - a*B - b*C)*(b*c - a*d))/2 + (b*(4*A*b^2*d - 3*a*A*(b*c - a*d) - (b*B - a*C)*(3*b*c + a*d)))/2))/2 - c*d*(-(a*(-2*a*(A*b^2 - a*(b*B - a*C))*d + (3*b*(A*b - a*B - b*C)*(b*c - a*d))/2)) + (b^2*(4*A*b^2*d - 3*a*A*(b*c - a*d) - (b*B - a*C)*(3*b*c + a*d)))/2)))*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
159,1,2018643,549,47.1701383,"\int \frac{(a+b \tan (e+f x))^{5/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \left(A d^2-B c d+c^2 C\right) (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{b \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left(2 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(d^4 (4 A+C)-2 B c^3 d-6 B c d^3+5 c^4 C+10 c^2 C d^2\right)\right)}{d^3 f \left(c^2+d^2\right)^2}-\frac{2 (a+b \tan (e+f x))^{3/2} \left(3 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (A-11 C)+5 A d^4-2 B c^3 d-8 B c d^3+5 c^4 C\right)\right)}{3 d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b)^{5/2} (i A+B-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)}{f (c-i d)^{5/2}}-\frac{(a+i b)^{5/2} (B-i (A-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)}{f (c+i d)^{5/2}}-\frac{b^{3/2} (-5 a C d-2 b B d+5 b c C) \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}",1,"Result too large to show","C",0
160,1,1347117,407,41.1207156,"\int \frac{(a+b \tan (e+f x))^{3/2} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \left(A d^2-B c d+c^2 C\right) (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{2 \sqrt{a+b \tan (e+f x)} \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right)\right)}{d^2 f \left(c^2+d^2\right)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b)^{3/2} (i A+B-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)}{f (c-i d)^{5/2}}-\frac{(a+i b)^{3/2} (B-i (A-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)}{f (c+i d)^{5/2}}+\frac{2 b^{3/2} C \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}",1,"Result too large to show","C",0
161,1,609,373,7.0168148,"\int \frac{\sqrt{a+b \tan (e+f x)} \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(Sqrt[a + b*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2),x]","-\frac{C \sqrt{a+b \tan (e+f x)}}{d f (c+d \tan (e+f x))^{3/2}}-\frac{-\frac{2 \sqrt{a+b \tan (e+f x)} \left(\frac{1}{2} d^2 (-a d (2 A-3 C)-b c C)-c \left(-\left(d^2 (a B+A b-b C)\right)-\frac{1}{2} c (a C d-2 b B d-b c C)\right)\right)}{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 \sqrt{a+b \tan (e+f x)} \left(-\frac{1}{2} d^2 (b c-a d) \left(3 a d (A c+B d-c C)+b \left(A d^2-B c d+c^2 C\right)\right)-c \left(\frac{1}{2} b c (b c-a d) \left(-d^2 (2 A-3 C)+2 B c d+c^2 C\right)-\frac{3}{2} d^2 (b c-a d) (-a A d+a B c+a C d+A b c+b B d-b c C)\right)\right)}{f \left(c^2+d^2\right) (a d-b c) \sqrt{c+d \tan (e+f x)}}-\frac{3 d (b c-a d)^2 \left(\frac{\sqrt{a+i b} (c-i d)^2 (B-i (A-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{c+i d}}+\frac{\sqrt{-a+i b} (c+i d)^2 (i A+B-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{-c+i d}}\right)}{2 f \left(c^2+d^2\right) (a d-b c)}\right)}{3 \left(c^2+d^2\right) (a d-b c)}}{d}","-\frac{2 \left(A d^2-B c d+c^2 C\right) \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{2 \sqrt{a+b \tan (e+f x)} \left(3 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)+b \left(-c^2 d^2 (5 A-7 C)+A d^4+2 B c^3 d-4 B c d^3+c^4 C\right)\right)}{3 d f \left(c^2+d^2\right)^2 (b c-a d) \sqrt{c+d \tan (e+f x)}}-\frac{\sqrt{a-i b} (i A+B-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)}{f (c-i d)^{5/2}}-\frac{\sqrt{a+i b} (B-i (A-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)}{f (c+i d)^{5/2}}",1,"-((C*Sqrt[a + b*Tan[e + f*x]])/(d*f*(c + d*Tan[e + f*x])^(3/2))) - ((-2*((d^2*(-(b*c*C) - a*(2*A - 3*C)*d))/2 - c*(-((A*b + a*B - b*C)*d^2) - (c*(-(b*c*C) - 2*b*B*d + a*C*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*d*(b*c - a*d)^2*((Sqrt[-a + I*b]*(I*A + B - I*C)*(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[-c + I*d] + (Sqrt[a + I*b]*(B - I*(A - C))*(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[c + I*d]))/(2*(-(b*c) + a*d)*(c^2 + d^2)*f) - (2*(-1/2*(d^2*(b*c - a*d)*(3*a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2))) - c*((-3*d^2*(b*c - a*d)*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d))/2 + (b*c*(b*c - a*d)*(c^2*C + 2*B*c*d - (2*A - 3*C)*d^2))/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)))/d","A",1
162,1,403,379,5.5912564,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)),x]","\frac{\frac{2 \left(c^2+d^2\right) (b c-a d) \left(A d^2-B c d+c^2 C\right) \sqrt{a+b \tan (e+f x)}}{(c+d \tan (e+f x))^{3/2}}+\frac{2 \sqrt{a+b \tan (e+f x)} \left(3 a d^2 \left(2 c d (C-A)+B \left(c^2-d^2\right)\right)+b \left(4 c^2 d^2 (2 A-C)+2 A d^4-5 B c^3 d+B c d^3+2 c^4 C\right)\right)}{\sqrt{c+d \tan (e+f x)}}+3 (b c-a d)^2 \left(\frac{i (c-i d)^2 (A+i B-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{(c+i d)^2 (i A+B-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)}{3 f \left(c^2+d^2\right)^2 (b c-a d)^2}","\frac{2 \left(A d^2-B c d+c^2 C\right) \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{2 \sqrt{a+b \tan (e+f x)} \left(b \left(4 c^2 d^2 (2 A-C)+2 A d^4-5 B c^3 d+B c d^3+2 c^4 C\right)-3 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\right)}{3 f \left(c^2+d^2\right)^2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{(B+i (A-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)}{f \sqrt{a-i b} (c-i d)^{5/2}}+\frac{(i A-B-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)}{f \sqrt{a+i b} (c+i d)^{5/2}}",1,"(3*(b*c - a*d)^2*(((I*A + B - I*C)*(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 + I*B - C)*(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*(b*c - a*d)*(c^2 + d^2)*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*Tan[e + f*x]])/(c + d*Tan[e + f*x])^(3/2) + (2*(b*(2*c^4*C - 5*B*c^3*d + 4*c^2*(2*A - C)*d^2 + B*c*d^3 + 2*A*d^4) + 3*a*d^2*(2*c*(-A + C)*d + B*(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
163,1,903,651,6.994101,"\int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}} \, dx","Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)),x]","-\frac{2 \left(A b^2-a (b B-a C)\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{1}{2} d^2 \left(4 A d b^2-a A (b c-a d)-(b B-a C) (b c+3 a d)\right)-c \left(\frac{1}{2} (A b-C b-a B) d (b c-a d)-2 c \left(A b^2-a (b B-a C)\right) d\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{3 (b c-a d)^3 \left(\frac{(i a+b) (A+i B-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)^2}{\sqrt{a+i b} \sqrt{c+i d}}+\frac{(a+i b) (i A+B-i C) (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)}{4 (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(\frac{1}{2} (A b-C b-a B) d (b c-a d)-2 c \left(A b^2-a (b B-a C)\right) d\right)+\frac{1}{2} \left(b d^2-\frac{3}{2} c (a d-b c)\right) \left(4 A d b^2-a A (b c-a d)-(b B-a C) (b c+3 a d)\right)\right)-c \left(\frac{3}{2} d (a d-b c) \left(-2 \left(A b^2-a (b B-a C)\right) d^2+\frac{1}{2} \left(4 A d b^2-a A (b c-a d)-(b B-a C) (b c+3 a d)\right) d-\frac{1}{2} c (A b-C b-a B) (b c-a d)\right)-b c \left(\frac{1}{2} d^2 \left(4 A d b^2-a A (b c-a d)-(b B-a C) (b c+3 a d)\right)-c \left(\frac{1}{2} (A b-C b-a B) d (b c-a d)-2 c \left(A b^2-a (b B-a C)\right) d\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)}","-\frac{2 d \sqrt{a+b \tan (e+f x)} \left(A \left(a^2 d^2+b^2 \left(3 c^2+4 d^2\right)\right)+a^2 \left(-B c d+4 c^2 C+3 C d^2\right)-3 a b B \left(c^2+d^2\right)+b^2 c (c C-B d)\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{2 \left(A b^2-a (b B-a C)\right)}{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 d \sqrt{a+b \tan (e+f x)} \left(3 a^3 d^2 \left(B \left(c^2-d^2\right)+2 c C d\right)+a^2 b \left(-8 B c^3 d-2 B c d^3+8 c^4 C+5 c^2 C d^2+3 C d^4\right)-A \left(6 a^3 c d^3-a^2 b d^2 \left(11 c^2+5 d^2\right)+6 a b^2 c d^3-\left(b^3 \left(3 c^4+17 c^2 d^2+8 d^4\right)\right)\right)+3 a b^2 \left(2 c C d^3-B \left(c^4+c^2 d^2+2 d^4\right)\right)+b^3 c \left(-8 B c^2 d-2 B d^3+5 c^3 C-c C d^2\right)\right)}{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 A+B-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)}{f (a-i b)^{3/2} (c-i d)^{5/2}}-\frac{(B-i (A-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)}{f (a+i b)^{3/2} (c+i d)^{5/2}}",1,"(-2*(A*b^2 - a*(b*B - a*C)))/((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*(-2*c*(A*b^2 - a*(b*B - a*C))*d + ((A*b - a*B - b*C)*d*(b*c - a*d))/2)) + (d^2*(4*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 3*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*(((a + I*b)*(I*A + B - I*C)*(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)*(A + I*B - C)*(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*(((b*c)/2 - (3*a*d)/2)*(-2*c*(A*b^2 - a*(b*B - a*C))*d + ((A*b - a*B - b*C)*d*(b*c - a*d))/2) + ((b*d^2 - (3*c*(-(b*c) + a*d))/2)*(4*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 3*a*d)))/2) - c*((3*d*(-(b*c) + a*d)*(-2*(A*b^2 - a*(b*B - a*C))*d^2 - (c*(A*b - a*B - b*C)*(b*c - a*d))/2 + (d*(4*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 3*a*d)))/2))/2 - b*c*(-(c*(-2*c*(A*b^2 - a*(b*B - a*C))*d + ((A*b - a*B - b*C)*d*(b*c - a*d))/2)) + (d^2*(4*A*b^2*d - a*A*(b*c - a*d) - (b*B - a*C)*(b*c + 3*a*d)))/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
164,0,0,376,25.4969988,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^n \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^n \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","-\frac{(B+i (A-C)) (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) (a-i b)}-\frac{(A+i B-C) (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{C (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} \, _2F_1\left(m+1,-n;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right)}{b f (m+1)}",1,"Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x]","F",-1
165,1,1390,560,6.4075288,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac{\frac{(3 b c C-3 a d C+b B d (m+4)) (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^{m+1}}{b f (m+3)}+\frac{\frac{d \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right) \tan (e+f x) (a+b \tan (e+f x))^{m+1}}{b f (m+2)}-\frac{\frac{\left(d \left(a \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)-b^3 \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) (m+2) (m+3) (m+4)\right)-b c (m+2) \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)\right) (a+b \tan (e+f x))^{m+1}}{b f (m+1)}+\frac{i \left(a d \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)+b c (m+2) \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)-b c (m+2) (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))-d \left(a \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)-b^3 \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) (m+2) (m+3) (m+4)\right)-i b (m+2) \left(c \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) (m+3) (m+4) b^2-d \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)+d (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{-i a-i b \tan (e+f x)}{b-i a}\right) (a+b \tan (e+f x))^{m+1}}{2 (a+i b) f (m+1)}-\frac{i \left(a d \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)+b c (m+2) \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)-b c (m+2) (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))-d \left(a \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)-b^3 \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) (m+2) (m+3) (m+4)\right)+i b (m+2) \left(c \left(2 c (A-C) d+B \left(c^2-d^2\right)\right) (m+3) (m+4) b^2-d \left(d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right)+d (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))\right)\right) \, _2F_1\left(1,m+1;m+2;-\frac{i a+i b \tan (e+f x)}{-i a-b}\right) (a+b \tan (e+f x))^{m+1}}{2 (a-i b) f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)}","\frac{d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left(b^2 d (m+3) (m+4) (d (A-C)+B c)-2 (b c-a d) (3 a C d-b (B d (m+4)+3 c C))\right)}{b^3 f (m+2) (m+3) (m+4)}+\frac{(a+b \tan (e+f x))^{m+1} \left(b c (m+2) \left(b^2 d (m+3) (m+4) (d (A-C)+B c)-2 (b c-a d) (3 a C d-b (B d (m+4)+3 c C))\right)+d \left(b^3 (m+2) (m+3) (m+4) \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)-a \left(b^2 d (m+3) (m+4) (d (A-C)+B c)-2 (b c-a d) (3 a C d-b (B d (m+4)+3 c C))\right)\right)\right)}{b^4 f (m+1) (m+2) (m+3) (m+4)}+\frac{(c-i d)^3 (A-i B-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) (b+i a)}-\frac{(c+i d)^3 (A+i B-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) (-b+i a)}-\frac{(c+d \tan (e+f x))^2 (3 a C d-b (B d (m+4)+3 c C)) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+3) (m+4)}+\frac{C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}",1,"(C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^3)/(b*f*(4 + m)) + (((3*b*c*C - 3*a*C*d + b*B*d*(4 + m))*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^2)/(b*f*(3 + m)) + ((d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))*Tan[e + f*x]*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(2 + m)) - (((-(b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))) + d*(-(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m)) + a*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))))*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m)) + ((I/2)*(a*d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) - b*c*(2 + m)*(-((2*a*d + b*c*(1 + m))*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(3 + m)*(A*b*c*(4 + m) - C*(3*a*d + b*c*(1 + m)))) - d*(-(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m)) + a*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))) - I*b*(2 + m)*(b^2*c*(2*c*(A - C)*d + B*(c^2 - d^2))*(3 + m)*(4 + m) - d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + d*(-((2*a*d + b*c*(1 + m))*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(3 + m)*(A*b*c*(4 + m) - C*(3*a*d + b*c*(1 + m))))))*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)*(a*d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) - b*c*(2 + m)*(-((2*a*d + b*c*(1 + m))*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(3 + m)*(A*b*c*(4 + m) - C*(3*a*d + b*c*(1 + m)))) - d*(-(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m)) + a*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m)))) + I*b*(2 + m)*(b^2*c*(2*c*(A - C)*d + B*(c^2 - d^2))*(3 + m)*(4 + m) - d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) + 2*(b*c - a*d)*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + d*(-((2*a*d + b*c*(1 + m))*(3*b*c*C - 3*a*C*d + b*B*d*(4 + m))) + b*c*(3 + m)*(A*b*c*(4 + m) - C*(3*a*d + b*c*(1 + m))))))*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)))/(b*(2 + m)))/(b*(3 + m)))/(b*(4 + m))","B",1
166,1,505,363,6.3493586,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{C (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^{m+1}}{b f (m+3)}+\frac{\frac{d \tan (e+f x) (-2 a C d+b B d (m+3)+2 b c C) (a+b \tan (e+f x))^{m+1}}{b f (m+2)}-\frac{\frac{i (a+b \tan (e+f x))^{m+1} \left(-\left(b^2 (m+2) (m+3) \left(A c^2-A d^2-2 B c d-c^2 C+C d^2\right)\right)-i b^2 (m+2) (m+3) \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right) \, _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 (a+b \tan (e+f x))^{m+1} \left(-b^2 (m+2) (m+3) \left(A c^2-A d^2-2 B c d-c^2 C+C d^2\right)+i b^2 (m+2) (m+3) \left(2 c d (A-C)+B \left(c^2-d^2\right)\right)\right) \, _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)}+\frac{(a+b \tan (e+f x))^{m+1} \left(-d \left(b^2 (m+2) (m+3) (d (A-C)+B c)-a (-2 a C d+b B d (m+3)+2 b c C)\right)-b c (m+2) (-2 a C d+b B d (m+3)+2 b c C)\right)}{b f (m+1)}}{b (m+2)}}{b (m+3)}","\frac{(a+b \tan (e+f x))^{m+1} \left(2 a^2 C d^2-a b d (m+3) (B d+2 c C)+b^2 (m+2) \left(d^2 (m+3) (A-C)+2 B c d (m+3)+2 c^2 C\right)\right)}{b^3 f (m+1) (m+2) (m+3)}+\frac{(c-i d)^2 (A-i B-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) (b+i a)}+\frac{(c+i d)^2 (i A-B-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)}-\frac{d \tan (e+f x) (2 a C d-b (B d (m+3)+2 c C)) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+2) (m+3)}+\frac{C (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^{m+1}}{b f (m+3)}",1,"(C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^2)/(b*f*(3 + m)) + ((d*(2*b*c*C - 2*a*C*d + b*B*d*(3 + m))*Tan[e + f*x]*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(2 + m)) - (((-(b*c*(2 + m)*(2*b*c*C - 2*a*C*d + b*B*d*(3 + m))) - d*(b^2*(B*c + (A - C)*d)*(2 + m)*(3 + m) - a*(2*b*c*C - 2*a*C*d + b*B*d*(3 + m))))*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m)) + ((I/2)*(-(b^2*(A*c^2 - c^2*C - 2*B*c*d - A*d^2 + C*d^2)*(2 + m)*(3 + m)) - I*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m))*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)*(-(b^2*(A*c^2 - c^2*C - 2*B*c*d - A*d^2 + C*d^2)*(2 + m)*(3 + m)) + I*b^2*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m))*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)))/(b*(2 + m)))/(b*(3 + m))","A",1
167,1,202,247,2.9982817,"\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{(a+b \tan (e+f x))^{m+1} \left(-\frac{i b (m+2) (c-i d) (A-i B-C) \, _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) (A+i B-C) \, _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{2 b (m+2) (B d+c C)-2 a C d}{b (m+1)}+2 C d \tan (e+f x)\right)}{2 b f (m+2)}","\frac{(c-i d) (A-i B-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) (b+i a)}-\frac{(c+i d) (A+i B-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) (-b+i a)}-\frac{(a C d-b (m+2) (B d+c C)) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+1) (m+2)}+\frac{C d \tan (e+f x) (a+b \tan (e+f x))^{m+1}}{b f (m+2)}",1,"((a + b*Tan[e + f*x])^(1 + m)*((-2*a*C*d + 2*b*(c*C + B*d)*(2 + m))/(b*(1 + m)) - (I*b*(A - I*B - C)*(c - I*d)*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/((a - I*b)*(1 + m)) + (I*b*(A + I*B - C)*(c + I*d)*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/((a + I*b)*(1 + m)) + 2*C*d*Tan[e + f*x]))/(2*b*f*(2 + m))","A",1
168,1,135,178,0.249695,"\int (a+b \tan (e+f x))^m \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right) \, dx","Integrate[(a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]","\frac{(a+b \tan (e+f x))^{m+1} \left(-\frac{i (A-i B-C) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{a-i b}+\frac{i (A+i B-C) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right)}{a+i b}+\frac{2 C}{b}\right)}{2 f (m+1)}","\frac{(A-i B-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) (b+i a)}+\frac{(i A-B-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)}+\frac{C (a+b \tan (e+f x))^{m+1}}{b f (m+1)}",1,"(((2*C)/b - (I*(A - I*B - C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/(a - I*b) + (I*(A + I*B - C)*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
169,1,204,258,1.1184056,"\int \frac{(a+b \tan (e+f x))^m \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{c+d \tan (e+f x)} \, dx","Integrate[((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]),x]","\frac{(a+b \tan (e+f x))^{m+1} \left(\frac{2 \left(A d^2-B c d+c^2 C\right) \, _2F_1\left(1,m+1;m+2;\frac{d (a+b \tan (e+f x))}{a d-b c}\right)}{b c-a d}+\frac{(d-i c) (A-i B-C) \, _2F_1\left(1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right)}{a-i b}+\frac{(d+i c) (A+i 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)}{2 f (m+1) \left(c^2+d^2\right)}","\frac{\left(A d^2-B c d+c^2 C\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) (b c-a d)}-\frac{(i A+B-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) (c-i d)}-\frac{(A+i B-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) (-b+i a) (c+i d)}",1,"((((A - I*B - C)*((-I)*c + d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)])/(a - I*b) + ((A + I*B - C)*(I*c + d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/(a + I*b) + (2*(c^2*C - B*c*d + A*d^2)*Hypergeometric2F1[1, 1 + m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)])/(b*c - a*d))*(a + b*Tan[e + f*x])^(1 + m))/(2*(c^2 + d^2)*f*(1 + m))","A",1
170,1,563,403,6.1964746,"\int \frac{(a+b \tan (e+f x))^m \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^2} \, dx","Integrate[((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2,x]","-\frac{\left(A d^2-c (B d-c C)\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{(a+b \tan (e+f x))^{m+1} \left(d^2 \left((c C-B d) (a d-b c (m+1))-A \left(a c d-b \left(c^2-d^2 m\right)\right)\right)-c d (b c-a d) (B c-d (A-C))-b c^2 m \left(A d^2-B c d+c^2 C\right)\right) \, _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(-\left((b c-a d) \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)\right)-i (b c-a d) \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\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(-(b c-a d) \left(-A \left(c^2-d^2\right)-2 B c d+c^2 C-C d^2\right)+i (b c-a d) \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)\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)}","-\frac{(a+b \tan (e+f x))^{m+1} \left(a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)-b \left(A d^2 \left(c^2 (2-m)-d^2 m\right)-B c d \left(c^2 (1-m)-d^2 (m+1)\right)+c^2 (-C) \left(c^2 m+d^2 (m+2)\right)\right)\right) \, _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{\left(A d^2-B c d+c^2 C\right) (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-i B-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) (b+i a) (c-i d)^2}+\frac{(i A-B-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) (c+i d)^2}",1,"-(((A*d^2 - c*(-(c*C) + B*d))*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))) - (-(((-(c*d*(b*c - a*d)*(B*c - (A - C)*d)) - b*c^2*(c^2*C - B*c*d + A*d^2)*m + d^2*((c*C - B*d)*(a*d - b*c*(1 + m)) - A*(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)*(-((b*c - a*d)*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2))) - I*(b*c - a*d)*(2*c*(A - C)*d - B*(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)*(-((b*c - a*d)*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2))) + I*(b*c - a*d)*(2*c*(A - C)*d - B*(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))","A",1
171,1,2238,702,6.2440396,"\int \frac{(a+b \tan (e+f x))^m \left(A+B \tan (e+f x)+C \tan ^2(e+f x)\right)}{(c+d \tan (e+f x))^3} \, dx","Integrate[((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3,x]","\text{Result too large to show}","\frac{(a+b \tan (e+f x))^{m+1} \left(2 a^2 d^3 \left(d (A-C) \left(3 c^2-d^2\right)-B \left(c^3-3 c d^2\right)\right)-2 a b d^2 \left(2 c d (A-C) \left(c^2 (3-m)-d^2 (m+1)\right)+B \left(-\left(c^4 (2-m)\right)+6 c^2 d^2-d^4 m\right)\right)-b^2 \left(A d^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)+B c d \left(c^4 \left(m^2-3 m+2\right)-2 c^2 d^2 \left(-m^2+m+3\right)+d^4 m (m+1)\right)+c^2 C \left(c^4 (1-m) m+2 c^2 d^2 \left(-m^2-m+3\right)-d^4 \left(m^2+3 m+2\right)\right)\right)\right) \, _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{\left(A d^2-B c d+c^2 C\right) (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} \left(2 a d^2 \left(2 c d (A-C)-B \left(c^2-d^2\right)\right)-b \left(c^2 d^2 (A (5-m)-C (m+3))+A d^4 (1-m)-B c^3 d (3-m)+B c d^3 (m+1)+c^4 C (1-m)\right)\right)}{2 f \left(c^2+d^2\right)^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{(A-i B-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) (b+i a) (c-i d)^3}+\frac{(A+i B-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) (-d+i c)^3}",1,"-1/2*((A*d^2 - c*(-(c*C) + B*d))*(a + b*Tan[e + f*x])^(1 + m))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-(((-(c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))) + d^2*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(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]))) - (-(((-(c*d*(-(b*c) + a*d)*(-2*c*(b*c - a*d)*(B*c - (A - C)*d) - b*d*(c^2*C - B*c*d + A*d^2)*(1 - m) + d*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))))) - b*c^2*m*(-(c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))) + d^2*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))) + d^2*((2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))*(-(a*d) + b*c*(1 + m)) + (-(c*(-(b*c) + a*d)) - b*d^2*m)*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + 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)*(d*(-(b*c) + a*d)*(-2*c*(b*c - a*d)*(B*c - (A - C)*d) - b*d*(c^2*C - B*c*d + A*d^2)*(1 - m) + d*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))) + c*((2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))*(-(a*d) + b*c*(1 + m)) + (-(c*(-(b*c) + a*d)) - b*d^2*m)*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))) + b*m*(-(c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))) + d^2*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))))) + I*(c*(-(b*c) + a*d)*(-2*c*(b*c - a*d)*(B*c - (A - C)*d) - b*d*(c^2*C - B*c*d + A*d^2)*(1 - m) + d*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))) - d*((2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))*(-(a*d) + b*c*(1 + m)) + (-(c*(-(b*c) + a*d)) - b*d^2*m)*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))) + b*m*(-(c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))) + d^2*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))))))*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)*(d*(-(b*c) + a*d)*(-2*c*(b*c - a*d)*(B*c - (A - C)*d) - b*d*(c^2*C - B*c*d + A*d^2)*(1 - m) + d*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))) + c*((2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))*(-(a*d) + b*c*(1 + m)) + (-(c*(-(b*c) + a*d)) - b*d^2*m)*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))) + b*m*(-(c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))) + d^2*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))))) - I*(c*(-(b*c) + a*d)*(-2*c*(b*c - a*d)*(B*c - (A - C)*d) - b*d*(c^2*C - B*c*d + A*d^2)*(1 - m) + d*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))) - d*((2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))*(-(a*d) + b*c*(1 + m)) + (-(c*(-(b*c) + a*d)) - b*d^2*m)*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m))) + b*m*(-(c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - b*c*(c^2*C - B*c*d + A*d^2)*(1 - m))) + d^2*(A*(2*c*(b*c - a*d) + b*d^2*(1 - m)) + (c*C - B*d)*(2*a*d - b*c*(1 + m)))))))*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))","B",1