Integrand size = 20, antiderivative size = 229 \[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}+\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}+\frac {2 b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {a b d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^2 \tan (e+f x)}{f} \]
-I*b^2*(d*x+c)^2/f+1/3*a^2*(d*x+c)^3/d+2/3*I*a*b*(d*x+c)^3/d-1/3*b^2*(d*x+ c)^3/d+2*b^2*d*(d*x+c)*ln(1+exp(2*I*(f*x+e)))/f^2-2*a*b*(d*x+c)^2*ln(1+exp (2*I*(f*x+e)))/f-I*b^2*d^2*polylog(2,-exp(2*I*(f*x+e)))/f^3+2*I*a*b*d*(d*x +c)*polylog(2,-exp(2*I*(f*x+e)))/f^2-a*b*d^2*polylog(3,-exp(2*I*(f*x+e)))/ f^3+b^2*(d*x+c)^2*tan(f*x+e)/f
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(649\) vs. \(2(229)=458\).
Time = 6.83 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.83 \[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=-\frac {i a b d^2 e^{-i e} \left (2 f^2 x^2 \left (2 f x-3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{-2 i (e+f x)}\right )\right )+6 \left (1+e^{2 i e}\right ) f x \operatorname {PolyLog}\left (2,-e^{-2 i (e+f x)}\right )-3 i \left (1+e^{2 i e}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (e+f x)}\right )\right ) \sec (e)}{6 f^3}+\frac {1}{3} x \left (3 c^2+3 c d x+d^2 x^2\right ) \sec (e) \left (a^2 \cos (e)-b^2 \cos (e)+2 a b \sin (e)\right )+\frac {2 b^2 c d \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f^2 \left (\cos ^2(e)+\sin ^2(e)\right )}-\frac {2 a b c^2 \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {b^2 d^2 \csc (e) \left (e^{-i \arctan (\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \arctan (\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\arctan (\cot (e))) \log \left (1-e^{2 i (f x-\arctan (\cot (e)))}\right )+\pi \log (\cos (f x))-2 \arctan (\cot (e)) \log (\sin (f x-\arctan (\cot (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x-\arctan (\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^3 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}-\frac {2 a b c d \csc (e) \left (e^{-i \arctan (\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \arctan (\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\arctan (\cot (e))) \log \left (1-e^{2 i (f x-\arctan (\cot (e)))}\right )+\pi \log (\cos (f x))-2 \arctan (\cot (e)) \log (\sin (f x-\arctan (\cot (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x-\arctan (\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^2 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}+\frac {\sec (e) \sec (e+f x) \left (b^2 c^2 \sin (f x)+2 b^2 c d x \sin (f x)+b^2 d^2 x^2 \sin (f x)\right )}{f} \]
((-1/6*I)*a*b*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^(( -2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f* x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e])/ (E^(I*e)*f^3) + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Sec[e]*(a^2*Cos[e] - b^2*Co s[e] + 2*a*b*Sin[e]))/3 + (2*b^2*c*d*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) - (2*a*b*c^2*S ec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f*(Co s[e]^2 + Sin[e]^2)) + (b^2*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (C ot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f* x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos [f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^ ((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^3*Sqrt[Cs c[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (2*a*b*c*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[ Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)* f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I* PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e]) /(f^2*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (Sec[e]*Sec[e + f*x]*(b^2*c^ 2*Sin[f*x] + 2*b^2*c*d*x*Sin[f*x] + b^2*d^2*x^2*Sin[f*x]))/f
Time = 0.60 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4205, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^2 (a+b \tan (e+f x))^2dx\) |
\(\Big \downarrow \) 4205 |
\(\displaystyle \int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \tan (e+f x)+b^2 (c+d x)^2 \tan ^2(e+f x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (c+d x)^3}{3 d}+\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {2 i a b (c+d x)^3}{3 d}-\frac {a b d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}+\frac {2 b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^2}{f}-\frac {b^2 (c+d x)^3}{3 d}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}\) |
((-I)*b^2*(c + d*x)^2)/f + (a^2*(c + d*x)^3)/(3*d) + (((2*I)/3)*a*b*(c + d *x)^3)/d - (b^2*(c + d*x)^3)/(3*d) + (2*b^2*d*(c + d*x)*Log[1 + E^((2*I)*( e + f*x))])/f^2 - (2*a*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f - (I* b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((2*I)*a*b*d*(c + d*x)*Pol yLog[2, -E^((2*I)*(e + f*x))])/f^2 - (a*b*d^2*PolyLog[3, -E^((2*I)*(e + f* x))])/f^3 + (b^2*(c + d*x)^2*Tan[e + f*x])/f
3.1.45.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (209 ) = 418\).
Time = 1.39 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.51
method | result | size |
risch | \(\frac {8 i b d c a e x}{f}+\frac {d^{2} a^{2} x^{3}}{3}+\frac {a^{2} c^{3}}{3 d}-\frac {d^{2} b^{2} x^{3}}{3}-b^{2} c^{2} x -\frac {b^{2} c^{3}}{3 d}+\frac {4 b^{2} e \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+d \,a^{2} c \,x^{2}+a^{2} c^{2} x -d \,b^{2} c \,x^{2}-\frac {2 b a \,c^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}-2 i a b \,c^{2} x -\frac {2 i a b \,c^{3}}{3 d}+\frac {2 i b^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {4 b a \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {2 b^{2} c d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {4 b^{2} c d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {2 b^{2} d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{f^{2}}-\frac {2 i b^{2} d^{2} x^{2}}{f}-\frac {2 i b^{2} d^{2} e^{2}}{f^{3}}+\frac {2 i d^{2} a b \,x^{3}}{3}-\frac {a b \,d^{2} \operatorname {Li}_{3}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}-\frac {i b^{2} d^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}-\frac {4 i b^{2} d^{2} e x}{f^{2}}+2 i d a b c \,x^{2}+\frac {4 b \,e^{2} a \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {2 b a \,d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{2}}{f}-\frac {8 i b a \,d^{2} e^{3}}{3 f^{3}}-\frac {8 b e a c d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {4 b d c a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{f}-\frac {4 i b a \,d^{2} e^{2} x}{f^{2}}+\frac {2 i b a \,d^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{2}}+\frac {2 i b d c a \,\operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{2}}+\frac {4 i b d c a \,e^{2}}{f^{2}}\) | \(575\) |
8*I/f*b*d*c*a*e*x+1/3*d^2*a^2*x^3+1/3/d*a^2*c^3-1/3*d^2*b^2*x^3-b^2*c^2*x- 1/3/d*b^2*c^3+4/f^3*b^2*e*d^2*ln(exp(I*(f*x+e)))-2/f*b*a*c^2*ln(exp(2*I*(f *x+e))+1)+d*a^2*c*x^2+a^2*c^2*x-d*b^2*c*x^2-2*I*a*b*c^2*x-2/3*I/d*a*b*c^3+ 2*I*b^2*(d^2*x^2+2*c*d*x+c^2)/f/(exp(2*I*(f*x+e))+1)+4/f*b*a*c^2*ln(exp(I* (f*x+e)))+2/f^2*b^2*c*d*ln(exp(2*I*(f*x+e))+1)-4/f^2*b^2*c*d*ln(exp(I*(f*x +e)))+2/f^2*b^2*d^2*ln(exp(2*I*(f*x+e))+1)*x-2*I/f*b^2*d^2*x^2-2*I/f^3*b^2 *d^2*e^2+2/3*I*d^2*a*b*x^3-a*b*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3+2*I*d* a*b*c*x^2-8/f^2*b*e*a*c*d*ln(exp(I*(f*x+e)))-4/f*b*d*c*a*ln(exp(2*I*(f*x+e ))+1)*x-4*I/f^2*b*a*d^2*e^2*x+2*I/f^2*b*a*d^2*polylog(2,-exp(2*I*(f*x+e))) *x+2*I/f^2*b*d*c*a*polylog(2,-exp(2*I*(f*x+e)))+4*I/f^2*b*d*c*a*e^2-I*b^2* d^2*polylog(2,-exp(2*I*(f*x+e)))/f^3+4/f^3*b*e^2*a*d^2*ln(exp(I*(f*x+e)))- 2/f*b*a*d^2*ln(exp(2*I*(f*x+e))+1)*x^2-8/3*I/f^3*b*a*d^2*e^3-4*I/f^2*b^2*d ^2*e*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (203) = 406\).
Time = 0.26 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.97 \[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=\frac {2 \, {\left (a^{2} - b^{2}\right )} d^{2} f^{3} x^{3} + 6 \, {\left (a^{2} - b^{2}\right )} c d f^{3} x^{2} + 6 \, {\left (a^{2} - b^{2}\right )} c^{2} f^{3} x - 3 \, a b d^{2} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, a b d^{2} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (2 i \, a b d^{2} f x + 2 i \, a b c d f - i \, b^{2} d^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 3 \, {\left (-2 i \, a b d^{2} f x - 2 i \, a b c d f + i \, b^{2} d^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \, {\left (a b d^{2} f^{2} x^{2} + a b c^{2} f^{2} - b^{2} c d f + {\left (2 \, a b c d f^{2} - b^{2} d^{2} f\right )} x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \, {\left (a b d^{2} f^{2} x^{2} + a b c^{2} f^{2} - b^{2} c d f + {\left (2 \, a b c d f^{2} - b^{2} d^{2} f\right )} x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} \tan \left (f x + e\right )}{6 \, f^{3}} \]
1/6*(2*(a^2 - b^2)*d^2*f^3*x^3 + 6*(a^2 - b^2)*c*d*f^3*x^2 + 6*(a^2 - b^2) *c^2*f^3*x - 3*a*b*d^2*polylog(3, (tan(f*x + e)^2 + 2*I*tan(f*x + e) - 1)/ (tan(f*x + e)^2 + 1)) - 3*a*b*d^2*polylog(3, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 3*(2*I*a*b*d^2*f*x + 2*I*a*b*c*d*f - I* b^2*d^2)*dilog(2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1) + 1) - 3*(-2*I* a*b*d^2*f*x - 2*I*a*b*c*d*f + I*b^2*d^2)*dilog(2*(-I*tan(f*x + e) - 1)/(ta n(f*x + e)^2 + 1) + 1) - 6*(a*b*d^2*f^2*x^2 + a*b*c^2*f^2 - b^2*c*d*f + (2 *a*b*c*d*f^2 - b^2*d^2*f)*x)*log(-2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 6*(a*b*d^2*f^2*x^2 + a*b*c^2*f^2 - b^2*c*d*f + (2*a*b*c*d*f^2 - b^2 *d^2*f)*x)*log(-2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) + 6*(b^2*d^2 *f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2)*tan(f*x + e))/f^3
\[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (203) = 406\).
Time = 0.91 (sec) , antiderivative size = 1263, normalized size of antiderivative = 5.52 \[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=\text {Too large to display} \]
1/3*(3*(f*x + e)*a^2*c^2 + (f*x + e)^3*a^2*d^2/f^2 - 3*(f*x + e)^2*a^2*d^2 *e/f^2 + 3*(f*x + e)*a^2*d^2*e^2/f^2 + 3*(f*x + e)^2*a^2*c*d/f - 6*(f*x + e)*a^2*c*d*e/f + 6*a*b*c^2*log(sec(f*x + e)) + 6*a*b*d^2*e^2*log(sec(f*x + e))/f^2 - 12*a*b*c*d*e*log(sec(f*x + e))/f + 3*((2*a*b + I*b^2)*(f*x + e) ^3*d^2 + 6*b^2*d^2*e^2 - 12*b^2*c*d*e*f + 6*b^2*c^2*f^2 - 3*((2*a*b + I*b^ 2)*d^2*e - (2*a*b + I*b^2)*c*d*f)*(f*x + e)^2 + 3*(I*b^2*d^2*e^2 - 2*I*b^2 *c*d*e*f + I*b^2*c^2*f^2)*(f*x + e) - 6*((f*x + e)^2*a*b*d^2 + b^2*d^2*e - b^2*c*d*f - (2*a*b*d^2*e - 2*a*b*c*d*f + b^2*d^2)*(f*x + e) + ((f*x + e)^ 2*a*b*d^2 + b^2*d^2*e - b^2*c*d*f - (2*a*b*d^2*e - 2*a*b*c*d*f + b^2*d^2)* (f*x + e))*cos(2*f*x + 2*e) - (-I*(f*x + e)^2*a*b*d^2 - I*b^2*d^2*e + I*b^ 2*c*d*f + (2*I*a*b*d^2*e - 2*I*a*b*c*d*f + I*b^2*d^2)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + ((2*a*b + I*b^2 )*(f*x + e)^3*d^2 - 3*(2*b^2*d^2 + (2*a*b + I*b^2)*d^2*e - (2*a*b + I*b^2) *c*d*f)*(f*x + e)^2 + 3*(I*b^2*d^2*e^2 + I*b^2*c^2*f^2 + 4*b^2*d^2*e + 2*( -I*b^2*c*d*e - 2*b^2*c*d)*f)*(f*x + e))*cos(2*f*x + 2*e) + 3*(2*(f*x + e)* a*b*d^2 - 2*a*b*d^2*e + 2*a*b*c*d*f - b^2*d^2 + (2*(f*x + e)*a*b*d^2 - 2*a *b*d^2*e + 2*a*b*c*d*f - b^2*d^2)*cos(2*f*x + 2*e) + (2*I*(f*x + e)*a*b*d^ 2 - 2*I*a*b*d^2*e + 2*I*a*b*c*d*f - I*b^2*d^2)*sin(2*f*x + 2*e))*dilog(-e^ (2*I*f*x + 2*I*e)) + 3*(I*(f*x + e)^2*a*b*d^2 + I*b^2*d^2*e - I*b^2*c*d*f + (-2*I*a*b*d^2*e + 2*I*a*b*c*d*f - I*b^2*d^2)*(f*x + e) + (I*(f*x + e)...
\[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (c+d x)^2 (a+b \tan (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]