Optimal. Leaf size=277 \[ \frac{3 i a^2 b d \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{i b^3 d \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{a^3 (c+d x)^2}{2 d}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}-3 a b^2 c x+\frac{3 a b^2 d \log (\cos (e+f x))}{f^2}-\frac{3}{2} a b^2 d x^2+\frac{b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}-\frac{i b^3 (c+d x)^2}{2 d}-\frac{b^3 d \tan (e+f x)}{2 f^2}+\frac{b^3 d x}{2 f} \]
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Rubi [A] time = 0.337686, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3722, 3719, 2190, 2279, 2391, 3720, 3475, 3473, 8} \[ -\frac{3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{3 i a^2 b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}+\frac{a^3 (c+d x)^2}{2 d}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}-3 a b^2 c x+\frac{3 a b^2 d \log (\cos (e+f x))}{f^2}-\frac{3}{2} a b^2 d x^2+\frac{b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}-\frac{i b^3 (c+d x)^2}{2 d}-\frac{i b^3 d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{b^3 d \tan (e+f x)}{2 f^2}+\frac{b^3 d x}{2 f} \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 3720
Rule 3475
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int (c+d x) (a+b \tan (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)+3 a^2 b (c+d x) \tan (e+f x)+3 a b^2 (c+d x) \tan ^2(e+f x)+b^3 (c+d x) \tan ^3(e+f x)\right ) \, dx\\ &=\frac{a^3 (c+d x)^2}{2 d}+\left (3 a^2 b\right ) \int (c+d x) \tan (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x) \tan ^2(e+f x) \, dx+b^3 \int (c+d x) \tan ^3(e+f x) \, dx\\ &=\frac{a^3 (c+d x)^2}{2 d}+\frac{3 i a^2 b (c+d x)^2}{2 d}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}-\left (6 i a^2 b\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx-\left (3 a b^2\right ) \int (c+d x) \, dx-b^3 \int (c+d x) \tan (e+f x) \, dx-\frac{\left (3 a b^2 d\right ) \int \tan (e+f x) \, dx}{f}-\frac{\left (b^3 d\right ) \int \tan ^2(e+f x) \, dx}{2 f}\\ &=-3 a b^2 c x-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}+\frac{3 i a^2 b (c+d x)^2}{2 d}-\frac{i b^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\cos (e+f x))}{f^2}-\frac{b^3 d \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}+\left (2 i b^3\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx+\frac{\left (3 a^2 b d\right ) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}+\frac{\left (b^3 d\right ) \int 1 \, dx}{2 f}\\ &=-3 a b^2 c x+\frac{b^3 d x}{2 f}-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}+\frac{3 i a^2 b (c+d x)^2}{2 d}-\frac{i b^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\cos (e+f x))}{f^2}-\frac{b^3 d \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}-\frac{\left (3 i a^2 b d\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^2}-\frac{\left (b^3 d\right ) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=-3 a b^2 c x+\frac{b^3 d x}{2 f}-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}+\frac{3 i a^2 b (c+d x)^2}{2 d}-\frac{i b^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\cos (e+f x))}{f^2}+\frac{3 i a^2 b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{b^3 d \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}+\frac{\left (i b^3 d\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^2}\\ &=-3 a b^2 c x+\frac{b^3 d x}{2 f}-\frac{3}{2} a b^2 d x^2+\frac{a^3 (c+d x)^2}{2 d}+\frac{3 i a^2 b (c+d x)^2}{2 d}-\frac{i b^3 (c+d x)^2}{2 d}-\frac{3 a^2 b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 a b^2 d \log (\cos (e+f x))}{f^2}+\frac{3 i a^2 b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{i b^3 d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{b^3 d \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x) \tan (e+f x)}{f}+\frac{b^3 (c+d x) \tan ^2(e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 3.53309, size = 277, normalized size = 1. \[ \frac{\cos (e+f x) (a+b \tan (e+f x))^3 \left (-i b d \left (b^2-3 a^2\right ) \cos ^2(e+f x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )+\cos ^2(e+f x) \left (-(e+f x) \left (-3 i a^2 b d (e+f x)+a^3 (d (e-f x)-2 c f)+3 a b^2 (2 c f-d e+d f x)+i b^3 d (e+f x)\right )+2 b \log (\cos (e+f x)) \left (3 a^2 (d e-c f)+3 a b d+b^2 (c f-d e)\right )+2 b d \left (b^2-3 a^2\right ) (e+f x) \log \left (1+e^{2 i (e+f x)}\right )\right )+\frac{1}{2} b^2 (\sin (2 (e+f x)) (6 a f (c+d x)-b d)+2 b f (c+d x))\right )}{2 f^2 (a \cos (e+f x)+b \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.136, size = 493, normalized size = 1.8 \begin{align*}{\frac{6\,ib{a}^{2}dex}{f}}-{\frac{{\frac{i}{2}}{b}^{3}d{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-3\,i{a}^{2}bcx+{\frac{{b}^{3}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) dx}{f}}-3\,{\frac{{a}^{2}bc\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{f}}+6\,{\frac{{a}^{2}bc\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}+3\,{\frac{a{b}^{2}d\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{{f}^{2}}}-6\,{\frac{a{b}^{2}d\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+2\,{\frac{{b}^{3}de\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-{\frac{i{b}^{3}d{e}^{2}}{{f}^{2}}}+{\frac{{a}^{3}d{x}^{2}}{2}}+{a}^{3}cx+{\frac{{\frac{3\,i}{2}}{a}^{2}bd{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-{\frac{i}{2}}{b}^{3}d{x}^{2}+{\frac{{b}^{3}c\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{f}}-2\,{\frac{{b}^{3}c\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}+i{b}^{3}cx-3\,{\frac{b\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ){a}^{2}dx}{f}}-6\,{\frac{b{a}^{2}de\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+{\frac{3\,i{a}^{2}bd{e}^{2}}{{f}^{2}}}-{\frac{2\,i{b}^{3}dex}{f}}+{\frac{{b}^{2} \left ( 6\,iadfx{{\rm e}^{2\,i \left ( fx+e \right ) }}+6\,iacf{{\rm e}^{2\,i \left ( fx+e \right ) }}+2\,bdfx{{\rm e}^{2\,i \left ( fx+e \right ) }}+6\,iadfx-ibd{{\rm e}^{2\,i \left ( fx+e \right ) }}+2\,bcf{{\rm e}^{2\,i \left ( fx+e \right ) }}+6\,iacf-ibd \right ) }{{f}^{2} \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) ^{2}}}-3\,a{b}^{2}cx-{\frac{3\,a{b}^{2}d{x}^{2}}{2}}+{\frac{3\,i}{2}}{a}^{2}bd{x}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.14418, size = 1782, normalized size = 6.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74405, size = 752, normalized size = 2.71 \begin{align*} \frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )} d f^{2} x^{2} - i \,{\left (3 \, a^{2} b - b^{3}\right )} d{\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 ) + i \,{\left (3 \, a^{2} b - b^{3}\right )} d{\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 ) + 2 \,{\left (b^{3} d f x + b^{3} c f\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (b^{3} d f + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c f^{2}\right )} x + 2 \,{\left (3 \, a b^{2} d -{\left (3 \, a^{2} b - b^{3}\right )} d f x -{\left (3 \, a^{2} b - b^{3}\right )} c f\right )} \log \left (-\frac{2 \,{\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (3 \, a b^{2} d -{\left (3 \, a^{2} b - b^{3}\right )} d f x -{\left (3 \, a^{2} b - b^{3}\right )} c f\right )} \log \left (-\frac{2 \,{\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (6 \, a b^{2} d f x + 6 \, a b^{2} c f - b^{3} d\right )} \tan \left (f x + e\right )}{4 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (e + f x \right )}\right )^{3} \left (c + d x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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