Optimal. Leaf size=84 \[ \frac{i b d \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i b (c+d x)^2}{2 d} \]
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Rubi [A] time = 0.119414, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3722, 3719, 2190, 2279, 2391} \[ \frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i b (c+d x)^2}{2 d}+\frac{i b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2} \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (c+d x) (a+b \tan (e+f x)) \, dx &=\int (a (c+d x)+b (c+d x) \tan (e+f x)) \, dx\\ &=\frac{a (c+d x)^2}{2 d}+b \int (c+d x) \tan (e+f x) \, dx\\ &=\frac{a (c+d x)^2}{2 d}+\frac{i b (c+d x)^2}{2 d}-(2 i b) \int \frac{e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx\\ &=\frac{a (c+d x)^2}{2 d}+\frac{i b (c+d x)^2}{2 d}-\frac{b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{(b d) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac{a (c+d x)^2}{2 d}+\frac{i b (c+d x)^2}{2 d}-\frac{b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{(i b d) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^2}\\ &=\frac{a (c+d x)^2}{2 d}+\frac{i b (c+d x)^2}{2 d}-\frac{b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}\\ \end{align*}
Mathematica [A] time = 0.0142465, size = 87, normalized size = 1.04 \[ \frac{i b d \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+a c x+\frac{1}{2} a d x^2-\frac{b c \log (\cos (e+f x))}{f}-\frac{b d x \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{1}{2} i b d x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 143, normalized size = 1.7 \begin{align*}{\frac{i}{2}}bd{x}^{2}-ibcx+{\frac{ad{x}^{2}}{2}}+acx-{\frac{bc\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{f}}+2\,{\frac{bc\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}+{\frac{2\,ibdex}{f}}+{\frac{ibd{e}^{2}}{{f}^{2}}}-{\frac{bd\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) x}{f}}+{\frac{{\frac{i}{2}}bd{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-2\,{\frac{bde\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63981, size = 176, normalized size = 2.1 \begin{align*} \frac{{\left (a + i \, b\right )} d f^{2} x^{2} + 2 \,{\left (a + i \, b\right )} c f^{2} x + i \, b d{\rm Li}_2\left (-e^{\left (2 i \, f x + 2 i \, e\right )}\right ) -{\left (2 i \, b d f x + 2 i \, b c f\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) -{\left (b d f x + b c f\right )} \log \left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}{2 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62954, size = 420, normalized size = 5. \begin{align*} \frac{2 \, a d f^{2} x^{2} + 4 \, a c f^{2} x - i \, b 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 \, b 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 d f x + b 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 (b d f x + b 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 )}{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 ) \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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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