Optimal. Leaf size=93 \[ \frac{i b d \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac{i b d \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+\frac{a (c+d x)^2}{2 d}-\frac{2 i b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f} \]
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Rubi [A] time = 0.0708628, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4190, 4181, 2279, 2391} \[ \frac{i b d \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac{i b d \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+\frac{a (c+d x)^2}{2 d}-\frac{2 i b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 4190
Rule 4181
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (c+d x) (a+b \sec (e+f x)) \, dx &=\int (a (c+d x)+b (c+d x) \sec (e+f x)) \, dx\\ &=\frac{a (c+d x)^2}{2 d}+b \int (c+d x) \sec (e+f x) \, dx\\ &=\frac{a (c+d x)^2}{2 d}-\frac{2 i b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}-\frac{(b d) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac{(b d) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f}\\ &=\frac{a (c+d x)^2}{2 d}-\frac{2 i b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac{(i b d) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2}-\frac{(i b d) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2}\\ &=\frac{a (c+d x)^2}{2 d}-\frac{2 i b (c+d x) \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac{i b d \text{Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac{i b d \text{Li}_2\left (i e^{i (e+f x)}\right )}{f^2}\\ \end{align*}
Mathematica [A] time = 0.0126583, size = 104, normalized size = 1.12 \[ \frac{i b d \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac{i b d \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+a c x+\frac{1}{2} a d x^2+\frac{b c \tanh ^{-1}(\sin (e+f x))}{f}-\frac{2 i b d x \tan ^{-1}\left (e^{i e+i f x}\right )}{f} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.072, size = 186, normalized size = 2. \begin{align*}{\frac{ad{x}^{2}}{2}}+acx-{\frac{2\,ibc\arctan \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}-{\frac{bd\ln \left ( 1+i{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{f}}-{\frac{bd\ln \left ( 1+i{{\rm e}^{i \left ( fx+e \right ) }} \right ) e}{{f}^{2}}}+{\frac{bd\ln \left ( 1-i{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{f}}+{\frac{bd\ln \left ( 1-i{{\rm e}^{i \left ( fx+e \right ) }} \right ) e}{{f}^{2}}}+{\frac{ibd{\it dilog} \left ( 1+i{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-{\frac{ibd{\it dilog} \left ( 1-i{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+{\frac{2\,ibde\arctan \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97448, size = 927, normalized size = 9.97 \begin{align*} \frac{a d f^{2} x^{2} + 2 \, a c f^{2} x - i \, b d{\rm Li}_2\left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - i \, b d{\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + i \, b d{\rm Li}_2\left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + i \, b d{\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) -{\left (b d e - b c f\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) +{\left (b d e - b c f\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) +{\left (b d f x + b d e\right )} \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) -{\left (b d f x + b d e\right )} \log \left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) +{\left (b d f x + b d e\right )} \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) -{\left (b d f x + b d e\right )} \log \left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) -{\left (b d e - b c f\right )} \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) +{\left (b d e - b c f\right )} \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right )}{2 \, 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 \sec{\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 \sec \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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