Optimal. Leaf size=136 \[ \frac{i a b d \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac{a^2 (c+d x)^2}{2 d}-\frac{2 a b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i a b (c+d x)^2}{d}+\frac{b^2 (c+d x) \tan (e+f x)}{f}-b^2 c x+\frac{b^2 d \log (\cos (e+f x))}{f^2}-\frac{1}{2} b^2 d x^2 \]
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Rubi [A] time = 0.190461, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3722, 3719, 2190, 2279, 2391, 3720, 3475} \[ \frac{a^2 (c+d x)^2}{2 d}-\frac{2 a b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i a b (c+d x)^2}{d}+\frac{i a b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac{b^2 (c+d x) \tan (e+f x)}{f}-b^2 c x+\frac{b^2 d \log (\cos (e+f x))}{f^2}-\frac{1}{2} b^2 d x^2 \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 3720
Rule 3475
Rubi steps
\begin{align*} \int (c+d x) (a+b \tan (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a b (c+d x) \tan (e+f x)+b^2 (c+d x) \tan ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \tan (e+f x) \, dx+b^2 \int (c+d x) \tan ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+\frac{i a b (c+d x)^2}{d}+\frac{b^2 (c+d x) \tan (e+f x)}{f}-(4 i a b) \int \frac{e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x) \, dx-\frac{\left (b^2 d\right ) \int \tan (e+f x) \, dx}{f}\\ &=-b^2 c x-\frac{1}{2} b^2 d x^2+\frac{a^2 (c+d x)^2}{2 d}+\frac{i a b (c+d x)^2}{d}-\frac{2 a b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^2 d \log (\cos (e+f x))}{f^2}+\frac{b^2 (c+d x) \tan (e+f x)}{f}+\frac{(2 a b d) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=-b^2 c x-\frac{1}{2} b^2 d x^2+\frac{a^2 (c+d x)^2}{2 d}+\frac{i a b (c+d x)^2}{d}-\frac{2 a b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^2 d \log (\cos (e+f x))}{f^2}+\frac{b^2 (c+d x) \tan (e+f x)}{f}-\frac{(i a b d) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^2}\\ &=-b^2 c x-\frac{1}{2} b^2 d x^2+\frac{a^2 (c+d x)^2}{2 d}+\frac{i a b (c+d x)^2}{d}-\frac{2 a b (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^2 d \log (\cos (e+f x))}{f^2}+\frac{i a b d \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac{b^2 (c+d x) \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 2.15947, size = 200, normalized size = 1.47 \[ \frac{\cos (e+f x) (a+b \tan (e+f x))^2 \left (2 i a b d \cos (e+f x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )+\cos (e+f x) \left (-(e+f x) \left (a^2 (-2 c f+d e-d f x)-2 i a b d (e+f x)+b^2 (2 c f-d e+d f x)\right )+2 b \log (\cos (e+f x)) (-2 a c f+2 a d e+b d)-4 a b d (e+f x) \log \left (1+e^{2 i (e+f x)}\right )\right )+2 b^2 f (c+d x) \sin (e+f x)\right )}{2 f^2 (a \cos (e+f x)+b \sin (e+f x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 238, normalized size = 1.8 \begin{align*}{\frac{2\,ibad{e}^{2}}{{f}^{2}}}+{\frac{2\,i{b}^{2} \left ( dx+c \right ) }{f \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }}+{\frac{{a}^{2}d{x}^{2}}{2}}-{\frac{{b}^{2}d{x}^{2}}{2}}+{a}^{2}cx-{b}^{2}cx+{\frac{iabd{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+{\frac{{b}^{2}d\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{{f}^{2}}}-2\,{\frac{{b}^{2}d\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+4\,{\frac{abc\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}-2\,{\frac{abc\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{f}}-4\,{\frac{abde\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+iabd{x}^{2}+{\frac{4\,ibadex}{f}}-2\,iabcx-2\,{\frac{b\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) adx}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.76626, size = 714, normalized size = 5.25 \begin{align*} \frac{2 \,{\left (f x + e\right )} a^{2} c + \frac{{\left (f x + e\right )}^{2} a^{2} d}{f} - \frac{2 \,{\left (f x + e\right )} a^{2} d e}{f} + 4 \, a b c \log \left (\sec \left (f x + e\right )\right ) - \frac{4 \, a b d e \log \left (\sec \left (f x + e\right )\right )}{f} + \frac{2 \,{\left ({\left (2 \, a b + i \, b^{2}\right )}{\left (f x + e\right )}^{2} d - 4 \, b^{2} d e + 4 \, b^{2} c f -{\left (2 i \, b^{2} d e - 2 i \, b^{2} c f\right )}{\left (f x + e\right )} -{\left (4 \,{\left (f x + e\right )} a b d - 2 \, b^{2} d + 2 \,{\left (2 \,{\left (f x + e\right )} a b d - b^{2} d\right )} \cos \left (2 \, f x + 2 \, e\right ) +{\left (4 i \,{\left (f x + e\right )} a b d - 2 i \, b^{2} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) +{\left ({\left (2 \, a b + i \, b^{2}\right )}{\left (f x + e\right )}^{2} d -{\left (2 i \, b^{2} d e - 2 i \, b^{2} c f + 4 \, b^{2} d\right )}{\left (f x + e\right )}\right )} \cos \left (2 \, f x + 2 \, e\right ) +{\left (2 \, a b d \cos \left (2 \, f x + 2 \, e\right ) + 2 i \, a b d \sin \left (2 \, f x + 2 \, e\right ) + 2 \, a b d\right )}{\rm Li}_2\left (-e^{\left (2 i \, f x + 2 i \, e\right )}\right ) -{\left (-2 i \,{\left (f x + e\right )} a b d + i \, b^{2} d +{\left (-2 i \,{\left (f x + e\right )} a b d + i \, b^{2} d\right )} \cos \left (2 \, f x + 2 \, e\right ) +{\left (2 \,{\left (f x + e\right )} a b d - b^{2} d\right )} \sin \left (2 \, f x + 2 \, e\right )\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 ) -{\left ({\left (-2 i \, a b + b^{2}\right )}{\left (f x + e\right )}^{2} d -{\left (2 \, b^{2} d e - 2 \, b^{2} c f - 4 i \, b^{2} d\right )}{\left (f x + e\right )}\right )} \sin \left (2 \, f x + 2 \, e\right )\right )}}{-2 i \, f \cos \left (2 \, f x + 2 \, e\right ) + 2 \, f \sin \left (2 \, f x + 2 \, e\right ) - 2 i \, f}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68379, size = 540, normalized size = 3.97 \begin{align*} \frac{{\left (a^{2} - b^{2}\right )} d f^{2} x^{2} + 2 \,{\left (a^{2} - b^{2}\right )} c f^{2} x - i \, a 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 \, a 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 ) -{\left (2 \, a b d f x + 2 \, a b c f - b^{2} d\right )} \log \left (-\frac{2 \,{\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) -{\left (2 \, a b d f x + 2 \, a b c f - b^{2} d\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^{2} d f x + b^{2} c f\right )} \tan \left (f x + e\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 \tan{\left (e + f x \right )}\right )^{2} \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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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