Optimal. Leaf size=77 \[ \frac{i x \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i x^3}{3} \]
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Rubi [A] time = 0.13408, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3719, 2190, 2531, 2282, 6589} \[ \frac{i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i x^3}{3} \]
Antiderivative was successfully verified.
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Rule 3719
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \tan (a+b x) \, dx &=\frac{i x^3}{3}-2 i \int \frac{e^{2 i (a+b x)} x^2}{1+e^{2 i (a+b x)}} \, dx\\ &=\frac{i x^3}{3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{2 \int x \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=\frac{i x^3}{3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{i \int \text{Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=\frac{i x^3}{3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^3}\\ &=\frac{i x^3}{3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.0079411, size = 77, normalized size = 1. \[ \frac{i x \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac{x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 103, normalized size = 1.3 \begin{align*}{\frac{i}{3}}{x}^{3}-{\frac{2\,i{a}^{2}x}{{b}^{2}}}-{\frac{{\frac{4\,i}{3}}{a}^{3}}{{b}^{3}}}-{\frac{{x}^{2}\ln \left ( 1+{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{b}}+{\frac{ix{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-{\frac{{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{2\,{b}^{3}}}+2\,{\frac{{a}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.67243, size = 219, normalized size = 2.84 \begin{align*} -\frac{-2 i \,{\left (b x + a\right )}^{3} + 6 i \,{\left (b x + a\right )}^{2} a - 6 i \, b x{\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, a^{2} \log \left (\sec \left (b x + a\right )\right ) +{\left (6 i \,{\left (b x + a\right )}^{2} - 12 i \,{\left (b x + a\right )} a\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \,{\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )})}{6 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.62134, size = 562, normalized size = 7.3 \begin{align*} -\frac{2 \, b^{2} x^{2} \log \left (-\frac{2 \,{\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 2 \, b^{2} x^{2} \log \left (-\frac{2 \,{\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1}\right ) + 2 i \, b x{\rm Li}_2\left (\frac{2 \,{\left (i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) - 2 i \, b x{\rm Li}_2\left (\frac{2 \,{\left (-i \, \tan \left (b x + a\right ) - 1\right )}}{\tan \left (b x + a\right )^{2} + 1} + 1\right ) +{\rm polylog}\left (3, \frac{\tan \left (b x + a\right )^{2} + 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right ) +{\rm polylog}\left (3, \frac{\tan \left (b x + a\right )^{2} - 2 i \, \tan \left (b x + a\right ) - 1}{\tan \left (b x + a\right )^{2} + 1}\right )}{4 \, b^{3}} \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 x^{2} \tan{\left (a + b 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 x^{2} \tan \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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