Optimal. Leaf size=98 \[ -\frac{3 i x \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac{3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac{x^3 \tan (a+b x)}{b}-\frac{i x^3}{b}-\frac{x^4}{4} \]
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Rubi [A] time = 0.16711, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {3720, 3719, 2190, 2531, 2282, 6589, 30} \[ -\frac{3 i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac{3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac{x^3 \tan (a+b x)}{b}-\frac{i x^3}{b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3719
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 30
Rubi steps
\begin{align*} \int x^3 \tan ^2(a+b x) \, dx &=\frac{x^3 \tan (a+b x)}{b}-\frac{3 \int x^2 \tan (a+b x) \, dx}{b}-\int x^3 \, dx\\ &=-\frac{i x^3}{b}-\frac{x^4}{4}+\frac{x^3 \tan (a+b x)}{b}+\frac{(6 i) \int \frac{e^{2 i (a+b x)} x^2}{1+e^{2 i (a+b x)}} \, dx}{b}\\ &=-\frac{i x^3}{b}-\frac{x^4}{4}+\frac{3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}+\frac{x^3 \tan (a+b x)}{b}-\frac{6 \int x \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{i x^3}{b}-\frac{x^4}{4}+\frac{3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac{3 i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{x^3 \tan (a+b x)}{b}+\frac{(3 i) \int \text{Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac{i x^3}{b}-\frac{x^4}{4}+\frac{3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac{3 i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{x^3 \tan (a+b x)}{b}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}\\ &=-\frac{i x^3}{b}-\frac{x^4}{4}+\frac{3 x^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^2}-\frac{3 i x \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^3}+\frac{3 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^4}+\frac{x^3 \tan (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.619273, size = 115, normalized size = 1.17 \[ \frac{6 i b x \text{PolyLog}\left (2,-e^{-2 i (a+b x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 i (a+b x)}\right )+2 b^2 x^2 \left (\frac{2 i b x}{1+e^{2 i a}}+3 \log \left (1+e^{-2 i (a+b x)}\right )\right )}{2 b^4}+\frac{x^3 \sec (a) \sin (b x) \sec (a+b x)}{b}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 133, normalized size = 1.4 \begin{align*} -{\frac{{x}^{4}}{4}}+{\frac{2\,i{x}^{3}}{b \left ( 1+{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }}-{\frac{2\,i{x}^{3}}{b}}+{\frac{6\,i{a}^{2}x}{{b}^{3}}}+{\frac{4\,i{a}^{3}}{{b}^{4}}}+3\,{\frac{{x}^{2}\ln \left ( 1+{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-{\frac{3\,ix{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+{\frac{3\,{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{2\,{b}^{4}}}-6\,{\frac{{a}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.78277, size = 864, normalized size = 8.82 \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 [C] time = 1.56632, size = 614, normalized size = 6.27 \begin{align*} -\frac{b^{4} x^{4} - 4 \, b^{3} x^{3} \tan \left (b x + a\right ) - 6 \, 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 ) - 6 \, 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 ) - 6 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 ) + 6 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 ) - 3 \,{\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 ) - 3 \,{\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^{4}} \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^{3} \tan ^{2}{\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^{3} \tan \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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