Optimal. Leaf size=113 \[ -\frac{6 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac{6 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^3}+\frac{6 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^4}-\frac{6 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^4}+\frac{6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{sech}(a+b x)}{b} \]
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Rubi [A] time = 0.105284, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5418, 4180, 2531, 2282, 6589} \[ -\frac{6 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac{6 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^3}+\frac{6 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^4}-\frac{6 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^4}+\frac{6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5418
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \text{sech}(a+b x) \tanh (a+b x) \, dx &=-\frac{x^3 \text{sech}(a+b x)}{b}+\frac{3 \int x^2 \text{sech}(a+b x) \, dx}{b}\\ &=\frac{6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{sech}(a+b x)}{b}-\frac{(6 i) \int x \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}+\frac{(6 i) \int x \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac{6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{6 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac{6 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac{x^3 \text{sech}(a+b x)}{b}+\frac{(6 i) \int \text{Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^3}-\frac{(6 i) \int \text{Li}_2\left (i e^{a+b x}\right ) \, dx}{b^3}\\ &=\frac{6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{6 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac{6 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac{x^3 \text{sech}(a+b x)}{b}+\frac{(6 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac{(6 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=\frac{6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{6 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac{6 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac{6 i \text{Li}_3\left (-i e^{a+b x}\right )}{b^4}-\frac{6 i \text{Li}_3\left (i e^{a+b x}\right )}{b^4}-\frac{x^3 \text{sech}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 2.19236, size = 130, normalized size = 1.15 \[ -\frac{x^3 \text{sech}(a+b x)}{b}+\frac{3 i \left (-2 b x \text{PolyLog}\left (2,-i e^{a+b x}\right )+2 b x \text{PolyLog}\left (2,i e^{a+b x}\right )+2 \text{PolyLog}\left (3,-i e^{a+b x}\right )-2 \text{PolyLog}\left (3,i e^{a+b x}\right )+b^2 x^2 \log \left (1-i e^{a+b x}\right )-b^2 x^2 \log \left (1+i e^{a+b x}\right )\right )}{b^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}\sinh \left ( bx+a \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x^{3} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} + 6 \, \int \frac{x^{2} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.02914, size = 1890, normalized size = 16.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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