Optimal. Leaf size=135 \[ \frac{2 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{2 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac{2 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^3}+\frac{2 \sinh (a+b x)}{b^3}-\frac{2 x \cosh (a+b x)}{b^2}-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^2 \sinh (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.129759, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5449, 3296, 2637, 4180, 2531, 2282, 6589} \[ \frac{2 i x \text{PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{PolyLog}\left (2,i e^{a+b x}\right )}{b^2}-\frac{2 i \text{PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac{2 i \text{PolyLog}\left (3,i e^{a+b x}\right )}{b^3}+\frac{2 \sinh (a+b x)}{b^3}-\frac{2 x \cosh (a+b x)}{b^2}-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^2 \sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5449
Rule 3296
Rule 2637
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \sinh (a+b x) \tanh (a+b x) \, dx &=\int x^2 \cosh (a+b x) \, dx-\int x^2 \text{sech}(a+b x) \, dx\\ &=-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^2 \sinh (a+b x)}{b}+\frac{(2 i) \int x \log \left (1-i e^{a+b x}\right ) \, dx}{b}-\frac{(2 i) \int x \log \left (1+i e^{a+b x}\right ) \, dx}{b}-\frac{2 \int x \sinh (a+b x) \, dx}{b}\\ &=-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{2 x \cosh (a+b x)}{b^2}+\frac{2 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac{x^2 \sinh (a+b x)}{b}-\frac{(2 i) \int \text{Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^2}+\frac{(2 i) \int \text{Li}_2\left (i e^{a+b x}\right ) \, dx}{b^2}+\frac{2 \int \cosh (a+b x) \, dx}{b^2}\\ &=-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{2 x \cosh (a+b x)}{b^2}+\frac{2 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac{2 \sinh (a+b x)}{b^3}+\frac{x^2 \sinh (a+b x)}{b}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac{2 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac{2 x \cosh (a+b x)}{b^2}+\frac{2 i x \text{Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac{2 i x \text{Li}_2\left (i e^{a+b x}\right )}{b^2}-\frac{2 i \text{Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac{2 i \text{Li}_3\left (i e^{a+b x}\right )}{b^3}+\frac{2 \sinh (a+b x)}{b^3}+\frac{x^2 \sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 1.42879, size = 153, normalized size = 1.13 \[ -\frac{i \left (-2 b x \text{PolyLog}\left (2,-i e^{a+b x}\right )+2 b x \text{PolyLog}\left (2,i e^{a+b x}\right )+2 \text{PolyLog}\left (3,-i e^{a+b x}\right )-2 \text{PolyLog}\left (3,i e^{a+b x}\right )+b^2 x^2 \log \left (1-i e^{a+b x}\right )-b^2 x^2 \log \left (1+i e^{a+b x}\right )+i b^2 x^2 \sinh (a+b x)+2 i \sinh (a+b x)-2 i b x \cosh (a+b x)\right )}{b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\rm sech} \left (bx+a\right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} -{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{3}} - 2 \, \int \frac{x^{2} e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.35461, size = 1345, normalized size = 9.96 \begin{align*} -\frac{b^{2} x^{2} -{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} - 2 \,{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) -{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x -{\left (-4 i \, b x \cosh \left (b x + a\right ) - 4 i \, b x \sinh \left (b x + a\right )\right )}{\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) -{\left (4 i \, b x \cosh \left (b x + a\right ) + 4 i \, b x \sinh \left (b x + a\right )\right )}{\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) -{\left (-2 i \, a^{2} \cosh \left (b x + a\right ) - 2 i \, a^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) -{\left (2 i \, a^{2} \cosh \left (b x + a\right ) + 2 i \, a^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) -{\left ({\left (2 i \, b^{2} x^{2} - 2 i \, a^{2}\right )} \cosh \left (b x + a\right ) +{\left (2 i \, b^{2} x^{2} - 2 i \, a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) -{\left ({\left (-2 i \, b^{2} x^{2} + 2 i \, a^{2}\right )} \cosh \left (b x + a\right ) +{\left (-2 i \, b^{2} x^{2} + 2 i \, a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) -{\left (4 i \, \cosh \left (b x + a\right ) + 4 i \, \sinh \left (b x + a\right )\right )}{\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) -{\left (-4 i \, \cosh \left (b x + a\right ) - 4 i \, \sinh \left (b x + a\right )\right )}{\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 2}{2 \,{\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh ^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]