Optimal. Leaf size=65 \[ \frac{\text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}+\frac{2 x \log \left (e^{2 (a+b x)}+1\right )}{b^2}-\frac{x^2 \tanh (a+b x)}{b}-\frac{x^2}{b}+\frac{x^3}{3} \]
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Rubi [A] time = 0.118195, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3720, 3718, 2190, 2279, 2391, 30} \[ \frac{\text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}+\frac{2 x \log \left (e^{2 (a+b x)}+1\right )}{b^2}-\frac{x^2 \tanh (a+b x)}{b}-\frac{x^2}{b}+\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int x^2 \tanh ^2(a+b x) \, dx &=-\frac{x^2 \tanh (a+b x)}{b}+\frac{2 \int x \tanh (a+b x) \, dx}{b}+\int x^2 \, dx\\ &=-\frac{x^2}{b}+\frac{x^3}{3}-\frac{x^2 \tanh (a+b x)}{b}+\frac{4 \int \frac{e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac{x^2}{b}+\frac{x^3}{3}+\frac{2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac{x^2 \tanh (a+b x)}{b}-\frac{2 \int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{x^2}{b}+\frac{x^3}{3}+\frac{2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac{x^2 \tanh (a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{b^3}\\ &=-\frac{x^2}{b}+\frac{x^3}{3}+\frac{2 x \log \left (1+e^{2 (a+b x)}\right )}{b^2}+\frac{\text{Li}_2\left (-e^{2 (a+b x)}\right )}{b^3}-\frac{x^2 \tanh (a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 3.20437, size = 168, normalized size = 2.58 \[ \frac{-3 \text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )-3 b^2 x^2 \text{sech}(a) \sinh (b x) \text{sech}(a+b x)-3 b^2 x^2 \tanh (a) \sqrt{-\text{csch}^2(a)} e^{-\tanh ^{-1}(\coth (a))}+6 b x \log \left (1-e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )+6 \tanh ^{-1}(\coth (a)) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}(\coth (a))+b x\right )\right )+b x\right )+b^3 x^3+3 i \pi b x-3 i \pi \log \left (e^{2 b x}+1\right )+3 i \pi \log (\cosh (b x))}{3 b^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.055, size = 99, normalized size = 1.5 \begin{align*}{\frac{{x}^{3}}{3}}+2\,{\frac{{x}^{2}}{b \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }}-2\,{\frac{{x}^{2}}{b}}-4\,{\frac{ax}{{b}^{2}}}-2\,{\frac{{a}^{2}}{{b}^{3}}}+2\,{\frac{x\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }{{b}^{2}}}+{\frac{{\it polylog} \left ( 2,-{{\rm e}^{2\,bx+2\,a}} \right ) }{{b}^{3}}}+4\,{\frac{a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3793, size = 113, normalized size = 1.74 \begin{align*} -\frac{2 \, x^{2}}{b} + \frac{b x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b x^{3} + 6 \, x^{2}}{3 \,{\left (b e^{\left (2 \, b x + 2 \, a\right )} + b\right )}} + \frac{2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.25351, size = 1424, normalized size = 21.91 \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^{2} \operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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