Optimal. Leaf size=12 \[ x^2-x \tanh (x)+\log (\cosh (x)) \]
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Rubi [A] time = 0.0358555, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5473, 3720, 3475, 30} \[ x^2-x \tanh (x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 5473
Rule 3720
Rule 3475
Rule 30
Rubi steps
\begin{align*} \int x \cosh (2 x) \text{sech}^2(x) \, dx &=\int \left (x+x \tanh ^2(x)\right ) \, dx\\ &=\frac{x^2}{2}+\int x \tanh ^2(x) \, dx\\ &=\frac{x^2}{2}-x \tanh (x)+\int x \, dx+\int \tanh (x) \, dx\\ &=x^2+\log (\cosh (x))-x \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0203948, size = 12, normalized size = 1. \[ x^2-x \tanh (x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 26, normalized size = 2.2 \begin{align*}{x}^{2}-2\,x+2\,{\frac{x}{{{\rm e}^{2\,x}}+1}}+\ln \left ({{\rm e}^{2\,x}}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.62947, size = 45, normalized size = 3.75 \begin{align*} \frac{x^{2} +{\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} + 1} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.82887, size = 297, normalized size = 24.75 \begin{align*} \frac{{\left (x^{2} - 2 \, x\right )} \cosh \left (x\right )^{2} + 2 \,{\left (x^{2} - 2 \, x\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (x^{2} - 2 \, x\right )} \sinh \left (x\right )^{2} + x^{2} +{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1} \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 \cosh{\left (2 x \right )} \operatorname{sech}^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14985, size = 63, normalized size = 5.25 \begin{align*} \frac{x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right )}{e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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