Optimal. Leaf size=12 \[ x^2-x \tanh (x)+\log (\cosh (x)) \]
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Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5473, 3720, 3475, 30} \[ x^2-x \tanh (x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 30
Rule 3475
Rule 3720
Rule 5473
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*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.00 \[ x^2-x \tanh (x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 91, normalized size = 7.58 \[ \frac {{\left (x^{2} - 2 \, x\right )} \cosh \relax (x)^{2} + 2 \, {\left (x^{2} - 2 \, x\right )} \cosh \relax (x) \sinh \relax (x) + {\left (x^{2} - 2 \, x\right )} \sinh \relax (x)^{2} + x^{2} + {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 47, normalized size = 3.92 \[ \frac {x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right )}{e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 26, normalized size = 2.17 \[ x^{2}-2 x +\frac {2 x}{1+{\mathrm e}^{2 x}}+\ln \left (1+{\mathrm e}^{2 x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 33, normalized size = 2.75 \[ \frac {x^{2} + {\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} + 1} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 25, normalized size = 2.08 \[ \ln \left ({\mathrm {e}}^{2\,x}+1\right )-2\,x+\frac {2\,x}{{\mathrm {e}}^{2\,x}+1}+x^2 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cosh {\left (2 x \right )} \operatorname {sech}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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