Optimal. Leaf size=4 \[ x+\tanh (x) \]
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Rubi [A] time = 0.07, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3657, 3473, 8} \[ x+\tanh (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3657
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x) \left (2-\tanh ^2(x)\right )}{1-\tanh ^2(x)} \, dx &=\int \left (2-\tanh ^2(x)\right ) \, dx\\ &=2 x-\int \tanh ^2(x) \, dx\\ &=2 x+\tanh (x)-\int 1 \, dx\\ &=x+\tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 4, normalized size = 1.00 \[ x+\tanh (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 14, normalized size = 3.50 \[ \frac {{\left (x - 1\right )} \cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 12, normalized size = 3.00 \[ x - \frac {2}{e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 34, normalized size = 8.50 \[ -\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {2 \tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 12, normalized size = 3.00 \[ x + \frac {2}{e^{\left (-2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.66, size = 12, normalized size = 3.00 \[ x-\frac {2}{{\mathrm {e}}^{2\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.81, size = 29, normalized size = 7.25 \[ - \frac {x \operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} - 1} - \frac {\tanh {\relax (x )} \operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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