Optimal. Leaf size=9 \[ \frac {\log (\cosh (x)+1)}{a} \]
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Rubi [A] time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3879, 31} \[ \frac {\log (\cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3879
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{a+a \text {sech}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a+a x} \, dx,x,\cosh (x)\right )\\ &=\frac {\log (1+\cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.33 \[ \frac {2 \log \left (\cosh \left (\frac {x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 16, normalized size = 1.78 \[ -\frac {x - 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 17, normalized size = 1.89 \[ -\frac {x}{a} + \frac {2 \, \log \left (e^{x} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 19, normalized size = 2.11 \[ \frac {\ln \left (1+\mathrm {sech}\relax (x )\right )}{a}-\frac {\ln \left (\mathrm {sech}\relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 18, normalized size = 2.00 \[ \frac {x}{a} + \frac {2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 14, normalized size = 1.56 \[ -\frac {x-2\,\ln \left ({\mathrm {e}}^x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 19, normalized size = 2.11 \[ \frac {x}{a} - \frac {\log {\left (\tanh {\relax (x )} + 1 \right )}}{a} + \frac {\log {\left (\operatorname {sech}{\relax (x )} + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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