Optimal. Leaf size=9 \[ \frac {\log (\cosh (x)+1)}{a} \]
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Rubi [A] time = 0.02, 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 = {2667, 31} \[ \frac {\log (\cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{a+a \cosh (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \cosh (x)\right )}{a}\\ &=\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 = 0.78, 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 = 0.16, 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 = 0.04, size = 12, normalized size = 1.33 \[ \frac {\ln \left (a +a \cosh \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 11, normalized size = 1.22 \[ \frac {\log \left (a \cosh \relax (x) + a\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 9, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {cosh}\relax (x)+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 7, normalized size = 0.78 \[ \frac {\log {\left (\cosh {\relax (x )} + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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