Optimal. Leaf size=18 \[ \frac {\log (\cosh (x))}{a}-\frac {\log (\cosh (x)+1)}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2707, 36, 29, 31} \[ \frac {\log (\cosh (x))}{a}-\frac {\log (\cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2707
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{a+a \cosh (x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,a \cosh (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,a \cosh (x)\right )}{a}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \cosh (x)\right )}{a}\\ &=\frac {\log (\cosh (x))}{a}-\frac {\log (1+\cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 0.67 \[ -\frac {2 \tanh ^{-1}(2 \cosh (x)+1)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 28, normalized size = 1.56 \[ \frac {\log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 22, normalized size = 1.22 \[ \frac {\log \left (e^{\left (2 \, x\right )} + 1\right )}{a} - \frac {2 \, \log \left (e^{x} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 19, normalized size = 1.06 \[ \frac {\ln \left (\cosh \relax (x )\right )}{a}-\frac {\ln \left (1+\cosh \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 24, normalized size = 1.33 \[ -\frac {2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 26, normalized size = 1.44 \[ -\frac {2\,\ln \left (36\,{\mathrm {e}}^x+36\right )-\ln \left (3\,{\mathrm {e}}^{2\,x}+3\right )}{a} \]
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 \[ \frac {\int \frac {\tanh {\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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