Optimal. Leaf size=23 \[ \frac {1}{2 (a \cosh (x)+a)}-\frac {\tanh ^{-1}(\cosh (x))}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2667, 44, 206} \[ \frac {1}{2 (a \cosh (x)+a)}-\frac {\tanh ^{-1}(\cosh (x))}{2 a} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx &=-\left (a \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^2} \, dx,x,a \cosh (x)\right )\right )\\ &=-\left (a \operatorname {Subst}\left (\int \left (\frac {1}{2 a (a+x)^2}+\frac {1}{2 a \left (a^2-x^2\right )}\right ) \, dx,x,a \cosh (x)\right )\right )\\ &=\frac {1}{2 (a+a \cosh (x))}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cosh (x)\right )\\ &=-\frac {\tanh ^{-1}(\cosh (x))}{2 a}+\frac {1}{2 (a+a \cosh (x))}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 1.83 \[ \frac {1-2 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )}{2 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.75, size = 103, normalized size = 4.48 \[ -\frac {{\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) - 2 \, \cosh \relax (x) - 2 \, \sinh \relax (x)}{2 \, {\left (a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + a\right )} \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 52, normalized size = 2.26 \[ -\frac {\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac {\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} + \frac {e^{\left (-x\right )} + e^{x} + 6}{4 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{4 a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 47, normalized size = 2.04 \[ \frac {e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 51, normalized size = 2.22 \[ \frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {\operatorname {csch}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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