Optimal. Leaf size=33 \[ \frac {\text {csch}^2(x)}{2 a}-\frac {\tanh ^{-1}(\cosh (x))}{2 a}-\frac {\coth (x) \text {csch}(x)}{2 a} \]
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Rubi [A] time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {3872, 2706, 2606, 30, 2611, 3770} \[ \frac {\text {csch}^2(x)}{2 a}-\frac {\tanh ^{-1}(\cosh (x))}{2 a}-\frac {\coth (x) \text {csch}(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2706
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\coth (x)}{-a-a \cosh (x)} \, dx\\ &=\frac {\int \coth ^2(x) \text {csch}(x) \, dx}{a}-\frac {\int \coth (x) \text {csch}^2(x) \, dx}{a}\\ &=-\frac {\coth (x) \text {csch}(x)}{2 a}+\frac {\int \text {csch}(x) \, dx}{2 a}-\frac {\operatorname {Subst}(\int x \, dx,x,-i \text {csch}(x))}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (x))}{2 a}-\frac {\coth (x) \text {csch}(x)}{2 a}+\frac {\text {csch}^2(x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 44, normalized size = 1.33 \[ -\frac {\text {sech}(x) \left (2 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+1\right )}{2 a (\text {sech}(x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 103, normalized size = 3.12 \[ -\frac {{\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (\cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) + 1\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 2 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, \cosh \relax (x) + 2 \, \sinh \relax (x)}{2 \, {\left (a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (a \cosh \relax (x) + a\right )} \sinh \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 52, normalized size = 1.58 \[ -\frac {\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac {\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} + \frac {e^{\left (-x\right )} + e^{x} - 2}{4 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 23, normalized size = 0.70 \[ \frac {\tanh ^{2}\left (\frac {x}{2}\right )}{4 a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 48, normalized size = 1.45 \[ -\frac {e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 51, normalized size = 1.55 \[ \frac {1}{a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {\operatorname {csch}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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