Integrand size = 11, antiderivative size = 23 \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=-\frac {\text {arctanh}(\cosh (x))}{2 a}+\frac {1}{2 (a+a \cosh (x))} \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2746, 46, 212} \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=\frac {1}{2 (a \cosh (x)+a)}-\frac {\text {arctanh}(\cosh (x))}{2 a} \]
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Rule 46
Rule 212
Rule 2746
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^2} \, dx,x,a \cosh (x)\right )\right ) \\ & = -\left (a \text {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} \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cosh (x)\right ) \\ & = -\frac {\text {arctanh}(\cosh (x))}{2 a}+\frac {1}{2 (a+a \cosh (x))} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=\frac {1-2 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )}{2 a (1+\cosh (x))} \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{2}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a}\) | \(20\) |
risch | \(\frac {{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right )^{2} a}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2 a}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2 a}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 4.48 \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{2 \, {\left (a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + a\right )}} \]
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\[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\operatorname {csch}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=\frac {e^{\left (-x\right )}}{2 \, a e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26 \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=-\frac {\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, a} + \frac {\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, a} + \frac {e^{\left (-x\right )} + e^{x} + 6}{4 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}} \]
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Time = 1.59 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {\text {csch}(x)}{a+a \cosh (x)} \, dx=\frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{\sqrt {-a^2}} \]
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