Integrand size = 11, antiderivative size = 20 \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=\frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2800, 36, 29, 31} \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=\frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \]
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Rule 29
Rule 31
Rule 36
Rule 2800
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \sinh (x)\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,b \sinh (x)\right )}{a}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{a} \\ & = \frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=\frac {\log (\sinh (x))}{a}-\frac {\log (a+b \sinh (x))}{a} \]
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Time = 0.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a}\) | \(33\) |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\) | \(36\) |
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none
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.00 \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=-\frac {\log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \]
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\[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=\int \frac {\coth {\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=-\frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a} \]
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Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=-\frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a} + \frac {\log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a} \]
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Time = 0.42 (sec) , antiderivative size = 195, normalized size of antiderivative = 9.75 \[ \int \frac {\coth (x)}{a+b \sinh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^2}+b\,{\mathrm {e}}^x\,\sqrt {-a^2}-2\,a\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^2}-b\,{\mathrm {e}}^{3\,x}\,\sqrt {-a^2}}{a^2}\right )}{\sqrt {-a^2}}-\frac {2\,\mathrm {atan}\left (\left (4\,a^4\,b\,\sqrt {-a^2}+4\,a^2\,b^3\,\sqrt {-a^2}\right )\,\left (\frac {1}{8\,a\,b\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^x\,\left (\frac {1}{16\,b^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^4\,b^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^3\,b\,{\left (a^2+b^2\right )}^2}\right )\right )}{\sqrt {-a^2}} \]
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