Integrand size = 13, antiderivative size = 23 \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=-\text {arctanh}(\cosh (x))+2 i \coth (x)+\frac {\coth (x)}{i+\sinh (x)} \]
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Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2847, 2827, 3852, 8, 3855} \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=-\text {arctanh}(\cosh (x))+2 i \coth (x)+\frac {\coth (x)}{\sinh (x)+i} \]
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Rule 8
Rule 2827
Rule 2847
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\coth (x)}{i+\sinh (x)}+\int \text {csch}^2(x) (-2 i+\sinh (x)) \, dx \\ & = \frac {\coth (x)}{i+\sinh (x)}-2 i \int \text {csch}^2(x) \, dx+\int \text {csch}(x) \, dx \\ & = -\text {arctanh}(\cosh (x))+\frac {\coth (x)}{i+\sinh (x)}-2 \text {Subst}(\int 1 \, dx,x,-i \coth (x)) \\ & = -\text {arctanh}(\cosh (x))+2 i \coth (x)+\frac {\coth (x)}{i+\sinh (x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=\text {sech}(x) \left (1-\text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)}+i \text {csch}(x)+2 i \sinh (x)\right ) \]
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Time = 2.72 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
method | result | size |
default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}\) | \(35\) |
risch | \(\frac {-4+2 i {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right ) \left ({\mathrm e}^{x}+i\right )}+\ln \left ({\mathrm e}^{x}-1\right )-\ln \left ({\mathrm e}^{x}+1\right )\) | \(42\) |
parallelrisch | \(\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2 i\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )+i \tanh \left (\frac {x}{2}\right )^{2}+6 i-\coth \left (\frac {x}{2}\right )}{2 \tanh \left (\frac {x}{2}\right )+2 i}\) | \(46\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35 \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=-\frac {{\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} + 1\right ) - {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 2 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 4}{e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i} \]
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\[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=\int \frac {\operatorname {csch}^{2}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=-\frac {2 \, {\left (-i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 2\right )}}{e^{\left (-x\right )} + i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=\frac {2 \, {\left (e^{\left (2 \, x\right )} + i \, e^{x} - 2\right )}}{e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 1.40 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {\text {csch}^2(x)}{i+\sinh (x)} \, dx=\ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+\frac {2\,{\mathrm {e}}^{2\,x}-4+{\mathrm {e}}^x\,2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}+{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x-\mathrm {i}} \]
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