Optimal. Leaf size=34 \[ -\frac {4 i \cosh (x)}{3 (\sinh (x)+i)}+\frac {\cosh (x)}{3 (\sinh (x)+i)^2}+\tanh ^{-1}(\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2766, 2978, 12, 3770} \[ -\frac {4 i \cosh (x)}{3 (\sinh (x)+i)}+\frac {\cosh (x)}{3 (\sinh (x)+i)^2}+\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2766
Rule 2978
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{(i+\sinh (x))^2} \, dx &=\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {1}{3} \int \frac {\text {csch}(x) (3 i-\sinh (x))}{i+\sinh (x)} \, dx\\ &=\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))}+\frac {1}{3} i \int 3 i \text {csch}(x) \, dx\\ &=\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))}-\int \text {csch}(x) \, dx\\ &=\tanh ^{-1}(\cosh (x))+\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.09, size = 91, normalized size = 2.68 \[ \frac {\cosh \left (\frac {x}{2}\right ) \left (6-9 \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )+\cosh \left (\frac {3 x}{2}\right ) \left (3 \log \left (\tanh \left (\frac {x}{2}\right )\right )-8\right )+6 i \sinh \left (\frac {x}{2}\right ) \left (2 \log \left (\tanh \left (\frac {x}{2}\right )\right )+\cosh (x) \log \left (\tanh \left (\frac {x}{2}\right )\right )-3\right )}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 82, normalized size = 2.41 \[ \frac {{\left (3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i\right )} \log \left (e^{x} + 1\right ) - {\left (3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i\right )} \log \left (e^{x} - 1\right ) - 6 \, e^{\left (2 \, x\right )} - 18 i \, e^{x} + 8}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 34, normalized size = 1.00 \[ -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 9 i \, e^{x} - 4\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} + \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 44, normalized size = 1.29 \[ \frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {4 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.40, size = 55, normalized size = 1.62 \[ \frac {2 \, {\left (-9 i \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - 4\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} + \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.28, size = 41, normalized size = 1.21 \[ \ln \left ({\mathrm {e}}^x+1\right )-\ln \left ({\mathrm {e}}^x-1\right )-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {2{}\mathrm {i}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^2}-\frac {4}{3\,{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}{\relax (x )}}{\left (\sinh {\relax (x )} + i\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________