Optimal. Leaf size=19 \[ \frac {\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2747, 2648, 3770} \[ \frac {\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2648
Rule 2747
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx &=-(i \int \text {csch}(x) \, dx)+i \int \frac {1}{i+\sinh (x)} \, dx\\ &=i \tanh ^{-1}(\cosh (x))+\frac {\cosh (x)}{i+\sinh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 30, normalized size = 1.58 \[ \text {sech}(x) \left (\sinh (x)+i \sqrt {\cosh ^2(x)} \tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right )-i\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 33, normalized size = 1.74 \[ \frac {{\left (i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + {\left (-i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 24, normalized size = 1.26 \[ -\frac {2 i}{e^{x} + i} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 21, normalized size = 1.11 \[ -i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 29, normalized size = 1.53 \[ -\frac {2 i}{e^{\left (-x\right )} - i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.49, size = 35, normalized size = 1.84 \[ -\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}-2{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}+2{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}{\relax (x )}}{\sinh {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________