Optimal. Leaf size=11 \[ -\tanh ^{-1}(\cosh (x))-i x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3888, 3770} \[ -\tanh ^{-1}(\cosh (x))-i x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rule 3888
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{i+\text {csch}(x)} \, dx &=\int (-i+\text {csch}(x)) \, dx\\ &=-i x+\int \text {csch}(x) \, dx\\ &=-i x-\tanh ^{-1}(\cosh (x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 13, normalized size = 1.18 \[ \log \left (\tanh \left (\frac {x}{2}\right )\right )-i x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.97, size = 16, normalized size = 1.45 \[ -i \, x - \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 17, normalized size = 1.55 \[ -i \, x - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.12, size = 27, normalized size = 2.45 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 20, normalized size = 1.82 \[ -i \, x - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 21, normalized size = 1.91 \[ \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )-x\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________