Optimal. Leaf size=14 \[ \frac{i \coth (x)}{\text{csch}(x)+i} \]
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Rubi [A] time = 0.0205694, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3794} \[ \frac{i \coth (x)}{\text{csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 3794
Rubi steps
\begin{align*} \int \frac{\text{csch}(x)}{i+\text{csch}(x)} \, dx &=\frac{i \coth (x)}{i+\text{csch}(x)}\\ \end{align*}
Mathematica [A] time = 0.0187179, size = 27, normalized size = 1.93 \[ \frac{2 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 12, normalized size = 0.9 \begin{align*} 2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02864, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{i \, e^{\left (-x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35672, size = 20, normalized size = 1.43 \begin{align*} \frac{2 i}{e^{x} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15148, size = 11, normalized size = 0.79 \begin{align*} \frac{2 i}{e^{x} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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