Optimal. Leaf size=14 \[ x-\frac{\cosh (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.0271906, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ x-\frac{\cosh (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{i+\sinh (x)} \, dx &=x-i \int \frac{1}{i+\sinh (x)} \, dx\\ &=x-\frac{\cosh (x)}{i+\sinh (x)}\\ \end{align*}
Mathematica [B] time = 0.0538288, size = 43, normalized size = 3.07 \[ i \text{sech}(x) \left (i \sinh (x)+2 \sqrt{\cosh ^2(x)} \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (x)}}{\sqrt{2}}\right )+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 29, normalized size = 2.1 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -2\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-1}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15381, size = 16, normalized size = 1.14 \begin{align*} x + \frac{2 i}{e^{\left (-x\right )} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07709, size = 42, normalized size = 3. \begin{align*} \frac{x e^{x} + i \, x + 2 i}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.157277, size = 8, normalized size = 0.57 \begin{align*} x + \frac{2 i}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29781, size = 14, normalized size = 1. \begin{align*} x + \frac{2 i}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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