Optimal. Leaf size=23 \[ B x-\frac{(B+i A) \cosh (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.037834, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2735, 2648} \[ B x-\frac{(B+i A) \cosh (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \sinh (x)}{i+\sinh (x)} \, dx &=B x-(-A+i B) \int \frac{1}{i+\sinh (x)} \, dx\\ &=B x-\frac{(i A+B) \cosh (x)}{i+\sinh (x)}\\ \end{align*}
Mathematica [B] time = 0.23284, size = 53, normalized size = 2.3 \[ \cosh (x) \left (\frac{2 i B \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (x)}}{\sqrt{2}}\right )}{\sqrt{\cosh ^2(x)}}-\frac{B+i A}{\sinh (x)+i}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 46, normalized size = 2. \begin{align*} B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -2\,{\frac{B}{\tanh \left ( x/2 \right ) +i}}-{2\,iA \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}}-B\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 = 1.26283, size = 35, normalized size = 1.52 \begin{align*} B{\left (x + \frac{2 i}{e^{\left (-x\right )} - i}\right )} - \frac{2 \, A}{e^{\left (-x\right )} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8033, size = 58, normalized size = 2.52 \begin{align*} \frac{B x e^{x} + i \, B x - 2 \, A + 2 i \, B}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.217398, size = 15, normalized size = 0.65 \begin{align*} B x + \frac{- 2 A + 2 i B}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3178, size = 23, normalized size = 1. \begin{align*} B x - \frac{2 \,{\left (A - i \, B\right )}}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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