Optimal. Leaf size=27 \[ -B x+\frac{(-B+i A) \cosh (x)}{-\sinh (x)+i} \]
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Rubi [A] time = 0.042611, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, 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 = 0.0753079, size = 59, normalized size = 2.19 \[ \frac{\left (-\sinh \left (\frac{x}{2}\right )+i \cosh \left (\frac{x}{2}\right )\right ) \left (B x \cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right ) (2 A+B (x+2 i))\right )}{\sinh (x)-i} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 46, normalized size = 1.7 \begin{align*} -B\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19698, size = 36, normalized size = 1.33 \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 = 1.7196, size = 59, normalized size = 2.19 \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 = 0.209746, size = 15, normalized size = 0.56 \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 = 1.29128, size = 24, normalized size = 0.89 \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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