Optimal. Leaf size=49 \[ \frac{(A-2 i B) \cosh (x)}{3 (-\sinh (x)+i)}+\frac{(-B+i A) \cosh (x)}{3 (-\sinh (x)+i)^2} \]
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Rubi [A] time = 0.0435185, antiderivative size = 49, 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 = {2750, 2648} \[ \frac{(A-2 i B) \cosh (x)}{3 (-\sinh (x)+i)}+\frac{(-B+i A) \cosh (x)}{3 (-\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \sinh (x)}{(i-\sinh (x))^2} \, dx &=\frac{(i A-B) \cosh (x)}{3 (i-\sinh (x))^2}+\frac{1}{3} (-i A-2 B) \int \frac{1}{i-\sinh (x)} \, dx\\ &=\frac{(i A-B) \cosh (x)}{3 (i-\sinh (x))^2}+\frac{(A-2 i B) \cosh (x)}{3 (i-\sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.0279167, size = 32, normalized size = 0.65 \[ \frac{\cosh (x) (-(A-2 i B) \sinh (x)+2 i A+B)}{3 (\sinh (x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 52, normalized size = 1.1 \begin{align*} -{(2\,iA-2\,B) \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}-2\,{\frac{A}{\tanh \left ( x/2 \right ) -i}}-{\frac{-4\,iB-4\,A}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.30827, size = 190, normalized size = 3.88 \begin{align*} -A{\left (\frac{6 \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} + \frac{2 i}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i}\right )} + \frac{1}{2} \, B{\left (\frac{12 i \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} + \frac{12 \, e^{\left (-2 \, x\right )}}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} - \frac{8}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8014, size = 119, normalized size = 2.43 \begin{align*} -\frac{6 \, B e^{\left (2 \, x\right )} + 6 \,{\left (A - i \, B\right )} e^{x} - 2 i \, A - 4 \, B}{3 \, e^{\left (3 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29165, size = 43, normalized size = 0.88 \begin{align*} -\frac{6 \, B e^{\left (2 \, x\right )} + 6 \, A e^{x} - 6 i \, B e^{x} - 2 i \, A - 4 \, B}{3 \,{\left (e^{x} - i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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