Optimal. Leaf size=43 \[ -\frac{(A+2 i B) \cosh (x)}{3 (\sinh (x)+i)}-\frac{(B+i A) \cosh (x)}{3 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.0422746, antiderivative size = 43, 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 = {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 = 0.030382, size = 32, normalized size = 0.74 \[ \frac{\cosh (x) (-(A+2 i B) \sinh (x)-2 i A+B)}{3 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 52, normalized size = 1.2 \begin{align*} -2\,{\frac{A}{\tanh \left ( x/2 \right ) +i}}-{(-2\,iA-2\,B) \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}-{\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 = 1.16712, size = 190, normalized size = 4.42 \begin{align*} -2 \, A{\left (\frac{3 \, 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{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 = 1.70625, size = 119, normalized size = 2.77 \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 = 1.26339, size = 43, normalized size = 1. \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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