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 = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 2648
Rule 2750
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.46, size = 46, normalized size = 1.07 \[ -\frac {6 \, B e^{\left (2 \, x\right )} + {\left (6 \, A + 6 i \, B\right )} e^{x} + 2 i \, A - 4 \, B}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 32, normalized size = 0.74 \[ -\frac {6 \, B e^{\left (2 \, x\right )} + 6 \, A e^{x} + 6 i \, B e^{x} + 2 i \, A - 4 \, B}{3 \, {\left (e^{x} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 1.21 \[ -\frac {-2 i A -2 B}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {2 A}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2 \left (2 i B -2 A \right )}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 141, normalized size = 3.28 \[ -2 \, A {\left (\frac {3 \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} - \frac {i}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i}\right )} + \frac {1}{2} \, B {\left (-\frac {12 i \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} + \frac {12 \, e^{\left (-2 \, x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} - \frac {8}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 39, normalized size = 0.91 \[ -\frac {\frac {2\,A}{3}+\frac {B\,4{}\mathrm {i}}{3}-{\mathrm {e}}^x\,\left (-2\,B+A\,2{}\mathrm {i}\right )-B\,{\mathrm {e}}^{2\,x}\,2{}\mathrm {i}}{{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 51, normalized size = 1.19 \[ \frac {2 i A + 6 B e^{2 x} - 4 B + \left (6 A + 6 i B\right ) e^{x}}{- 3 e^{3 x} - 9 i e^{2 x} + 9 e^{x} + 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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