Optimal. Leaf size=23 \[ B x-\frac {(B+i A) \cosh (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.04, antiderivative size = 23, 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 = {2735, 2648} \[ B x-\frac {(B+i A) \cosh (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
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*}
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Mathematica [B] time = 0.25, size = 53, normalized size = 2.30 \[ \cosh (x) \left (\frac {2 i B \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}-\frac {B+i A}{\sinh (x)+i}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 23, normalized size = 1.00 \[ \frac {B x e^{x} + i \, B x - 2 \, A + 2 i \, B}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 17, normalized size = 0.74 \[ B x - \frac {2 \, {\left (A - i \, B\right )}}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 46, normalized size = 2.00 \[ -\frac {2 B}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2 i A}{\tanh \left (\frac {x}{2}\right )+i}-B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 26, normalized size = 1.13 \[ B {\left (x + \frac {2 i}{e^{\left (-x\right )} - i}\right )} - \frac {2 \, A}{e^{\left (-x\right )} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 21, normalized size = 0.91 \[ B\,x-\frac {2\,A-B\,2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.83 \[ B x + \frac {- 2 i A - 2 B}{i e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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