Optimal. Leaf size=22 \[ -i x+\cosh (x)+\frac {i \cosh (x)}{\sinh (x)+i} \]
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Rubi [A] time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 2735, 2648} \[ -i x+\cosh (x)+\frac {i \cosh (x)}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx &=\cosh (x)-i \int \frac {\sinh (x)}{i+\sinh (x)} \, dx\\ &=-i x+\cosh (x)-\int \frac {1}{i+\sinh (x)} \, dx\\ &=-i x+\cosh (x)+\frac {i \cosh (x)}{i+\sinh (x)}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 79, normalized size = 3.59 \[ \frac {\cosh (x) \left (\sinh (x)+\frac {2 \sinh (x) \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}+\frac {2 i \sin ^{-1}\left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}+2 i\right )}{\sinh (x)+i} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 38, normalized size = 1.73 \[ \frac {{\left (-2 i \, x + i\right )} e^{\left (2 \, x\right )} + {\left (2 \, x + 5\right )} e^{x} + e^{\left (3 \, x\right )} + i}{2 \, e^{\left (2 \, x\right )} + 2 i \, e^{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 26, normalized size = 1.18 \[ -i \, x + \frac {{\left (5 \, e^{x} + i\right )} e^{\left (-x\right )}}{2 \, {\left (e^{x} + i\right )}} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 52, normalized size = 2.36 \[ i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 33, normalized size = 1.50 \[ -i \, x + \frac {10 \, e^{\left (-x\right )} - 2 i}{4 \, {\left (-i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )}\right )}} + \frac {1}{2} \, e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 24, normalized size = 1.09 \[ \frac {{\mathrm {e}}^{-x}}{2}-x\,1{}\mathrm {i}+\frac {{\mathrm {e}}^x}{2}+\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.91 \[ - i x + \frac {e^{x}}{2} + \frac {e^{- x}}{2} + \frac {2}{e^{x} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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