Optimal. Leaf size=20 \[ x-2 i \cosh (x)-\frac {\cosh (x)}{\text {csch}(x)+i} \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3819, 3787, 2638, 8} \[ x-2 i \cosh (x)-\frac {\cosh (x)}{\text {csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2638
Rule 3787
Rule 3819
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{i+\text {csch}(x)} \, dx &=-\frac {\cosh (x)}{i+\text {csch}(x)}+\int (-2 i+\text {csch}(x)) \sinh (x) \, dx\\ &=-\frac {\cosh (x)}{i+\text {csch}(x)}-2 i \int \sinh (x) \, dx+\int 1 \, dx\\ &=x-2 i \cosh (x)-\frac {\cosh (x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 35, normalized size = 1.75 \[ x-i \cosh (x)-\frac {2 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 40, normalized size = 2.00 \[ \frac {{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + {\left (-2 i \, x - 5 i\right )} e^{x} - i \, e^{\left (3 \, x\right )} - 1}{2 \, e^{\left (2 \, x\right )} - 2 i \, e^{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 26, normalized size = 1.30 \[ x + \frac {{\left (5 \, e^{x} - i\right )} e^{\left (-x\right )}}{2 \, {\left (i \, e^{x} + 1\right )}} - \frac {1}{2} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 51, normalized size = 2.55 \[ \frac {i}{\tanh \left (\frac {x}{2}\right )-1}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {i}{\tanh \left (\frac {x}{2}\right )+1}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 31, normalized size = 1.55 \[ x - \frac {5 i \, e^{\left (-x\right )} - 1}{2 \, {\left (i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )}\right )}} - \frac {1}{2} i \, e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 24, normalized size = 1.20 \[ x-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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