Optimal. Leaf size=16 \[ \log (-\sinh (x)+i)-i \sinh (x) \]
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Rubi [A] time = 0.0584481, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3872, 2833, 43} \[ \log (-\sinh (x)+i)-i \sinh (x) \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{i+\text{csch}(x)} \, dx &=i \int \frac{\cosh (x) \sinh (x)}{i-\sinh (x)} \, dx\\ &=i \operatorname{Subst}\left (\int \frac{x}{i+x} \, dx,x,-\sinh (x)\right )\\ &=i \operatorname{Subst}\left (\int \left (1-\frac{i}{i+x}\right ) \, dx,x,-\sinh (x)\right )\\ &=\log (i-\sinh (x))-i \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0093764, size = 16, normalized size = 1. \[ \log (-\sinh (x)+i)-i \sinh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 20, normalized size = 1.3 \begin{align*} -\ln \left ({\rm csch} \left (x\right ) \right ) -{\frac{i}{{\rm csch} \left (x\right )}}+\ln \left ( i+{\rm csch} \left (x\right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02466, size = 28, normalized size = 1.75 \begin{align*} x + \frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} + 2 \, \log \left (e^{\left (-x\right )} + i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74491, size = 81, normalized size = 5.06 \begin{align*} -\frac{1}{2} \,{\left (2 \, x e^{x} - 4 \, e^{x} \log \left (e^{x} - i\right ) + i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1312, size = 34, normalized size = 2.12 \begin{align*} \frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} - \log \left (-i \, e^{x}\right ) + 2 \, \log \left (e^{x} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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