Optimal. Leaf size=16 \[ \log (-\sinh (x)+i)-i \sinh (x) \]
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Rubi [A] time = 0.06, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 43
Rule 2833
Rule 3872
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*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \log (-\sinh (x)+i)-i \sinh (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 28, normalized size = 1.75 \[ -\frac {1}{2} \, {\left (2 \, x e^{x} - 4 \, e^{x} \log \left (e^{x} - i\right ) + i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 25, normalized size = 1.56 \[ \frac {1}{2} i \, e^{\left (-x\right )} - \frac {1}{2} i \, e^{x} - \log \left (-i \, e^{x}\right ) + 2 \, \log \left (e^{x} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 20, normalized size = 1.25 \[ \ln \left (i+\mathrm {csch}\relax (x )\right )-\ln \left (\mathrm {csch}\relax (x )\right )-\frac {i}{\mathrm {csch}\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 21, normalized size = 1.31 \[ x + \frac {1}{2} i \, e^{\left (-x\right )} - \frac {1}{2} i \, e^{x} + 2 \, \log \left (e^{\left (-x\right )} + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 12, normalized size = 0.75 \[ \ln \left (\mathrm {sinh}\relax (x)-\mathrm {i}\right )-\mathrm {sinh}\relax (x)\,1{}\mathrm {i} \]
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 \[ \int \frac {\cosh {\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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