Optimal. Leaf size=19 \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 19, 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 = {2707, 36, 29, 31} \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2707
Rubi steps
\begin {align*} \int \frac {\coth (x)}{i+\sinh (x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (i+x)} \, dx,x,\sinh (x)\right )\\ &=-\left (i \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\sinh (x)\right )\right )+i \operatorname {Subst}\left (\int \frac {1}{i+x} \, dx,x,\sinh (x)\right )\\ &=-i \log (\sinh (x))+i \log (i+\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ i \log (\sinh (x)+i)-i \log (\sinh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 17, normalized size = 0.89 \[ -i \, \log \left (e^{\left (2 \, x\right )} - 1\right ) + 2 i \, \log \left (e^{x} + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 23, normalized size = 1.21 \[ -i \, \log \left (e^{x} + 1\right ) + 2 i \, \log \left (e^{x} + i\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 0.89 \[ -i \ln \left (\sinh \relax (x )\right )+i \ln \left (i+\sinh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 28, normalized size = 1.47 \[ -i \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 i \, \log \left (e^{\left (-x\right )} - i\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 24, normalized size = 1.26 \[ \ln \left (-36\,{\mathrm {e}}^x-36{}\mathrm {i}\right )\,2{}\mathrm {i}-\ln \left (3-3\,{\mathrm {e}}^{2\,x}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 15, normalized size = 0.79 \[ - 2 \log {\left (e^{x} + i \right )} + \log {\left (e^{2 x} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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