Optimal. Leaf size=24 \[ -\frac {i}{2 (\sinh (x)+i)}-\frac {1}{2} i \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 24, 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 = {2667, 44, 203} \[ -\frac {i}{2 (\sinh (x)+i)}-\frac {1}{2} i \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 2667
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{i+\sinh (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(i-x) (i+x)^2} \, dx,x,\sinh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {i}{2 (i+x)^2}+\frac {i}{2 \left (1+x^2\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=-\frac {i}{2 (i+\sinh (x))}-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{2} i \tan ^{-1}(\sinh (x))-\frac {i}{2 (i+\sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 0.75 \[ -\frac {1}{2} i \left (\tan ^{-1}(\sinh (x))+\frac {1}{\sinh (x)+i}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 53, normalized size = 2.21 \[ \frac {{\left (e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )} \log \left (e^{x} + i\right ) - {\left (e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1\right )} \log \left (e^{x} - i\right ) - 2 i \, e^{x}}{2 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 51, normalized size = 2.12 \[ -\frac {e^{\left (-x\right )} - e^{x} - 6 i}{4 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}} + \frac {1}{4} \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {1}{4} \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 43, normalized size = 1.79 \[ \frac {i}{\tanh \left (\frac {x}{2}\right )+i}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{2}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 41, normalized size = 1.71 \[ \frac {2 i \, e^{\left (-x\right )}}{-4 i \, e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} - 2} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + i\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 46, normalized size = 1.92 \[ \frac {\ln \left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{2}-\frac {1}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {1{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
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 \[ \int \frac {\operatorname {sech}{\relax (x )}}{\sinh {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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