Optimal. Leaf size=26 \[ -\frac {2 i}{1-i \sinh (x)}-i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.05, antiderivative size = 26, 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 = {4391, 2667, 43} \[ -\frac {2 i}{1-i \sinh (x)}-i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4391
Rubi steps
\begin {align*} \int (\text {sech}(x)+i \tanh (x))^3 \, dx &=\int \text {sech}^3(x) (1+i \sinh (x))^3 \, dx\\ &=-\left (i \operatorname {Subst}\left (\int \frac {1+x}{(1-x)^2} \, dx,x,i \sinh (x)\right )\right )\\ &=-\left (i \operatorname {Subst}\left (\int \left (\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,i \sinh (x)\right )\right )\\ &=-i \log (i+\sinh (x))-\frac {2 i}{1-i \sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.50 \[ \frac {1}{2} i \tanh ^2(x)-\frac {3}{2} i \text {sech}^2(x)-\tan ^{-1}(\sinh (x))-i \log (\cosh (x))+2 \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 49, normalized size = 1.88 \[ \frac {i \, x e^{\left (2 \, x\right )} - 2 \, {\left (x - 2\right )} e^{x} + {\left (-2 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} + 2 i\right )} \log \left (e^{x} + i\right ) - i \, x}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 21, normalized size = 0.81 \[ i \, x + \frac {4 \, e^{x}}{{\left (e^{x} + i\right )}^{2}} - 2 i \, \log \left (e^{x} + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 41, normalized size = 1.58 \[ -\mathrm {sech}\relax (x ) \tanh \relax (x )-2 \arctan \left ({\mathrm e}^{x}\right )-\frac {3 i}{2 \cosh \relax (x )^{2}}+\frac {3 \sinh \relax (x )}{\cosh \relax (x )^{2}}-i \ln \left (\cosh \relax (x )\right )+\frac {i \left (\tanh ^{2}\relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 73, normalized size = 2.81 \[ \frac {3}{2} i \, \tanh \relax (x)^{2} - i \, x + \frac {4 \, {\left (e^{\left (-x\right )} - e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \frac {2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \arctan \left (e^{\left (-x\right )}\right ) - i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 39, normalized size = 1.50 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,2{}\mathrm {i}-\frac {4{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {4}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
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 \[ \int \left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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