Optimal. Leaf size=28 \[ \frac {2 i}{1+i \sinh (x)}+i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.05, antiderivative size = 28, 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 \frac {1}{(\text {sech}(x)+i \tanh (x))^3} \, dx &=\int \frac {\cosh ^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 {1}{-1-x}+\frac {2}{(1+x)^2}\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 = 0.04, size = 40, normalized size = 1.43 \[ -2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )+i \log (\cosh (x))+\frac {2 i}{\left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 49, normalized size = 1.75 \[ \frac {-i \, x e^{\left (2 \, x\right )} - 2 \, {\left (x - 2\right )} e^{x} + {\left (2 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} - 2 i\right )} \log \left (e^{x} - i\right ) + i \, x}{e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 27, normalized size = 0.96 \[ \frac {4 \, e^{x}}{{\left (e^{x} - i\right )}^{2}} - i \, \log \left (i \, e^{x}\right ) + 2 i \, \log \left (-i \, e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 56, normalized size = 2.00 \[ -i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+2 i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )-\frac {4 i}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}}-\frac {4}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 33, normalized size = 1.18 \[ i \, x - \frac {4 \, e^{\left (-x\right )}}{2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} + 2 i \, \log \left (e^{\left (-x\right )} + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 41, normalized size = 1.46 \[ -x\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,2{}\mathrm {i}-\frac {4{}\mathrm {i}}{1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}}+\frac {4}{{\mathrm {e}}^x-\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.80, size = 432, normalized size = 15.43 \[ - \frac {2 i x \tanh ^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} - \frac {4 x \tanh {\relax (x )} \operatorname {sech}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {2 i x \operatorname {sech}^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} - \frac {2 i \log {\left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )} \right )} \tanh ^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} - \frac {4 \log {\left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )} \right )} \tanh {\relax (x )} \operatorname {sech}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {2 i \log {\left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )} \right )} \operatorname {sech}^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {2 i \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh ^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {4 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )} \operatorname {sech}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} - \frac {2 i \log {\left (\tanh {\relax (x )} + 1 \right )} \operatorname {sech}^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {i \tanh ^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {i \operatorname {sech}^{2}{\relax (x )}}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} + \frac {i}{- 2 \tanh ^{2}{\relax (x )} + 4 i \tanh {\relax (x )} \operatorname {sech}{\relax (x )} + 2 \operatorname {sech}^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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