Optimal. Leaf size=40 \[ \frac {4 i}{1-i \sinh (x)}-\frac {2 i}{(1-i \sinh (x))^2}+i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.06, antiderivative size = 40, 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} \[ \frac {4 i}{1-i \sinh (x)}-\frac {2 i}{(1-i \sinh (x))^2}+i \log (\sinh (x)+i) \]
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))^5 \, dx &=\int \text {sech}^5(x) (1+i \sinh (x))^5 \, dx\\ &=-\left (i \operatorname {Subst}\left (\int \frac {(1+x)^2}{(1-x)^3} \, dx,x,i \sinh (x)\right )\right )\\ &=-\left (i \operatorname {Subst}\left (\int \left (\frac {1}{1-x}-\frac {4}{(-1+x)^3}-\frac {4}{(-1+x)^2}\right ) \, dx,x,i \sinh (x)\right )\right )\\ &=i \log (i+\sinh (x))-\frac {2 i}{(1-i \sinh (x))^2}+\frac {4 i}{1-i \sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 62, normalized size = 1.55 \[ -\frac {11}{4} i \tanh ^4(x)-\frac {1}{2} i \tanh ^2(x)-\frac {5}{4} i \text {sech}^4(x)+\tan ^{-1}(\sinh (x))+i \log (\cosh (x))-\tanh (x) \text {sech}^3(x)-5 \tanh ^3(x) \text {sech}(x)+\tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 92, normalized size = 2.30 \[ \frac {-i \, x e^{\left (4 \, x\right )} + 4 \, {\left (x - 2\right )} e^{\left (3 \, x\right )} + {\left (6 i \, x - 8 i\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x - 2\right )} e^{x} + {\left (2 i \, e^{\left (4 \, x\right )} - 8 \, e^{\left (3 \, x\right )} - 12 i \, e^{\left (2 \, x\right )} + 8 \, e^{x} + 2 i\right )} \log \left (e^{x} + i\right ) - i \, x}{e^{\left (4 \, x\right )} + 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 0.85 \[ -i \, x - \frac {8 \, {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x}\right )}}{{\left (e^{x} + i\right )}^{4}} + 2 i \, \log \left (e^{x} + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 78, normalized size = 1.95 \[ \frac {8 \left (\frac {\mathrm {sech}\relax (x )^{3}}{4}+\frac {3 \,\mathrm {sech}\relax (x )}{8}\right ) \tanh \relax (x )}{3}+2 \arctan \left ({\mathrm e}^{x}\right )+\frac {5 i}{4 \cosh \relax (x )^{4}}-\frac {5 \sinh \relax (x )}{3 \cosh \relax (x )^{4}}+\frac {5 i \left (\sinh ^{2}\relax (x )\right )}{\cosh \relax (x )^{4}}-\frac {5 \left (\sinh ^{3}\relax (x )\right )}{\cosh \relax (x )^{4}}+i \ln \left (\cosh \relax (x )\right )-\frac {i \left (\tanh ^{2}\relax (x )\right )}{2}-\frac {i \left (\tanh ^{4}\relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 235, normalized size = 5.88 \[ -\frac {5}{2} i \, \tanh \relax (x)^{4} + i \, x - \frac {5 \, {\left (5 \, e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} - 5 \, e^{\left (-7 \, x\right )}\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac {5 \, {\left (e^{\left (-x\right )} - 7 \, e^{\left (-3 \, x\right )} + 7 \, e^{\left (-5 \, x\right )} - e^{\left (-7 \, x\right )}\right )}}{2 \, {\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {4 i \, {\left (e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1} - \frac {20 i}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} - 2 \, \arctan \left (e^{\left (-x\right )}\right ) + i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 90, normalized size = 2.25 \[ -x\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,2{}\mathrm {i}+\frac {16{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {8{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1+{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}-{\mathrm {e}}^x\,4{}\mathrm {i}}-\frac {8}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {16}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \left (i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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