Optimal. Leaf size=38 \[ x+\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)} \]
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Rubi [A] time = 0.11, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4391, 2670, 2680, 8} \[ x+\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2670
Rule 2680
Rule 4391
Rubi steps
\begin {align*} \int (\text {sech}(x)-i \tanh (x))^4 \, dx &=\int \text {sech}^4(x) (1-i \sinh (x))^4 \, dx\\ &=\int \frac {\cosh ^4(x)}{(1+i \sinh (x))^4} \, dx\\ &=\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\int \frac {\cosh ^2(x)}{(1+i \sinh (x))^2} \, dx\\ &=\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)}+\int 1 \, dx\\ &=x+\frac {2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac {2 i \cosh (x)}{1+i \sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 1.97 \[ \frac {3 (3 x+8 i) \cosh \left (\frac {x}{2}\right )-(3 x+16 i) \cosh \left (\frac {3 x}{2}\right )+6 i \sinh \left (\frac {x}{2}\right ) (2 x+x \cosh (x)+4 i)}{6 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 52, normalized size = 1.37 \[ \frac {3 \, x e^{\left (3 \, x\right )} + {\left (-9 i \, x - 24 i\right )} e^{\left (2 \, x\right )} - 3 \, {\left (3 \, x + 8\right )} e^{x} + 3 i \, x + 16 i}{3 \, e^{\left (3 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 22, normalized size = 0.58 \[ x - \frac {24 i \, e^{\left (2 \, x\right )} + 24 \, e^{x} - 16 i}{3 \, {\left (e^{x} - i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 60, normalized size = 1.58 \[ -2 \left (\frac {2}{3}+\frac {\mathrm {sech}\relax (x )^{2}}{3}\right ) \tanh \relax (x )+\frac {4 i}{3 \cosh \relax (x )^{3}}+\frac {3 \sinh \relax (x )}{\cosh \relax (x )^{3}}+4 i \left (-\frac {\sinh ^{2}\relax (x )}{\cosh \relax (x )^{3}}-\frac {2}{3 \cosh \relax (x )^{3}}\right )+x -\tanh \relax (x )-\frac {\left (\tanh ^{3}\relax (x )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 181, normalized size = 4.76 \[ -2 \, \tanh \relax (x)^{3} + x - \frac {4 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} - \frac {8 i \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} - \frac {16 i \, e^{\left (-3 \, x\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} - \frac {8 i \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac {4}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} + \frac {32 i}{3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 67, normalized size = 1.76 \[ x+\frac {\frac {{\mathrm {e}}^{2\,x}\,8{}\mathrm {i}}{3}-\frac {8}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x-\mathrm {i}}-\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x-\mathrm {i}\right )}+\frac {{\mathrm {e}}^x\,8{}\mathrm {i}}{3\,\left (1-{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,2{}\mathrm {i}\right )} \]
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 \[ \int \left (- i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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