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 = 0.05, antiderivative size = 38, 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 = {4476, 2759, 8}
\begin {gather*} x-\frac {2 i \cosh ^3(x)}{3 (1-i \sinh (x))^3}+\frac {2 i \cosh (x)}{1-i \sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2759
Rule 4476
Rubi steps
\begin {align*} \int \frac {1}{(\text {sech}(x)-i \tanh (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 = 0.01, size = 74, normalized size = 1.95 \begin {gather*} \frac {3 (-8 i+3 x) \cosh \left (\frac {x}{2}\right )+(16 i-3 x) \cosh \left (\frac {3 x}{2}\right )-6 i (-4 i+2 x+x \cosh (x)) \sinh \left (\frac {x}{2}\right )}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.90, size = 41, normalized size = 1.08
method | result | size |
risch | \(x +\frac {8 i \left (3 i {\mathrm e}^{x}+3 \,{\mathrm e}^{2 x}-2\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}\) | \(26\) |
default | \(\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {8 i}{\left (i+\tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {16}{3 \left (i+\tanh \left (\frac {x}{2}\right )\right )^{3}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 40, normalized size = 1.05 \begin {gather*} x - \frac {8 \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - 2 i\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 52, normalized size = 1.37 \begin {gather*} \frac {3 \, x e^{\left (3 \, x\right )} - 3 \, {\left (-3 i \, x - 8 i\right )} e^{\left (2 \, x\right )} - 3 \, {\left (3 \, x + 8\right )} e^{x} - 3 i \, x - 16 i}{3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (- i \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 22, normalized size = 0.58 \begin {gather*} x - \frac {8 \, {\left (-3 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + 2 i\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.61, size = 65, normalized size = 1.71 \begin {gather*} 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 {{\mathrm {e}}^x\,8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}\right )}+\frac {8{}\mathrm {i}}{3\,\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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