Optimal. Leaf size=26 \[ -i \log (i+\sinh (x))-\frac {2 i}{1-i \sinh (x)} \]
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Rubi [A]
time = 0.04, 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 = {4476, 2746, 45}
\begin {gather*} -\frac {2 i}{1-i \sinh (x)}-i \log (\sinh (x)+i) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rule 4476
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\\ &=i \text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,-i \sinh (x)\right )\\ &=i \text {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,-i \sinh (x)\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 = 27, normalized size = 1.04 \begin {gather*} -2 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )-i \log (\cosh (x))+\frac {2}{i+\sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 55 vs. \(2 (22 ) = 44\).
time = 2.25, size = 56, normalized size = 2.15
method | result | size |
risch | \(i x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}+i\right )^{2}}-2 i \ln \left ({\mathrm e}^{x}+i\right )\) | \(26\) |
default | \(i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\frac {4 i}{\left (i+\tanh \left (\frac {x}{2}\right )\right )^{2}}-2 i \ln \left (i+\tanh \left (\frac {x}{2}\right )\right )-\frac {4}{i+\tanh \left (\frac {x}{2}\right )}+i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 33, normalized size = 1.27 \begin {gather*} -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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 50 vs. \(2 (18) = 36\).
time = 0.41, size = 50, normalized size = 1.92 \begin {gather*} \frac {i \, x e^{\left (2 \, x\right )} - 2 \, {\left (x - 2\right )} e^{x} - 2 \, {\left (i \, e^{\left (2 \, x\right )} - 2 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) - i \, x}{e^{\left (2 \, x\right )} + 2 i \, e^{x} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 432 vs. \(2 (19) = 38\).
time = 0.82, size = 432, normalized size = 16.62 \begin {gather*} \frac {2 i x \tanh ^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {4 x \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {2 i x \operatorname {sech}^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {2 i \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} + \frac {4 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} + \frac {2 i \log {\left (\tanh {\left (x \right )} + 1 \right )} \operatorname {sech}^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} + \frac {2 i \log {\left (\tanh {\left (x \right )} + i \operatorname {sech}{\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {4 \log {\left (\tanh {\left (x \right )} + i \operatorname {sech}{\left (x \right )} \right )} \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {2 i \log {\left (\tanh {\left (x \right )} + i \operatorname {sech}{\left (x \right )} \right )} \operatorname {sech}^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {i \tanh ^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {i \operatorname {sech}^{2}{\left (x \right )}}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} - \frac {i}{- 2 \tanh ^{2}{\left (x \right )} - 4 i \tanh {\left (x \right )} \operatorname {sech}{\left (x \right )} + 2 \operatorname {sech}^{2}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 27, normalized size = 1.04 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 39, normalized size = 1.50 \begin {gather*} 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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