Optimal. Leaf size=37 \[ \frac{4}{15} i \tanh ^3(x)-\frac{4}{5} i \tanh (x)-\frac{i \text{sech}^3(x)}{5 (\sinh (x)+i)} \]
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Rubi [A] time = 0.0437954, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2672, 3767} \[ \frac{4}{15} i \tanh ^3(x)-\frac{4}{5} i \tanh (x)-\frac{i \text{sech}^3(x)}{5 (\sinh (x)+i)} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{\text{sech}^4(x)}{i+\sinh (x)} \, dx &=-\frac{i \text{sech}^3(x)}{5 (i+\sinh (x))}-\frac{4}{5} i \int \text{sech}^4(x) \, dx\\ &=-\frac{i \text{sech}^3(x)}{5 (i+\sinh (x))}+\frac{4}{5} \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (x)\right )\\ &=-\frac{i \text{sech}^3(x)}{5 (i+\sinh (x))}-\frac{4}{5} i \tanh (x)+\frac{4}{15} i \tanh ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0642261, size = 35, normalized size = 0.95 \[ -\frac{1}{15} i \left (8 \tanh ^3(x)+\frac{3 \text{sech}^3(x)}{\sinh (x)+i}+12 \tanh (x) \text{sech}^2(x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 93, normalized size = 2.5 \begin{align*}{{\frac{i}{6}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}}-{{\frac{5\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}+{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}-{{\frac{2\,i}{5}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-5}}+{{\frac{5\,i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-{{\frac{11\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-4}-{\frac{3}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21073, size = 277, normalized size = 7.49 \begin{align*} -\frac{32 \, e^{\left (-x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac{32 i \, e^{\left (-2 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} - \frac{96 \, e^{\left (-3 \, x\right )}}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} + \frac{16 i}{-30 i \, e^{\left (-x\right )} - 30 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} - 90 i \, e^{\left (-5 \, x\right )} + 30 \, e^{\left (-6 \, x\right )} - 30 i \, e^{\left (-7 \, x\right )} + 15 \, e^{\left (-8 \, x\right )} - 15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78141, size = 197, normalized size = 5.32 \begin{align*} -\frac{96 \, e^{\left (3 \, x\right )} + 32 i \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16 i}{15 \, e^{\left (8 \, x\right )} + 30 i \, e^{\left (7 \, x\right )} + 30 \, e^{\left (6 \, x\right )} + 90 i \, e^{\left (5 \, x\right )} + 90 i \, e^{\left (3 \, x\right )} - 30 \, e^{\left (2 \, x\right )} + 30 i \, e^{x} - 15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27, size = 72, normalized size = 1.95 \begin{align*} \frac{9 \, e^{\left (2 \, x\right )} - 24 i \, e^{x} - 11}{24 \,{\left (e^{x} - i\right )}^{3}} - \frac{45 \, e^{\left (4 \, x\right )} + 240 i \, e^{\left (3 \, x\right )} - 490 \, e^{\left (2 \, x\right )} - 320 i \, e^{x} + 73}{120 \,{\left (e^{x} + i\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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