Optimal. Leaf size=37 \[ -\frac{2 \tanh (x)}{5}-\frac{\text{sech}(x)}{5 (\sinh (x)+i)}-\frac{i \text{sech}(x)}{5 (\sinh (x)+i)^2} \]
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Rubi [A] time = 0.0708574, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ -\frac{2 \tanh (x)}{5}-\frac{\text{sech}(x)}{5 (\sinh (x)+i)}-\frac{i \text{sech}(x)}{5 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{(i+\sinh (x))^2} \, dx &=-\frac{i \text{sech}(x)}{5 (i+\sinh (x))^2}-\frac{3}{5} i \int \frac{\text{sech}^2(x)}{i+\sinh (x)} \, dx\\ &=-\frac{i \text{sech}(x)}{5 (i+\sinh (x))^2}-\frac{\text{sech}(x)}{5 (i+\sinh (x))}-\frac{2}{5} \int \text{sech}^2(x) \, dx\\ &=-\frac{i \text{sech}(x)}{5 (i+\sinh (x))^2}-\frac{\text{sech}(x)}{5 (i+\sinh (x))}-\frac{2}{5} i \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=-\frac{i \text{sech}(x)}{5 (i+\sinh (x))^2}-\frac{\text{sech}(x)}{5 (i+\sinh (x))}-\frac{2 \tanh (x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0218572, size = 31, normalized size = 0.84 \[ -\frac{\text{sech}(x) (-5 \sinh (x)+\sinh (3 x)+4 i \cosh (2 x))}{10 (\sinh (x)+i)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 70, normalized size = 1.9 \begin{align*} -{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}-{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-4}}+{{\frac{5\,i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}-{\frac{4}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-5}}+3\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-3}-{\frac{7}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.31682, size = 158, normalized size = 4.27 \begin{align*} -\frac{16 i \, e^{\left (-x\right )}}{20 i \, e^{\left (-x\right )} - 25 \, e^{\left (-2 \, x\right )} - 25 \, e^{\left (-4 \, x\right )} - 20 i \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-6 \, x\right )} + 5} + \frac{20 \, e^{\left (-2 \, x\right )}}{20 i \, e^{\left (-x\right )} - 25 \, e^{\left (-2 \, x\right )} - 25 \, e^{\left (-4 \, x\right )} - 20 i \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-6 \, x\right )} + 5} - \frac{4}{20 i \, e^{\left (-x\right )} - 25 \, e^{\left (-2 \, x\right )} - 25 \, e^{\left (-4 \, x\right )} - 20 i \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-6 \, x\right )} + 5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73613, size = 132, normalized size = 3.57 \begin{align*} -\frac{4 \,{\left (5 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} - 1\right )}}{5 \, e^{\left (6 \, x\right )} + 20 i \, e^{\left (5 \, x\right )} - 25 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} - 20 i \, e^{x} + 5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{\left (\sinh{\left (x \right )} + i\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23532, size = 55, normalized size = 1.49 \begin{align*} -\frac{i}{4 \,{\left (e^{x} - i\right )}} - \frac{-5 i \, e^{\left (4 \, x\right )} + 30 \, e^{\left (3 \, x\right )} + 80 i \, e^{\left (2 \, x\right )} - 50 \, e^{x} - 11 i}{20 \,{\left (e^{x} + i\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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