Optimal. Leaf size=25 \[ -\frac{2}{3} i \tanh (x)-\frac{i \text{sech}(x)}{3 (\sinh (x)+i)} \]
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Rubi [A] time = 0.0413912, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ -\frac{2}{3} i \tanh (x)-\frac{i \text{sech}(x)}{3 (\sinh (x)+i)} \]
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)} \, dx &=-\frac{i \text{sech}(x)}{3 (i+\sinh (x))}-\frac{2}{3} i \int \text{sech}^2(x) \, dx\\ &=-\frac{i \text{sech}(x)}{3 (i+\sinh (x))}+\frac{2}{3} \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=-\frac{i \text{sech}(x)}{3 (i+\sinh (x))}-\frac{2}{3} i \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0322801, size = 22, normalized size = 0.88 \[ -\frac{1}{3} i \left (2 \tanh (x)+\frac{\text{sech}(x)}{\sinh (x)+i}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 49, normalized size = 2. \begin{align*}{-{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}- \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}+{{\frac{2\,i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-3}}-{{\frac{3\,i}{2}} \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 = 1.10298, size = 72, normalized size = 2.88 \begin{align*} -\frac{8 \, e^{\left (-x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} + \frac{4 i}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63876, size = 76, normalized size = 3.04 \begin{align*} -\frac{8 \, e^{x} + 4 i}{3 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 6 i \, e^{x} - 3} \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 )}}{\sinh{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.272, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{2 \,{\left (e^{x} - i\right )}} - \frac{3 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} - 5}{6 \,{\left (e^{x} + i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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