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 = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 8
Rule 2672
Rule 3767
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.40, size = 28, normalized size = 1.12 \[ -\frac {8 \, e^{x} + 4 i}{3 \, e^{\left (4 \, x\right )} + 6 i \, e^{\left (3 \, x\right )} + 6 i \, e^{x} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 29, normalized size = 1.16 \[ \frac {1}{2 \, {\left (e^{x} - i\right )}} - \frac {3 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} - 5}{6 \, {\left (e^{x} + i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 49, normalized size = 1.96 \[ -\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {3 i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )}-\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 53, normalized size = 2.12 \[ -\frac {8 \, e^{\left (-x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} + \frac {4 i}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 63, normalized size = 2.52 \[ -\frac {8\,{\mathrm {e}}^x}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {8\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{2\,x}-1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}+\frac {{\mathrm {e}}^{2\,x}\,16{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {\left ({\mathrm {e}}^{2\,x}-1\right )\,4{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\sinh {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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