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 = 0.07, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 8
Rule 2672
Rule 3767
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.64, size = 46, normalized size = 1.24 \[ -\frac {20 \, e^{\left (2 \, x\right )} + 16 i \, e^{x} - 4}{5 \, e^{\left (6 \, x\right )} + 20 i \, e^{\left (5 \, x\right )} - 25 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} - 20 i \, e^{x} + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 41, normalized size = 1.11 \[ -\frac {i}{4 \, {\left (e^{x} - i\right )}} - \frac {-5 i \, e^{\left (4 \, x\right )} + 30 \, e^{\left (3 \, x\right )} + 80 i \, e^{\left (2 \, x\right )} - 50 \, e^{x} - 11 i}{20 \, {\left (e^{x} + i\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 70, normalized size = 1.89 \[ -\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{4}}+\frac {5 i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4}{5 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{5}}+\frac {3}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {7}{4 \left (\tanh \left (\frac {x}{2}\right )+i\right )}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 117, normalized size = 3.16 \[ -\frac {16 i \, e^{\left (-x\right )}}{20 i \, e^{\left (-x\right )} - 25 \, e^{\left (-2 \, x\right )} - 25 \, e^{\left (-4 \, x\right )} - 20 i \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-6 \, x\right )} + 5} + \frac {20 \, e^{\left (-2 \, x\right )}}{20 i \, e^{\left (-x\right )} - 25 \, e^{\left (-2 \, x\right )} - 25 \, e^{\left (-4 \, x\right )} - 20 i \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-6 \, x\right )} + 5} - \frac {4}{20 i \, e^{\left (-x\right )} - 25 \, e^{\left (-2 \, x\right )} - 25 \, e^{\left (-4 \, x\right )} - 20 i \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-6 \, x\right )} + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 109, normalized size = 2.95 \[ -\frac {16\,{\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )}{5\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^5}-\frac {\left (4\,{\mathrm {e}}^{2\,x}-\frac {4}{5}\right )\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^5}-\frac {{\mathrm {e}}^x\,\left ({\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1\right )\,16{}\mathrm {i}}{5\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^5}+\frac {\left (4\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^x\right )\,\left (4\,{\mathrm {e}}^{2\,x}-\frac {4}{5}\right )\,1{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^5} \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\left (\sinh {\relax (x )} + i\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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