Optimal. Leaf size=19 \[ -\frac {1}{3} \text {sech}^3(x)-\frac {1}{3} i \tanh ^3(x) \]
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Rubi [A] time = 0.11, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3872, 2839, 2606, 30, 2607} \[ -\frac {1}{3} \text {sech}^3(x)-\frac {1}{3} i \tanh ^3(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2607
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{i+\text {csch}(x)} \, dx &=i \int \frac {\text {sech}(x) \tanh (x)}{i-\sinh (x)} \, dx\\ &=-\left (i \int \text {sech}^2(x) \tanh ^2(x) \, dx\right )+\int \text {sech}^3(x) \tanh (x) \, dx\\ &=-\operatorname {Subst}\left (\int x^2 \, dx,x,\text {sech}(x)\right )+\operatorname {Subst}\left (\int x^2 \, dx,x,i \tanh (x)\right )\\ &=-\frac {1}{3} \text {sech}^3(x)-\frac {1}{3} i \tanh ^3(x)\\ \end {align*}
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Mathematica [B] time = 0.06, size = 64, normalized size = 3.37 \[ \frac {-2 i \sinh (x)+\cosh (x)+\cosh (2 x)+i \sinh (x) \cosh (x)-3}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 33, normalized size = 1.74 \[ \frac {6 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} - 2 i}{3 \, e^{\left (4 \, x\right )} - 6 i \, e^{\left (3 \, x\right )} - 6 i \, e^{x} - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 27, normalized size = 1.42 \[ -\frac {i}{2 \, {\left (i \, e^{x} - 1\right )}} + \frac {3 \, e^{\left (2 \, x\right )} - 1}{6 \, {\left (e^{x} - i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 49, normalized size = 2.58 \[ -\frac {i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )-2 i}-\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )-i\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 81, normalized size = 4.26 \[ \frac {8 \, e^{\left (-x\right )}}{12 i \, e^{\left (-x\right )} + 12 i \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} - 6} - \frac {12 i \, e^{\left (-2 \, x\right )}}{12 i \, e^{\left (-x\right )} + 12 i \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} - 6} + \frac {4 i}{12 i \, e^{\left (-x\right )} + 12 i \, e^{\left (-3 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} - 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 31, normalized size = 1.63 \[ -\frac {\frac {2}{3}-2\,{\mathrm {e}}^{2\,x}+\frac {{\mathrm {e}}^x\,4{}\mathrm {i}}{3}}{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,{\left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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