Optimal. Leaf size=14 \[ \frac {i \coth (x)}{\text {csch}(x)+i} \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3794} \[ \frac {i \coth (x)}{\text {csch}(x)+i} \]
Antiderivative was successfully verified.
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Rule 3794
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{i+\text {csch}(x)} \, dx &=\frac {i \coth (x)}{i+\text {csch}(x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.93 \[ \frac {2 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 8, normalized size = 0.57 \[ \frac {2 i}{e^{x} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 8, normalized size = 0.57 \[ \frac {2 i}{e^{x} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 12, normalized size = 0.86 \[ \frac {2}{\tanh \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 12, normalized size = 0.86 \[ -\frac {2}{i \, e^{\left (-x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 10, normalized size = 0.71 \[ \frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]
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 \[ \int \frac {\operatorname {csch}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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