Optimal. Leaf size=9 \[ \log (\sin (x))-i x \]
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Rubi [A] time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3487, 31} \[ \log (\sin (x))-i x \]
Antiderivative was successfully verified.
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Rule 31
Rule 3487
Rubi steps
\begin {align*} \int \frac {\csc ^2(x)}{i+\cot (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{i+x} \, dx,x,\cot (x)\right )\\ &=-i x+\log (\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 9, normalized size = 1.00 \[ \log (\sin (x))-i x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 11, normalized size = 1.22 \[ -2 i \, x + \log \left (e^{\left (2 i \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 15, normalized size = 1.67 \[ -2 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - i\right ) + \log \left (\tan \left (\frac {1}{2} \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 9, normalized size = 1.00 \[ -\ln \left (i+\cot \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 7, normalized size = 0.78 \[ -\log \left (\cot \relax (x) + i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 11, normalized size = 1.22 \[ -\mathrm {atan}\left (2\,\mathrm {tan}\relax (x)-\mathrm {i}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.82, size = 7, normalized size = 0.78 \[ - \log {\left (\cot {\relax (x )} + i \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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