Optimal. Leaf size=9 \[ \log (\sin (x))-i x \]
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Rubi [A] time = 0.032013, antiderivative size = 9, normalized size of antiderivative = 1., 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 3487
Rule 31
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*}
Mathematica [A] time = 0.0032833, size = 9, normalized size = 1. \[ \log (\sin (x))-i x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 9, normalized size = 1. \begin{align*} -\ln \left ( i+\cot \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21216, size = 9, normalized size = 1. \begin{align*} -\log \left (\cot \left (x\right ) + i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (e^{\left (2 i \, x\right )} - 1\right )} e^{\left (2 i \, x\right )}{\rm integral}\left (\frac{2 i \, e^{\left (-2 i \, x\right )}}{e^{\left (2 i \, x\right )} - 1}, x\right ) - e^{\left (2 i \, x\right )} + 1\right )} e^{\left (-2 i \, x\right )}}{e^{\left (2 i \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25351, size = 22, normalized size = 2.44 \begin{align*} -2 \, \log \left (\tan \left (\frac{1}{2} \, x\right ) - i\right ) + \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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