Optimal. Leaf size=18 \[ i x+i \cot (x)+\log (\tan (x))+\log (\cos (x)) \]
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Rubi [A] time = 0.0333998, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3516, 44} \[ i x+i \cot (x)+\log (\tan (x))+\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3516
Rule 44
Rubi steps
\begin{align*} \int \frac{\csc ^2(x)}{i+\tan (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^2 (i+x)} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{-i-x}-\frac{i}{x^2}+\frac{1}{x}\right ) \, dx,x,\tan (x)\right )\\ &=i x+i \cot (x)+\log (\cos (x))+\log (\tan (x))\\ \end{align*}
Mathematica [A] time = 0.0191565, size = 15, normalized size = 0.83 \[ i x+i \cot (x)+\log (\sin (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 20, normalized size = 1.1 \begin{align*} -\ln \left ( i+\tan \left ( x \right ) \right ) +\ln \left ( \tan \left ( x \right ) \right ) +{\frac{i}{\tan \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40432, size = 23, normalized size = 1.28 \begin{align*} \frac{i}{\tan \left (x\right )} - \log \left (\tan \left (x\right ) + i\right ) + \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02021, size = 78, normalized size = 4.33 \begin{align*} \frac{{\left (e^{\left (2 i \, x\right )} - 1\right )} \log \left (e^{\left (2 i \, x\right )} - 1\right ) - 2}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (x \right )}}{\tan{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32035, size = 24, normalized size = 1.33 \begin{align*} \frac{i}{\tan \left (x\right )} - \log \left (\tan \left (x\right ) + i\right ) + \log \left ({\left | \tan \left (x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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