Optimal. Leaf size=14 \[ \frac{i \csc (x)}{\cot (x)+i} \]
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Rubi [A] time = 0.0187808, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {3488} \[ \frac{i \csc (x)}{\cot (x)+i} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin{align*} \int \frac{\csc (x)}{i+\cot (x)} \, dx &=\frac{i \csc (x)}{i+\cot (x)}\\ \end{align*}
Mathematica [A] time = 0.0035542, size = 9, normalized size = 0.64 \[ \sin (x)+i \cos (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 12, normalized size = 0.9 \begin{align*} 2\, \left ( \tan \left ( x/2 \right ) -i \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17676, size = 20, normalized size = 1.43 \begin{align*} \frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - i} \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*}{\rm integral}\left (\frac{{\left (e^{\left (3 i \, x\right )} - e^{\left (i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{e^{\left (2 i \, x\right )} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0056, size = 8, normalized size = 0.57 \begin{align*} \frac{i \csc{\left (x \right )}}{\cot{\left (x \right )} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23846, size = 14, normalized size = 1. \begin{align*} \frac{2}{\tan \left (\frac{1}{2} \, x\right ) - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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