Optimal. Leaf size=14 \[ \frac {i \csc (x)}{\cot (x)+i} \]
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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 = {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*}
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Mathematica [A] time = 0.00, size = 9, normalized size = 0.64 \[ \sin (x)+i \cos (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 6, normalized size = 0.43 \[ i \, e^{\left (-i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 10, normalized size = 0.71 \[ \frac {2}{\tan \left (\frac {1}{2} \, x\right ) - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 12, normalized size = 0.86 \[ \frac {2}{\tan \left (\frac {x}{2}\right )-i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 15, normalized size = 1.07 \[ \frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 14, normalized size = 1.00 \[ \frac {2{}\mathrm {i}}{1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 8, normalized size = 0.57 \[ \frac {i \csc {\relax (x )}}{\cot {\relax (x )} + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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