Optimal. Leaf size=12 \[ -\frac {\cot (x)}{a \csc (x)+a} \]
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Rubi [A] time = 0.02, antiderivative size = 12, 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 = {3794} \[ -\frac {\cot (x)}{a \csc (x)+a} \]
Antiderivative was successfully verified.
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Rule 3794
Rubi steps
\begin {align*} \int \frac {\csc (x)}{a+a \csc (x)} \, dx &=-\frac {\cot (x)}{a+a \csc (x)}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 26, normalized size = 2.17 \[ \frac {2 \sin \left (\frac {x}{2}\right )}{a \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 22, normalized size = 1.83 \[ -\frac {\cos \relax (x) - \sin \relax (x) + 1}{a \cos \relax (x) + a \sin \relax (x) + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 13, normalized size = 1.08 \[ -\frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 14, normalized size = 1.17 \[ -\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 16, normalized size = 1.33 \[ -\frac {2}{a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 13, normalized size = 1.08 \[ -\frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc {\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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