Optimal. Leaf size=12 \[ -\frac{\cot (x)}{a \csc (x)+a} \]
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Rubi [A] time = 0.0198414, antiderivative size = 12, 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 = {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*}
Mathematica [B] time = 0.0239544, 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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Maple [A] time = 0.024, size = 14, normalized size = 1.2 \begin{align*} -2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01956, size = 22, normalized size = 1.83 \begin{align*} -\frac{2}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.444167, size = 68, normalized size = 5.67 \begin{align*} -\frac{\cos \left (x\right ) - \sin \left (x\right ) + 1}{a \cos \left (x\right ) + a \sin \left (x\right ) + a} \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*} \frac{\int \frac{\csc{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3368, size = 18, normalized size = 1.5 \begin{align*} -\frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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