Optimal. Leaf size=24 \[ \frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a} \]
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Rubi [A] time = 0.046257, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ \frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\csc ^2(x)}{a+a \cos (x)} \, dx &=\frac{\csc (x)}{3 (a+a \cos (x))}+\frac{2 \int \csc ^2(x) \, dx}{3 a}\\ &=\frac{\csc (x)}{3 (a+a \cos (x))}-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{3 a}\\ &=-\frac{2 \cot (x)}{3 a}+\frac{\csc (x)}{3 (a+a \cos (x))}\\ \end{align*}
Mathematica [A] time = 0.0480526, size = 30, normalized size = 1.25 \[ -\frac{(2 \cos (x)+\cos (2 x)) \csc \left (\frac{x}{2}\right ) \sec ^3\left (\frac{x}{2}\right )}{12 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 29, normalized size = 1.2 \begin{align*}{\frac{1}{4\,a} \left ({\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+2\,\tan \left ( x/2 \right ) - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1243, size = 55, normalized size = 2.29 \begin{align*} \frac{\frac{6 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{12 \, a} - \frac{\cos \left (x\right ) + 1}{4 \, a \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57161, size = 78, normalized size = 3.25 \begin{align*} -\frac{2 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) - 1}{3 \,{\left (a \cos \left (x\right ) + a\right )} \sin \left (x\right )} \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 ^{2}{\left (x \right )}}{\cos{\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 = 1.13177, size = 50, normalized size = 2.08 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, x\right )^{3} + 6 \, a^{2} \tan \left (\frac{1}{2} \, x\right )}{12 \, a^{3}} - \frac{1}{4 \, a \tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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