Optimal. Leaf size=30 \[ -\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a} \]
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Rubi [A] time = 0.0799864, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2607, 30, 2606} \[ -\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2606
Rubi steps
\begin{align*} \int \frac{\cot ^2(x)}{a+a \cos (x)} \, dx &=-\frac{\int \cot ^3(x) \csc (x) \, dx}{a}+\frac{\int \cot ^2(x) \csc ^2(x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (x)\right )}{a}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (x)\right )}{a}\\ &=-\frac{\cot ^3(x)}{3 a}-\frac{\csc (x)}{a}+\frac{\csc ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0523389, size = 25, normalized size = 0.83 \[ \frac{(-4 \cos (x)+\cos (2 x)-3) \csc (x)}{6 a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 29, normalized size = 1. \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 [A] time = 1.00078, size = 57, normalized size = 1.9 \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 = 1.38026, size = 74, normalized size = 2.47 \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 2}{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{\cot ^{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 = 1.2764, size = 50, normalized size = 1.67 \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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