Optimal. Leaf size=19 \[ -\frac{\cot ^3(x)}{3 a}-\frac{\cot (x)}{a} \]
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Rubi [A] time = 0.0477914, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ -\frac{\cot ^3(x)}{3 a}-\frac{\cot (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin{align*} \int \frac{\csc ^2(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \csc ^4(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{a}\\ &=-\frac{\cot (x)}{a}-\frac{\cot ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0032647, size = 21, normalized size = 1.11 \[ \frac{-\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 18, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{3\, \left ( \tan \left ( x \right ) \right ) ^{3}}}- \left ( \tan \left ( x \right ) \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941513, size = 23, normalized size = 1.21 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, a \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82509, size = 76, normalized size = 4. \begin{align*} -\frac{2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \,{\left (a \cos \left (x\right )^{2} - 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 ^{2}{\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 = 1.17476, size = 23, normalized size = 1.21 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, a \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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