Optimal. Leaf size=19 \[ -\frac {\cot ^3(x)}{3 a}-\frac {\cot (x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.47, size = 29, normalized size = 1.53 \[ -\frac {2 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)}{3 \, {\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 17, normalized size = 0.89 \[ -\frac {3 \, \tan \relax (x)^{2} + 1}{3 \, a \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 18, normalized size = 0.95 \[ \frac {-\frac {1}{3 \tan \relax (x )^{3}}-\frac {1}{\tan \relax (x )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 17, normalized size = 0.89 \[ -\frac {3 \, \tan \relax (x)^{2} + 1}{3 \, a \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.03, size = 13, normalized size = 0.68 \[ -\frac {\mathrm {cot}\relax (x)\,\left ({\mathrm {cot}\relax (x)}^2+3\right )}{3\,a} \]
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 \[ - \frac {\int \frac {\csc ^{2}{\relax (x )}}{\cos ^{2}{\relax (x )} - 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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