Optimal. Leaf size=13 \[ -\frac {1}{3} \cot ^3(x)-\cot (x) \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3175, 3767} \[ -\frac {1}{3} \cot ^3(x)-\cot (x) \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{\left (1-\cos ^2(x)\right )^2} \, dx &=\int \csc ^4(x) \, dx\\ &=-\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )\\ &=-\cot (x)-\frac {\cot ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.31 \[ -\frac {2 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 25, normalized size = 1.92 \[ -\frac {2 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)}{3 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 14, normalized size = 1.08 \[ -\frac {3 \, \tan \relax (x)^{2} + 1}{3 \, \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 14, normalized size = 1.08 \[ -\frac {1}{3 \tan \relax (x )^{3}}-\frac {1}{\tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 14, normalized size = 1.08 \[ -\frac {3 \, \tan \relax (x)^{2} + 1}{3 \, \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.25, size = 10, normalized size = 0.77 \[ -\frac {\mathrm {cot}\relax (x)\,\left ({\mathrm {cot}\relax (x)}^2+3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.14, size = 34, normalized size = 2.62 \[ \frac {\tan ^{3}{\left (\frac {x}{2} \right )}}{24} + \frac {3 \tan {\left (\frac {x}{2} \right )}}{8} - \frac {3}{8 \tan {\left (\frac {x}{2} \right )}} - \frac {1}{24 \tan ^{3}{\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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