Optimal. Leaf size=21 \[ -\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x) \]
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Rubi [A] time = 0.0171482, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3175, 3767} \[ -\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x) \]
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 )^3} \, dx &=\int \csc ^6(x) \, dx\\ &=-\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (x)\right )\\ &=-\cot (x)-\frac{2 \cot ^3(x)}{3}-\frac{\cot ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.003592, size = 27, normalized size = 1.29 \[ -\frac{8 \cot (x)}{15}-\frac{1}{5} \cot (x) \csc ^4(x)-\frac{4}{15} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 20, normalized size = 1. \begin{align*} - \left ( \tan \left ( x \right ) \right ) ^{-1}-{\frac{2}{3\, \left ( \tan \left ( x \right ) \right ) ^{3}}}-{\frac{1}{5\, \left ( \tan \left ( x \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951776, size = 27, normalized size = 1.29 \begin{align*} -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58894, size = 112, normalized size = 5.33 \begin{align*} -\frac{8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )}{15 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.11888, size = 54, normalized size = 2.57 \begin{align*} \frac{\tan ^{5}{\left (\frac{x}{2} \right )}}{160} + \frac{5 \tan ^{3}{\left (\frac{x}{2} \right )}}{96} + \frac{5 \tan{\left (\frac{x}{2} \right )}}{16} - \frac{5}{16 \tan{\left (\frac{x}{2} \right )}} - \frac{5}{96 \tan ^{3}{\left (\frac{x}{2} \right )}} - \frac{1}{160 \tan ^{5}{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16443, size = 27, normalized size = 1.29 \begin{align*} -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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