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 = 0.02, antiderivative size = 21, 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} \[ -\frac {1}{5} \cot ^5(x)-\frac {2 \cot ^3(x)}{3}-\cot (x) \]
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*}
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Mathematica [A] time = 0.00, 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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fricas [B] time = 0.58, size = 37, normalized size = 1.76 \[ -\frac {8 \, \cos \relax (x)^{5} - 20 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)}{15 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 20, normalized size = 0.95 \[ -\frac {15 \, \tan \relax (x)^{4} + 10 \, \tan \relax (x)^{2} + 3}{15 \, \tan \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 0.95 \[ -\frac {2}{3 \tan \relax (x )^{3}}-\frac {1}{5 \tan \relax (x )^{5}}-\frac {1}{\tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 0.95 \[ -\frac {15 \, \tan \relax (x)^{4} + 10 \, \tan \relax (x)^{2} + 3}{15 \, \tan \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 17, normalized size = 0.81 \[ -\frac {{\mathrm {cot}\relax (x)}^5}{5}-\frac {2\,{\mathrm {cot}\relax (x)}^3}{3}-\mathrm {cot}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.01, size = 54, normalized size = 2.57 \[ \frac {\tan ^{5}{\left (\frac {x}{2} \right )}}{160} + \frac {5 \tan ^{3}{\left (\frac {x}{2} \right )}}{96} + \frac {5 \tan {\left (\frac {x}{2} \right )}}{16} - \frac {5}{16 \tan {\left (\frac {x}{2} \right )}} - \frac {5}{96 \tan ^{3}{\left (\frac {x}{2} \right )}} - \frac {1}{160 \tan ^{5}{\left (\frac {x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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