Optimal. Leaf size=17 \[ -\frac {1}{5} \cot ^5(x)-\frac {\cot ^3(x)}{3} \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2607, 14} \[ -\frac {1}{5} \cot ^5(x)-\frac {\cot ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rubi steps
\begin {align*} \int \cot ^2(x) \csc ^4(x) \, dx &=\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (x)\right )\\ &=\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (x)\right )\\ &=-\frac {1}{3} \cot ^3(x)-\frac {\cot ^5(x)}{5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.59 \[ \frac {2 \cot (x)}{15}-\frac {1}{5} \cot (x) \csc ^4(x)+\frac {1}{15} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 33, normalized size = 1.94 \[ \frac {2 \, \cos \relax (x)^{5} - 5 \, \cos \relax (x)^{3}}{15 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 14, normalized size = 0.82 \[ -\frac {5 \, \tan \relax (x)^{2} + 3}{15 \, \tan \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 22, normalized size = 1.29 \[ -\frac {\cos ^{3}\relax (x )}{5 \sin \relax (x )^{5}}-\frac {2 \left (\cos ^{3}\relax (x )\right )}{15 \sin \relax (x )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 14, normalized size = 0.82 \[ -\frac {5 \, \tan \relax (x)^{2} + 3}{15 \, \tan \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 19, normalized size = 1.12 \[ -{\cos \relax (x)}^3\,\left (\frac {2}{15\,{\sin \relax (x)}^3}+\frac {1}{5\,{\sin \relax (x)}^5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.07, size = 29, normalized size = 1.71 \[ \frac {2 \cos {\relax (x )}}{15 \sin {\relax (x )}} + \frac {\cos {\relax (x )}}{15 \sin ^{3}{\relax (x )}} - \frac {\cos {\relax (x )}}{5 \sin ^{5}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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