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 = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3767} \[ -\frac {1}{5} \cot ^5(x)-\frac {2 \cot ^3(x)}{3}-\cot (x) \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^6(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.48, 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 = 0.31, size = 18, normalized size = 0.86
method | result | size |
default | \(\left (-\frac {8}{15}-\frac {\left (\csc ^{4}\relax (x )\right )}{5}-\frac {4 \left (\csc ^{2}\relax (x )\right )}{15}\right ) \cot \relax (x )\) | \(18\) |
risch | \(-\frac {16 i \left (10 \,{\mathrm e}^{4 i x}-5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}-1\right )^{5}}\) | \(29\) |
norman | \(\frac {-\frac {1}{160}-\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{96}-\frac {5 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {5 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {5 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{96}+\frac {\left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{160}}{\tan \left (\frac {x}{2}\right )^{5}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, 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 = 0.20, size = 27, normalized size = 1.29 \[ -\frac {8\,\cos \relax (x)\,{\sin \relax (x)}^4+4\,\cos \relax (x)\,{\sin \relax (x)}^2+3\,\cos \relax (x)}{15\,{\sin \relax (x)}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 32, normalized size = 1.52 \[ - \frac {8 \cos {\relax (x )}}{15 \sin {\relax (x )}} - \frac {4 \cos {\relax (x )}}{15 \sin ^{3}{\relax (x )}} - \frac {\cos {\relax (x )}}{5 \sin ^{5}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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