Optimal. Leaf size=26 \[ -\frac {3}{8} \tanh ^{-1}(\cos (x))-\frac {1}{4} \cot (x) \csc ^3(x)-\frac {3}{8} \cot (x) \csc (x) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3768, 3770} \[ -\frac {3}{8} \tanh ^{-1}(\cos (x))-\frac {1}{4} \cot (x) \csc ^3(x)-\frac {3}{8} \cot (x) \csc (x) \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^5(x) \, dx &=-\frac {1}{4} \cot (x) \csc ^3(x)+\frac {3}{4} \int \csc ^3(x) \, dx\\ &=-\frac {3}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x)+\frac {3}{8} \int \csc (x) \, dx\\ &=-\frac {3}{8} \tanh ^{-1}(\cos (x))-\frac {3}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x)\\ \end {align*}
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Mathematica [B] time = 0.01, size = 71, normalized size = 2.73 \[ -\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {3}{32} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {3}{32} \sec ^2\left (\frac {x}{2}\right )+\frac {3}{8} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {3}{8} \log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.97, size = 69, normalized size = 2.65 \[ \frac {6 \, \cos \relax (x)^{3} - 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 10 \, \cos \relax (x)}{16 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 38, normalized size = 1.46 \[ \frac {3 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)}{8 \, {\left (\cos \relax (x)^{2} - 1\right )}^{2}} - \frac {3}{16} \, \log \left (\cos \relax (x) + 1\right ) + \frac {3}{16} \, \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 26, normalized size = 1.00
method | result | size |
default | \(\left (-\frac {\left (\csc ^{3}\relax (x )\right )}{4}-\frac {3 \csc \relax (x )}{8}\right ) \cot \relax (x )+\frac {3 \ln \left (\csc \relax (x )-\cot \relax (x )\right )}{8}\) | \(26\) |
norman | \(\frac {-\frac {1}{64}-\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {\left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{8}+\frac {\left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{64}}{\tan \left (\frac {x}{2}\right )^{4}}+\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{8}\) | \(42\) |
risch | \(\frac {3 \,{\mathrm e}^{7 i x}-11 \,{\mathrm e}^{5 i x}-11 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{8}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{8}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 42, normalized size = 1.62 \[ \frac {3 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)}{8 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} - \frac {3}{16} \, \log \left (\cos \relax (x) + 1\right ) + \frac {3}{16} \, \log \left (\cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 33, normalized size = 1.27 \[ -\frac {3\,\mathrm {atanh}\left (\cos \relax (x)\right )}{8}-\frac {\frac {5\,\cos \relax (x)}{8}-\frac {3\,{\cos \relax (x)}^3}{8}}{{\cos \relax (x)}^4-2\,{\cos \relax (x)}^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 46, normalized size = 1.77 \[ \frac {3 \cos ^{3}{\relax (x )} - 5 \cos {\relax (x )}}{8 \cos ^{4}{\relax (x )} - 16 \cos ^{2}{\relax (x )} + 8} + \frac {3 \log {\left (\cos {\relax (x )} - 1 \right )}}{16} - \frac {3 \log {\left (\cos {\relax (x )} + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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