Optimal. Leaf size=26 \[ \frac {1}{8} \tanh ^{-1}(\cos (x))-\frac {1}{4} \cot (x) \csc ^3(x)+\frac {1}{8} \cot (x) \csc (x) \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ \frac {1}{8} \tanh ^{-1}(\cos (x))-\frac {1}{4} \cot (x) \csc ^3(x)+\frac {1}{8} \cot (x) \csc (x) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(x) \csc ^3(x) \, dx &=-\frac {1}{4} \cot (x) \csc ^3(x)-\frac {1}{4} \int \csc ^3(x) \, dx\\ &=\frac {1}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x)-\frac {1}{8} \int \csc (x) \, dx\\ &=\frac {1}{8} \tanh ^{-1}(\cos (x))+\frac {1}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x)\\ \end {align*}
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Mathematica [B] time = 0.02, size = 71, normalized size = 2.73 \[ -\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )+\frac {1}{32} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )-\frac {1}{32} \sec ^2\left (\frac {x}{2}\right )-\frac {1}{8} \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {1}{8} \log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^2(x) \csc ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.73, size = 68, normalized size = 2.62 \[ -\frac {2 \, \cos \relax (x)^{3} - {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, \cos \relax (x)}{16 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 47, normalized size = 1.81 \[ -\frac {\frac {1}{\cos \relax (x)} + \cos \relax (x)}{8 \, {\left ({\left (\frac {1}{\cos \relax (x)} + \cos \relax (x)\right )}^{2} - 4\right )}} + \frac {1}{32} \, \log \left ({\left | \frac {1}{\cos \relax (x)} + \cos \relax (x) + 2 \right |}\right ) - \frac {1}{32} \, \log \left ({\left | \frac {1}{\cos \relax (x)} + \cos \relax (x) - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 36, normalized size = 1.38
method | result | size |
default | \(-\frac {\cos ^{3}\relax (x )}{4 \sin \relax (x )^{4}}-\frac {\cos ^{3}\relax (x )}{8 \sin \relax (x )^{2}}-\frac {\cos \relax (x )}{8}-\frac {\ln \left (\csc \relax (x )-\cot \relax (x )\right )}{8}\) | \(36\) |
risch | \(-\frac {{\mathrm e}^{7 i x}+7 \,{\mathrm e}^{5 i x}+7 \,{\mathrm e}^{3 i x}+{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{8}-\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{8}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 38, normalized size = 1.46 \[ -\frac {\cos \relax (x)^{3} + \cos \relax (x)}{8 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} + \frac {1}{16} \, \log \left (\cos \relax (x) + 1\right ) - \frac {1}{16} \, \log \left (\cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 24, normalized size = 0.92 \[ \frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{64}-\frac {1}{64\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 41, normalized size = 1.58 \[ \frac {- \cos ^{3}{\relax (x )} - \cos {\relax (x )}}{8 \cos ^{4}{\relax (x )} - 16 \cos ^{2}{\relax (x )} + 8} - \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{16} + \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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