Optimal. Leaf size=20 \[ -\frac {2}{1-\cos (x)}-\log (1-\cos (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4392, 2667, 43} \[ -\frac {2}{1-\cos (x)}-\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4392
Rubi steps
\begin {align*} \int (\cot (x)+\csc (x))^3 \, dx &=\int (1+\cos (x))^3 \csc ^3(x) \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1+x}{(1-x)^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac {2}{1-\cos (x)}-\log (1-\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.04, size = 20, normalized size = 1.00 \[ -\csc ^2\left (\frac {x}{2}\right )-2 \log \left (\sin \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.68, size = 22, normalized size = 1.10 \[ -\frac {{\left (\cos \relax (x) - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2}{\cos \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 18, normalized size = 0.90 \[ \frac {2}{\cos \relax (x) - 1} - \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 49, normalized size = 2.45 \[ -\frac {\left (\cot ^{2}\relax (x )\right )}{2}-\ln \left (\sin \relax (x )\right )-\frac {3 \left (\cos ^{3}\relax (x )\right )}{2 \sin \relax (x )^{2}}-\frac {3 \cos \relax (x )}{2}-\ln \left (\csc \relax (x )-\cot \relax (x )\right )-\frac {3}{2 \sin \relax (x )^{2}}-\frac {\cot \relax (x ) \csc \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 46, normalized size = 2.30 \[ -\frac {3}{2} \, \cot \relax (x)^{2} + \frac {2 \, \cos \relax (x)}{\cos \relax (x)^{2} - 1} - \frac {1}{2 \, \sin \relax (x)^{2}} - \frac {1}{2} \, \log \left (\sin \relax (x)^{2}\right ) + \frac {1}{2} \, \log \left (\cos \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 25, normalized size = 1.25 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\frac {1}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.64, size = 46, normalized size = 2.30 \[ - \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} + \frac {\log {\left (- \csc ^{2}{\relax (x )} \right )}}{2} - 2 \csc ^{2}{\relax (x )} + \frac {4 \cos {\relax (x )}}{2 \cos ^{2}{\relax (x )} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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