Optimal. Leaf size=28 \[ \frac {4}{1-\cos (x)}-\frac {2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 28, 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} \[ \frac {4}{1-\cos (x)}-\frac {2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rule 4392
Rubi steps
\begin {align*} \int (\cot (x)+\csc (x))^5 \, dx &=\int (1+\cos (x))^5 \csc ^5(x) \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(1+x)^2}{(1-x)^3} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{1-x}-\frac {4}{(-1+x)^3}-\frac {4}{(-1+x)^2}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac {2}{(1-\cos (x))^2}+\frac {4}{1-\cos (x)}+\log (1-\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.07, size = 32, normalized size = 1.14 \[ -\frac {1}{2} \csc ^4\left (\frac {x}{2}\right )+2 \csc ^2\left (\frac {x}{2}\right )+2 \log \left (\sin \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.58, size = 37, normalized size = 1.32 \[ \frac {{\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 4 \, \cos \relax (x) + 2}{\cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 22, normalized size = 0.79 \[ -\frac {2 \, {\left (2 \, \cos \relax (x) - 1\right )}}{{\left (\cos \relax (x) - 1\right )}^{2}} + \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 105, normalized size = 3.75 \[ -\frac {\left (\cot ^{4}\relax (x )\right )}{4}+\frac {\left (\cot ^{2}\relax (x )\right )}{2}+\ln \left (\sin \relax (x )\right )-\frac {5 \left (\cos ^{5}\relax (x )\right )}{4 \sin \relax (x )^{4}}+\frac {5 \left (\cos ^{5}\relax (x )\right )}{8 \sin \relax (x )^{2}}+\frac {5 \left (\cos ^{3}\relax (x )\right )}{8}+\frac {5 \cos \relax (x )}{8}+\ln \left (\csc \relax (x )-\cot \relax (x )\right )-\frac {5 \left (\cos ^{4}\relax (x )\right )}{2 \sin \relax (x )^{4}}-\frac {5 \left (\cos ^{3}\relax (x )\right )}{2 \sin \relax (x )^{4}}-\frac {5 \left (\cos ^{3}\relax (x )\right )}{4 \sin \relax (x )^{2}}-\frac {5}{4 \sin \relax (x )^{4}}+\left (-\frac {\left (\csc ^{3}\relax (x )\right )}{4}-\frac {3 \csc \relax (x )}{8}\right ) \cot \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 125, normalized size = 4.46 \[ -\frac {5}{2} \, \cot \relax (x)^{4} - \frac {5 \, {\left (5 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )}}{8 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} + \frac {3 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)}{8 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} - \frac {5 \, {\left (\cos \relax (x)^{3} + \cos \relax (x)\right )}}{4 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )}} + \frac {4 \, \sin \relax (x)^{2} - 1}{4 \, \sin \relax (x)^{4}} - \frac {5}{4 \, \sin \relax (x)^{4}} + \frac {1}{2} \, \log \left (\sin \relax (x)^{2}\right ) - \frac {1}{2} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.42, size = 34, normalized size = 1.21 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2-\frac {1}{2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 107.21, size = 68, normalized size = 2.43 \[ \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} - \frac {\log {\left (\csc ^{2}{\relax (x )} \right )}}{2} - \frac {5 \cot ^{4}{\relax (x )}}{2} - \frac {3 \csc ^{4}{\relax (x )}}{2} + \csc ^{2}{\relax (x )} - \frac {32 \cos ^{3}{\relax (x )}}{8 \cos ^{4}{\relax (x )} - 16 \cos ^{2}{\relax (x )} + 8} \]
Verification of antiderivative is not currently implemented for this CAS.
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