Optimal. Leaf size=22 \[ \frac {\sin ^2(x)}{2}-\frac {1}{2} \csc ^2(x)-2 \log (\sin (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2590, 266, 43} \[ \frac {\sin ^2(x)}{2}-\frac {1}{2} \csc ^2(x)-2 \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2590
Rubi steps
\begin {align*} \int \cos ^2(x) \cot ^3(x) \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^3} \, dx,x,-\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1-x)^2}{x^2} \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {2}{x}\right ) \, dx,x,\sin ^2(x)\right )\\ &=-\frac {1}{2} \csc ^2(x)-2 \log (\sin (x))+\frac {\sin ^2(x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.91 \[ \frac {1}{2} \left (\sin ^2(x)-\csc ^2(x)-4 \log (\sin (x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 37, normalized size = 1.68 \[ -\frac {2 \, \cos \relax (x)^{4} - 3 \, \cos \relax (x)^{2} + 8 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \relax (x)\right ) - 1}{4 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 28, normalized size = 1.27 \[ -\frac {1}{2} \, \cos \relax (x)^{2} + \frac {1}{2 \, {\left (\cos \relax (x)^{2} - 1\right )}} - \log \left (-\cos \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 1.32 \[ -\frac {\cos ^{6}\relax (x )}{2 \sin \relax (x )^{2}}-\frac {\left (\cos ^{4}\relax (x )\right )}{2}-\left (\cos ^{2}\relax (x )\right )-2 \ln \left (\sin \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 20, normalized size = 0.91 \[ \frac {1}{2} \, \sin \relax (x)^{2} - \frac {1}{2 \, \sin \relax (x)^{2}} - \log \left (\sin \relax (x)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 32, normalized size = 1.45 \[ \ln \left ({\mathrm {tan}\relax (x)}^2+1\right )-2\,\ln \left (\mathrm {tan}\relax (x)\right )-\frac {{\mathrm {tan}\relax (x)}^2+\frac {1}{2}}{{\mathrm {tan}\relax (x)}^4+{\mathrm {tan}\relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 20, normalized size = 0.91 \[ - 2 \log {\left (\sin {\relax (x )} \right )} + \frac {\sin ^{2}{\relax (x )}}{2} - \frac {1}{2 \sin ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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