Optimal. Leaf size=12 \[ \cos (x)+2 \log (1-\cos (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 45}
\begin {gather*} \cos (x)+2 \log (1-\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{(1-\cos (x))^2} \, dx &=\text {Subst}\left (\int \frac {1-x}{1+x} \, dx,x,-\cos (x)\right )\\ &=\text {Subst}\left (\int \left (-1+\frac {2}{1+x}\right ) \, dx,x,-\cos (x)\right )\\ &=\cos (x)+2 \log (1-\cos (x))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 13, normalized size = 1.08 \begin {gather*} -1+\cos (x)+4 \log \left (\sin \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 11, normalized size = 0.92
method | result | size |
derivativedivides | \(\cos \left (x \right )+2 \ln \left (-1+\cos \left (x \right )\right )\) | \(11\) |
default | \(\cos \left (x \right )+2 \ln \left (-1+\cos \left (x \right )\right )\) | \(11\) |
risch | \(-2 i x +\frac {{\mathrm e}^{i x}}{2}+\frac {{\mathrm e}^{-i x}}{2}+4 \ln \left ({\mathrm e}^{i x}-1\right )\) | \(30\) |
norman | \(\frac {-2 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )-\frac {4 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3} \tan \left (\frac {x}{2}\right )^{3}}+4 \ln \left (\tan \left (\frac {x}{2}\right )\right )-2 \ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 10, normalized size = 0.83 \begin {gather*} \cos \left (x\right ) + 2 \, \log \left (\cos \left (x\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 12, normalized size = 1.00 \begin {gather*} \cos \left (x\right ) + 2 \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (10) = 20\).
time = 0.20, size = 58, normalized size = 4.83 \begin {gather*} \frac {2 \log {\left (\cos {\left (x \right )} - 1 \right )} \cos {\left (x \right )}}{\cos {\left (x \right )} - 1} - \frac {2 \log {\left (\cos {\left (x \right )} - 1 \right )}}{\cos {\left (x \right )} - 1} + \frac {\sin ^{2}{\left (x \right )}}{\cos {\left (x \right )} - 1} + \frac {2 \cos ^{2}{\left (x \right )}}{\cos {\left (x \right )} - 1} - \frac {2}{\cos {\left (x \right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 12, normalized size = 1.00 \begin {gather*} \cos \left (x\right ) + 2 \, \log \left (-\cos \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 10, normalized size = 0.83 \begin {gather*} 2\,\ln \left (\cos \left (x\right )-1\right )+\cos \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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