Optimal. Leaf size=20 \[ -\frac {2}{1-\cos (x)}-\log (1-\cos (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 20, 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*} -\frac {2}{1-\cos (x)}-\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))^3} \, dx &=\text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,-\cos (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,-\cos (x)\right )\\ &=-\frac {2}{1-\cos (x)}-\log (1-\cos (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.45 \begin {gather*} -\cot ^2\left (\frac {x}{2}\right )-2 \log \left (\cos \left (\frac {x}{2}\right )\right )-2 \log \left (\tan \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 17, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {2}{-1+\cos \left (x \right )}-\ln \left (-1+\cos \left (x \right )\right )\) | \(17\) |
default | \(\frac {2}{-1+\cos \left (x \right )}-\ln \left (-1+\cos \left (x \right )\right )\) | \(17\) |
risch | \(i x +\frac {4 \,{\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}-1\right )^{2}}-2 \ln \left ({\mathrm e}^{i x}-1\right )\) | \(32\) |
norman | \(\frac {\tan ^{11}\left (\frac {x}{2}\right )+2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )-2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )-\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3} \tan \left (\frac {x}{2}\right )^{5}}-2 \ln \left (\tan \left (\frac {x}{2}\right )\right )+\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 16, normalized size = 0.80 \begin {gather*} \frac {2}{\cos \left (x\right ) - 1} - \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 = 0.36, size = 22, normalized size = 1.10 \begin {gather*} -\frac {{\left (\cos \left (x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2}{\cos \left (x\right ) - 1} \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. 126 vs.
\(2 (14) = 28\).
time = 0.25, size = 126, normalized size = 6.30 \begin {gather*} - \frac {2 \log {\left (\cos {\left (x \right )} - 1 \right )} \cos ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} + \frac {4 \log {\left (\cos {\left (x \right )} - 1 \right )} \cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} - \frac {2 \log {\left (\cos {\left (x \right )} - 1 \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} - \frac {\sin ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} + \frac {2 \cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} - \frac {2}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 18, normalized size = 0.90 \begin {gather*} \frac {2}{\cos \left (x\right ) - 1} - \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 = 16, normalized size = 0.80 \begin {gather*} \frac {2}{\cos \left (x\right )-1}-\ln \left (\cos \left (x\right )-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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