Optimal. Leaf size=12 \[ -\frac {1}{2 (1-\cos (x))^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2746, 32}
\begin {gather*} -\frac {1}{2 (1-\cos (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx &=\text {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,-\cos (x)\right )\\ &=-\frac {1}{2 (1-\cos (x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{8} \csc ^4\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 11, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {1}{2 \left (1-\cos \left (x \right )\right )^{2}}\) | \(11\) |
default | \(-\frac {1}{2 \left (1-\cos \left (x \right )\right )^{2}}\) | \(11\) |
risch | \(-\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{i x}-1\right )^{4}}\) | \(17\) |
norman | \(\frac {-\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}-\frac {3 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tan \left (\frac {x}{2}\right )}{8}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right ) \tan \left (\frac {x}{2}\right )^{5}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{2 \, {\left (\cos \left (x\right ) - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 14, normalized size = 1.17 \begin {gather*} -\frac {1}{2 \, {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 15, normalized size = 1.25 \begin {gather*} - \frac {1}{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 = 0.45, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{2 \, {\left (\cos \left (x\right ) - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{2\,{\left (\cos \left (x\right )-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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