Integrand size = 11, antiderivative size = 12 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{2 (1-\cos (x))^2} \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2746, 32} \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{2 (1-\cos (x))^2} \]
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Rule 32
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,-\cos (x)\right ) \\ & = -\frac {1}{2 (1-\cos (x))^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{8} \csc ^4\left (\frac {x}{2}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 11, normalized size of antiderivative = 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 | \(-\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{i x}-1\right )^{4}}\) | \(17\) |
parallelrisch | \(-\frac {\left (\cot ^{2}\left (\frac {x}{2}\right )\right ) \left (\cot ^{2}\left (\frac {x}{2}\right )+2\right )}{8}\) | \(17\) |
norman | \(\frac {-\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}-\frac {3 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{8}-\frac {\tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {x}{2}\right )^{5}}\) | \(41\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{2 \, {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=- \frac {1}{2 \cos ^{2}{\left (x \right )} - 4 \cos {\left (x \right )} + 2} \]
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none
Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{2 \, {\left (\cos \left (x\right ) - 1\right )}^{2}} \]
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none
Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{2 \, {\left (\cos \left (x\right ) - 1\right )}^{2}} \]
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Time = 12.95 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\sin (x)}{(1-\cos (x))^3} \, dx=-\frac {1}{2\,{\left (\cos \left (x\right )-1\right )}^2} \]
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