Integrand size = 9, antiderivative size = 6 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{1+\cos (x)} \]
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Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2746, 32} \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{\cos (x)+1} \]
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Rule 32
Rule 2746
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\cos (x)\right ) \\ & = \frac {1}{1+\cos (x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 2.00 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{2} \sec ^2\left (\frac {x}{2}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {1}{\cos \left (x \right )+1}\) | \(7\) |
default | \(\frac {1}{\cos \left (x \right )+1}\) | \(7\) |
norman | \(\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}\) | \(9\) |
parallelrisch | \(\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}\) | \(9\) |
risch | \(\frac {2 \,{\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+1\right )^{2}}\) | \(17\) |
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none
Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{\cos \left (x\right ) + 1} \]
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Time = 0.16 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{\cos {\left (x \right )} + 1} \]
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none
Time = 0.22 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{\cos \left (x\right ) + 1} \]
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none
Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{\cos \left (x\right ) + 1} \]
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Time = 13.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{(1+\cos (x))^2} \, dx=\frac {1}{\cos \left (x\right )+1} \]
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