Integrand size = 13, antiderivative size = 7 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-2 \log (1+\cos (x)) \]
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Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 31} \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-2 \log (\cos (x)+1) \]
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Rule 12
Rule 31
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {2}{1+x} \, dx,x,\cos (x)\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\cos (x)\right )\right ) \\ & = -2 \log (1+\cos (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-4 \log \left (\cos \left (\frac {x}{2}\right )\right ) \]
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Time = 2.50 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(-2 \ln \left (\cos \left (x \right )+1\right )\) | \(8\) |
default | \(-2 \ln \left (\cos \left (x \right )+1\right )\) | \(8\) |
risch | \(2 i x -4 \ln \left ({\mathrm e}^{i x}+1\right )\) | \(16\) |
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Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-2 \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \]
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Time = 1.34 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.14 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=- 2 \log {\left (\cos {\left (x \right )} + 1 \right )} \]
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Time = 0.20 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-2 \, \log \left (\cos \left (x\right ) + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-2 \, \log \left (\cos \left (x\right ) + 1\right ) \]
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Time = 0.07 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \sin (2 x)}{1+\cos (x)} \, dx=-2\,\ln \left (\cos \left (x\right )+1\right ) \]
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