Integrand size = 15, antiderivative size = 20 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\sqrt {5} \arctan \left (\frac {\cos (x)}{\sqrt {5}}\right )-\cos (x) \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4420, 327, 209} \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\sqrt {5} \arctan \left (\frac {\cos (x)}{\sqrt {5}}\right )-\cos (x) \]
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Rule 209
Rule 327
Rule 4420
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2}{5+x^2} \, dx,x,\cos (x)\right ) \\ & = -\cos (x)+5 \text {Subst}\left (\int \frac {1}{5+x^2} \, dx,x,\cos (x)\right ) \\ & = \sqrt {5} \arctan \left (\frac {\cos (x)}{\sqrt {5}}\right )-\cos (x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(20)=40\).
Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.10 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\frac {1}{20} \left (-\sqrt {5} \arctan \left (\frac {\cos (x)}{\sqrt {5}}\right )+21 \sqrt {5} \arctan \left (\frac {1}{\sqrt {5}}-\sqrt {\frac {6}{5}} \tan \left (\frac {x}{2}\right )\right )+21 \sqrt {5} \arctan \left (\frac {1}{\sqrt {5}}+\sqrt {\frac {6}{5}} \tan \left (\frac {x}{2}\right )\right )-20 \cos (x)\right ) \]
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Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\cos \left (x \right )+\arctan \left (\frac {\cos \left (x \right ) \sqrt {5}}{5}\right ) \sqrt {5}\) | \(18\) |
default | \(-\cos \left (x \right )+\arctan \left (\frac {\cos \left (x \right ) \sqrt {5}}{5}\right ) \sqrt {5}\) | \(18\) |
risch | \(-\frac {{\mathrm e}^{i x}}{2}-\frac {{\mathrm e}^{-i x}}{2}-\frac {i \sqrt {5}\, \ln \left ({\mathrm e}^{2 i x}-2 i \sqrt {5}\, {\mathrm e}^{i x}+1\right )}{2}+\frac {i \sqrt {5}\, \ln \left ({\mathrm e}^{2 i x}+2 i \sqrt {5}\, {\mathrm e}^{i x}+1\right )}{2}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} \cos \left (x\right )\right ) - \cos \left (x\right ) \]
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Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=- \cos {\left (x \right )} + \sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} \cos {\left (x \right )}}{5} \right )} \]
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} \cos \left (x\right )\right ) - \cos \left (x\right ) \]
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Time = 0.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} \cos \left (x\right )\right ) - \cos \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx=\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\cos \left (x\right )}{5}\right )-\cos \left (x\right ) \]
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