Integrand size = 9, antiderivative size = 34 \[ \int \cos ^4(x) \sin ^2(x) \, dx=\frac {x}{16}+\frac {1}{16} \cos (x) \sin (x)+\frac {1}{24} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^5(x) \sin (x) \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2648, 2715, 8} \[ \int \cos ^4(x) \sin ^2(x) \, dx=\frac {x}{16}-\frac {1}{6} \sin (x) \cos ^5(x)+\frac {1}{24} \sin (x) \cos ^3(x)+\frac {1}{16} \sin (x) \cos (x) \]
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Rule 8
Rule 2648
Rule 2715
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6} \cos ^5(x) \sin (x)+\frac {1}{6} \int \cos ^4(x) \, dx \\ & = \frac {1}{24} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^5(x) \sin (x)+\frac {1}{8} \int \cos ^2(x) \, dx \\ & = \frac {1}{16} \cos (x) \sin (x)+\frac {1}{24} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^5(x) \sin (x)+\frac {\int 1 \, dx}{16} \\ & = \frac {x}{16}+\frac {1}{16} \cos (x) \sin (x)+\frac {1}{24} \cos ^3(x) \sin (x)-\frac {1}{6} \cos ^5(x) \sin (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \cos ^4(x) \sin ^2(x) \, dx=\frac {x}{16}+\frac {1}{64} \sin (2 x)-\frac {1}{64} \sin (4 x)-\frac {1}{192} \sin (6 x) \]
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Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {x}{16}-\frac {\sin \left (6 x \right )}{192}-\frac {\sin \left (4 x \right )}{64}+\frac {\sin \left (2 x \right )}{64}\) | \(23\) |
parallelrisch | \(\frac {x}{16}-\frac {\sin \left (6 x \right )}{192}-\frac {\sin \left (4 x \right )}{64}+\frac {\sin \left (2 x \right )}{64}\) | \(23\) |
default | \(-\frac {\left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )}{6}+\frac {\left (\cos ^{3}\left (x \right )+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{24}+\frac {x}{16}\) | \(26\) |
norman | \(\frac {\frac {x}{16}+\frac {47 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{24}-\frac {13 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}+\frac {13 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4}-\frac {47 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {15 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {5 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{4}+\frac {15 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{16}+\frac {3 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{8}+\frac {x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{16}-\frac {\tan \left (\frac {x}{2}\right )}{8}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{6}}\) | \(116\) |
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \cos ^4(x) \sin ^2(x) \, dx=-\frac {1}{48} \, {\left (8 \, \cos \left (x\right )^{5} - 2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {1}{16} \, x \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \cos ^4(x) \sin ^2(x) \, dx=\frac {x}{16} - \frac {\sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{6} + \frac {\sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{24} + \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{16} \]
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Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.53 \[ \int \cos ^4(x) \sin ^2(x) \, dx=\frac {1}{48} \, \sin \left (2 \, x\right )^{3} + \frac {1}{16} \, x - \frac {1}{64} \, \sin \left (4 \, x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \cos ^4(x) \sin ^2(x) \, dx=\frac {1}{16} \, x - \frac {1}{192} \, \sin \left (6 \, x\right ) - \frac {1}{64} \, \sin \left (4 \, x\right ) + \frac {1}{64} \, \sin \left (2 \, x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \cos ^4(x) \sin ^2(x) \, dx=\left (\frac {{\cos \left (x\right )}^3}{6}+\frac {\cos \left (x\right )}{8}\right )\,{\sin \left (x\right )}^3-\frac {\cos \left (x\right )\,\sin \left (x\right )}{16}+\frac {x}{16} \]
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