Integrand size = 4, antiderivative size = 19 \[ \int \cos ^5(x) \, dx=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \[ \int \cos ^5(x) \, dx=\frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right ) \\ & = \sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos ^5(x) \, dx=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \]
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Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{5}\) | \(17\) |
risch | \(\frac {5 \sin \left (x \right )}{8}+\frac {\sin \left (5 x \right )}{80}+\frac {5 \sin \left (3 x \right )}{48}\) | \(18\) |
parallelrisch | \(\frac {5 \sin \left (x \right )}{8}+\frac {\sin \left (5 x \right )}{80}+\frac {5 \sin \left (3 x \right )}{48}\) | \(18\) |
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \cos ^5(x) \, dx=\frac {1}{15} \, {\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \cos ^5(x) \, dx=\frac {\sin ^{5}{\left (x \right )}}{5} - \frac {2 \sin ^{3}{\left (x \right )}}{3} + \sin {\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \cos ^5(x) \, dx=\frac {1}{5} \, \sin \left (x\right )^{5} - \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \cos ^5(x) \, dx=\frac {1}{5} \, \sin \left (x\right )^{5} - \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \cos ^5(x) \, dx=\frac {\sin \left (x\right )\,{\cos \left (x\right )}^4}{5}+\frac {4\,\sin \left (x\right )\,{\cos \left (x\right )}^2}{15}+\frac {8\,\sin \left (x\right )}{15} \]
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