Integrand size = 8, antiderivative size = 83 \[ \int x^2 \cos ^5(x) \, dx=\frac {16}{15} x \cos (x)+\frac {8}{45} x \cos ^3(x)+\frac {2}{25} x \cos ^5(x)-\frac {298 \sin (x)}{225}+\frac {8}{15} x^2 \sin (x)+\frac {4}{15} x^2 \cos ^2(x) \sin (x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)+\frac {76 \sin ^3(x)}{675}-\frac {2 \sin ^5(x)}{125} \]
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Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3392, 3377, 2717, 2713} \[ \int x^2 \cos ^5(x) \, dx=\frac {8}{15} x^2 \sin (x)+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {4}{15} x^2 \sin (x) \cos ^2(x)-\frac {2 \sin ^5(x)}{125}+\frac {76 \sin ^3(x)}{675}-\frac {298 \sin (x)}{225}+\frac {2}{25} x \cos ^5(x)+\frac {8}{45} x \cos ^3(x)+\frac {16}{15} x \cos (x) \]
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Rule 2713
Rule 2717
Rule 3377
Rule 3392
Rubi steps \begin{align*} \text {integral}& = \frac {2}{25} x \cos ^5(x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)-\frac {2}{25} \int \cos ^5(x) \, dx+\frac {4}{5} \int x^2 \cos ^3(x) \, dx \\ & = \frac {8}{45} x \cos ^3(x)+\frac {2}{25} x \cos ^5(x)+\frac {4}{15} x^2 \cos ^2(x) \sin (x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)+\frac {2}{25} \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )-\frac {8}{45} \int \cos ^3(x) \, dx+\frac {8}{15} \int x^2 \cos (x) \, dx \\ & = \frac {8}{45} x \cos ^3(x)+\frac {2}{25} x \cos ^5(x)-\frac {2 \sin (x)}{25}+\frac {8}{15} x^2 \sin (x)+\frac {4}{15} x^2 \cos ^2(x) \sin (x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)+\frac {4 \sin ^3(x)}{75}-\frac {2 \sin ^5(x)}{125}+\frac {8}{45} \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )-\frac {16}{15} \int x \sin (x) \, dx \\ & = \frac {16}{15} x \cos (x)+\frac {8}{45} x \cos ^3(x)+\frac {2}{25} x \cos ^5(x)-\frac {58 \sin (x)}{225}+\frac {8}{15} x^2 \sin (x)+\frac {4}{15} x^2 \cos ^2(x) \sin (x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)+\frac {76 \sin ^3(x)}{675}-\frac {2 \sin ^5(x)}{125}-\frac {16}{15} \int \cos (x) \, dx \\ & = \frac {16}{15} x \cos (x)+\frac {8}{45} x \cos ^3(x)+\frac {2}{25} x \cos ^5(x)-\frac {298 \sin (x)}{225}+\frac {8}{15} x^2 \sin (x)+\frac {4}{15} x^2 \cos ^2(x) \sin (x)+\frac {1}{5} x^2 \cos ^4(x) \sin (x)+\frac {76 \sin ^3(x)}{675}-\frac {2 \sin ^5(x)}{125} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int x^2 \cos ^5(x) \, dx=\frac {5}{4} x \cos (x)+\frac {5}{72} x \cos (3 x)+\frac {1}{200} x \cos (5 x)+\frac {5}{8} \left (-2+x^2\right ) \sin (x)+\frac {5}{432} \left (-2+9 x^2\right ) \sin (3 x)+\frac {\left (-2+25 x^2\right ) \sin (5 x)}{2000} \]
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Time = 0.53 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {5 x \cos \left (x \right )}{4}+\frac {5 \left (x^{2}-2\right ) \sin \left (x \right )}{8}+\frac {x \cos \left (5 x \right )}{200}+\frac {\left (25 x^{2}-2\right ) \sin \left (5 x \right )}{2000}+\frac {5 x \cos \left (3 x \right )}{72}+\frac {5 \left (9 x^{2}-2\right ) \sin \left (3 x \right )}{432}\) | \(56\) |
parallelrisch | \(\frac {\left (5625 x^{2}-1250\right ) \sin \left (3 x \right )}{54000}+\frac {\left (675 x^{2}-54\right ) \sin \left (5 x \right )}{54000}+\frac {5 x^{2} \sin \left (x \right )}{8}+\frac {5 x \cos \left (x \right )}{4}+\frac {5 x \cos \left (3 x \right )}{72}+\frac {x \cos \left (5 x \right )}{200}-\frac {5 \sin \left (x \right )}{4}\) | \(58\) |
default | \(\frac {x^{2} \left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{5}+\frac {2 \left (\cos ^{5}\left (x \right )\right ) x}{25}-\frac {2 \left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{125}+\frac {8 x \left (\cos ^{3}\left (x \right )\right )}{45}-\frac {8 \left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{135}-\frac {16 \sin \left (x \right )}{15}+\frac {16 x \cos \left (x \right )}{15}\) | \(70\) |
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Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69 \[ \int x^2 \cos ^5(x) \, dx=\frac {2}{25} \, x \cos \left (x\right )^{5} + \frac {8}{45} \, x \cos \left (x\right )^{3} + \frac {16}{15} \, x \cos \left (x\right ) + \frac {1}{3375} \, {\left (27 \, {\left (25 \, x^{2} - 2\right )} \cos \left (x\right )^{4} + 4 \, {\left (225 \, x^{2} - 68\right )} \cos \left (x\right )^{2} + 1800 \, x^{2} - 4144\right )} \sin \left (x\right ) \]
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Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int x^2 \cos ^5(x) \, dx=\frac {8 x^{2} \sin ^{5}{\left (x \right )}}{15} + \frac {4 x^{2} \sin ^{3}{\left (x \right )} \cos ^{2}{\left (x \right )}}{3} + x^{2} \sin {\left (x \right )} \cos ^{4}{\left (x \right )} + \frac {16 x \sin ^{4}{\left (x \right )} \cos {\left (x \right )}}{15} + \frac {104 x \sin ^{2}{\left (x \right )} \cos ^{3}{\left (x \right )}}{45} + \frac {298 x \cos ^{5}{\left (x \right )}}{225} - \frac {4144 \sin ^{5}{\left (x \right )}}{3375} - \frac {1712 \sin ^{3}{\left (x \right )} \cos ^{2}{\left (x \right )}}{675} - \frac {298 \sin {\left (x \right )} \cos ^{4}{\left (x \right )}}{225} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66 \[ \int x^2 \cos ^5(x) \, dx=\frac {1}{200} \, x \cos \left (5 \, x\right ) + \frac {5}{72} \, x \cos \left (3 \, x\right ) + \frac {5}{4} \, x \cos \left (x\right ) + \frac {1}{2000} \, {\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac {5}{432} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {5}{8} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66 \[ \int x^2 \cos ^5(x) \, dx=\frac {1}{200} \, x \cos \left (5 \, x\right ) + \frac {5}{72} \, x \cos \left (3 \, x\right ) + \frac {5}{4} \, x \cos \left (x\right ) + \frac {1}{2000} \, {\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac {5}{432} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {5}{8} \, {\left (x^{2} - 2\right )} \sin \left (x\right ) \]
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Time = 0.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int x^2 \cos ^5(x) \, dx=\frac {8\,x\,{\cos \left (x\right )}^3}{45}-\frac {4144\,\sin \left (x\right )}{3375}+\frac {2\,x\,{\cos \left (x\right )}^5}{25}+\frac {8\,x^2\,\sin \left (x\right )}{15}-\frac {272\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{3375}-\frac {2\,{\cos \left (x\right )}^4\,\sin \left (x\right )}{125}+\frac {16\,x\,\cos \left (x\right )}{15}+\frac {4\,x^2\,{\cos \left (x\right )}^2\,\sin \left (x\right )}{15}+\frac {x^2\,{\cos \left (x\right )}^4\,\sin \left (x\right )}{5} \]
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