Optimal. Leaf size=83 \[ \frac{8}{15} x^2 \sin (x)+\frac{1}{5} x^2 \sin (x) \cos ^4(x)+\frac{4}{15} x^2 \sin (x) \cos ^2(x)-\frac{2 \sin ^5(x)}{125}+\frac{76 \sin ^3(x)}{675}-\frac{298 \sin (x)}{225}+\frac{2}{25} x \cos ^5(x)+\frac{8}{45} x \cos ^3(x)+\frac{16}{15} x \cos (x) \]
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Rubi [A] time = 0.0937162, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3311, 3296, 2637, 2633} \[ \frac{8}{15} x^2 \sin (x)+\frac{1}{5} x^2 \sin (x) \cos ^4(x)+\frac{4}{15} x^2 \sin (x) \cos ^2(x)-\frac{2 \sin ^5(x)}{125}+\frac{76 \sin ^3(x)}{675}-\frac{298 \sin (x)}{225}+\frac{2}{25} x \cos ^5(x)+\frac{8}{45} x \cos ^3(x)+\frac{16}{15} x \cos (x) \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 2637
Rule 2633
Rubi steps
\begin{align*} \int x^2 \cos ^5(x) \, dx &=\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} \operatorname{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} \operatorname{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*}
Mathematica [A] time = 0.0539509, size = 67, normalized size = 0.81 \[ \frac{5}{8} \left (x^2-2\right ) \sin (x)+\frac{5}{432} \left (9 x^2-2\right ) \sin (3 x)+\frac{\left (25 x^2-2\right ) \sin (5 x)}{2000}+\frac{5}{4} x \cos (x)+\frac{5}{72} x \cos (3 x)+\frac{1}{200} x \cos (5 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 70, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}\sin \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) }-{\frac{16\,\sin \left ( x \right ) }{15}}+{\frac{16\,x\cos \left ( x \right ) }{15}}+{\frac{2\,x \left ( \cos \left ( x \right ) \right ) ^{5}}{25}}-{\frac{2\,\sin \left ( x \right ) }{125} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) }+{\frac{8\,x \left ( \cos \left ( x \right ) \right ) ^{3}}{45}}-{\frac{ \left ( 16+8\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{135}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960549, size = 74, normalized size = 0.89 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31121, size = 190, normalized size = 2.29 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.46872, size = 112, normalized size = 1.35 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07878, size = 74, normalized size = 0.89 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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