Optimal. Leaf size=83 \[ \frac {8}{15} x^2 \sin (x)+\frac {1}{5} x^2 \sin (x) \cos ^4(x)+\frac {4}{15} x^2 \sin (x) \cos ^2(x)-\frac {2 \sin ^5(x)}{125}+\frac {76 \sin ^3(x)}{675}-\frac {298 \sin (x)}{225}+\frac {2}{25} x \cos ^5(x)+\frac {8}{45} x \cos ^3(x)+\frac {16}{15} x \cos (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.
[In]
[Out]
Rule 2633
Rule 2637
Rule 3296
Rule 3311
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 = 0.05, 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.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^2 \cos ^5(x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.12, size = 57, normalized size = 0.69 \[ \frac {2}{25} \, x \cos \relax (x)^{5} + \frac {8}{45} \, x \cos \relax (x)^{3} + \frac {16}{15} \, x \cos \relax (x) + \frac {1}{3375} \, {\left (27 \, {\left (25 \, x^{2} - 2\right )} \cos \relax (x)^{4} + 4 \, {\left (225 \, x^{2} - 68\right )} \cos \relax (x)^{2} + 1800 \, x^{2} - 4144\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.58, size = 55, normalized size = 0.66 \[ \frac {1}{200} \, x \cos \left (5 \, x\right ) + \frac {5}{72} \, x \cos \left (3 \, x\right ) + \frac {5}{4} \, x \cos \relax (x) + \frac {1}{2000} \, {\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac {5}{432} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {5}{8} \, {\left (x^{2} - 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.38, size = 56, normalized size = 0.67
method | result | size |
risch | \(\frac {5 x \cos \relax (x )}{4}+\frac {5 \left (x^{2}-2\right ) \sin \relax (x )}{8}+\frac {x \cos \left (5 x \right )}{200}+\frac {\left (25 x^{2}-2\right ) \sin \left (5 x \right )}{2000}+\frac {5 x \cos \left (3 x \right )}{72}+\frac {5 \left (9 x^{2}-2\right ) \sin \left (3 x \right )}{432}\) | \(56\) |
default | \(\frac {x^{2} \left (\frac {8}{3}+\cos ^{4}\relax (x )+\frac {4 \left (\cos ^{2}\relax (x )\right )}{3}\right ) \sin \relax (x )}{5}-\frac {16 \sin \relax (x )}{15}+\frac {16 x \cos \relax (x )}{15}+\frac {2 x \left (\cos ^{5}\relax (x )\right )}{25}-\frac {2 \left (\frac {8}{3}+\cos ^{4}\relax (x )+\frac {4 \left (\cos ^{2}\relax (x )\right )}{3}\right ) \sin \relax (x )}{125}+\frac {8 x \left (\cos ^{3}\relax (x )\right )}{45}-\frac {8 \left (2+\cos ^{2}\relax (x )\right ) \sin \relax (x )}{135}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 55, normalized size = 0.66 \[ \frac {1}{200} \, x \cos \left (5 \, x\right ) + \frac {5}{72} \, x \cos \left (3 \, x\right ) + \frac {5}{4} \, x \cos \relax (x) + \frac {1}{2000} \, {\left (25 \, x^{2} - 2\right )} \sin \left (5 \, x\right ) + \frac {5}{432} \, {\left (9 \, x^{2} - 2\right )} \sin \left (3 \, x\right ) + \frac {5}{8} \, {\left (x^{2} - 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 69, normalized size = 0.83 \[ \frac {8\,x\,{\cos \relax (x)}^3}{45}-\frac {4144\,\sin \relax (x)}{3375}+\frac {2\,x\,{\cos \relax (x)}^5}{25}+\frac {8\,x^2\,\sin \relax (x)}{15}-\frac {272\,{\cos \relax (x)}^2\,\sin \relax (x)}{3375}-\frac {2\,{\cos \relax (x)}^4\,\sin \relax (x)}{125}+\frac {16\,x\,\cos \relax (x)}{15}+\frac {4\,x^2\,{\cos \relax (x)}^2\,\sin \relax (x)}{15}+\frac {x^2\,{\cos \relax (x)}^4\,\sin \relax (x)}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.49, size = 112, normalized size = 1.35 \[ \frac {8 x^{2} \sin ^{5}{\relax (x )}}{15} + \frac {4 x^{2} \sin ^{3}{\relax (x )} \cos ^{2}{\relax (x )}}{3} + x^{2} \sin {\relax (x )} \cos ^{4}{\relax (x )} + \frac {16 x \sin ^{4}{\relax (x )} \cos {\relax (x )}}{15} + \frac {104 x \sin ^{2}{\relax (x )} \cos ^{3}{\relax (x )}}{45} + \frac {298 x \cos ^{5}{\relax (x )}}{225} - \frac {4144 \sin ^{5}{\relax (x )}}{3375} - \frac {1712 \sin ^{3}{\relax (x )} \cos ^{2}{\relax (x )}}{675} - \frac {298 \sin {\relax (x )} \cos ^{4}{\relax (x )}}{225} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________