Integrand size = 17, antiderivative size = 37 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=\frac {\cos \left (x^2\right )}{2}-\frac {1}{6} \cos ^3\left (x^2\right )+\frac {\sin \left (x^2\right )}{2}-\frac {1}{6} \sin ^3\left (x^2\right ) \] Output:
1/2*cos(x^2)-1/6*cos(x^2)^3+1/2*sin(x^2)-1/6*sin(x^2)^3
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=\frac {3 \cos \left (x^2\right )}{8}-\frac {1}{24} \cos \left (3 x^2\right )+\frac {\sin \left (x^2\right )}{2}-\frac {1}{6} \sin ^3\left (x^2\right ) \] Input:
Integrate[x*(Cos[x^2]^3 - Sin[x^2]^3),x]
Output:
(3*Cos[x^2])/8 - Cos[3*x^2]/24 + Sin[x^2]/2 - Sin[x^2]^3/6
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2010, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (x \cos ^3\left (x^2\right )-x \sin ^3\left (x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{6} \sin ^3\left (x^2\right )+\frac {\sin \left (x^2\right )}{2}-\frac {1}{6} \cos ^3\left (x^2\right )+\frac {\cos \left (x^2\right )}{2}\) |
Input:
Int[x*(Cos[x^2]^3 - Sin[x^2]^3),x]
Output:
Cos[x^2]/2 - Cos[x^2]^3/6 + Sin[x^2]/2 - Sin[x^2]^3/6
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.89 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\left (2+\cos \left (x^{2}\right )^{2}\right ) \sin \left (x^{2}\right )}{6}+\frac {\left (2+\sin \left (x^{2}\right )^{2}\right ) \cos \left (x^{2}\right )}{6}\) | \(30\) |
default | \(\frac {\left (2+\cos \left (x^{2}\right )^{2}\right ) \sin \left (x^{2}\right )}{6}+\frac {\left (2+\sin \left (x^{2}\right )^{2}\right ) \cos \left (x^{2}\right )}{6}\) | \(30\) |
risch | \(\frac {3 \cos \left (x^{2}\right )}{8}+\frac {3 \sin \left (x^{2}\right )}{8}-\frac {\cos \left (3 x^{2}\right )}{24}+\frac {\sin \left (3 x^{2}\right )}{24}\) | \(30\) |
parts | \(\frac {\left (2+\cos \left (x^{2}\right )^{2}\right ) \sin \left (x^{2}\right )}{6}+\frac {\left (2+\sin \left (x^{2}\right )^{2}\right ) \cos \left (x^{2}\right )}{6}\) | \(30\) |
norman | \(\frac {\tan \left (\frac {x^{2}}{2}\right )^{5}+2 \tan \left (\frac {x^{2}}{2}\right )^{2}+\frac {2 \tan \left (\frac {x^{2}}{2}\right )^{3}}{3}+\frac {2}{3}+\tan \left (\frac {x^{2}}{2}\right )}{{\left (1+\tan \left (\frac {x^{2}}{2}\right )^{2}\right )}^{3}}\) | \(50\) |
parallelrisch | \(\frac {2+3 \tan \left (\frac {x^{2}}{2}\right )^{5}+2 \tan \left (\frac {x^{2}}{2}\right )^{3}+6 \tan \left (\frac {x^{2}}{2}\right )^{2}+3 \tan \left (\frac {x^{2}}{2}\right )}{3 {\left (1+\tan \left (\frac {x^{2}}{2}\right )^{2}\right )}^{3}}\) | \(55\) |
orering | \(\frac {5 \left (8 x^{4}+3\right ) \left (\cos \left (x^{2}\right )^{3}-\sin \left (x^{2}\right )^{3}\right )}{144 x^{6}}-\frac {5 \left (8 x^{4}+3\right ) \left (\cos \left (x^{2}\right )^{3}-\sin \left (x^{2}\right )^{3}+x \left (-6 \cos \left (x^{2}\right )^{2} x \sin \left (x^{2}\right )-6 \sin \left (x^{2}\right )^{2} x \cos \left (x^{2}\right )\right )\right )}{144 x^{6}}+\frac {-12 \cos \left (x^{2}\right )^{2} x \sin \left (x^{2}\right )-12 \sin \left (x^{2}\right )^{2} x \cos \left (x^{2}\right )+x \left (24 \cos \left (x^{2}\right ) \sin \left (x^{2}\right )^{2} x^{2}-6 \cos \left (x^{2}\right )^{2} \sin \left (x^{2}\right )-12 \cos \left (x^{2}\right )^{3} x^{2}-24 \cos \left (x^{2}\right )^{2} \sin \left (x^{2}\right ) x^{2}-6 \sin \left (x^{2}\right )^{2} \cos \left (x^{2}\right )+12 \sin \left (x^{2}\right )^{3} x^{2}\right )}{24 x^{5}}-\frac {72 \cos \left (x^{2}\right ) \sin \left (x^{2}\right )^{2} x^{2}-18 \cos \left (x^{2}\right )^{2} \sin \left (x^{2}\right )-36 \cos \left (x^{2}\right )^{3} x^{2}-72 \cos \left (x^{2}\right )^{2} \sin \left (x^{2}\right ) x^{2}-18 \sin \left (x^{2}\right )^{2} \cos \left (x^{2}\right )+36 \sin \left (x^{2}\right )^{3} x^{2}+x \left (-48 x^{3} \sin \left (x^{2}\right )^{3}+168 \cos \left (x^{2}\right )^{2} \sin \left (x^{2}\right ) x^{3}+72 \sin \left (x^{2}\right )^{2} x \cos \left (x^{2}\right )-36 x \cos \left (x^{2}\right )^{3}+168 \cos \left (x^{2}\right ) \sin \left (x^{2}\right )^{2} x^{3}-48 \cos \left (x^{2}\right )^{3} x^{3}-72 \cos \left (x^{2}\right )^{2} x \sin \left (x^{2}\right )+36 x \sin \left (x^{2}\right )^{3}\right )}{144 x^{4}}\) | \(377\) |
Input:
int(x*(cos(x^2)^3-sin(x^2)^3),x,method=_RETURNVERBOSE)
Output:
1/6*(2+cos(x^2)^2)*sin(x^2)+1/6*(2+sin(x^2)^2)*cos(x^2)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=-\frac {1}{6} \, \cos \left (x^{2}\right )^{3} + \frac {1}{6} \, {\left (\cos \left (x^{2}\right )^{2} + 2\right )} \sin \left (x^{2}\right ) + \frac {1}{2} \, \cos \left (x^{2}\right ) \] Input:
integrate(x*(cos(x^2)^3-sin(x^2)^3),x, algorithm="fricas")
Output:
-1/6*cos(x^2)^3 + 1/6*(cos(x^2)^2 + 2)*sin(x^2) + 1/2*cos(x^2)
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=\frac {\sin ^{3}{\left (x^{2} \right )}}{3} + \frac {\sin ^{2}{\left (x^{2} \right )} \cos {\left (x^{2} \right )}}{2} + \frac {\sin {\left (x^{2} \right )} \cos ^{2}{\left (x^{2} \right )}}{2} + \frac {\cos ^{3}{\left (x^{2} \right )}}{3} \] Input:
integrate(x*(cos(x**2)**3-sin(x**2)**3),x)
Output:
sin(x**2)**3/3 + sin(x**2)**2*cos(x**2)/2 + sin(x**2)*cos(x**2)**2/2 + cos (x**2)**3/3
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=-\frac {1}{24} \, \cos \left (3 \, x^{2}\right ) + \frac {3}{8} \, \cos \left (x^{2}\right ) + \frac {1}{24} \, \sin \left (3 \, x^{2}\right ) + \frac {3}{8} \, \sin \left (x^{2}\right ) \] Input:
integrate(x*(cos(x^2)^3-sin(x^2)^3),x, algorithm="maxima")
Output:
-1/24*cos(3*x^2) + 3/8*cos(x^2) + 1/24*sin(3*x^2) + 3/8*sin(x^2)
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=-\frac {1}{6} \, \cos \left (x^{2}\right )^{3} - \frac {1}{6} \, \sin \left (x^{2}\right )^{3} + \frac {1}{2} \, \cos \left (x^{2}\right ) + \frac {1}{2} \, \sin \left (x^{2}\right ) \] Input:
integrate(x*(cos(x^2)^3-sin(x^2)^3),x, algorithm="giac")
Output:
-1/6*cos(x^2)^3 - 1/6*sin(x^2)^3 + 1/2*cos(x^2) + 1/2*sin(x^2)
Time = 16.81 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=-\frac {{\cos \left (x^2\right )}^3}{6}+\frac {\sin \left (x^2\right )\,{\cos \left (x^2\right )}^2}{6}+\frac {\cos \left (x^2\right )}{2}+\frac {\sin \left (x^2\right )}{3} \] Input:
int(x*(cos(x^2)^3 - sin(x^2)^3),x)
Output:
cos(x^2)/2 + sin(x^2)/3 + (cos(x^2)^2*sin(x^2))/6 - cos(x^2)^3/6
Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int x \left (\cos ^3\left (x^2\right )-\sin ^3\left (x^2\right )\right ) \, dx=\frac {\cos \left (x^{2}\right )^{2} \sin \left (x^{2}\right )}{2}+\frac {\cos \left (x^{2}\right ) \sin \left (x^{2}\right )^{2}}{6}+\frac {\cos \left (x^{2}\right )}{3}+\frac {\sin \left (x^{2}\right )^{3}}{3}-\frac {1}{3} \] Input:
int(x*(cos(x^2)^3-sin(x^2)^3),x)
Output:
(3*cos(x**2)**2*sin(x**2) + cos(x**2)*sin(x**2)**2 + 2*cos(x**2) + 2*sin(x **2)**3 - 2)/6