3.30 \(\int x^3 \sin \left (x^2\right ) \, dx\)

Optimal. Leaf size=20 \[ \frac{\sin \left (x^2\right )}{2}-\frac{1}{2} x^2 \cos \left (x^2\right ) \]

[Out]

-(x^2*Cos[x^2])/2 + Sin[x^2]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0267647, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{\sin \left (x^2\right )}{2}-\frac{1}{2} x^2 \cos \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sin[x^2],x]

[Out]

-(x^2*Cos[x^2])/2 + Sin[x^2]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.41254, size = 15, normalized size = 0.75 \[ - \frac{x^{2} \cos{\left (x^{2} \right )}}{2} + \frac{\sin{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*sin(x**2),x)

[Out]

-x**2*cos(x**2)/2 + sin(x**2)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00476135, size = 20, normalized size = 1. \[ \frac{\sin \left (x^2\right )}{2}-\frac{1}{2} x^2 \cos \left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sin[x^2],x]

[Out]

-(x^2*Cos[x^2])/2 + Sin[x^2]/2

_______________________________________________________________________________________

Maple [A]  time = 0., size = 17, normalized size = 0.9 \[ -{\frac{{x}^{2}\cos \left ({x}^{2} \right ) }{2}}+{\frac{\sin \left ({x}^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*sin(x^2),x)

[Out]

-1/2*x^2*cos(x^2)+1/2*sin(x^2)

_______________________________________________________________________________________

Maxima [A]  time = 1.33502, size = 22, normalized size = 1.1 \[ -\frac{1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*sin(x^2),x, algorithm="maxima")

[Out]

-1/2*x^2*cos(x^2) + 1/2*sin(x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.229118, size = 22, normalized size = 1.1 \[ -\frac{1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*sin(x^2),x, algorithm="fricas")

[Out]

-1/2*x^2*cos(x^2) + 1/2*sin(x^2)

_______________________________________________________________________________________

Sympy [A]  time = 0.776329, size = 15, normalized size = 0.75 \[ - \frac{x^{2} \cos{\left (x^{2} \right )}}{2} + \frac{\sin{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*sin(x**2),x)

[Out]

-x**2*cos(x**2)/2 + sin(x**2)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.198359, size = 22, normalized size = 1.1 \[ -\frac{1}{2} \, x^{2} \cos \left (x^{2}\right ) + \frac{1}{2} \, \sin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*sin(x^2),x, algorithm="giac")

[Out]

-1/2*x^2*cos(x^2) + 1/2*sin(x^2)