3.99 \(\int x \cos ^2(x) \, dx\)

Optimal. Leaf size=25 \[ \frac{x^2}{4}+\frac{\cos ^2(x)}{4}+\frac{1}{2} x \sin (x) \cos (x) \]

[Out]

x^2/4 + Cos[x]^2/4 + (x*Cos[x]*Sin[x])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0245117, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^2}{4}+\frac{\cos ^2(x)}{4}+\frac{1}{2} x \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[x*Cos[x]^2,x]

[Out]

x^2/4 + Cos[x]^2/4 + (x*Cos[x]*Sin[x])/2

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x \sin{\left (x \right )} \cos{\left (x \right )}}{2} + \frac{\cos ^{2}{\left (x \right )}}{4} + \frac{\int x\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*cos(x)**2,x)

[Out]

x*sin(x)*cos(x)/2 + cos(x)**2/4 + Integral(x, x)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00512229, size = 25, normalized size = 1. \[ \frac{x^2}{4}+\frac{1}{4} x \sin (2 x)+\frac{1}{8} \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Cos[x]^2,x]

[Out]

x^2/4 + Cos[2*x]/8 + (x*Sin[2*x])/4

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 25, normalized size = 1. \[ x \left ({\frac{x}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \right ) -{\frac{{x}^{2}}{4}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*cos(x)^2,x)

[Out]

x*(1/2*x+1/2*cos(x)*sin(x))-1/4*x^2-1/4*sin(x)^2

_______________________________________________________________________________________

Maxima [A]  time = 1.41086, size = 26, normalized size = 1.04 \[ \frac{1}{4} \, x^{2} + \frac{1}{4} \, x \sin \left (2 \, x\right ) + \frac{1}{8} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^2 + 1/4*x*sin(2*x) + 1/8*cos(2*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.218863, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, x \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4} \, x^{2} + \frac{1}{4} \, \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*cos(x)*sin(x) + 1/4*x^2 + 1/4*cos(x)^2

_______________________________________________________________________________________

Sympy [A]  time = 0.405291, size = 36, normalized size = 1.44 \[ \frac{x^{2} \sin ^{2}{\left (x \right )}}{4} + \frac{x^{2} \cos ^{2}{\left (x \right )}}{4} + \frac{x \sin{\left (x \right )} \cos{\left (x \right )}}{2} - \frac{\sin ^{2}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(x)**2,x)

[Out]

x**2*sin(x)**2/4 + x**2*cos(x)**2/4 + x*sin(x)*cos(x)/2 - sin(x)**2/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.200048, size = 26, normalized size = 1.04 \[ \frac{1}{4} \, x^{2} + \frac{1}{4} \, x \sin \left (2 \, x\right ) + \frac{1}{8} \, \cos \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^2 + 1/4*x*sin(2*x) + 1/8*cos(2*x)