3.96 \(\int x \sin ^3(x) \, dx\)

Optimal. Leaf size=33 \[ \frac{\sin ^3(x)}{9}+\frac{2 \sin (x)}{3}-\frac{2}{3} x \cos (x)-\frac{1}{3} x \sin ^2(x) \cos (x) \]

[Out]

(-2*x*Cos[x])/3 + (2*Sin[x])/3 - (x*Cos[x]*Sin[x]^2)/3 + Sin[x]^3/9

_______________________________________________________________________________________

Rubi [A]  time = 0.0338782, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{\sin ^3(x)}{9}+\frac{2 \sin (x)}{3}-\frac{2}{3} x \cos (x)-\frac{1}{3} x \sin ^2(x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[x*Sin[x]^3,x]

[Out]

(-2*x*Cos[x])/3 + (2*Sin[x])/3 - (x*Cos[x]*Sin[x]^2)/3 + Sin[x]^3/9

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.36319, size = 32, normalized size = 0.97 \[ - \frac{x \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{3} - \frac{2 x \cos{\left (x \right )}}{3} + \frac{\sin ^{3}{\left (x \right )}}{9} + \frac{2 \sin{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*sin(x)**3,x)

[Out]

-x*sin(x)**2*cos(x)/3 - 2*x*cos(x)/3 + sin(x)**3/9 + 2*sin(x)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0063939, size = 31, normalized size = 0.94 \[ \frac{3 \sin (x)}{4}-\frac{1}{36} \sin (3 x)-\frac{3}{4} x \cos (x)+\frac{1}{12} x \cos (3 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sin[x]^3,x]

[Out]

(-3*x*Cos[x])/4 + (x*Cos[3*x])/12 + (3*Sin[x])/4 - Sin[3*x]/36

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 23, normalized size = 0.7 \[ -{\frac{x \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{9}}+{\frac{2\,\sin \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*sin(x)^3,x)

[Out]

-1/3*x*(2+sin(x)^2)*cos(x)+1/9*sin(x)^3+2/3*sin(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.42682, size = 31, normalized size = 0.94 \[ \frac{1}{12} \, x \cos \left (3 \, x\right ) - \frac{3}{4} \, x \cos \left (x\right ) - \frac{1}{36} \, \sin \left (3 \, x\right ) + \frac{3}{4} \, \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x*cos(3*x) - 3/4*x*cos(x) - 1/36*sin(3*x) + 3/4*sin(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.228395, size = 31, normalized size = 0.94 \[ \frac{1}{3} \, x \cos \left (x\right )^{3} - x \cos \left (x\right ) - \frac{1}{9} \,{\left (\cos \left (x\right )^{2} - 7\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x*cos(x)^3 - x*cos(x) - 1/9*(cos(x)^2 - 7)*sin(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.817574, size = 39, normalized size = 1.18 \[ - x \sin ^{2}{\left (x \right )} \cos{\left (x \right )} - \frac{2 x \cos ^{3}{\left (x \right )}}{3} + \frac{7 \sin ^{3}{\left (x \right )}}{9} + \frac{2 \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x)**3,x)

[Out]

-x*sin(x)**2*cos(x) - 2*x*cos(x)**3/3 + 7*sin(x)**3/9 + 2*sin(x)*cos(x)**2/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.201063, size = 31, normalized size = 0.94 \[ \frac{1}{12} \, x \cos \left (3 \, x\right ) - \frac{3}{4} \, x \cos \left (x\right ) - \frac{1}{36} \, \sin \left (3 \, x\right ) + \frac{3}{4} \, \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x*cos(3*x) - 3/4*x*cos(x) - 1/36*sin(3*x) + 3/4*sin(x)