3.101 \(\int x \cos ^3(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0358407, 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 \[ \frac{2}{3} x \sin (x)+\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}+\frac{1}{3} x \sin (x) \cos ^2(x) \]

Antiderivative was successfully verified.

[In]  Int[x*Cos[x]^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 1.33991, size = 32, normalized size = 0.97 \[ \frac{x \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{3} + \frac{2 x \sin{\left (x \right )}}{3} + \frac{\cos ^{3}{\left (x \right )}}{9} + \frac{2 \cos{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*cos(x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.00656925, size = 31, normalized size = 0.94 \[ \frac{3}{4} x \sin (x)+\frac{1}{12} x \sin (3 x)+\frac{3 \cos (x)}{4}+\frac{1}{36} \cos (3 x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Cos[x]^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 23, normalized size = 0.7 \[{\frac{x \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{9}}+{\frac{2\,\cos \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*cos(x)^3,x)

[Out]

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

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Maxima [A]  time = 1.41896, size = 31, normalized size = 0.94 \[ \frac{1}{12} \, x \sin \left (3 \, x\right ) + \frac{3}{4} \, x \sin \left (x\right ) + \frac{1}{36} \, \cos \left (3 \, x\right ) + \frac{3}{4} \, \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221458, size = 34, normalized size = 1.03 \[ \frac{1}{9} \, \cos \left (x\right )^{3} + \frac{1}{3} \,{\left (x \cos \left (x\right )^{2} + 2 \, x\right )} \sin \left (x\right ) + \frac{2}{3} \, \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.81658, size = 39, normalized size = 1.18 \[ \frac{2 x \sin ^{3}{\left (x \right )}}{3} + x \sin{\left (x \right )} \cos ^{2}{\left (x \right )} + \frac{2 \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{3} + \frac{7 \cos ^{3}{\left (x \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*cos(x)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.198689, size = 31, normalized size = 0.94 \[ \frac{1}{12} \, x \sin \left (3 \, x\right ) + \frac{3}{4} \, x \sin \left (x\right ) + \frac{1}{36} \, \cos \left (3 \, x\right ) + \frac{3}{4} \, \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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