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

Optimal. Leaf size=11 \[ \sin (x)-\frac{\sin ^3(x)}{3} \]

[Out]

Sin[x] - Sin[x]^3/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0107204, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \sin (x)-\frac{\sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^3,x]

[Out]

Sin[x] - Sin[x]^3/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.671478, size = 8, normalized size = 0.73 \[ - \frac{\sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**3,x)

[Out]

-sin(x)**3/3 + sin(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00293136, size = 15, normalized size = 1.36 \[ \frac{3 \sin (x)}{4}+\frac{1}{12} \sin (3 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^3,x]

[Out]

(3*Sin[x])/4 + Sin[3*x]/12

_______________________________________________________________________________________

Maple [A]  time = 0.04, size = 11, normalized size = 1. \[{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35495, size = 12, normalized size = 1.09 \[ -\frac{1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sin(x)^3 + sin(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.206491, size = 14, normalized size = 1.27 \[ \frac{1}{3} \,{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(cos(x)^2 + 2)*sin(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.040248, size = 8, normalized size = 0.73 \[ - \frac{\sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**3,x)

[Out]

-sin(x)**3/3 + sin(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234835, size = 12, normalized size = 1.09 \[ -\frac{1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sin(x)^3 + sin(x)