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

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

[Out]

sin(x)-1/3*sin(x)^3

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2633} \[ \sin (x)-\frac {\sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3,x]

[Out]

Sin[x] - Sin[x]^3/3

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \cos ^3(x) \, dx &=-\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )\\ &=\sin (x)-\frac {\sin ^3(x)}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, 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

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fricas [A]  time = 0.41, size = 10, normalized size = 0.91 \[ \frac {1}{3} \, {\left (\cos \relax (x)^{2} + 2\right )} \sin \relax (x) \]

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)

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giac [A]  time = 0.01, size = 9, normalized size = 0.82 \[ -\frac {1}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]

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)

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maple [A]  time = 0.09, size = 11, normalized size = 1.00 \[ \frac {\left (\cos ^{2}\relax (x )+2\right ) \sin \relax (x )}{3} \]

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)

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maxima [A]  time = 0.58, size = 9, normalized size = 0.82 \[ -\frac {1}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]

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)

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mupad [B]  time = 0.03, size = 9, normalized size = 0.82 \[ \sin \relax (x)-\frac {{\sin \relax (x)}^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3,x)

[Out]

sin(x) - sin(x)^3/3

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sympy [A]  time = 0.07, size = 8, normalized size = 0.73 \[ - \frac {\sin ^{3}{\relax (x )}}{3} + \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3,x)

[Out]

-sin(x)**3/3 + sin(x)

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