3.4.31 \(\int \cos ^3(x) \, dx\) [331]

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 = {2713} \begin {gather*} \sin (x)-\frac {\sin ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3,x]

[Out]

Sin[x] - Sin[x]^3/3

Rule 2713

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 &=-\text {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 \begin {gather*} \frac {3 \sin (x)}{4}+\frac {1}{12} \sin (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3,x]

[Out]

(3*Sin[x])/4 + Sin[3*x]/12

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Maple [A]
time = 0.06, size = 11, normalized size = 1.00

method	result	size
default	\(\frac {\left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{3}\)	\(11\)
risch	\(\frac {3 \sin \left (x \right )}{4}+\frac {\sin \left (3 x \right )}{12}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.22, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end {gather*}

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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Fricas [A]
time = 0.74, size = 10, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, {\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \end {gather*}

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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Sympy [A]
time = 0.01, size = 8, normalized size = 0.73 \begin {gather*} - \frac {\sin ^{3}{\left (x \right )}}{3} + \sin {\left (x \right )} \end {gather*}

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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Giac [A]
time = 1.33, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end {gather*}

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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Mupad [B]
time = 0.03, size = 9, normalized size = 0.82 \begin {gather*} \sin \left (x\right )-\frac {{\sin \left (x\right )}^3}{3} \end {gather*}

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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