3.371 \(\int \cos ^3(x) \sin ^3(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\sin ^4(x)}{4}-\frac{\sin ^6(x)}{6} \]

[Out]

Sin[x]^4/4 - Sin[x]^6/6

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Rubi [A]  time = 0.0222371, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2564, 14} \[ \frac{\sin ^4(x)}{4}-\frac{\sin ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Sin[x]^3,x]

[Out]

Sin[x]^4/4 - Sin[x]^6/6

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cos ^3(x) \sin ^3(x) \, dx &=\operatorname{Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (x)\right )\\ &=\frac{\sin ^4(x)}{4}-\frac{\sin ^6(x)}{6}\\ \end{align*}

Mathematica [A]  time = 0.006301, size = 17, normalized size = 1. \[ \frac{1}{192} \cos (6 x)-\frac{3}{64} \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Sin[x]^3,x]

[Out]

(-3*Cos[2*x])/64 + Cos[6*x]/192

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Maple [A]  time = 0.006, size = 18, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \sin \left ( x \right ) \right ) ^{2}}{6}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x)^3,x)

[Out]

-1/6*cos(x)^4*sin(x)^2-1/12*cos(x)^4

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Maxima [A]  time = 0.923391, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{6} \, \sin \left (x\right )^{6} + \frac{1}{4} \, \sin \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^3,x, algorithm="maxima")

[Out]

-1/6*sin(x)^6 + 1/4*sin(x)^4

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Fricas [A]  time = 2.00521, size = 39, normalized size = 2.29 \begin{align*} \frac{1}{6} \, \cos \left (x\right )^{6} - \frac{1}{4} \, \cos \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^3,x, algorithm="fricas")

[Out]

1/6*cos(x)^6 - 1/4*cos(x)^4

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Sympy [A]  time = 0.058797, size = 12, normalized size = 0.71 \begin{align*} - \frac{\sin ^{6}{\left (x \right )}}{6} + \frac{\sin ^{4}{\left (x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(x)**6/6 + sin(x)**4/4

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Giac [A]  time = 1.05459, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{6} \, \cos \left (x\right )^{6} - \frac{1}{4} \, \cos \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^3,x, algorithm="giac")

[Out]

1/6*cos(x)^6 - 1/4*cos(x)^4