3.91 \(\int \cos (x) \sin ^2(x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\sin ^3(x)}{3} \]

[Out]

Sin[x]^3/3

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Rubi [A]  time = 0.0127217, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2564, 30} \[ \frac{\sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sin[x]^2,x]

[Out]

Sin[x]^3/3

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0010916, size = 8, normalized size = 1. \[ \frac{\sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sin[x]^2,x]

[Out]

Sin[x]^3/3

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Maple [A]  time = 0.002, size = 7, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(x)^2,x)

[Out]

1/3*sin(x)^3

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Maxima [A]  time = 0.919453, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \sin \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x)^2,x, algorithm="maxima")

[Out]

1/3*sin(x)^3

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Fricas [A]  time = 2.07203, size = 38, normalized size = 4.75 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x)^2,x, algorithm="fricas")

[Out]

-1/3*(cos(x)^2 - 1)*sin(x)

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Sympy [A]  time = 0.057543, size = 5, normalized size = 0.62 \begin{align*} \frac{\sin ^{3}{\left (x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x)**2,x)

[Out]

sin(x)**3/3

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Giac [A]  time = 1.05907, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \sin \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x)^2,x, algorithm="giac")

[Out]

1/3*sin(x)^3