3.215 \(\int a \cos (5+3 x) \sin ^2(5+3 x) \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{9} a \sin ^3(3 x+5) \]

[Out]

(a*Sin[5 + 3*x]^3)/9

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Rubi [A]  time = 0.0174402, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {12, 2564, 30} \[ \frac{1}{9} a \sin ^3(3 x+5) \]

Antiderivative was successfully verified.

[In]

Int[a*Cos[5 + 3*x]*Sin[5 + 3*x]^2,x]

[Out]

(a*Sin[5 + 3*x]^3)/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 a \cos (5+3 x) \sin ^2(5+3 x) \, dx &=a \int \cos (5+3 x) \sin ^2(5+3 x) \, dx\\ &=\frac{1}{3} a \operatorname{Subst}\left (\int x^2 \, dx,x,\sin (5+3 x)\right )\\ &=\frac{1}{9} a \sin ^3(5+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0070728, size = 13, normalized size = 1. \[ \frac{1}{9} a \sin ^3(3 x+5) \]

Antiderivative was successfully verified.

[In]

Integrate[a*Cos[5 + 3*x]*Sin[5 + 3*x]^2,x]

[Out]

(a*Sin[5 + 3*x]^3)/9

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Maple [A]  time = 0.006, size = 12, normalized size = 0.9 \begin{align*}{\frac{a \left ( \sin \left ( 5+3\,x \right ) \right ) ^{3}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*cos(5+3*x)*sin(5+3*x)^2,x)

[Out]

1/9*a*sin(5+3*x)^3

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Maxima [A]  time = 0.943559, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{9} \, a \sin \left (3 \, x + 5\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*cos(5+3*x)*sin(5+3*x)^2,x, algorithm="maxima")

[Out]

1/9*a*sin(3*x + 5)^3

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Fricas [A]  time = 1.65026, size = 57, normalized size = 4.38 \begin{align*} -\frac{1}{9} \,{\left (a \cos \left (3 \, x + 5\right )^{2} - a\right )} \sin \left (3 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*cos(5+3*x)*sin(5+3*x)^2,x, algorithm="fricas")

[Out]

-1/9*(a*cos(3*x + 5)^2 - a)*sin(3*x + 5)

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Sympy [A]  time = 0.303857, size = 10, normalized size = 0.77 \begin{align*} \frac{a \sin ^{3}{\left (3 x + 5 \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*cos(5+3*x)*sin(5+3*x)**2,x)

[Out]

a*sin(3*x + 5)**3/9

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Giac [A]  time = 1.10725, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{9} \, a \sin \left (3 \, x + 5\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*cos(5+3*x)*sin(5+3*x)^2,x, algorithm="giac")

[Out]

1/9*a*sin(3*x + 5)^3