3.254 \(\int \cos ^3(x) \sin ^{\frac{3}{2}}(x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{5} \sin ^{\frac{5}{2}}(x)-\frac{2}{9} \sin ^{\frac{9}{2}}(x) \]

[Out]

(2*Sin[x]^(5/2))/5 - (2*Sin[x]^(9/2))/9

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Rubi [A]  time = 0.0243724, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2564, 14} \[ \frac{2}{5} \sin ^{\frac{5}{2}}(x)-\frac{2}{9} \sin ^{\frac{9}{2}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Sin[x]^(3/2),x]

[Out]

(2*Sin[x]^(5/2))/5 - (2*Sin[x]^(9/2))/9

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 ^{\frac{3}{2}}(x) \, dx &=\operatorname{Subst}\left (\int x^{3/2} \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^{3/2}-x^{7/2}\right ) \, dx,x,\sin (x)\right )\\ &=\frac{2}{5} \sin ^{\frac{5}{2}}(x)-\frac{2}{9} \sin ^{\frac{9}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0126511, size = 18, normalized size = 0.86 \[ \frac{1}{45} \sin ^{\frac{5}{2}}(x) (5 \cos (2 x)+13) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Sin[x]^(3/2),x]

[Out]

((13 + 5*Cos[2*x])*Sin[x]^(5/2))/45

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Maple [A]  time = 0.04, size = 14, normalized size = 0.7 \begin{align*}{\frac{2}{5} \left ( \sin \left ( x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{2}{9} \left ( \sin \left ( x \right ) \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x)^(3/2),x)

[Out]

2/5*sin(x)^(5/2)-2/9*sin(x)^(9/2)

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Maxima [A]  time = 0.964323, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{9} \, \sin \left (x\right )^{\frac{9}{2}} + \frac{2}{5} \, \sin \left (x\right )^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(3/2),x, algorithm="maxima")

[Out]

-2/9*sin(x)^(9/2) + 2/5*sin(x)^(5/2)

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Fricas [A]  time = 2.22841, size = 65, normalized size = 3.1 \begin{align*} -\frac{2}{45} \,{\left (5 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{2} - 4\right )} \sqrt{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(3/2),x, algorithm="fricas")

[Out]

-2/45*(5*cos(x)^4 - cos(x)^2 - 4)*sqrt(sin(x))

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Sympy [A]  time = 68.453, size = 24, normalized size = 1.14 \begin{align*} \frac{8 \sin ^{\frac{9}{2}}{\left (x \right )}}{45} + \frac{2 \sin ^{\frac{5}{2}}{\left (x \right )} \cos ^{2}{\left (x \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3*sin(x)**(3/2),x)

[Out]

8*sin(x)**(9/2)/45 + 2*sin(x)**(5/2)*cos(x)**2/5

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Giac [A]  time = 1.10614, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{9} \, \sin \left (x\right )^{\frac{9}{2}} + \frac{2}{5} \, \sin \left (x\right )^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(3/2),x, algorithm="giac")

[Out]

-2/9*sin(x)^(9/2) + 2/5*sin(x)^(5/2)