3.74 \(\int \sqrt{\cos (x)} \sin ^3(x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{7} \cos ^{\frac{7}{2}}(x)-\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

[Out]

(-2*Cos[x]^(3/2))/3 + (2*Cos[x]^(7/2))/7

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Rubi [A]  time = 0.0224993, 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 = {2565, 14} \[ \frac{2}{7} \cos ^{\frac{7}{2}}(x)-\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[x]^(3/2))/3 + (2*Cos[x]^(7/2))/7

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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 \sqrt{\cos (x)} \sin ^3(x) \, dx &=-\operatorname{Subst}\left (\int \sqrt{x} \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\sqrt{x}-x^{5/2}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{2}{3} \cos ^{\frac{3}{2}}(x)+\frac{2}{7} \cos ^{\frac{7}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0634669, size = 34, normalized size = 1.62 \[ \frac{(3 \cos (2 x)-11) \cos ^2(x)+8 \sqrt [4]{\cos ^2(x)}}{21 \sqrt{\cos (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(Cos[x]^2)^(1/4) + Cos[x]^2*(-11 + 3*Cos[2*x]))/(21*Sqrt[Cos[x]])

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Maple [B]  time = 0.033, size = 39, normalized size = 1.9 \begin{align*} -{\frac{8}{21}\sqrt{-2\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+1} \left ( 6\, \left ( \sin \left ( x/2 \right ) \right ) ^{6}-9\, \left ( \sin \left ( x/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3*cos(x)^(1/2),x)

[Out]

-8/21*(-2*sin(1/2*x)^2+1)^(1/2)*(6*sin(1/2*x)^6-9*sin(1/2*x)^4+sin(1/2*x)^2+1)

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Maxima [A]  time = 0.929643, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{7} \, \cos \left (x\right )^{\frac{7}{2}} - \frac{2}{3} \, \cos \left (x\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3*cos(x)^(1/2),x, algorithm="maxima")

[Out]

2/7*cos(x)^(7/2) - 2/3*cos(x)^(3/2)

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Fricas [A]  time = 2.06191, size = 58, normalized size = 2.76 \begin{align*} \frac{2}{21} \,{\left (3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )\right )} \sqrt{\cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3*cos(x)^(1/2),x, algorithm="fricas")

[Out]

2/21*(3*cos(x)^3 - 7*cos(x))*sqrt(cos(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3*cos(x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.05324, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{7} \, \cos \left (x\right )^{\frac{7}{2}} - \frac{2}{3} \, \cos \left (x\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3*cos(x)^(1/2),x, algorithm="giac")

[Out]

2/7*cos(x)^(7/2) - 2/3*cos(x)^(3/2)