3.754 \(\int \sqrt{\cos (x)} \sin (x) \, dx\)

Optimal. Leaf size=10 \[ -\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

[Out]

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

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Rubi [A]  time = 0.0121782, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2565, 30} \[ -\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 30

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

Rubi steps

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

Mathematica [A]  time = 0.002313, size = 10, normalized size = 1. \[ -\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 7, normalized size = 0.7 \begin{align*} -{\frac{2}{3} \left ( \cos \left ( x \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*cos(x)^(3/2)

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Maxima [A]  time = 0.961796, size = 8, normalized size = 0.8 \begin{align*} -\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)*cos(x)^(1/2),x, algorithm="maxima")

[Out]

-2/3*cos(x)^(3/2)

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Fricas [A]  time = 2.07166, size = 26, normalized size = 2.6 \begin{align*} -\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)*cos(x)^(1/2),x, algorithm="fricas")

[Out]

-2/3*cos(x)^(3/2)

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Sympy [A]  time = 0.283816, size = 10, normalized size = 1. \begin{align*} - \frac{2 \cos ^{\frac{3}{2}}{\left (x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)**(3/2)/3

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Giac [A]  time = 1.10316, size = 8, normalized size = 0.8 \begin{align*} -\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)*cos(x)^(1/2),x, algorithm="giac")

[Out]

-2/3*cos(x)^(3/2)