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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 30

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

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])

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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ -\frac {2}{3} \cos ^{\frac {3}{2}}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 6, normalized size = 0.60 \[ -\frac {2}{3} \, \cos \relax (x)^{\frac {3}{2}} \]

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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giac [A]  time = 0.14, size = 6, normalized size = 0.60 \[ -\frac {2}{3} \, \cos \relax (x)^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.02, size = 7, normalized size = 0.70 \[ -\frac {2 \left (\cos ^{\frac {3}{2}}\relax (x )\right )}{3} \]

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 = 0.32, size = 6, normalized size = 0.60 \[ -\frac {2}{3} \, \cos \relax (x)^{\frac {3}{2}} \]

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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mupad [B]  time = 0.07, size = 6, normalized size = 0.60 \[ -\frac {2\,{\cos \relax (x)}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.25, size = 10, normalized size = 1.00 \[ - \frac {2 \cos ^{\frac {3}{2}}{\relax (x )}}{3} \]

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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