\(\int \sqrt {\cos (x)} \sin (x) \, dx\) [754]

Optimal result

Integrand size = 9, antiderivative size = 10 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2}{3} \cos ^{\frac {3}{2}}(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2645, 30} \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2}{3} \cos ^{\frac {3}{2}}(x) \]

[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 2645

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2}{3} \cos ^{\frac {3}{2}}(x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {2 \cos \left (x \right )^{\frac {3}{2}}}{3}\)	\(7\)
default	\(-\frac {2 \cos \left (x \right )^{\frac {3}{2}}}{3}\)	\(7\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2}{3} \, \cos \left (x\right )^{\frac {3}{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=- \frac {2 \cos ^{\frac {3}{2}}{\left (x \right )}}{3} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2}{3} \, \cos \left (x\right )^{\frac {3}{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2}{3} \, \cos \left (x\right )^{\frac {3}{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \sqrt {\cos (x)} \sin (x) \, dx=-\frac {2\,{\cos \left (x\right )}^{3/2}}{3} \]

[In]

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

[Out]

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