3.64 \(\int \sqrt{x} \cos (x) \, dx\)

Optimal. Leaf size=36 \[ \sqrt{x} \sin (x)-\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right ) \]

[Out]

-(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[x]]) + Sqrt[x]*Sin[x]

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Rubi [A]  time = 0.034927, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3296, 3305, 3351} \[ \sqrt{x} \sin (x)-\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[x]]) + Sqrt[x]*Sin[x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int \sqrt{x} \cos (x) \, dx &=\sqrt{x} \sin (x)-\frac{1}{2} \int \frac{\sin (x)}{\sqrt{x}} \, dx\\ &=\sqrt{x} \sin (x)-\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=-\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )+\sqrt{x} \sin (x)\\ \end{align*}

Mathematica [C]  time = 0.0061006, size = 48, normalized size = 1.33 \[ \frac{\sqrt{-i x} \text{Gamma}\left (\frac{3}{2},-i x\right )+\sqrt{i x} \text{Gamma}\left (\frac{3}{2},i x\right )}{2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(-I)*x]*Gamma[3/2, (-I)*x] + Sqrt[I*x]*Gamma[3/2, I*x])/(2*Sqrt[x])

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Maple [A]  time = 0.028, size = 27, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{2}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{x}} \right ) }+\sin \left ( x \right ) \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*FresnelS(2^(1/2)/Pi^(1/2)*x^(1/2))*2^(1/2)*Pi^(1/2)+sin(x)*x^(1/2)

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Maxima [C]  time = 1.74941, size = 90, normalized size = 2.5 \begin{align*} -\frac{1}{16} \, \sqrt{\pi }{\left (\left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) + \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (\sqrt{-i} \sqrt{x}\right ) + \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\left (-1\right )^{\frac{1}{4}} \sqrt{x}\right )\right )} + \sqrt{x} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(pi)*((I + 1)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*sqrt(x)) + (I - 1)*sqrt(2)*erf((1/2*I - 1/2)*sqrt(2)
*sqrt(x)) - (I - 1)*sqrt(2)*erf(sqrt(-I)*sqrt(x)) + (I + 1)*sqrt(2)*erf((-1)^(1/4)*sqrt(x))) + sqrt(x)*sin(x)

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Fricas [A]  time = 1.62922, size = 105, normalized size = 2.92 \begin{align*} -\frac{1}{2} \, \sqrt{2} \sqrt{\pi } \operatorname{S}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) + \sqrt{x} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*sqrt(pi)*fresnel_sin(sqrt(2)*sqrt(x)/sqrt(pi)) + sqrt(x)*sin(x)

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Sympy [A]  time = 1.19293, size = 61, normalized size = 1.69 \begin{align*} \frac{3 \sqrt{x} \sin{\left (x \right )} \Gamma \left (\frac{3}{4}\right )}{4 \Gamma \left (\frac{7}{4}\right )} - \frac{3 \sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{3}{4}\right )}{8 \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*sqrt(x)*sin(x)*gamma(3/4)/(4*gamma(7/4)) - 3*sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*sqrt(x)/sqrt(pi))*gamma(3/4)/
(8*gamma(7/4))

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Giac [C]  time = 1.13463, size = 72, normalized size = 2. \begin{align*} -\left (\frac{1}{8} i - \frac{1}{8}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) + \left (\frac{1}{8} i + \frac{1}{8}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) - \frac{1}{2} i \, \sqrt{x} e^{\left (i \, x\right )} + \frac{1}{2} i \, \sqrt{x} e^{\left (-i \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/8*I - 1/8)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(x)) + (1/8*I + 1/8)*sqrt(2)*sqrt(pi)*erf(-(1/2*
I + 1/2)*sqrt(2)*sqrt(x)) - 1/2*I*sqrt(x)*e^(I*x) + 1/2*I*sqrt(x)*e^(-I*x)