3.65 \(\int \frac{\cos (x)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=24 \[ \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right ) \]

[Out]

Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[x]]

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Rubi [A]  time = 0.019832, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3304, 3352} \[ \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[x],x]

[Out]

Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[x]]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(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 3352

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

Rubi steps

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

Mathematica [C]  time = 0.0066152, size = 51, normalized size = 2.12 \[ -\frac{i \left (\sqrt{-i x} \text{Gamma}\left (\frac{1}{2},-i x\right )-\sqrt{i x} \text{Gamma}\left (\frac{1}{2},i x\right )\right )}{2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/Sqrt[x],x]

[Out]

((-I/2)*(Sqrt[(-I)*x]*Gamma[1/2, (-I)*x] - Sqrt[I*x]*Gamma[1/2, I*x]))/Sqrt[x]

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Maple [A]  time = 0.026, size = 19, normalized size = 0.8 \begin{align*}{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{x}} \right ) \sqrt{2}\sqrt{\pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

FresnelC(2^(1/2)/Pi^(1/2)*x^(1/2))*2^(1/2)*Pi^(1/2)

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Maxima [C]  time = 1.8776, size = 81, normalized size = 3.38 \begin{align*} -\frac{1}{8} \, \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*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)))

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Fricas [A]  time = 1.6563, size = 76, normalized size = 3.17 \begin{align*} \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt(pi)*fresnel_cos(sqrt(2)*sqrt(x)/sqrt(pi))

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Sympy [A]  time = 0.996044, size = 37, normalized size = 1.54 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)*sqrt(x)/sqrt(pi))*gamma(1/4)/(4*gamma(5/4))

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Giac [C]  time = 1.13746, size = 47, normalized size = 1.96 \begin{align*} -\left (\frac{1}{4} i + \frac{1}{4}\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}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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