3.66 \(\int \frac{\cos (x)}{x^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ -2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )-\frac{2 \cos (x)}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0366712, antiderivative size = 35, 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 = {3297, 3305, 3351} \[ -2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )-\frac{2 \cos (x)}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/x^(3/2),x]

[Out]

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

Rule 3297

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

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 \frac{\cos (x)}{x^{3/2}} \, dx &=-\frac{2 \cos (x)}{\sqrt{x}}-2 \int \frac{\sin (x)}{\sqrt{x}} \, dx\\ &=-\frac{2 \cos (x)}{\sqrt{x}}-4 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \cos (x)}{\sqrt{x}}-2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0423232, size = 63, normalized size = 1.8 \[ \frac{\sqrt{-i x} \text{Gamma}\left (\frac{1}{2},-i x\right )+\sqrt{i x} \text{Gamma}\left (\frac{1}{2},i x\right )-e^{-i x} \left (1+e^{2 i x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]/x^(3/2),x]

[Out]

(-((1 + E^((2*I)*x))/E^(I*x)) + 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 = 0.028, size = 28, normalized size = 0.8 \begin{align*} -2\,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{x}}{\sqrt{\pi }}} \right ) \sqrt{2}\sqrt{\pi }-2\,{\frac{\cos \left ( x \right ) }{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/x^(3/2),x)

[Out]

-2*FresnelS(2^(1/2)/Pi^(1/2)*x^(1/2))*2^(1/2)*Pi^(1/2)-2*cos(x)/x^(1/2)

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Maxima [C]  time = 2.43777, size = 28, normalized size = 0.8 \begin{align*} -\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2} \Gamma \left (-\frac{1}{2}, i \, x\right ) + \left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \Gamma \left (-\frac{1}{2}, -i \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/x^(3/2),x, algorithm="maxima")

[Out]

-(1/4*I + 1/4)*sqrt(2)*gamma(-1/2, I*x) + (1/4*I - 1/4)*sqrt(2)*gamma(-1/2, -I*x)

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Fricas [A]  time = 1.66506, size = 111, normalized size = 3.17 \begin{align*} -\frac{2 \,{\left (\sqrt{2} \sqrt{\pi } x \operatorname{S}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) + \sqrt{x} \cos \left (x\right )\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/x^(3/2),x, algorithm="fricas")

[Out]

-2*(sqrt(2)*sqrt(pi)*x*fresnel_sin(sqrt(2)*sqrt(x)/sqrt(pi)) + sqrt(x)*cos(x))/x

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Sympy [A]  time = 4.17337, size = 61, normalized size = 1.74 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (- \frac{1}{4}\right )}{2 \Gamma \left (\frac{3}{4}\right )} + \frac{\cos{\left (x \right )} \Gamma \left (- \frac{1}{4}\right )}{2 \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/x**(3/2),x)

[Out]

sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*sqrt(x)/sqrt(pi))*gamma(-1/4)/(2*gamma(3/4)) + cos(x)*gamma(-1/4)/(2*sqrt(x)
*gamma(3/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (x\right )}{x^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)/x^(3/2),x, algorithm="giac")

[Out]

integrate(cos(x)/x^(3/2), x)