3.1.66 \(\int \frac {\cos (x)}{x^{3/2}} \, dx\) [66]

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

[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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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {3378, 3386, 3432} \begin {gather*} -2 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right )-\frac {2 \cos (x)}{\sqrt {x}} \end {gather*}

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 3378

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 3386

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 3432

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 63, normalized size = 1.80 \begin {gather*} \frac {-e^{-i x} \left (1+e^{2 i x}\right )+\sqrt {-i x} \text {Gamma}\left (\frac {1}{2},-i x\right )+\sqrt {i x} \text {Gamma}\left (\frac {1}{2},i x\right )}{\sqrt {x}} \end {gather*}

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 = 0.04, size = 28, normalized size = 0.80

method	result	size
derivativedivides	\(-2 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }-\frac {2 \cos \left (x \right )}{\sqrt {x}}\)	\(28\)
default	\(-2 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }-\frac {2 \cos \left (x \right )}{\sqrt {x}}\)	\(28\)
meijerg	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \left (-\frac {4 \sqrt {2}\, \cos \left (x \right )}{\sqrt {\pi }\, \sqrt {x}}-8 \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )\right )}{4}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/x^(3/2),x,method=_RETURNVERBOSE)

[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] Result contains complex when optimal does not.
time = 0.56, size = 21, normalized size = 0.60 \begin {gather*} -\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 {gather*}

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 = 0.41, size = 31, normalized size = 0.89 \begin {gather*} -\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 {gather*}

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 = 1.01, size = 61, normalized size = 1.74 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\cos \left (x\right )}{x^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(cos(x)/x^(3/2), x)

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