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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [C]  time = 0.01, size = 51, normalized size = 2.12 \[ -\frac {i \left (\sqrt {-i x} \Gamma \left (\frac {1}{2},-i x\right )-\sqrt {i x} \Gamma \left (\frac {1}{2},i x\right )\right )}{2 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 18, normalized size = 0.75 \[ \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\pi }}\right ) \]

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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giac [C]  time = 0.34, size = 35, normalized size = 1.46 \[ -\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) \]

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))

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maple [A]  time = 0.03, size = 19, normalized size = 0.79 \[ \FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi } \]

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 = 0.75, size = 60, normalized size = 2.50 \[ -\frac {1}{8} \, \sqrt {\pi } {\left (\left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) + \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) - \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\sqrt {-i} \sqrt {x}\right ) + \left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} \sqrt {x}\right )\right )} \]

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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mupad [B]  time = 0.03, size = 18, normalized size = 0.75 \[ \sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {x}}{\sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2^(1/2)*pi^(1/2)*fresnelc((2^(1/2)*x^(1/2))/pi^(1/2))

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sympy [A]  time = 0.72, size = 37, normalized size = 1.54 \[ \frac {\sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{4 \Gamma \left (\frac {5}{4}\right )} \]

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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