3.62 \(\int \frac{\sin (f x)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{f}} \]

[Out]

(Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[f])

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Rubi [A]  time = 0.0344381, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3305, 3351} \[ \frac{\sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[f*x]/Sqrt[d*x],x]

[Out]

(Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[f])

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{\sin (f x)}{\sqrt{d x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \sin \left (\frac{f x^2}{d}\right ) \, dx,x,\sqrt{d x}\right )}{d}\\ &=\frac{\sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{f}}\\ \end{align*}

Mathematica [C]  time = 0.0076831, size = 59, normalized size = 1.28 \[ \frac{-\sqrt{-i f x} \text{Gamma}\left (\frac{1}{2},-i f x\right )-\sqrt{i f x} \text{Gamma}\left (\frac{1}{2},i f x\right )}{2 f \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[f*x]/Sqrt[d*x],x]

[Out]

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

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Maple [A]  time = 0.007, size = 42, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{d}{\it FresnelS} \left ({\frac{\sqrt{2}f}{\sqrt{\pi }d}\sqrt{dx}{\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{f}{d}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x)/(d*x)^(1/2),x)

[Out]

1/d*2^(1/2)*Pi^(1/2)/(1/d*f)^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(1/d*f)^(1/2)*(d*x)^(1/2)/d*f)

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Maxima [C]  time = 1.66492, size = 344, normalized size = 7.48 \begin{align*} \frac{{\left (i \, \sqrt{\pi } \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) + i \, \sqrt{\pi } \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) + \sqrt{\pi } \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) - \sqrt{\pi } \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right )\right )} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{i \, f}{d}}\right ) +{\left (-i \, \sqrt{\pi } \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) - i \, \sqrt{\pi } \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) + \sqrt{\pi } \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) - \sqrt{\pi } \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, f\right ) + \frac{1}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right )\right )} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{i \, f}{d}}\right )}{4 \, d \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(1/2),x, algorithm="maxima")

[Out]

1/4*((I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + I*sqrt(pi)*cos(-1/4*pi + 1/2*
arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt
(d^2))) - sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))))*erf(sqrt(d*x)*sqrt(I*f/d))
+ (-I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) - I*sqrt(pi)*cos(-1/4*pi + 1/2*ar
ctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d
^2))) - sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))))*erf(sqrt(d*x)*sqrt(-I*f/d)))/
(d*sqrt(abs(f)/abs(d)))

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Fricas [A]  time = 2.04256, size = 101, normalized size = 2.2 \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{f}{\pi d}} \operatorname{S}\left (\sqrt{2} \sqrt{d x} \sqrt{\frac{f}{\pi d}}\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(1/2),x, algorithm="fricas")

[Out]

sqrt(2)*pi*sqrt(f/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d)))/f

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Sympy [A]  time = 1.38134, size = 54, normalized size = 1.17 \begin{align*} \frac{3 \sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{3}{4}\right )}{4 \sqrt{d} \sqrt{f} \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)**(1/2),x)

[Out]

3*sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(3/4)/(4*sqrt(d)*sqrt(f)*gamma(7/4))

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Giac [C]  time = 1.15571, size = 184, normalized size = 4. \begin{align*} -\frac{\frac{i \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d f} \sqrt{d x}{\left (\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt{d f}{\left (\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}} - \frac{i \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{d f} \sqrt{d x}{\left (-\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt{d f}{\left (-\frac{i \, d f}{\sqrt{d^{2} f^{2}}} + 1\right )}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(1/2),x, algorithm="giac")

[Out]

-1/2*(I*sqrt(2)*sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(d*f)*sqrt(d*x)*(I*d*f/sqrt(d^2*f^2) + 1)/d)/(sqrt(d*f)*(I*d*f
/sqrt(d^2*f^2) + 1)) - I*sqrt(2)*sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(d*f)*sqrt(d*x)*(-I*d*f/sqrt(d^2*f^2) + 1)/d)
/(sqrt(d*f)*(-I*d*f/sqrt(d^2*f^2) + 1)))/d