3.63 \(\int \frac{\sin (f x)}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=64 \[ \frac{2 \sqrt{2 \pi } \sqrt{f} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \sin (f x)}{d \sqrt{d x}} \]

[Out]

(2*Sqrt[f]*Sqrt[2*Pi]*FresnelC[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*Sin[f*x])/(d*Sqrt[d*x])

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Rubi [A]  time = 0.0649724, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3297, 3304, 3352} \[ \frac{2 \sqrt{2 \pi } \sqrt{f} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \sin (f x)}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[f*x]/(d*x)^(3/2),x]

[Out]

(2*Sqrt[f]*Sqrt[2*Pi]*FresnelC[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*Sin[f*x])/(d*Sqrt[d*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 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{\sin (f x)}{(d x)^{3/2}} \, dx &=-\frac{2 \sin (f x)}{d \sqrt{d x}}+\frac{(2 f) \int \frac{\cos (f x)}{\sqrt{d x}} \, dx}{d}\\ &=-\frac{2 \sin (f x)}{d \sqrt{d x}}+\frac{(4 f) \operatorname{Subst}\left (\int \cos \left (\frac{f x^2}{d}\right ) \, dx,x,\sqrt{d x}\right )}{d^2}\\ &=\frac{2 \sqrt{f} \sqrt{2 \pi } C\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{2 \sin (f x)}{d \sqrt{d x}}\\ \end{align*}

Mathematica [C]  time = 0.0226545, size = 64, normalized size = 1. \[ \frac{x \left (-i \sqrt{-i f x} \text{Gamma}\left (\frac{1}{2},-i f x\right )+i \sqrt{i f x} \text{Gamma}\left (\frac{1}{2},i f x\right )-2 \sin (f x)\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[f*x]/(d*x)^(3/2),x]

[Out]

(x*((-I)*Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x] + I*Sqrt[I*f*x]*Gamma[1/2, I*f*x] - 2*Sin[f*x]))/(d*x)^(3/2)

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Maple [A]  time = 0.007, size = 60, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ( -{\frac{\sin \left ( fx \right ) }{\sqrt{dx}}}+{\frac{f\sqrt{2}\sqrt{\pi }}{d}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{dx}f}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x)/(d*x)^(3/2),x)

[Out]

2/d*(-sin(f*x)/(d*x)^(1/2)+1/d*f*2^(1/2)*Pi^(1/2)/(1/d*f)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)/(1/d*f)^(1/2)*(d*x)^
(1/2)/d*f))

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Maxima [C]  time = 1.15665, size = 231, normalized size = 3.61 \begin{align*} \frac{\sqrt{\frac{d x{\left | f \right |}}{{\left | d \right |}}}{\left ({\left (-i \, \Gamma \left (-\frac{1}{2}, i \, f x\right ) + i \, \Gamma \left (-\frac{1}{2}, -i \, f x\right )\right )} \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 ) +{\left (-i \, \Gamma \left (-\frac{1}{2}, i \, f x\right ) + i \, \Gamma \left (-\frac{1}{2}, -i \, f x\right )\right )} \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 ) +{\left (\Gamma \left (-\frac{1}{2}, i \, f x\right ) + \Gamma \left (-\frac{1}{2}, -i \, f x\right )\right )} \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 ) -{\left (\Gamma \left (-\frac{1}{2}, i \, f x\right ) + \Gamma \left (-\frac{1}{2}, -i \, f x\right )\right )} \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 )}}{4 \, \sqrt{d x} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.22915, size = 149, normalized size = 2.33 \begin{align*} \frac{2 \,{\left (\sqrt{2} \pi d x \sqrt{\frac{f}{\pi d}} \operatorname{C}\left (\sqrt{2} \sqrt{d x} \sqrt{\frac{f}{\pi d}}\right ) - \sqrt{d x} \sin \left (f x\right )\right )}}{d^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(3/2),x, algorithm="fricas")

[Out]

2*(sqrt(2)*pi*d*x*sqrt(f/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d))) - sqrt(d*x)*sin(f*x))/(d^2*x)

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Sympy [A]  time = 7.55038, size = 80, normalized size = 1.25 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } \sqrt{f} C\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{2 d^{\frac{3}{2}} \Gamma \left (\frac{5}{4}\right )} - \frac{\sin{\left (f x \right )} \Gamma \left (\frac{1}{4}\right )}{2 d^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)**(3/2),x)

[Out]

sqrt(2)*sqrt(pi)*sqrt(f)*fresnelc(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(1/4)/(2*d**(3/2)*gamma(5/4)) - sin(f
*x)*gamma(1/4)/(2*d**(3/2)*sqrt(x)*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sin(f*x)/(d*x)^(3/2), x)