3.64 \(\int \frac{\sin (f x)}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{4 \sqrt{2 \pi } f^{3/2} S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}} \]

[Out]

(-4*f*Cos[f*x])/(3*d^2*Sqrt[d*x]) - (4*f^(3/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(3
*d^(5/2)) - (2*Sin[f*x])/(3*d*(d*x)^(3/2))

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Rubi [A]  time = 0.0926116, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3297, 3305, 3351} \[ -\frac{4 \sqrt{2 \pi } f^{3/2} S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[f*x]/(d*x)^(5/2),x]

[Out]

(-4*f*Cos[f*x])/(3*d^2*Sqrt[d*x]) - (4*f^(3/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(3
*d^(5/2)) - (2*Sin[f*x])/(3*d*(d*x)^(3/2))

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{\sin (f x)}{(d x)^{5/2}} \, dx &=-\frac{2 \sin (f x)}{3 d (d x)^{3/2}}+\frac{(2 f) \int \frac{\cos (f x)}{(d x)^{3/2}} \, dx}{3 d}\\ &=-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}}-\frac{\left (4 f^2\right ) \int \frac{\sin (f x)}{\sqrt{d x}} \, dx}{3 d^2}\\ &=-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}}-\frac{\left (8 f^2\right ) \operatorname{Subst}\left (\int \sin \left (\frac{f x^2}{d}\right ) \, dx,x,\sqrt{d x}\right )}{3 d^3}\\ &=-\frac{4 f \cos (f x)}{3 d^2 \sqrt{d x}}-\frac{4 f^{3/2} \sqrt{2 \pi } S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{2 \sin (f x)}{3 d (d x)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0849214, size = 111, normalized size = 1.28 \[ -\frac{2 x \sin (f x)}{3 (d x)^{5/2}}+\frac{2 f x^{5/2} \left (\frac{\sqrt{i f x} \text{Gamma}\left (\frac{1}{2},i f x\right )-e^{-i f x}}{\sqrt{x}}-\frac{e^{i f x}-\sqrt{-i f x} \text{Gamma}\left (\frac{1}{2},-i f x\right )}{\sqrt{x}}\right )}{3 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[f*x]/(d*x)^(5/2),x]

[Out]

(2*f*x^(5/2)*(-((E^(I*f*x) - Sqrt[(-I)*f*x]*Gamma[1/2, (-I)*f*x])/Sqrt[x]) + (-E^((-I)*f*x) + Sqrt[I*f*x]*Gamm
a[1/2, I*f*x])/Sqrt[x]))/(3*(d*x)^(5/2)) - (2*x*Sin[f*x])/(3*(d*x)^(5/2))

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Maple [A]  time = 0.006, size = 79, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{d} \left ( -1/3\,{\frac{\sin \left ( fx \right ) }{ \left ( dx \right ) ^{3/2}}}+2/3\,{\frac{f}{d} \left ( -{\frac{\cos \left ( fx \right ) }{\sqrt{dx}}}-{\frac{f\sqrt{2}\sqrt{\pi }}{d}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx}f}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{f}{d}}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x)/(d*x)^(5/2),x)

[Out]

2/d*(-1/3*sin(f*x)/(d*x)^(3/2)+2/3/d*f*(-1/(d*x)^(1/2)*cos(f*x)-1/d*f*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 = 1.17412, size = 231, normalized size = 2.66 \begin{align*} \frac{\left (\frac{d x{\left | f \right |}}{{\left | d \right |}}\right )^{\frac{3}{2}}{\left ({\left (-i \, \Gamma \left (-\frac{3}{2}, i \, f x\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \cos \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) +{\left (-i \, \Gamma \left (-\frac{3}{2}, i \, f x\right ) + i \, \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \cos \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) +{\left (\Gamma \left (-\frac{3}{2}, i \, f x\right ) + \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \sin \left (\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right ) -{\left (\Gamma \left (-\frac{3}{2}, i \, f x\right ) + \Gamma \left (-\frac{3}{2}, -i \, f x\right )\right )} \sin \left (-\frac{3}{4} \, \pi + \frac{3}{2} \, \arctan \left (0, f\right ) + \frac{3}{2} \, \arctan \left (0, \frac{d}{\sqrt{d^{2}}}\right )\right )\right )}}{4 \, \left (d x\right )^{\frac{3}{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

1/4*(d*x*abs(f)/abs(d))^(3/2)*((-I*gamma(-3/2, I*f*x) + I*gamma(-3/2, -I*f*x))*cos(3/4*pi + 3/2*arctan2(0, f)
+ 3/2*arctan2(0, d/sqrt(d^2))) + (-I*gamma(-3/2, I*f*x) + I*gamma(-3/2, -I*f*x))*cos(-3/4*pi + 3/2*arctan2(0,
f) + 3/2*arctan2(0, d/sqrt(d^2))) + (gamma(-3/2, I*f*x) + gamma(-3/2, -I*f*x))*sin(3/4*pi + 3/2*arctan2(0, f)
+ 3/2*arctan2(0, d/sqrt(d^2))) - (gamma(-3/2, I*f*x) + gamma(-3/2, -I*f*x))*sin(-3/4*pi + 3/2*arctan2(0, f) +
3/2*arctan2(0, d/sqrt(d^2))))/((d*x)^(3/2)*d)

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Fricas [A]  time = 2.28062, size = 189, normalized size = 2.17 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2} \pi d f x^{2} \sqrt{\frac{f}{\pi d}} \operatorname{S}\left (\sqrt{2} \sqrt{d x} \sqrt{\frac{f}{\pi d}}\right ) +{\left (2 \, f x \cos \left (f x\right ) + \sin \left (f x\right )\right )} \sqrt{d x}\right )}}{3 \, d^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(2*sqrt(2)*pi*d*f*x^2*sqrt(f/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d))) + (2*f*x*cos(f*x) + si
n(f*x))*sqrt(d*x))/(d^3*x^2)

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Sympy [A]  time = 145.307, size = 114, normalized size = 1.31 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } f^{\frac{3}{2}} S\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (- \frac{1}{4}\right )}{3 d^{\frac{5}{2}} \Gamma \left (\frac{3}{4}\right )} + \frac{f \cos{\left (f x \right )} \Gamma \left (- \frac{1}{4}\right )}{3 d^{\frac{5}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} + \frac{\sin{\left (f x \right )} \Gamma \left (- \frac{1}{4}\right )}{6 d^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)**(5/2),x)

[Out]

sqrt(2)*sqrt(pi)*f**(3/2)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(-1/4)/(3*d**(5/2)*gamma(3/4)) + f*c
os(f*x)*gamma(-1/4)/(3*d**(5/2)*sqrt(x)*gamma(3/4)) + sin(f*x)*gamma(-1/4)/(6*d**(5/2)*x**(3/2)*gamma(3/4))

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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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x)/(d*x)^(5/2),x, algorithm="giac")

[Out]

integrate(sin(f*x)/(d*x)^(5/2), x)