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]

-2/3*sin(f*x)/d/(d*x)^(3/2)-4/3*f^(3/2)*FresnelS(f^(1/2)*2^(1/2)/Pi^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2
)/d^(5/2)-4/3*f*cos(f*x)/d^2/(d*x)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [C]  time = 0.09, size = 111, normalized size = 1.28 \[ -\frac {2 x \sin (f x)}{3 (d x)^{5/2}}+\frac {2 f x^{5/2} \left (\frac {\sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )-e^{-i f x}}{\sqrt {x}}-\frac {e^{i f x}-\sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )}{\sqrt {x}}\right )}{3 (d x)^{5/2}} \]

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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fricas [A]  time = 0.60, size = 69, normalized size = 0.79 \[ -\frac {2 \, {\left (2 \, \sqrt {2} \pi d f x^{2} \sqrt {\frac {f}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) + {\left (2 \, f x \cos \left (f x\right ) + \sin \left (f x\right )\right )} \sqrt {d x}\right )}}{3 \, d^{3} x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (f x\right )}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.01, size = 79, normalized size = 0.91 \[ \frac {-\frac {2 \sin \left (f x \right )}{3 \left (d x \right )^{\frac {3}{2}}}+\frac {4 f \left (-\frac {\cos \left (f x \right )}{\sqrt {d x}}-\frac {f \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x}\, f}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\right )}{3 d}}{d} \]

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 = 1.07, size = 38, normalized size = 0.44 \[ -\frac {\left (f x\right )^{\frac {3}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, f x\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, f x\right )\right )}}{4 \, \left (d x\right )^{\frac {3}{2}} d} \]

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*(f*x)^(3/2)*(-(I + 1)*sqrt(2)*gamma(-3/2, I*f*x) + (I - 1)*sqrt(2)*gamma(-3/2, -I*f*x))/((d*x)^(3/2)*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (f\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sin(f*x)/(d*x)^(5/2), x)

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sympy [A]  time = 24.50, size = 114, normalized size = 1.31 \[ \frac {\sqrt {2} \sqrt {\pi } f^{\frac {3}{2}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {f \cos {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{3 d^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {\sin {\left (f x \right )} \Gamma \left (- \frac {1}{4}\right )}{6 d^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]

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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