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

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

[Out]

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

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Rubi [A]  time = 0.109295, 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 = {3296, 3305, 3351} \[ -\frac{3 \sqrt{\frac{\pi }{2}} d^{3/2} S\left (\frac{\sqrt{f} \sqrt{\frac{2}{\pi }} \sqrt{d x}}{\sqrt{d}}\right )}{2 f^{5/2}}+\frac{3 d \sqrt{d x} \sin (f x)}{2 f^2}-\frac{(d x)^{3/2} \cos (f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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

Mathematica [C]  time = 0.0131099, size = 60, normalized size = 0.69 \[ \frac{d^2 \left (\sqrt{-i f x} \text{Gamma}\left (\frac{5}{2},-i f x\right )+\sqrt{i f x} \text{Gamma}\left (\frac{5}{2},i f x\right )\right )}{2 f^3 \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(Sqrt[(-I)*f*x]*Gamma[5/2, (-I)*f*x] + Sqrt[I*f*x]*Gamma[5/2, I*f*x]))/(2*f^3*Sqrt[d*x])

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Maple [A]  time = 0.01, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( -1/2\,{\frac{d \left ( dx \right ) ^{3/2}\cos \left ( fx \right ) }{f}}+3/2\,{\frac{d}{f} \left ( 1/2\,{\frac{\sqrt{dx}\sin \left ( fx \right ) d}{f}}-1/4\,{\frac{d\sqrt{2}\sqrt{\pi }}{f}{\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((d*x)^(3/2)*sin(f*x),x)

[Out]

2/d*(-1/2*d/f*(d*x)^(3/2)*cos(f*x)+3/2*d/f*(1/2*d/f*(d*x)^(1/2)*sin(f*x)-1/4*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 = 1.76174, size = 423, normalized size = 4.86 \begin{align*} -\frac{16 \, \left (d x\right )^{\frac{3}{2}} d f \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}} \cos \left (f x\right ) -{\left (-3 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 ) - 3 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 ) - 3 \, \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 ) + 3 \, \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 )} d^{2} \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{i \, f}{d}}\right ) -{\left (3 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 ) + 3 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 ) - 3 \, \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 ) + 3 \, \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 )} d^{2} \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{i \, f}{d}}\right ) - 24 \, \sqrt{d x} d^{2} \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}} \sin \left (f x\right )}{16 \, d f^{2} \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(16*(d*x)^(3/2)*d*f*sqrt(abs(f)/abs(d))*cos(f*x) - (-3*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, f) + 1/2*a
rctan2(0, d/sqrt(d^2))) - 3*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) - 3*sqrt
(pi)*sin(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + 3*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, f
) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*erf(sqrt(d*x)*sqrt(I*f/d)) - (3*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, f
) + 1/2*arctan2(0, d/sqrt(d^2))) + 3*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2)))
- 3*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + 3*sqrt(pi)*sin(-1/4*pi + 1/2*arc
tan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*erf(sqrt(d*x)*sqrt(-I*f/d)) - 24*sqrt(d*x)*d^2*sqrt(abs(f)/abs(
d))*sin(f*x))/(d*f^2*sqrt(abs(f)/abs(d)))

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Fricas [A]  time = 2.29349, size = 192, normalized size = 2.21 \begin{align*} -\frac{3 \, \sqrt{2} \pi d^{2} \sqrt{\frac{f}{\pi d}} \operatorname{S}\left (\sqrt{2} \sqrt{d x} \sqrt{\frac{f}{\pi d}}\right ) + 2 \,{\left (2 \, d f^{2} x \cos \left (f x\right ) - 3 \, d f \sin \left (f x\right )\right )} \sqrt{d x}}{4 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*sqrt(2)*pi*d^2*sqrt(f/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d))) + 2*(2*d*f^2*x*cos(f*x) -
3*d*f*sin(f*x))*sqrt(d*x))/f^3

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Sympy [A]  time = 126.238, size = 117, normalized size = 1.34 \begin{align*} - \frac{7 d^{\frac{3}{2}} x^{\frac{3}{2}} \cos{\left (f x \right )} \Gamma \left (\frac{7}{4}\right )}{4 f \Gamma \left (\frac{11}{4}\right )} + \frac{21 d^{\frac{3}{2}} \sqrt{x} \sin{\left (f x \right )} \Gamma \left (\frac{7}{4}\right )}{8 f^{2} \Gamma \left (\frac{11}{4}\right )} - \frac{21 \sqrt{2} \sqrt{\pi } d^{\frac{3}{2}} S\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{7}{4}\right )}{16 f^{\frac{5}{2}} \Gamma \left (\frac{11}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*d**(3/2)*x**(3/2)*cos(f*x)*gamma(7/4)/(4*f*gamma(11/4)) + 21*d**(3/2)*sqrt(x)*sin(f*x)*gamma(7/4)/(8*f**2*g
amma(11/4)) - 21*sqrt(2)*sqrt(pi)*d**(3/2)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(7/4)/(16*f**(5/2)*
gamma(11/4))

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Giac [C]  time = 1.15531, size = 286, normalized size = 3.29 \begin{align*} -\frac{-\frac{3 i \, \sqrt{2} \sqrt{\pi } d^{3} \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 )} f^{2}} + \frac{3 i \, \sqrt{2} \sqrt{\pi } d^{3} \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 )} f^{2}} - \frac{2 i \,{\left (2 i \, \sqrt{d x} d^{2} f x - 3 \, \sqrt{d x} d^{2}\right )} e^{\left (i \, f x\right )}}{f^{2}} - \frac{2 i \,{\left (2 i \, \sqrt{d x} d^{2} f x + 3 \, \sqrt{d x} d^{2}\right )} e^{\left (-i \, f x\right )}}{f^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-3*I*sqrt(2)*sqrt(pi)*d^3*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)*f^2) + 3*I*sqrt(2)*sqrt(pi)*d^3*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)*f^2) - 2*I*(2*I*sqrt(d*x)*d^2*f*x - 3*sqrt(d*x)*d^2)*e^(I*
f*x)/f^2 - 2*I*(2*I*sqrt(d*x)*d^2*f*x + 3*sqrt(d*x)*d^2)*e^(-I*f*x)/f^2)/d