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

Optimal. Leaf size=65 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{f^{3/2}}-\frac{\sqrt{d x} \cos (f x)}{f} \]

[Out]

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

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Rubi [A]  time = 0.0579075, antiderivative size = 65, 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 = {3296, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{d} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{f} \sqrt{d x}}{\sqrt{d}}\right )}{f^{3/2}}-\frac{\sqrt{d x} \cos (f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.0114526, size = 69, normalized size = 1.06 \[ -\frac{\sqrt{d x} \text{Gamma}\left (\frac{3}{2},-i f x\right )}{2 f \sqrt{-i f x}}-\frac{\sqrt{d x} \text{Gamma}\left (\frac{3}{2},i f x\right )}{2 f \sqrt{i f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 65, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( -1/2\,{\frac{d\sqrt{dx}\cos \left ( fx \right ) }{f}}+1/4\,{\frac{d\sqrt{2}\sqrt{\pi }}{f}{\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((d*x)^(1/2)*sin(f*x),x)

[Out]

2/d*(-1/2*d/f*(d*x)^(1/2)*cos(f*x)+1/4*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 = 1.7341, size = 379, normalized size = 5.83 \begin{align*} -\frac{8 \, \sqrt{d x} d \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}} \cos \left (f x\right ) -{\left (\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 } \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 } \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 ) + i \, \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 \operatorname{erf}\left (\sqrt{d x} \sqrt{\frac{i \, f}{d}}\right ) -{\left (\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 } \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 } \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 ) - i \, \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 \operatorname{erf}\left (\sqrt{d x} \sqrt{-\frac{i \, f}{d}}\right )}{8 \, d f \sqrt{\frac{{\left | f \right |}}{{\left | d \right |}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(8*sqrt(d*x)*d*sqrt(abs(f)/abs(d))*cos(f*x) - (sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d
/sqrt(d^2))) + sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) - I*sqrt(pi)*sin(1/4*pi
+ 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2
(0, d/sqrt(d^2))))*d*erf(sqrt(d*x)*sqrt(I*f/d)) - (sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/
sqrt(d^2))) + sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) + I*sqrt(pi)*sin(1/4*pi
+ 1/2*arctan2(0, f) + 1/2*arctan2(0, d/sqrt(d^2))) - I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, f) + 1/2*arctan2(
0, d/sqrt(d^2))))*d*erf(sqrt(d*x)*sqrt(-I*f/d)))/(d*f*sqrt(abs(f)/abs(d)))

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Fricas [A]  time = 2.34448, size = 149, normalized size = 2.29 \begin{align*} \frac{\sqrt{2} \pi d \sqrt{\frac{f}{\pi d}} \operatorname{C}\left (\sqrt{2} \sqrt{d x} \sqrt{\frac{f}{\pi d}}\right ) - 2 \, \sqrt{d x} f \cos \left (f x\right )}{2 \, f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*pi*d*sqrt(f/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d))) - 2*sqrt(d*x)*f*cos(f*x))/f^2

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Sympy [A]  time = 3.11813, size = 85, normalized size = 1.31 \begin{align*} - \frac{5 \sqrt{d} \sqrt{x} \cos{\left (f x \right )} \Gamma \left (\frac{5}{4}\right )}{4 f \Gamma \left (\frac{9}{4}\right )} + \frac{5 \sqrt{2} \sqrt{\pi } \sqrt{d} C\left (\frac{\sqrt{2} \sqrt{f} \sqrt{x}}{\sqrt{\pi }}\right ) \Gamma \left (\frac{5}{4}\right )}{8 f^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*sqrt(d)*sqrt(x)*cos(f*x)*gamma(5/4)/(4*f*gamma(9/4)) + 5*sqrt(2)*sqrt(pi)*sqrt(d)*fresnelc(sqrt(2)*sqrt(f)*
sqrt(x)/sqrt(pi))*gamma(5/4)/(8*f**(3/2)*gamma(9/4))

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Giac [C]  time = 1.15394, size = 238, normalized size = 3.66 \begin{align*} -\frac{\frac{\sqrt{2} \sqrt{\pi } d^{2} \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} + \frac{\sqrt{2} \sqrt{\pi } d^{2} \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} + \frac{2 \, \sqrt{d x} d e^{\left (i \, f x\right )}}{f} + \frac{2 \, \sqrt{d x} d e^{\left (-i \, f x\right )}}{f}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2)*sqrt(pi)*d^2*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) + sqrt(2)*sqrt(pi)*d^2*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*sqrt(d*x)*d*e^(I*f*x)/f + 2*sqrt(d*x)*d*e^(-I*f*x)/f)/d