\(\int \frac {\sin (f x)}{\sqrt {d x}} \, dx\) [62]

Optimal result

Integrand size = 12, antiderivative size = 46 \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {f}} \]

[Out]

FresnelS(f^(1/2)*2^(1/2)/Pi^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(1/2)/f^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3386, 3432} \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {f}} \]

[In]

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

[Out]

(Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[f])

Rule 3386

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \sin \left (\frac {f x^2}{d}\right ) \, dx,x,\sqrt {d x}\right )}{d} \\ & = \frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {f}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {-\sqrt {-i f x} \Gamma \left (\frac {1}{2},-i f x\right )-\sqrt {i f x} \Gamma \left (\frac {1}{2},i f x\right )}{2 f \sqrt {d x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72

method	result	size
meijerg	\(\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )}{\sqrt {d x}\, \sqrt {f}}\)	\(33\)
derivativedivides	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\)	\(42\)
default	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{d \sqrt {\frac {f}{d}}}\)	\(42\)

[In]

int(sin(f*x)/(d*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

Pi^(1/2)/(d*x)^(1/2)*x^(1/2)/f^(1/2)*2^(1/2)*FresnelS(1/Pi^(1/2)*2^(1/2)*x^(1/2)*f^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {\sqrt {2} \pi \sqrt {\frac {f}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right )}{f} \]

[In]

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

[Out]

sqrt(2)*pi*sqrt(f/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x)*sqrt(f/(pi*d)))/f

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.17 \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {3 \sqrt {2} \sqrt {\pi } S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \sqrt {d} \sqrt {f} \Gamma \left (\frac {7}{4}\right )} \]

[In]

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

[Out]

3*sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(3/4)/(4*sqrt(d)*sqrt(f)*gamma(7/4))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {\sqrt {2} {\left (\left (i + 1\right ) \, \sqrt {\pi } \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {i \, f}{d}}\right ) - \left (i - 1\right ) \, \sqrt {\pi } \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {i \, f}{d}}\right )\right )}}{4 \, f} \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.91 \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {d f} \sqrt {d x} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}} + \frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {d f} \sqrt {d x} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}}{2 \, d} \]

[In]

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

[Out]

1/2*(sqrt(2)*sqrt(pi)*d*erf(-1/2*I*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)) + sqrt(2)*sqrt(pi)*d*erf(1/2*I*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)))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (f x)}{\sqrt {d x}} \, dx=\int \frac {\sin \left (f\,x\right )}{\sqrt {d\,x}} \,d x \]

[In]

int(sin(f*x)/(d*x)^(1/2),x)

[Out]

int(sin(f*x)/(d*x)^(1/2), x)