\(\int \operatorname {FresnelS}(b x) \, dx\) [8]

Optimal result

Integrand size = 4, antiderivative size = 26 \[ \int \operatorname {FresnelS}(b x) \, dx=\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+x \operatorname {FresnelS}(b x) \]

[Out]

cos(1/2*b^2*Pi*x^2)/b/Pi+x*FresnelS(b*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6553} \[ \int \operatorname {FresnelS}(b x) \, dx=\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x \operatorname {FresnelS}(b x) \]

[In]

Int[FresnelS[b*x],x]

[Out]

Cos[(b^2*Pi*x^2)/2]/(b*Pi) + x*FresnelS[b*x]

Rule 6553

Int[FresnelS[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(FresnelS[a + b*x]/b), x] + Simp[Cos[(Pi/2)*(a +
b*x)^2]/(b*Pi), x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+x \operatorname {FresnelS}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \operatorname {FresnelS}(b x) \, dx=\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+x \operatorname {FresnelS}(b x) \]

[In]

Integrate[FresnelS[b*x],x]

[Out]

Cos[(b^2*Pi*x^2)/2]/(b*Pi) + x*FresnelS[b*x]

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
parts	\(\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b \pi }+x \,\operatorname {FresnelS}\left (b x \right )\)	\(25\)
derivativedivides	\(\frac {\operatorname {FresnelS}\left (b x \right ) b x +\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }}{b}\)	\(27\)
default	\(\frac {\operatorname {FresnelS}\left (b x \right ) b x +\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }}{b}\)	\(27\)
meijerg	\(\frac {b^{3} \pi \,x^{4} \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [\frac {3}{2}, \frac {7}{4}, 2\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{24}\)	\(29\)

[In]

int(FresnelS(b*x),x,method=_RETURNVERBOSE)

[Out]

cos(1/2*b^2*Pi*x^2)/b/Pi+x*FresnelS(b*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \operatorname {FresnelS}(b x) \, dx=\frac {\pi b x \operatorname {S}\left (b x\right ) + \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{\pi b} \]

[In]

integrate(fresnel_sin(b*x),x, algorithm="fricas")

[Out]

(pi*b*x*fresnel_sin(b*x) + cos(1/2*pi*b^2*x^2))/(pi*b)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).

Time = 0.57 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \operatorname {FresnelS}(b x) \, dx=\frac {3 x S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {3 \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{4 \pi b \Gamma \left (\frac {7}{4}\right )} \]

[In]

integrate(fresnels(b*x),x)

[Out]

3*x*fresnels(b*x)*gamma(3/4)/(4*gamma(7/4)) + 3*cos(pi*b**2*x**2/2)*gamma(3/4)/(4*pi*b*gamma(7/4))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \operatorname {FresnelS}(b x) \, dx=\frac {b x \operatorname {S}\left (b x\right ) + \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{\pi }}{b} \]

[In]

integrate(fresnel_sin(b*x),x, algorithm="maxima")

[Out]

(b*x*fresnel_sin(b*x) + cos(1/2*pi*b^2*x^2)/pi)/b

Giac [F]

\[ \int \operatorname {FresnelS}(b x) \, dx=\int { \operatorname {S}\left (b x\right ) \,d x } \]

[In]

integrate(fresnel_sin(b*x),x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int \operatorname {FresnelS}(b x) \, dx=\int \mathrm {FresnelS}\left (b\,x\right ) \,d x \]

[In]

int(FresnelS(b*x),x)

[Out]

int(FresnelS(b*x), x)