\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x)}{x^2} \, dx\) [101]

Optimal result

Integrand size = 20, antiderivative size = 48 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x}-\frac {1}{2} b \pi \operatorname {FresnelS}(b x)^2+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right ) \] Output:

-cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x-1/2*b*Pi*FresnelS(b*x)^2+1/4*b*Si(b^2 
*Pi*x^2)
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x}-\frac {1}{2} b \pi \operatorname {FresnelS}(b x)^2+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right ) \] Input:

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

Output:

-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x) - (b*Pi*FresnelS[b*x]^2)/2 + (b*S 
inIntegral[b^2*Pi*x^2])/4
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7018, 3856, 6994, 15}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2} \, dx\)

\(\Big \downarrow \) 7018

\(\displaystyle -\pi b^2 \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx+\frac {1}{2} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\)

\(\Big \downarrow \) 3856

\(\displaystyle -\pi b^2 \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )\)

\(\Big \downarrow \) 6994

\(\displaystyle -\pi b \int \operatorname {FresnelS}(b x)d\operatorname {FresnelS}(b x)-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{2} \pi b \operatorname {FresnelS}(b x)^2\)

Input:

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

Output:

-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x) - (b*Pi*FresnelS[b*x]^2)/2 + (b*S 
inIntegral[b^2*Pi*x^2])/4
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 3856

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] / 
; FreeQ[{d, n}, x]
 

rule 6994

Int[FresnelS[(b_.)*(x_)]^(n_.)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[Pi*(b/( 
2*d))   Subst[Int[x^n, x], x, FresnelS[b*x]], x] /; FreeQ[{b, d, n}, x] && 
EqQ[d^2, (Pi^2/4)*b^4]
 

rule 7018

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m + 1)*Cos[d*x^2]*(FresnelS[b*x]/(m + 1)), x] + (Simp[2*(d/(m + 1))   Int[x 
^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x], x] - Simp[d/(Pi*b*(m + 1))   Int[x^( 
m + 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] 
&& ILtQ[m, -1]
 

Maple [F]

\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=-\frac {2 \, \pi b x \operatorname {S}\left (b x\right )^{2} - b x \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 4 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{4 \, x} \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^2,x, algorithm="fricas")
 

Output:

-1/4*(2*pi*b*x*fresnel_sin(b*x)^2 - b*x*sin_integral(pi*b^2*x^2) + 4*cos(1 
/2*pi*b^2*x^2)*fresnel_sin(b*x))/x
 

Sympy [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{2}}\, dx \] Input:

integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**2,x)
 

Output:

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**2, x)
 

Maxima [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^2} \,d x \] Input:

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^2,x)
 

Output:

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^2, x)
 

Reduce [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx=\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )}{x^{2}}d x \] Input:

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

Output:

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