\(\int \frac {\operatorname {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^2} \, dx\) [210]

Optimal result

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

1/2*b*Pi*FresnelC(b*x)^2-FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x+1/4*b*Si(b^2* 
Pi*x^2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\frac {1}{2} b \pi \operatorname {FresnelC}(b x)^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right ) \] Input:

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2,x]
 

Output:

(b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x + (b*SinI 
ntegral[b^2*Pi*x^2])/4
 

Rubi [A] (verified)

Time = 0.31 (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 = {7019, 3856, 6995, 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.

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3856

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

\(\Big \downarrow \) 6995

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

\(\Big \downarrow \) 15

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

Input:

Int[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2,x]
 

Output:

(b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x + (b*SinI 
ntegral[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 6995

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

rule 7019

Int[FresnelC[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[x^( 
m + 1)*Sin[d*x^2]*(FresnelC[b*x]/(m + 1)), x] + (-Simp[2*(d/(m + 1))   Int[ 
x^(m + 2)*Cos[d*x^2]*FresnelC[b*x], x], x] - Simp[b/(2*(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]

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

Input:

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)
 

Output:

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\frac {2 \, \pi b x \operatorname {C}\left (b x\right )^{2} + b x \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 4 \, \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, x} \] Input:

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

Output:

1/4*(2*pi*b*x*fresnel_cos(b*x)^2 + b*x*sin_integral(pi*b^2*x^2) - 4*fresne 
l_cos(b*x)*sin(1/2*pi*b^2*x^2))/x
 

Sympy [F]

\[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{2}}\, dx \] Input:

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2)/x**2,x)
 

Output:

Integral(sin(pi*b**2*x**2/2)*fresnelc(b*x)/x**2, x)
 

Maxima [F]

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

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

Output:

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^2, x)
 

Giac [F]

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

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

Output:

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((FresnelC(b*x)*sin((Pi*b^2*x^2)/2))/x^2,x)
 

Output:

int((FresnelC(b*x)*sin((Pi*b^2*x^2)/2))/x^2, x)
 

Reduce [F]

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

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)
 

Output:

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)