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

3.3.10.1 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+1/4*b*Si(b^2*Pi*x^2)-FresnelC(b*x)*sin(1/2*b^2*Pi 
*x^2)/x
 

3.3.10.2 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
 

3.3.10.3 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
 

3.3.10.3.1 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]
 

3.3.10.4 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)
 

3.3.10.5 Fricas [A] (verification not implemented)

Time = 0.26 (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
 

3.3.10.6 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)
 

3.3.10.7 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)
 

3.3.10.8 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)
 

3.3.10.9 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)