\(\int \cos (c+\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x) \, dx\) [64]

Optimal result

Integrand size = 19, antiderivative size = 101 \[ \int \cos \left (c+\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {\cos (c) \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}-\frac {1}{8} i b x^2 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \cos (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )-\frac {\operatorname {FresnelS}(b x)^2 \sin (c)}{2 b} \] Output:

1/2*cos(c)*FresnelC(b*x)*FresnelS(b*x)/b-1/8*I*b*x^2*cos(c)*hypergeom([1, 
1],[3/2, 2],-1/2*I*b^2*Pi*x^2)+1/8*I*b*x^2*cos(c)*hypergeom([1, 1],[3/2, 2 
],1/2*I*b^2*Pi*x^2)-1/2*FresnelS(b*x)^2*sin(c)/b
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {7002, 6994, 15, 7000}

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.

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

\(\Big \downarrow \) 7002

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

\(\Big \downarrow \) 6994

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7000

\(\displaystyle -\frac {\sin (c) \operatorname {FresnelS}(b x)^2}{2 b}+\cos (c) \left (-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}\right )\)

Input:

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

Output:

Cos[c]*((FresnelC[b*x]*FresnelS[b*x])/(2*b) - (I/8)*b*x^2*HypergeometricPF 
Q[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2] + (I/8)*b*x^2*HypergeometricPFQ[{ 
1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2]) - (FresnelS[b*x]^2*Sin[c])/(2*b)
 

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 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 7000

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)], x_Symbol] :> Simp[FresnelC[b*x] 
*(FresnelS[b*x]/(2*b)), x] + (-Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, 
 {3/2, 2}, (-2^(-1))*I*b^2*Pi*x^2], x] + Simp[(1/8)*I*b*x^2*HypergeometricP 
FQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2], x]) /; FreeQ[{b, d}, x] && EqQ[d^ 
2, (Pi^2/4)*b^4]
 

rule 7002

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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