\(\int \cos (c+\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x) \, dx\) [173]

Optimal result

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

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6997, 6995, 15, 7001}

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.

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

\(\Big \downarrow \) 6997

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

\(\Big \downarrow \) 6995

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7001

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

Input:

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

Output:

(Cos[c]*FresnelC[b*x]^2)/(2*b) - ((FresnelC[b*x]*FresnelS[b*x])/(2*b) + (I 
/8)*b*x^2*HypergeometricPFQ[{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])*Sin[c]
 

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 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 6997

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

rule 7001

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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