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

Optimal result

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

[Out]

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*hypergeom([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)*sin(c)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6578, 6576, 30, 6582} \[ \int \cos \left (c+\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \sin (c) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )-\frac {\sin (c) \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}+\frac {\cos (c) \operatorname {FresnelC}(b x)^2}{2 b} \]

[In]

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

[Out]

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

Rule 30

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

Rule 6576

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

Rule 6578

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

Rule 6582

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]

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = -\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)+\frac {\cos (c) \text {Subst}(\int x \, dx,x,\operatorname {FresnelC}(b x))}{b} \\ & = \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) \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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