(* ::Package:: *)

(* ::Title:: *)
(*Integration Problems Involving Fresnel Integral Functions*)


(* ::Subsection::Closed:: *)
(*Fresnel integral S function*)


{FresnelS[a + b*x]/x, x, 0, Int[FresnelS[a + b*x]/x, x]}
{FresnelS[a + b*x], x, 1, Cos[(1/2)*Pi*(a + b*x)^2]/(b*Pi) + ((a + b*x)*FresnelS[a + b*x])/b}
{x*FresnelS[a + b*x], x, 9, -((a*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^2*Pi)) + (x*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b*Pi) - FresnelC[a + b*x]/(2*b^2*Pi) - (1/2)*(a^2/b^2 - x^2)*FresnelS[a + b*x]}
{x^2*FresnelS[a + b*x], x, 12, -((a*x*Cos[(1/2)*Pi*(a + b*x)^2])/(b^2*Pi)) + ((a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi) + (a*FresnelC[a + b*x])/(b^3*Pi) + (1/3)*(a^3/b^3 + x^3)*FresnelS[a + b*x] - (2*Sin[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi^2)}
{x^3*FresnelS[a + b*x], x, 15, (a^3*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^4*Pi) + (3*a^2*x*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^3*Pi) - (a*(a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) + ((a + b*x)^3*Cos[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi) - (3*a^2*FresnelC[a + b*x])/(2*b^4*Pi) - (1/4)*(a^4/b^4 - 3/(b^4*Pi^2) - x^4)*FresnelS[a + b*x] + (5*a*Sin[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi^2) - (3*x*Sin[(1/2)*Pi*(a + b*x)^2])/(4*b^3*Pi^2)}


{FresnelS[b*x]^2/x^10, x, 0, Int[FresnelS[b*x]^2/x^10, x]}
{FresnelS[b*x]^2/x^9, x, 17, -(b^2/(336*x^6)) + (b^6*Pi^2)/(1680*x^2) + (b^2*Cos[b^2*Pi*x^2])/(336*x^6) - (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2) + (1/840)*(b^8*Pi^4 - 105/x^8)*FresnelS[b*x]^2 - (b*FresnelS[b*x]*(b^2*Pi*x^2*(3 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + (15 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(420*x^7) - (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) - (1/280)*b^8*Pi^3*SinIntegral[b^2*Pi*x^2]}
{FresnelS[b*x]^2/x^8, x, 0, Int[FresnelS[b*x]^2/x^8, x]}
{FresnelS[b*x]^2/x^7, x, 0, Int[FresnelS[b*x]^2/x^7, x]}
{FresnelS[b*x]^2/x^6, x, 0, Int[FresnelS[b*x]^2/x^6, x]}
{FresnelS[b*x]^2/x^5, x, 8, -(b^2/(24*x^2)) + (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - ((3 + b^4*Pi^2*x^4)*FresnelS[b*x]^2)/(12*x^4) - (b*FresnelS[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] + Sin[(1/2)*b^2*Pi*x^2]))/(6*x^3) + (1/12)*b^4*Pi*SinIntegral[b^2*Pi*x^2]}
{FresnelS[b*x]^2/x^4, x, 0, Int[FresnelS[b*x]^2/x^4, x]}
{FresnelS[b*x]^2/x^3, x, 0, Int[FresnelS[b*x]^2/x^3, x]}
{FresnelS[b*x]^2/x^2, x, 0, Int[FresnelS[b*x]^2/x^2, x]}
{FresnelS[b*x]^2/x, x, 0, Int[FresnelS[b*x]^2/x, x]}
{FresnelS[b*x]^2, x, 4, (2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b*Pi) + x*FresnelS[b*x]^2 - FresnelS[Sqrt[2]*b*x]/(Sqrt[2]*b*Pi)}
{x*FresnelS[b*x]^2, x, 0, Int[x*FresnelS[b*x]^2, x]}
{x^2*FresnelS[b*x]^2, x, 6, (2*x)/(3*b^2*Pi^2) + (x*Cos[b^2*Pi*x^2])/(6*b^2*Pi^2) - (5*FresnelC[Sqrt[2]*b*x])/(6*Sqrt[2]*b^3*Pi^2) + (1/3)*x^3*FresnelS[b*x]^2 + (2*FresnelS[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - 2*Sin[(1/2)*b^2*Pi*x^2]))/(3*b^3*Pi^2)}
{x^3*FresnelS[b*x]^2, x, 10, (3*x^2)/(8*b^2*Pi^2) + (x^2*Cos[b^2*Pi*x^2])/(8*b^2*Pi^2) + (1/4)*(3/(b^4*Pi^2) + x^4)*FresnelS[b*x]^2 + (x*FresnelS[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - 3*Sin[(1/2)*b^2*Pi*x^2]))/(2*b^3*Pi^2) - Sin[b^2*Pi*x^2]/(2*b^4*Pi^3)}
{x^4*FresnelS[b*x]^2, x, 10, (4*x^3)/(15*b^2*Pi^2) + (x^3*Cos[b^2*Pi*x^2])/(10*b^2*Pi^2) + (1/5)*x^5*FresnelS[b*x]^2 + (43*FresnelS[Sqrt[2]*b*x])/(20*Sqrt[2]*b^5*Pi^3) - (2*FresnelS[b*x]*((8 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + 4*b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(5*b^5*Pi^3) - (11*x*Sin[b^2*Pi*x^2])/(20*b^4*Pi^3)}
{x^5*FresnelS[b*x]^2, x, 0, Int[x^5*FresnelS[b*x]^2, x]}
{x^6*FresnelS[b*x]^2, x, 15, -((48*x)/(7*b^6*Pi^4)) + (6*x^5)/(35*b^2*Pi^2) - (21*x*Cos[b^2*Pi*x^2])/(8*b^6*Pi^4) + (x^5*Cos[b^2*Pi*x^2])/(14*b^2*Pi^2) + (531*FresnelC[Sqrt[2]*b*x])/(56*Sqrt[2]*b^7*Pi^4) + (1/7)*x^7*FresnelS[b*x]^2 - (2*FresnelS[b*x]*(b^2*Pi*x^2*(24 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - 6*(8 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(7*b^7*Pi^4) - (17*x^3*Sin[b^2*Pi*x^2])/(28*b^4*Pi^3)}
{x^7*FresnelS[b*x]^2, x, 21, -((105*x^2)/(16*b^6*Pi^4)) + (7*x^6)/(48*b^2*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/(16*b^6*Pi^4) + (x^6*Cos[b^2*Pi*x^2])/(16*b^2*Pi^2) - (1/8)*(105/(b^8*Pi^4) - x^8)*FresnelS[b*x]^2 - (x*FresnelS[b*x]*(b^2*Pi*x^2*(35 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - 7*(15 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(4*b^7*Pi^4) + (10*Sin[b^2*Pi*x^2])/(b^8*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)}


{FresnelS[a + b*x]^2/x, x, 0, Int[FresnelS[a + b*x]^2/x, x]}
{FresnelS[a + b*x]^2, x, 4, (2*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(b*Pi) + ((a + b*x)*FresnelS[a + b*x]^2)/b - FresnelS[Sqrt[2]*(a + b*x)]/(Sqrt[2]*b*Pi)}
{x*FresnelS[a + b*x]^2, x, 0, Int[x*FresnelS[a + b*x]^2, x]}


{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^10, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^10, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^9, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^9, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^8, x, 16, -(b/(84*x^6)) + (b^5*Pi^2)/(420*x^2) + (b*Cos[b^2*Pi*x^2])/(84*x^6) - (b^5*Pi^2*Cos[b^2*Pi*x^2])/(84*x^2) + (1/210)*b^7*Pi^4*FresnelS[b*x]^2 - (FresnelS[b*x]*(b^2*Pi*x^2*(3 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + (15 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(105*x^7) - (b^3*Pi*Sin[b^2*Pi*x^2])/(105*x^4) - (1/70)*b^7*Pi^3*SinIntegral[b^2*Pi*x^2]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^7, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^7, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^6, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^6, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^5, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^5, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^4, x, 7, -(b/(12*x^2)) + (b*Cos[b^2*Pi*x^2])/(12*x^2) - (1/6)*b^3*Pi^2*FresnelS[b*x]^2 - (FresnelS[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] + Sin[(1/2)*b^2*Pi*x^2]))/(3*x^3) + (1/6)*b^3*Pi*SinIntegral[b^2*Pi*x^2]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^3, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^3, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^2, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x, x, 0, Int[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]}
{Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 2, FresnelS[b*x]^2/(2*b)}
{x*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 2, -((Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi)) + FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)}
{x^2*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 1, (x^2*FresnelS[b*x]^2)/(2*b) - Int[x*FresnelS[b*x]^2, x]/b}
{x^3*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 5, -(x/(b^3*Pi^2)) - (x*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (5*FresnelC[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) - (FresnelS[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - 2*Sin[(1/2)*b^2*Pi*x^2]))/(b^4*Pi^2)}
{x^4*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 9, -((3*x^2)/(4*b^3*Pi^2)) - (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (3*FresnelS[b*x]^2)/(2*b^5*Pi^2) - (x*FresnelS[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - 3*Sin[(1/2)*b^2*Pi*x^2]))/(b^4*Pi^2) + Sin[b^2*Pi*x^2]/(b^5*Pi^3)}
{x^5*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 9, -((2*x^3)/(3*b^3*Pi^2)) - (x^3*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (43*FresnelS[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (FresnelS[b*x]*((8 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + 4*b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(b^6*Pi^3) + (11*x*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)}
{x^6*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 1, (x^6*FresnelS[b*x]^2)/(2*b) - (3*Int[x^5*FresnelS[b*x]^2, x])/b}
{x^7*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 14, (24*x)/(b^7*Pi^4) - (3*x^5)/(5*b^3*Pi^2) + (147*x*Cos[b^2*Pi*x^2])/(16*b^7*Pi^4) - (x^5*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (531*FresnelC[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) + (FresnelS[b*x]*(b^2*Pi*x^2*(24 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - 6*(8 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^8*Pi^4) + (17*x^3*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)}
(* {x^8*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 20, (105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/(4*b^7*Pi^4) - (x^6*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (105*FresnelS[b*x]^2)/(2*b^9*Pi^4) + (x*FresnelS[b*x]*(b^2*Pi*x^2*(35 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - 7*(15 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^8*Pi^4) - (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(2*b^5*Pi^3)} *)


{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^10, x, 22, (b^3*Pi)/(756*x^6) - (b^7*Pi^3)/(3780*x^2) - (11*b^3*Pi*Cos[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Cos[b^2*Pi*x^2])/(2016*x^2) - (b^9*Pi^5*FresnelS[b*x]^2)/1890 - (FresnelS[b*x]*((105 - 3*b^4*Pi^2*x^4 + b^8*Pi^4*x^8)*Cos[(1/2)*b^2*Pi*x^2] - b^2*Pi*x^2*(15 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(945*x^9) - (b*Sin[b^2*Pi*x^2])/(144*x^8) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(30240*x^4) + (83*b^9*Pi^4*SinIntegral[b^2*Pi*x^2])/30240}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^9, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x^9, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^8, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x^8, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^7, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x^7, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^6, x, 11, (b^3*Pi)/(60*x^2) - (b^3*Pi*Cos[b^2*Pi*x^2])/(24*x^2) + (1/30)*b^5*Pi^3*FresnelS[b*x]^2 - (FresnelS[b*x]*((3 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(15*x^5) - (b*Sin[b^2*Pi*x^2])/(40*x^4) - (7/120)*b^5*Pi^2*SinIntegral[b^2*Pi*x^2]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^5, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x^5, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^4, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x^4, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^3, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x^3, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^2, x, 4, -((Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x) - (1/2)*b*Pi*FresnelS[b*x]^2 + (1/4)*b*SinIntegral[b^2*Pi*x^2]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]}
{Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 0, Int[Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x], x]}
{x*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 2, -(x/(2*b*Pi)) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi) + (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi)}
{x^2*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 5, -(x^2/(4*b*Pi)) - FresnelS[b*x]^2/(2*b^3*Pi) + (x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) + Sin[b^2*Pi*x^2]/(4*b^3*Pi^2)}
{x^3*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 5, -(x^3/(6*b*Pi)) - (5*FresnelS[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) + (FresnelS[b*x]*(2*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(b^4*Pi^2) + (x*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^4*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 0, Int[x^4*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x], x]}
{x^5*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 9, (4*x)/(b^5*Pi^3) - x^5/(10*b*Pi) + (11*x*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) - (43*FresnelC[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (FresnelS[b*x]*(4*b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - (8 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^6*Pi^3) + (x^3*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^6*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 14, (15*x^2)/(4*b^5*Pi^3) - x^6/(12*b*Pi) + (7*x^2*Cos[b^2*Pi*x^2])/(4*b^5*Pi^3) + (15*FresnelS[b*x]^2)/(2*b^7*Pi^3) + (x*FresnelS[b*x]*(5*b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - (15 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^6*Pi^3) - (11*Sin[b^2*Pi*x^2])/(2*b^7*Pi^4) + (x^4*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^7*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 14, (4*x^3)/(b^5*Pi^3) - x^7/(14*b*Pi) + (17*x^3*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) + (531*FresnelS[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) - (FresnelS[b*x]*(6*(8 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*(24 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^8*Pi^4) - (147*x*Sin[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^8*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 0, Int[x^8*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x], x]}


(* ::Subsection::Closed:: *)
(*Fresnel integral C function*)


{FresnelC[a + b*x]/x, x, 0, Int[FresnelC[a + b*x]/x, x]}
{FresnelC[a + b*x], x, 1, ((a + b*x)*FresnelC[a + b*x])/b - Sin[(1/2)*Pi*(a + b*x)^2]/(b*Pi)}
{x*FresnelC[a + b*x], x, 9, (-(1/2))*(a^2/b^2 - x^2)*FresnelC[a + b*x] + FresnelS[a + b*x]/(2*b^2*Pi) + (a*Sin[(1/2)*Pi*(a + b*x)^2])/(2*b^2*Pi) - (x*Sin[(1/2)*Pi*(a + b*x)^2])/(2*b*Pi)}
{x^2*FresnelC[a + b*x], x, 12, -((2*Cos[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi^2)) + (1/3)*(a^3/b^3 + x^3)*FresnelC[a + b*x] - (a*FresnelS[a + b*x])/(b^3*Pi) + (a*x*Sin[(1/2)*Pi*(a + b*x)^2])/(b^2*Pi) - ((a + b*x)^2*Sin[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi)}
{x^3*FresnelC[a + b*x], x, 15, (5*a*Cos[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi^2) - (3*x*Cos[(1/2)*Pi*(a + b*x)^2])/(4*b^3*Pi^2) - (1/4)*(a^4/b^4 - 3/(b^4*Pi^2) - x^4)*FresnelC[a + b*x] + (3*a^2*FresnelS[a + b*x])/(2*b^4*Pi) - (a^3*Sin[(1/2)*Pi*(a + b*x)^2])/(2*b^4*Pi) - (3*a^2*x*Sin[(1/2)*Pi*(a + b*x)^2])/(2*b^3*Pi) + (a*(a + b*x)^2*Sin[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) - ((a + b*x)^3*Sin[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi)}


{FresnelC[b*x]^2/x^10, x, 0, Int[FresnelC[b*x]^2/x^10, x]}
{FresnelC[b*x]^2/x^9, x, 17, -(b^2/(336*x^6)) + (b^6*Pi^2)/(1680*x^2) - (b^2*Cos[b^2*Pi*x^2])/(336*x^6) + (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2) + (1/840)*(b^8*Pi^4 - 105/x^8)*FresnelC[b*x]^2 - (b*FresnelC[b*x]*((15 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - b^2*Pi*x^2*(3 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(420*x^7) + (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) + (1/280)*b^8*Pi^3*SinIntegral[b^2*Pi*x^2]}
{FresnelC[b*x]^2/x^8, x, 0, Int[FresnelC[b*x]^2/x^8, x]}
{FresnelC[b*x]^2/x^7, x, 0, Int[FresnelC[b*x]^2/x^7, x]}
{FresnelC[b*x]^2/x^6, x, 0, Int[FresnelC[b*x]^2/x^6, x]}
{FresnelC[b*x]^2/x^5, x, 8, -(b^2/(24*x^2)) - (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - ((3 + b^4*Pi^2*x^4)*FresnelC[b*x]^2)/(12*x^4) - (b*FresnelC[b*x]*(Cos[(1/2)*b^2*Pi*x^2] - b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(6*x^3) - (1/12)*b^4*Pi*SinIntegral[b^2*Pi*x^2]}
{FresnelC[b*x]^2/x^4, x, 0, Int[FresnelC[b*x]^2/x^4, x]}
{FresnelC[b*x]^2/x^3, x, 0, Int[FresnelC[b*x]^2/x^3, x]}
{FresnelC[b*x]^2/x^2, x, 0, Int[FresnelC[b*x]^2/x^2, x]}
{FresnelC[b*x]^2/x, x, 0, Int[FresnelC[b*x]^2/x, x]}
{FresnelC[b*x]^2, x, 4, x*FresnelC[b*x]^2 + FresnelS[Sqrt[2]*b*x]/(Sqrt[2]*b*Pi) - (2*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b*Pi)}
{x*FresnelC[b*x]^2, x, 0, Int[x*FresnelC[b*x]^2, x]}
{x^2*FresnelC[b*x]^2, x, 6, (2*x)/(3*b^2*Pi^2) - (x*Cos[b^2*Pi*x^2])/(6*b^2*Pi^2) + (1/3)*x^3*FresnelC[b*x]^2 + (5*FresnelC[Sqrt[2]*b*x])/(6*Sqrt[2]*b^3*Pi^2) - (2*FresnelC[b*x]*(2*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(3*b^3*Pi^2)}
{x^3*FresnelC[b*x]^2, x, 10, (3*x^2)/(8*b^2*Pi^2) - (x^2*Cos[b^2*Pi*x^2])/(8*b^2*Pi^2) + (1/4)*(3/(b^4*Pi^2) + x^4)*FresnelC[b*x]^2 - (x*FresnelC[b*x]*(3*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(2*b^3*Pi^2) + Sin[b^2*Pi*x^2]/(2*b^4*Pi^3)}
{x^4*FresnelC[b*x]^2, x, 10, (4*x^3)/(15*b^2*Pi^2) - (x^3*Cos[b^2*Pi*x^2])/(10*b^2*Pi^2) + (1/5)*x^5*FresnelC[b*x]^2 - (43*FresnelS[Sqrt[2]*b*x])/(20*Sqrt[2]*b^5*Pi^3) - (2*FresnelC[b*x]*(4*b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - (8 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(5*b^5*Pi^3) + (11*x*Sin[b^2*Pi*x^2])/(20*b^4*Pi^3)}
{x^5*FresnelC[b*x]^2, x, 0, Int[x^5*FresnelC[b*x]^2, x]}
{x^6*FresnelC[b*x]^2, x, 15, -((48*x)/(7*b^6*Pi^4)) + (6*x^5)/(35*b^2*Pi^2) + (21*x*Cos[b^2*Pi*x^2])/(8*b^6*Pi^4) - (x^5*Cos[b^2*Pi*x^2])/(14*b^2*Pi^2) + (1/7)*x^7*FresnelC[b*x]^2 - (531*FresnelC[Sqrt[2]*b*x])/(56*Sqrt[2]*b^7*Pi^4) + (2*FresnelC[b*x]*(6*(8 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*(24 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(7*b^7*Pi^4) + (17*x^3*Sin[b^2*Pi*x^2])/(28*b^4*Pi^3)}
{x^7*FresnelC[b*x]^2, x, 21, -((105*x^2)/(16*b^6*Pi^4)) + (7*x^6)/(48*b^2*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/(16*b^6*Pi^4) - (x^6*Cos[b^2*Pi*x^2])/(16*b^2*Pi^2) - (1/8)*(105/(b^8*Pi^4) - x^8)*FresnelC[b*x]^2 + (x*FresnelC[b*x]*(7*(15 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*(35 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(4*b^7*Pi^4) - (10*Sin[b^2*Pi*x^2])/(b^8*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)}


{FresnelC[a + b*x]^2/x, x, 0, Int[FresnelC[a + b*x]^2/x, x]}
{FresnelC[a + b*x]^2, x, 4, ((a + b*x)*FresnelC[a + b*x]^2)/b + FresnelS[Sqrt[2]*(a + b*x)]/(Sqrt[2]*b*Pi) - (2*FresnelC[a + b*x]*Sin[(1/2)*Pi*(a + b*x)^2])/(b*Pi)}
{x*FresnelC[a + b*x]^2, x, 0, Int[x*FresnelC[a + b*x]^2, x]}


{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^10, x, 22, -((b^3*Pi)/(756*x^6)) + (b^7*Pi^3)/(3780*x^2) - (11*b^3*Pi*Cos[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Cos[b^2*Pi*x^2])/(2016*x^2) + (b^9*Pi^5*FresnelC[b*x]^2)/1890 - (FresnelC[b*x]*(b^2*Pi*x^2*(15 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + (105 - 3*b^4*Pi^2*x^4 + b^8*Pi^4*x^8)*Sin[(1/2)*b^2*Pi*x^2]))/(945*x^9) - (b*Sin[b^2*Pi*x^2])/(144*x^8) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(30240*x^4) + (83*b^9*Pi^4*SinIntegral[b^2*Pi*x^2])/30240}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^9, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^9, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^8, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^8, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^7, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^7, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^6, x, 11, -((b^3*Pi)/(60*x^2)) - (b^3*Pi*Cos[b^2*Pi*x^2])/(24*x^2) - (1/30)*b^5*Pi^3*FresnelC[b*x]^2 - (FresnelC[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] + (3 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(15*x^5) - (b*Sin[b^2*Pi*x^2])/(40*x^4) - (7/120)*b^5*Pi^2*SinIntegral[b^2*Pi*x^2]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^5, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^5, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^4, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^4, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^3, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^3, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x^2, x, 4, (1/2)*b*Pi*FresnelC[b*x]^2 - (FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x + (1/4)*b*SinIntegral[b^2*Pi*x^2]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x]/x, x, 0, Int[(FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]}
{Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 0, Int[FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2], x]}
{x*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 2, x/(2*b*Pi) - (Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^2*Pi) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)}
{x^2*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 5, x^2/(4*b*Pi) - (x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^2*Pi) + FresnelC[b*x]^2/(2*b^3*Pi) + Sin[b^2*Pi*x^2]/(4*b^3*Pi^2)}
{x^3*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 5, x^3/(6*b*Pi) - (5*FresnelS[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) - (FresnelC[b*x]*(b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - 2*Sin[(1/2)*b^2*Pi*x^2]))/(b^4*Pi^2) + (x*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^4*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 0, Int[x^4*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2], x]}
{x^5*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 9, -((4*x)/(b^5*Pi^3)) + x^5/(10*b*Pi) + (11*x*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) - (43*FresnelC[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (FresnelC[b*x]*((8 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + 4*b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(b^6*Pi^3) + (x^3*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^6*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 14, -((15*x^2)/(4*b^5*Pi^3)) + x^6/(12*b*Pi) + (7*x^2*Cos[b^2*Pi*x^2])/(4*b^5*Pi^3) - (15*FresnelC[b*x]^2)/(2*b^7*Pi^3) + (x*FresnelC[b*x]*((15 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + 5*b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(b^6*Pi^3) - (11*Sin[b^2*Pi*x^2])/(2*b^7*Pi^4) + (x^4*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^7*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 14, -((4*x^3)/(b^5*Pi^3)) + x^7/(14*b*Pi) + (17*x^3*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) + (531*FresnelS[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) + (FresnelC[b*x]*(b^2*Pi*x^2*(24 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - 6*(8 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^8*Pi^4) - (147*x*Sin[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)}
{x^8*Sin[Pi/2*b^2*x^2]*FresnelC[b*x], x, 0, Int[x^8*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2], x]}


{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^10, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^10, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^9, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^9, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^8, x, 16, -(b/(84*x^6)) + (b^5*Pi^2)/(420*x^2) - (b*Cos[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Cos[b^2*Pi*x^2])/(84*x^2) + (1/210)*b^7*Pi^4*FresnelC[b*x]^2 - (FresnelC[b*x]*((15 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] - b^2*Pi*x^2*(3 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(105*x^7) + (b^3*Pi*Sin[b^2*Pi*x^2])/(105*x^4) + (1/70)*b^7*Pi^3*SinIntegral[b^2*Pi*x^2]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^7, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^7, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^6, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^6, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^5, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^5, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^4, x, 7, -(b/(12*x^2)) - (b*Cos[b^2*Pi*x^2])/(12*x^2) - (1/6)*b^3*Pi^2*FresnelC[b*x]^2 - (FresnelC[b*x]*(Cos[(1/2)*b^2*Pi*x^2] - b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(3*x^3) - (1/6)*b^3*Pi*SinIntegral[b^2*Pi*x^2]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^3, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^3, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x^2, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x^2, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]/x, x, 0, Int[(Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/x, x]}
{Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 2, FresnelC[b*x]^2/(2*b)}
{x*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 2, -(FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)) + (FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi)}
{x^2*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 1, (x^2*FresnelC[b*x]^2)/(2*b) - Int[x*FresnelC[b*x]^2, x]/b}
{x^3*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 5, -(x/(b^3*Pi^2)) + (x*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (5*FresnelC[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) + (FresnelC[b*x]*(2*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(b^4*Pi^2)}
{x^4*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 9, -((3*x^2)/(4*b^3*Pi^2)) + (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (3*FresnelC[b*x]^2)/(2*b^5*Pi^2) + (x*FresnelC[b*x]*(3*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*Sin[(1/2)*b^2*Pi*x^2]))/(b^4*Pi^2) - Sin[b^2*Pi*x^2]/(b^5*Pi^3)}
{x^5*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 9, -((2*x^3)/(3*b^3*Pi^2)) + (x^3*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (43*FresnelS[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (FresnelC[b*x]*(4*b^2*Pi*x^2*Cos[(1/2)*b^2*Pi*x^2] - (8 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^6*Pi^3) - (11*x*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)}
{x^6*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 1, (x^6*FresnelC[b*x]^2)/(2*b) - (3*Int[x^5*FresnelC[b*x]^2, x])/b}
{x^7*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 14, (24*x)/(b^7*Pi^4) - (3*x^5)/(5*b^3*Pi^2) - (147*x*Cos[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (531*FresnelC[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) - (FresnelC[b*x]*(6*(8 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*(24 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^8*Pi^4) - (17*x^3*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)}
(* {x^8*Cos[Pi/2*b^2*x^2]*FresnelC[b*x], x, 20, (105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (105*FresnelC[b*x]^2)/(2*b^9*Pi^4) - (x*FresnelC[b*x]*(7*(15 - b^4*Pi^2*x^4)*Cos[(1/2)*b^2*Pi*x^2] + b^2*Pi*x^2*(35 - b^4*Pi^2*x^4)*Sin[(1/2)*b^2*Pi*x^2]))/(b^8*Pi^4) + (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(2*b^5*Pi^3)} *)
