\(\int x^2 \operatorname {FresnelC}(b x)^2 \, dx\) [145]

Optimal result

Integrand size = 10, antiderivative size = 124 \[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\frac {2 x}{3 b^2 \pi ^2}-\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{3 b^3 \pi ^2}+\frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2+\frac {5 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}-\frac {2 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b \pi } \] Output:

2/3*x/b^2/Pi^2-1/6*x*cos(b^2*Pi*x^2)/b^2/Pi^2-4/3*cos(1/2*b^2*Pi*x^2)*Fres 
nelC(b*x)/b^3/Pi^2+1/3*x^3*FresnelC(b*x)^2+5/12*FresnelC(2^(1/2)*b*x)*2^(1 
/2)/b^3/Pi^2-2/3*x^2*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b/Pi
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\frac {-2 b x \left (-4+\cos \left (b^2 \pi x^2\right )\right )+4 b^3 \pi ^2 x^3 \operatorname {FresnelC}(b x)^2+5 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )-8 \operatorname {FresnelC}(b x) \left (2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )}{12 b^3 \pi ^2} \] Input:

Integrate[x^2*FresnelC[b*x]^2,x]
 

Output:

(-2*b*x*(-4 + Cos[b^2*Pi*x^2]) + 4*b^3*Pi^2*x^3*FresnelC[b*x]^2 + 5*Sqrt[2 
]*FresnelC[Sqrt[2]*b*x] - 8*FresnelC[b*x]*(2*Cos[(b^2*Pi*x^2)/2] + b^2*Pi* 
x^2*Sin[(b^2*Pi*x^2)/2]))/(12*b^3*Pi^2)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6985, 7009, 3866, 3833, 7015, 3839, 2009}

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.

\(\displaystyle \int x^2 \operatorname {FresnelC}(b x)^2 \, dx\)

\(\Big \downarrow \) 6985

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx\)

\(\Big \downarrow \) 7009

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \left (-\frac {2 \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}+\frac {x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \left (-\frac {2 \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}+\frac {x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \left (-\frac {2 \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 7015

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \left (-\frac {2 \left (\frac {\int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3839

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \left (-\frac {2 \left (\frac {\int \left (\frac {1}{2} \cos \left (b^2 \pi x^2\right )+\frac {1}{2}\right )dx}{\pi b}-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(b x)^2-\frac {2}{3} b \left (\frac {x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b}+\frac {x}{2}}{\pi b}-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

Input:

Int[x^2*FresnelC[b*x]^2,x]
 

Output:

(x^3*FresnelC[b*x]^2)/3 - (2*b*(-1/2*(-1/2*(x*Cos[b^2*Pi*x^2])/(b^2*Pi) + 
FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^3*Pi))/(b*Pi) - (2*(-((Cos[(b^2*Pi*x^2) 
/2]*FresnelC[b*x])/(b^2*Pi)) + (x/2 + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b)) 
/(b*Pi)))/(b^2*Pi) + (x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)))/3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3839

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_), x_Sy 
mbol] :> Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
 

rule 3866

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^ 
(n - 1))*(e*x)^(m - n + 1)*(Cos[c + d*x^n]/(d*n)), x] + Simp[e^n*((m - n + 
1)/(d*n))   Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] 
 && IGtQ[n, 0] && LtQ[n, m + 1]
 

rule 6985

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel 
C[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1))   Int[x^(m + 1)*Cos[(Pi/2)*b^2*x 
^2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
 

rule 7009

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m - 1)*Sin[d*x^2]*(FresnelC[b*x]/(2*d)), x] + (-Simp[(m - 1)/(2*d)   Int[x^ 
(m - 2)*Sin[d*x^2]*FresnelC[b*x], x], x] - Simp[b/(4*d)   Int[x^(m - 1)*Sin 
[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m, 
 1]
 

rule 7015

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

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right )^{2} b^{3} x^{3}}{3}-2 \,\operatorname {FresnelC}\left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}\right )+\frac {2 b x}{3 \pi ^{2}}+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{3 \pi ^{2}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4 \pi }}{3 \pi }}{b^{3}}\)	\(122\)
default	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right )^{2} b^{3} x^{3}}{3}-2 \,\operatorname {FresnelC}\left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}\right )+\frac {2 b x}{3 \pi ^{2}}+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{3 \pi ^{2}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4 \pi }}{3 \pi }}{b^{3}}\)	\(122\)

Input:

int(x^2*FresnelC(b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b^3*(1/3*FresnelC(b*x)^2*b^3*x^3-2*FresnelC(b*x)*(1/3/Pi*b^2*x^2*sin(1/2 
*b^2*Pi*x^2)+2/3/Pi^2*cos(1/2*b^2*Pi*x^2))+2/3*b*x/Pi^2+1/3/Pi^2*2^(1/2)*F 
resnelC(2^(1/2)*b*x)+1/3/Pi*(-1/2/Pi*b*x*cos(b^2*Pi*x^2)+1/4/Pi*2^(1/2)*Fr 
esnelC(2^(1/2)*b*x)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\frac {4 \, \pi ^{2} b^{4} x^{3} \operatorname {C}\left (b x\right )^{2} - 8 \, \pi b^{3} x^{2} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 10 \, b^{2} x - 16 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + 5 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi ^{2} b^{4}} \] Input:

integrate(x^2*fresnel_cos(b*x)^2,x, algorithm="fricas")
 

Output:

1/12*(4*pi^2*b^4*x^3*fresnel_cos(b*x)^2 - 8*pi*b^3*x^2*fresnel_cos(b*x)*si 
n(1/2*pi*b^2*x^2) - 4*b^2*x*cos(1/2*pi*b^2*x^2)^2 + 10*b^2*x - 16*b*cos(1/ 
2*pi*b^2*x^2)*fresnel_cos(b*x) + 5*sqrt(2)*sqrt(b^2)*fresnel_cos(sqrt(2)*s 
qrt(b^2)*x))/(pi^2*b^4)
 

Sympy [F]

\[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{2} C^{2}\left (b x\right )\, dx \] Input:

integrate(x**2*fresnelc(b*x)**2,x)
 

Output:

Integral(x**2*fresnelc(b*x)**2, x)
 

Maxima [F]

\[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{2} \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^2*fresnel_cos(b*x)^2,x, algorithm="maxima")
 

Output:

integrate(x^2*fresnel_cos(b*x)^2, x)
 

Giac [F]

\[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{2} \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^2*fresnel_cos(b*x)^2,x, algorithm="giac")
 

Output:

integrate(x^2*fresnel_cos(b*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\int x^2\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \] Input:

int(x^2*FresnelC(b*x)^2,x)
 

Output:

int(x^2*FresnelC(b*x)^2, x)
 

Reduce [F]

\[ \int x^2 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{2} \mathrm {FresnelC}\left (b x \right )^{2}d x \] Input:

int(x^2*FresnelC(b*x)^2,x)
 

Output:

int(x^2*FresnelC(b*x)^2,x)