\(\int x \operatorname {FresnelC}(b x)^2 \, dx\) [146]

Optimal result

Integrand size = 8, antiderivative size = 144 \[ \int x \operatorname {FresnelC}(b x)^2 \, dx=-\frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {1}{2} x^2 \operatorname {FresnelC}(b x)^2+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^2 \pi }+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b \pi } \] Output:

-1/4*cos(b^2*Pi*x^2)/b^2/Pi^2+1/2*x^2*FresnelC(b*x)^2+1/2*FresnelC(b*x)*Fr 
esnelS(b*x)/b^2/Pi+1/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/ 
Pi-1/8*I*x^2*hypergeom([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)/Pi-x*FresnelC(b*x 
)*sin(1/2*b^2*Pi*x^2)/b/Pi
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6985, 7009, 3860, 3042, 3118, 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.

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

\(\Big \downarrow \) 6985

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3860

\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelC}(b x)^2-b \left (-\frac {\int \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelC}(b x)^2-b \left (-\frac {\int \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}+\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelC}(b x)^2-b \left (-\frac {\int \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )\)

\(\Big \downarrow \) 7001

\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelC}(b x)^2-b \left (-\frac {\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}}{\pi b^2}+\frac {x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

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 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]
 

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]
 

Maple [F]

\[\int x \operatorname {FresnelC}\left (b x \right )^{2}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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