\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx\) [152]

Optimal result

Integrand size = 10, antiderivative size = 127 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)^2-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}-\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \] Output:

-1/24*b^2/x^2-1/24*b^2*cos(b^2*Pi*x^2)/x^2-1/6*b*cos(1/2*b^2*Pi*x^2)*Fresn 
elC(b*x)/x^3-1/12*b^4*Pi^2*FresnelC(b*x)^2-1/4*FresnelC(b*x)^2/x^4+1/6*b^3 
*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x-1/12*b^4*Pi*Si(b^2*Pi*x^2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)^2-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}-\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \] Input:

Integrate[FresnelC[b*x]^2/x^5,x]
 

Output:

-1/24*b^2/x^2 - (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - (b*Cos[(b^2*Pi*x^2)/2]*Fr 
esnelC[b*x])/(6*x^3) - (b^4*Pi^2*FresnelC[b*x]^2)/12 - FresnelC[b*x]^2/(4* 
x^4) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(6*x) - (b^4*Pi*SinInteg 
ral[b^2*Pi*x^2])/12
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6985, 7011, 3861, 3042, 3778, 25, 3042, 3780, 7019, 3856, 6995, 15}

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.

\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx\)

\(\Big \downarrow \) 6985

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

\(\Big \downarrow \) 7011

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx+\frac {1}{6} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 3861

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx+\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^4}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx+\frac {1}{12} b \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx+\frac {1}{12} b \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}+\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 7019

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \left (\pi b^2 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx+\frac {1}{2} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}+\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 3856

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \left (\pi b^2 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}+\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 6995

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \left (\pi b \int \operatorname {FresnelC}(b x)d\operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}+\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} b \left (-\frac {1}{3} \pi b^2 \left (-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )+\frac {1}{2} \pi b \operatorname {FresnelC}(b x)^2\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}+\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}\)

Input:

Int[FresnelC[b*x]^2/x^5,x]
 

Output:

-1/4*FresnelC[b*x]^2/x^4 + (b*(-1/12*b/x^2 - (Cos[(b^2*Pi*x^2)/2]*FresnelC 
[b*x])/(3*x^3) - (b^2*Pi*((b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b 
^2*Pi*x^2)/2])/x + (b*SinIntegral[b^2*Pi*x^2])/4))/3 + (b*(-(Cos[b^2*Pi*x^ 
2]/x^2) - b^2*Pi*SinIntegral[b^2*Pi*x^2]))/12))/2
 

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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3778

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

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3856

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] / 
; FreeQ[{d, n}, x]
 

rule 3861

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[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 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 7011

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

rule 7019

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

Maple [F]

\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{5}}d x\]

Input:

int(FresnelC(b*x)^2/x^5,x)
 

Output:

int(FresnelC(b*x)^2/x^5,x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {\pi b^{4} x^{4} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi b^{3} x^{3} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )^{2}}{12 \, x^{4}} \] Input:

integrate(fresnel_cos(b*x)^2/x^5,x, algorithm="fricas")
 

Output:

-1/12*(pi*b^4*x^4*sin_integral(pi*b^2*x^2) - 2*pi*b^3*x^3*fresnel_cos(b*x) 
*sin(1/2*pi*b^2*x^2) + b^2*x^2*cos(1/2*pi*b^2*x^2)^2 + 2*b*x*cos(1/2*pi*b^ 
2*x^2)*fresnel_cos(b*x) + (pi^2*b^4*x^4 + 3)*fresnel_cos(b*x)^2)/x^4
 

Sympy [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{5}}\, dx \] Input:

integrate(fresnelc(b*x)**2/x**5,x)
 

Output:

Integral(fresnelc(b*x)**2/x**5, x)
 

Maxima [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{5}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^5,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x)^2/x^5, x)
 

Giac [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{5}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^5,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)^2/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^5} \,d x \] Input:

int(FresnelC(b*x)^2/x^5,x)
 

Output:

int(FresnelC(b*x)^2/x^5, x)
 

Reduce [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int \frac {\mathrm {FresnelC}\left (b x \right )^{2}}{x^{5}}d x \] Input:

int(FresnelC(b*x)^2/x^5,x)
 

Output:

int(FresnelC(b*x)^2/x^5,x)