3.1.46 \(\int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx\) [46]

3.1.46.1 Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{84 x^4}-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}-\frac {b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{72 \sqrt {2}}-\frac {2}{315} \sqrt {2} b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac {1}{168} b^7 \pi ^3 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x},x\right ) \]

output

-1/210*b^2/x^5+1/336*b^6*Pi^2/x+1/210*b^2*cos(b^2*Pi*x^2)/x^5-67/5040*b^6* 
Pi^2*cos(b^2*Pi*x^2)/x-1/84*b^3*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^4-1 
/7*FresnelS(b*x)^2/x^7-1/21*b*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^6+1/168* 
b^5*Pi^2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^2-13/2520*b^4*Pi*sin(b^2*Pi*x 
^2)/x^3-67/5040*b^7*Pi^3*FresnelS(b*x*2^(1/2))*2^(1/2)-1/168*b^7*Pi^3*Unin 
tegrable(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x,x)
 

3.1.46.2 Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx \]

input

Integrate[FresnelS[b*x]^2/x^8,x]
 

output

Integrate[FresnelS[b*x]^2/x^8, x]
 

3.1.46.3 Rubi [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6984, 7010, 3869, 3868, 3869, 3832, 7018, 3868, 3869, 3832, 7010, 3869, 3832, 7020}

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.

\(\displaystyle \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx\)

\(\Big \downarrow \) 6984

\(\displaystyle \frac {2}{7} b \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7}dx-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7010

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^5}dx-\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^5}dx-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^5}dx-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^5}dx-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^5}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7018

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx+\frac {1}{8} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7010

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x}dx-\frac {1}{4} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x}dx-\frac {1}{4} b \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {1}{4} b \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7020

\(\displaystyle \frac {2}{7} b \left (\frac {1}{6} \pi b^2 \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {1}{4} b \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelS}(b x)^2}{7 x^7}\)

input

Int[FresnelS[b*x]^2/x^8,x]
 

output

$Aborted
 

3.1.46.3.1 Defintions of rubi rules used

rule 3832

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

rule 3868

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

rule 3869

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

rule 6984

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

rule 7010

Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[x^( 
m + 1)*Sin[d*x^2]*(FresnelS[b*x]/(m + 1)), x] + (-Simp[d*(x^(m + 2)/(Pi*b*( 
m + 1)*(m + 2))), x] - Simp[2*(d/(m + 1))   Int[x^(m + 2)*Cos[d*x^2]*Fresne 
lS[b*x], x], x] + Simp[d/(Pi*b*(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 7018

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m + 1)*Cos[d*x^2]*(FresnelS[b*x]/(m + 1)), x] + (Simp[2*(d/(m + 1))   Int[x 
^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x], x] - Simp[d/(Pi*b*(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]
 

rule 7020

Int[Cos[(c_.) + (d_.)*(x_)^2]*FresnelS[(a_.) + (b_.)*(x_)]^(n_.)*((e_.)*(x_ 
))^(m_.), x_Symbol] :> Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelS[a + b*x 
]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

3.1.46.4 Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{8}}d x\]

input

int(FresnelS(b*x)^2/x^8,x)
 

output

int(FresnelS(b*x)^2/x^8,x)
 

3.1.46.5 Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{8}} \,d x } \]

input

integrate(fresnel_sin(b*x)^2/x^8,x, algorithm="fricas")
 

output

integral(fresnel_sin(b*x)^2/x^8, x)
 

3.1.46.6 Sympy [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{8}}\, dx \]

input

integrate(fresnels(b*x)**2/x**8,x)
 

output

Integral(fresnels(b*x)**2/x**8, x)
 

3.1.46.7 Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{8}} \,d x } \]

input

integrate(fresnel_sin(b*x)^2/x^8,x, algorithm="maxima")
 

output

integrate(fresnel_sin(b*x)^2/x^8, x)
 

3.1.46.8 Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{8}} \,d x } \]

input

integrate(fresnel_sin(b*x)^2/x^8,x, algorithm="giac")
 

output

integrate(fresnel_sin(b*x)^2/x^8, x)
 

3.1.46.9 Mupad [N/A]

Not integrable

Time = 4.75 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^8} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^8} \,d x \]

input

int(FresnelS(b*x)^2/x^8,x)
 

output

int(FresnelS(b*x)^2/x^8, x)