Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=-\frac {b}{60 x^5}+\frac {b^5 \pi ^2}{96 x}-\frac {b \cos \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{1440 x}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^2}+\frac {7 b^6 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{144 \sqrt {2}}+\frac {1}{45} \sqrt {2} b^6 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}+\frac {13 b^3 \pi \sin \left (b^2 \pi x^2\right )}{720 x^3}+\frac {1}{48} b^6 \pi ^3 \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]
-1/60*b/x^5+1/96*b^5*Pi^2/x-1/60*b*cos(b^2*Pi*x^2)/x^5+67/1440*b^5*Pi^2*co s(b^2*Pi*x^2)/x-1/6*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6+1/48*b^4*Pi^2*co s(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2+1/24*b^2*Pi*FresnelC(b*x)*sin(1/2*b^2* Pi*x^2)/x^4+13/720*b^3*Pi*sin(b^2*Pi*x^2)/x^3+67/1440*b^6*Pi^3*FresnelS(b* x*2^(1/2))*2^(1/2)+1/48*b^6*Pi^3*Unintegrable(FresnelC(b*x)*sin(1/2*b^2*Pi *x^2)/x,x)
Not integrable
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx \]
Not integrable
Time = 1.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {7011, 3869, 3868, 3869, 3832, 7019, 3868, 3869, 3832, 7011, 3869, 3832, 7021}
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.
| \(\displaystyle \int \frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^7} \, dx\) |
| \(\Big \downarrow \) 7011 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3869 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3868 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3869 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 7019 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx+\frac {1}{8} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3868 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3869 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 7011 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3869 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}+\frac {1}{4} b \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
| \(\Big \downarrow \) 7021 |
| \(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}+\frac {1}{4} b \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\) |
3.2.95.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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]
Int[FresnelC[(a_.) + (b_.)*(x_)]^(n_.)*((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.) *(x_)^2], x_Symbol] :> Unintegrable[(e*x)^m*FresnelC[a + b*x]^n*Sin[c + d*x ^2], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Not integrable
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelC}\left (b x \right )}{x^{7}}d x\]
Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{7}} \,d x } \]
Not integrable
Time = 11.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{7}}\, dx \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{7}} \,d x } \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{7}} \,d x } \]
Not integrable
Time = 4.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^7} \,d x \]