Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=-\frac {b^2}{60 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{10 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}-\frac {7 b^5 \pi ^2 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{60 \sqrt {2}}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^2}+\frac {7 b^4 \pi \sin \left (b^2 \pi x^2\right )}{120 x}-\frac {1}{20} b^5 \pi ^2 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x},x\right ) \]
-1/60*b^2/x^3-1/60*b^2*cos(b^2*Pi*x^2)/x^3-1/10*b*cos(1/2*b^2*Pi*x^2)*Fres nelC(b*x)/x^4-1/5*FresnelC(b*x)^2/x^5+1/20*b^3*Pi*FresnelC(b*x)*sin(1/2*b^ 2*Pi*x^2)/x^2+7/120*b^4*Pi*sin(b^2*Pi*x^2)/x-7/120*b^5*Pi^2*FresnelC(b*x*2 ^(1/2))*2^(1/2)-1/20*b^5*Pi^2*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b* x)/x,x)
Not integrable
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx \]
Not integrable
Time = 0.93 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6985, 7011, 3869, 3868, 3833, 7019, 3868, 3833, 7013}
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.
\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx\) |
\(\Big \downarrow \) 6985 |
\(\displaystyle \frac {2}{5} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5}dx-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 7011 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx+\frac {1}{8} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 7019 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x}dx+\frac {1}{4} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x}dx+\frac {1}{4} b \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
\(\Big \downarrow \) 7013 |
\(\displaystyle \frac {2}{5} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {b}{24 x^3}\right )-\frac {\operatorname {FresnelC}(b x)^2}{5 x^5}\) |
3.2.53.3.1 Defintions of rubi rules used
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]
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[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]
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[Cos[(c_.) + (d_.)*(x_)^2]*FresnelC[(a_.) + (b_.)*(x_)]^(n_.)*((e_.)*(x_ ))^(m_.), x_Symbol] :> Unintegrable[(e*x)^m*Cos[c + d*x^2]*FresnelC[a + b*x ]^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
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]
Not integrable
Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{6}}d x\]
Not integrable
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{6}} \,d x } \]
Not integrable
Time = 1.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{6}}\, dx \]
Not integrable
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{6}} \,d x } \]
Not integrable
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{6}} \,d x } \]
Not integrable
Time = 4.90 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^6} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^6} \,d x \]