Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=-\frac {b^2}{120 x^4}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {1}{72} b^6 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{15 x^5}-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} b^5 \pi ^2 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2},x\right ) \]
-1/120*b^2/x^4-1/72*b^6*Pi^2*Ci(b^2*Pi*x^2)-1/120*b^2*cos(b^2*Pi*x^2)/x^4- 1/15*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^5-1/6*FresnelC(b*x)^2/x^6+1/45* b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3+1/72*b^4*Pi*sin(b^2*Pi*x^2)/x ^2-1/45*b^5*Pi^2*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2,x)
Not integrable
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx \]
Not integrable
Time = 1.37 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6985, 7011, 3861, 3042, 3778, 25, 3042, 3778, 3042, 3783, 7019, 3860, 3042, 3778, 3042, 3783, 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.
\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx\) |
\(\Big \downarrow \) 6985 |
\(\displaystyle \frac {1}{3} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 7011 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{10} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3861 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 7019 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{6} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
\(\Big \downarrow \) 7013 |
\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\) |
3.2.54.3.1 Defintions of rubi rules used
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]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
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]))
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]))
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
\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{7}}d x\]
Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{7}} \,d x } \]
Not integrable
Time = 1.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{7}}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{7}} \,d x } \]
Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{7}} \,d x } \]
Not integrable
Time = 4.95 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^7} \,d x \]