Integrand size = 8, antiderivative size = 77 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {1}{80} b^5 \pi ^2 \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{40 x^2} \] Output:
-1/20*b*cos(1/2*b^2*Pi*x^2)/x^4-1/80*b^5*Pi^2*Ci(1/2*b^2*Pi*x^2)-1/5*Fresn elC(b*x)/x^5+1/40*b^3*Pi*sin(1/2*b^2*Pi*x^2)/x^2
Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{20 x^4}-\frac {1}{80} b^5 \pi ^2 \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{40 x^2} \] Input:
Integrate[FresnelC[b*x]/x^6,x]
Output:
-1/20*(b*Cos[(b^2*Pi*x^2)/2])/x^4 - (b^5*Pi^2*CosIntegral[(b^2*Pi*x^2)/2]) /80 - FresnelC[b*x]/(5*x^5) + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(40*x^2)
Time = 0.45 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {6981, 3861, 3042, 3778, 25, 3042, 3778, 3042, 3783}
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.
| \(\displaystyle \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx\) |
| \(\Big \downarrow \) 6981 |
| \(\displaystyle \frac {1}{5} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3861 |
| \(\displaystyle \frac {1}{10} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{10} b \left (\frac {1}{4} \pi b^2 \int -\frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} b \left (-\frac {1}{4} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} b \left (-\frac {1}{4} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {1}{10} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {1}{10} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{2} \pi b^2 \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x)}{5 x^5}\) |
Input:
Int[FresnelC[b*x]/x^6,x]
Output:
-1/5*FresnelC[b*x]/x^5 + (b*(-1/2*Cos[(b^2*Pi*x^2)/2]/x^4 - (b^2*Pi*((b^2* Pi*CosIntegral[(b^2*Pi*x^2)/2])/2 - Sin[(b^2*Pi*x^2)/2]/x^2))/4))/10
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[((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_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1 )*(FresnelC[b*x]/(d*(m + 1))), x] - Simp[b/(d*(m + 1)) Int[(d*x)^(m + 1)* Cos[(Pi/2)*b^2*x^2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
Time = 0.56 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88
| method | result | size |
| parts | \(-\frac {\operatorname {FresnelC}\left (b x \right )}{5 x^{5}}+\frac {b \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 x^{4}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 x^{2}}+\frac {b^{2} \pi \,\operatorname {Ci}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{5}\) | \(68\) |
| derivativedivides | \(b^{5} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{5 b^{5} x^{5}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{20 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \,\operatorname {Ci}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{20}\right )\) | \(71\) |
| default | \(b^{5} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{5 b^{5} x^{5}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{20 b^{4} x^{4}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}+\frac {\pi \,\operatorname {Ci}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{20}\right )\) | \(71\) |
| meijerg | \(\frac {\pi ^{\frac {5}{2}} b^{5} \left (-\frac {64}{\pi ^{\frac {5}{2}} x^{4} b^{4}}-\frac {8 \left (-\frac {19}{5}+2 \gamma -2 \ln \left (2\right )+4 \ln \left (x \right )+2 \ln \left (\pi \right )+4 \ln \left (b \right )\right )}{5 \sqrt {\pi }}+\frac {\pi ^{\frac {3}{2}} x^{4} b^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {9}{4}\right ], \left [2, \frac {5}{2}, 3, \frac {13}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{54}\right )}{256}\) | \(79\) |
Input:
int(FresnelC(b*x)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*FresnelC(b*x)/x^5+1/5*b*(-1/4/x^4*cos(1/2*b^2*Pi*x^2)-1/4*b^2*Pi*(-1/ 2*sin(1/2*b^2*Pi*x^2)/x^2+1/4*b^2*Pi*Ci(1/2*b^2*Pi*x^2)))
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=-\frac {\pi ^{2} b^{5} x^{5} \operatorname {Ci}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 16 \, \operatorname {C}\left (b x\right )}{80 \, x^{5}} \] Input:
integrate(fresnel_cos(b*x)/x^6,x, algorithm="fricas")
Output:
-1/80*(pi^2*b^5*x^5*cos_integral(1/2*pi*b^2*x^2) - 2*pi*b^3*x^3*sin(1/2*pi *b^2*x^2) + 4*b*x*cos(1/2*pi*b^2*x^2) + 16*fresnel_cos(b*x))/x^5
Time = 1.37 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=\frac {\pi ^{4} b^{9} x^{4} \Gamma \left (\frac {9}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {9}{4} \\ 2, \frac {5}{2}, 3, \frac {13}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{6144 \Gamma \left (\frac {13}{4}\right )} - \frac {\pi ^{2} b^{5} \log {\left (b^{4} x^{4} \right )}}{160} - \frac {b}{4 x^{4}} \] Input:
integrate(fresnelc(b*x)/x**6,x)
Output:
pi**4*b**9*x**4*gamma(9/4)*hyper((1, 1, 9/4), (2, 5/2, 3, 13/4), -pi**2*b* *4*x**4/16)/(6144*gamma(13/4)) - pi**2*b**5*log(b**4*x**4)/160 - b/(4*x**4 )
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=\frac {1}{80} \, {\left (\pi ^{2} \Gamma \left (-2, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \pi ^{2} \Gamma \left (-2, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{5} - \frac {\operatorname {C}\left (b x\right )}{5 \, x^{5}} \] Input:
integrate(fresnel_cos(b*x)/x^6,x, algorithm="maxima")
Output:
1/80*(pi^2*gamma(-2, 1/2*I*pi*b^2*x^2) + pi^2*gamma(-2, -1/2*I*pi*b^2*x^2) )*b^5 - 1/5*fresnel_cos(b*x)/x^5
\[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{6}} \,d x } \] Input:
integrate(fresnel_cos(b*x)/x^6,x, algorithm="giac")
Output:
integrate(fresnel_cos(b*x)/x^6, x)
Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^6} \,d x \] Input:
int(FresnelC(b*x)/x^6,x)
Output:
int(FresnelC(b*x)/x^6, x)
\[ \int \frac {\operatorname {FresnelC}(b x)}{x^6} \, dx=\int \frac {\mathrm {FresnelC}\left (b x \right )}{x^{6}}d x \] Input:
int(FresnelC(b*x)/x^6,x)
Output:
int(FresnelC(b*x)/x^6,x)