\(\int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx\) [126]

Optimal result

Integrand size = 8, antiderivative size = 119 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{56 x^7}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{840 x^3}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelC}(b x)}{8 x^8}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{280 x^5}-\frac {b^7 \pi ^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{840 x} \] Output:

-1/56*b*cos(1/2*b^2*Pi*x^2)/x^7+1/840*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)/x^3+1/8 
40*b^8*Pi^4*FresnelC(b*x)-1/8*FresnelC(b*x)/x^8+1/280*b^3*Pi*sin(1/2*b^2*P 
i*x^2)/x^5-1/840*b^7*Pi^3*sin(1/2*b^2*Pi*x^2)/x
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=\frac {b x \left (-15+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-105+b^8 \pi ^4 x^8\right ) \operatorname {FresnelC}(b x)+b^3 \pi x^3 \left (3-b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{840 x^8} \] Input:

Integrate[FresnelC[b*x]/x^9,x]
 

Output:

(b*x*(-15 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + (-105 + b^8*Pi^4*x^8)*Fres 
nelC[b*x] + b^3*Pi*x^3*(3 - b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(840*x^8)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6981, 3869, 3868, 3869, 3868, 3833}

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.

\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx\)

\(\Big \downarrow \) 6981

\(\displaystyle \frac {1}{8} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx-\frac {\operatorname {FresnelC}(b x)}{8 x^8}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {1}{8} b \left (-\frac {1}{7} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelC}(b x)}{8 x^8}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} b \left (-\frac {1}{7} \pi b^2 \left (\frac {1}{5} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelC}(b x)}{8 x^8}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {1}{8} b \left (-\frac {1}{7} \pi b^2 \left (\frac {1}{5} \pi b^2 \left (-\frac {1}{3} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelC}(b x)}{8 x^8}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} b \left (-\frac {1}{7} \pi b^2 \left (\frac {1}{5} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\pi b^2 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelC}(b x)}{8 x^8}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{8} b \left (-\frac {1}{7} \pi b^2 \left (\frac {1}{5} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\pi b \operatorname {FresnelC}(b x)-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelC}(b x)}{8 x^8}\)

Input:

Int[FresnelC[b*x]/x^9,x]
 

Output:

-1/8*FresnelC[b*x]/x^8 + (b*(-1/7*Cos[(b^2*Pi*x^2)/2]/x^7 - (b^2*Pi*(-1/5* 
Sin[(b^2*Pi*x^2)/2]/x^5 + (b^2*Pi*(-1/3*Cos[(b^2*Pi*x^2)/2]/x^3 - (b^2*Pi* 
(b*Pi*FresnelC[b*x] - Sin[(b^2*Pi*x^2)/2]/x))/3))/5))/7))/8
 

Defintions of rubi rules used

rule 3833

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]
 

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 6981

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.50 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.22

method	result	size
meijerg	\(-\frac {b \operatorname {hypergeom}\left (\left [-\frac {7}{4}, \frac {1}{4}\right ], \left [-\frac {3}{4}, \frac {1}{2}, \frac {5}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{7 x^{7}}\)	\(26\)
derivativedivides	\(b^{8} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{8 b^{8} x^{8}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 b^{7} x^{7}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{3}\right )}{5}\right )}{56}\right )\)	\(108\)
default	\(b^{8} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{8 b^{8} x^{8}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 b^{7} x^{7}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{3}\right )}{5}\right )}{56}\right )\)	\(108\)
parts	\(-\frac {\operatorname {FresnelC}\left (b x \right )}{8 x^{8}}+\frac {b \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 x^{7}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 x^{5}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}+\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{5}\right )}{7}\right )}{8}\)	\(126\)

Input:

int(FresnelC(b*x)/x^9,x,method=_RETURNVERBOSE)
 

Output:

-1/7*b/x^7*hypergeom([-7/4,1/4],[-3/4,1/2,5/4],-1/16*x^4*Pi^2*b^4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.68 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=\frac {{\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right ) - {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \] Input:

integrate(fresnel_cos(b*x)/x^9,x, algorithm="fricas")
 

Output:

1/840*((pi^2*b^5*x^5 - 15*b*x)*cos(1/2*pi*b^2*x^2) + (pi^4*b^8*x^8 - 105)* 
fresnel_cos(b*x) - (pi^3*b^7*x^7 - 3*pi*b^3*x^3)*sin(1/2*pi*b^2*x^2))/x^8
 

Sympy [A] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.55 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=\frac {\pi ^{4} b^{8} C\left (b x\right ) \Gamma \left (- \frac {7}{4}\right )}{2560 \Gamma \left (\frac {5}{4}\right )} - \frac {\pi ^{3} b^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{2560 x \Gamma \left (\frac {5}{4}\right )} + \frac {\pi ^{2} b^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{2560 x^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {3 \pi b^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{2560 x^{5} \Gamma \left (\frac {5}{4}\right )} - \frac {3 b \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{512 x^{7} \Gamma \left (\frac {5}{4}\right )} - \frac {21 C\left (b x\right ) \Gamma \left (- \frac {7}{4}\right )}{512 x^{8} \Gamma \left (\frac {5}{4}\right )} \] Input:

integrate(fresnelc(b*x)/x**9,x)
 

Output:

pi**4*b**8*fresnelc(b*x)*gamma(-7/4)/(2560*gamma(5/4)) - pi**3*b**7*sin(pi 
*b**2*x**2/2)*gamma(-7/4)/(2560*x*gamma(5/4)) + pi**2*b**5*cos(pi*b**2*x** 
2/2)*gamma(-7/4)/(2560*x**3*gamma(5/4)) + 3*pi*b**3*sin(pi*b**2*x**2/2)*ga 
mma(-7/4)/(2560*x**5*gamma(5/4)) - 3*b*cos(pi*b**2*x**2/2)*gamma(-7/4)/(51 
2*x**7*gamma(5/4)) - 21*fresnelc(b*x)*gamma(-7/4)/(512*x**8*gamma(5/4))
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.51 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {7}{2}} {\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{8}}{512 \, x^{7}} - \frac {\operatorname {C}\left (b x\right )}{8 \, x^{8}} \] Input:

integrate(fresnel_cos(b*x)/x^9,x, algorithm="maxima")
 

Output:

-1/512*sqrt(1/2)*(pi*x^2)^(7/2)*(-(I - 1)*sqrt(2)*gamma(-7/2, 1/2*I*pi*b^2 
*x^2) + (I + 1)*sqrt(2)*gamma(-7/2, -1/2*I*pi*b^2*x^2))*b^8/x^7 - 1/8*fres 
nel_cos(b*x)/x^8
 

Giac [F]

\[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{9}} \,d x } \] Input:

integrate(fresnel_cos(b*x)/x^9,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)/x^9, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^9} \,d x \] Input:

int(FresnelC(b*x)/x^9,x)
 

Output:

int(FresnelC(b*x)/x^9, x)
 

Reduce [F]

\[ \int \frac {\operatorname {FresnelC}(b x)}{x^9} \, dx=\int \frac {\mathrm {FresnelC}\left (b x \right )}{x^{9}}d x \] Input:

int(FresnelC(b*x)/x^9,x)
 

Output:

int(FresnelC(b*x)/x^9,x)