Integrand size = 8, antiderivative size = 119 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{280 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{840 x}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelS}(b x)-\frac {\operatorname {FresnelS}(b x)}{8 x^8}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{56 x^7}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{840 x^3} \]
-1/280*b^3*Pi*cos(1/2*b^2*Pi*x^2)/x^5+1/840*b^7*Pi^3*cos(1/2*b^2*Pi*x^2)/x +1/840*b^8*Pi^4*FresnelS(b*x)-1/8*FresnelS(b*x)/x^8-1/56*b*sin(1/2*b^2*Pi* x^2)/x^7+1/840*b^5*Pi^2*sin(1/2*b^2*Pi*x^2)/x^3
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.71 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=\frac {b^3 \pi x^3 \left (-3+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-105+b^8 \pi ^4 x^8\right ) \operatorname {FresnelS}(b x)+b x \left (-15+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{840 x^8} \]
(b^3*Pi*x^3*(-3 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] + (-105 + b^8*Pi^4*x^8 )*FresnelS[b*x] + b*x*(-15 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/(840*x^8)
Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6980, 3868, 3869, 3868, 3869, 3832}
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.
\(\displaystyle \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx\) |
\(\Big \downarrow \) 6980 |
\(\displaystyle \frac {1}{8} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx-\frac {\operatorname {FresnelS}(b x)}{8 x^8}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {1}{8} b \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x)}{8 x^8}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle \frac {1}{8} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x)}{8 x^8}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {1}{8} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x)}{8 x^8}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle \frac {1}{8} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\pi b^2 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x)}{8 x^8}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {1}{8} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}-\pi b \operatorname {FresnelS}(b x)\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x)}{8 x^8}\) |
-1/8*FresnelS[b*x]/x^8 + (b*(-1/7*Sin[(b^2*Pi*x^2)/2]/x^7 + (b^2*Pi*(-1/5* Cos[(b^2*Pi*x^2)/2]/x^5 - (b^2*Pi*((b^2*Pi*(-(Cos[(b^2*Pi*x^2)/2]/x) - b*P i*FresnelS[b*x]))/3 - Sin[(b^2*Pi*x^2)/2]/(3*x^3)))/5))/7))/8
3.1.17.3.1 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[FresnelS[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1 )*(FresnelS[b*x]/(d*(m + 1))), x] - Simp[b/(d*(m + 1)) Int[(d*x)^(m + 1)* Sin[(Pi/2)*b^2*x^2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.24
method | result | size |
meijerg | \(-\frac {\pi \,b^{3} \operatorname {hypergeom}\left (\left [-\frac {5}{4}, \frac {3}{4}\right ], \left [-\frac {1}{4}, \frac {3}{2}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{30 x^{5}}\) | \(29\) |
derivativedivides | \(b^{8} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{8 b^{8} x^{8}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 b^{7} x^{7}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 b^{5} x^{5}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\operatorname {FresnelS}\left (b x \right )\right )}{3}\right )}{5}\right )}{56}\right )\) | \(109\) |
default | \(b^{8} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{8 b^{8} x^{8}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 b^{7} x^{7}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 b^{5} x^{5}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\operatorname {FresnelS}\left (b x \right )\right )}{3}\right )}{5}\right )}{56}\right )\) | \(109\) |
parts | \(-\frac {\operatorname {FresnelS}\left (b x \right )}{8 x^{8}}+\frac {b \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 x^{7}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 x^{5}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}-\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{5}\right )}{7}\right )}{8}\) | \(127\) |
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=\frac {{\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {S}\left (b x\right ) + {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \]
1/840*((pi^3*b^7*x^7 - 3*pi*b^3*x^3)*cos(1/2*pi*b^2*x^2) + (pi^4*b^8*x^8 - 105)*fresnel_sin(b*x) + (pi^2*b^5*x^5 - 15*b*x)*sin(1/2*pi*b^2*x^2))/x^8
Time = 1.72 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.55 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=\frac {\pi ^{4} b^{8} S\left (b x\right ) \Gamma \left (- \frac {5}{4}\right )}{3584 \Gamma \left (\frac {7}{4}\right )} + \frac {\pi ^{3} b^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {5}{4}\right )}{3584 x \Gamma \left (\frac {7}{4}\right )} + \frac {\pi ^{2} b^{5} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {5}{4}\right )}{3584 x^{3} \Gamma \left (\frac {7}{4}\right )} - \frac {3 \pi b^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {5}{4}\right )}{3584 x^{5} \Gamma \left (\frac {7}{4}\right )} - \frac {15 b \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {5}{4}\right )}{3584 x^{7} \Gamma \left (\frac {7}{4}\right )} - \frac {15 S\left (b x\right ) \Gamma \left (- \frac {5}{4}\right )}{512 x^{8} \Gamma \left (\frac {7}{4}\right )} \]
pi**4*b**8*fresnels(b*x)*gamma(-5/4)/(3584*gamma(7/4)) + pi**3*b**7*cos(pi *b**2*x**2/2)*gamma(-5/4)/(3584*x*gamma(7/4)) + pi**2*b**5*sin(pi*b**2*x** 2/2)*gamma(-5/4)/(3584*x**3*gamma(7/4)) - 3*pi*b**3*cos(pi*b**2*x**2/2)*ga mma(-5/4)/(3584*x**5*gamma(7/4)) - 15*b*sin(pi*b**2*x**2/2)*gamma(-5/4)/(3 584*x**7*gamma(7/4)) - 15*fresnels(b*x)*gamma(-5/4)/(512*x**8*gamma(7/4))
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.51 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {7}{2}} {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{8}}{512 \, x^{7}} - \frac {\operatorname {S}\left (b x\right )}{8 \, x^{8}} \]
-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*fresn el_sin(b*x)/x^8
\[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right )}{x^{9}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)}{x^9} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^9} \,d x \]