Integrand size = 8, antiderivative size = 94 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {1}{90} b^6 \pi ^3 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelS}(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x} \]
-1/90*b^3*Pi*cos(1/2*b^2*Pi*x^2)/x^3-1/90*b^6*Pi^3*FresnelC(b*x)-1/6*Fresn elS(b*x)/x^6-1/30*b*sin(1/2*b^2*Pi*x^2)/x^5+1/90*b^5*Pi^2*sin(1/2*b^2*Pi*x ^2)/x
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\frac {1}{90} \left (-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}-b^6 \pi ^3 \operatorname {FresnelC}(b x)-\frac {15 \operatorname {FresnelS}(b x)}{x^6}+\frac {b \left (-3+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}\right ) \]
(-((b^3*Pi*Cos[(b^2*Pi*x^2)/2])/x^3) - b^6*Pi^3*FresnelC[b*x] - (15*Fresne lS[b*x])/x^6 + (b*(-3 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2])/x^5)/90
Time = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6980, 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.
\(\displaystyle \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx\) |
\(\Big \downarrow \) 6980 |
\(\displaystyle \frac {1}{6} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelS}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {1}{6} b \left (\frac {1}{5} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle \frac {1}{6} b \left (\frac {1}{5} \pi b^2 \left (-\frac {1}{3} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {1}{6} b \left (\frac {1}{5} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\pi b^2 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {1}{6} b \left (\frac {1}{5} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\pi b \operatorname {FresnelC}(b x)-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x)}{6 x^6}\) |
-1/6*FresnelS[b*x]/x^6 + (b*(-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))/6
3.1.15.3.1 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[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 = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.31
method | result | size |
meijerg | \(-\frac {\pi \,b^{3} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {1}{4}, \frac {3}{2}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{18 x^{3}}\) | \(29\) |
derivativedivides | \(b^{6} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(86\) |
default | \(b^{6} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(86\) |
parts | \(-\frac {\operatorname {FresnelS}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 x^{5}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}+\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{5}\right )}{6}\) | \(104\) |
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {\pi ^{3} \sqrt {b^{2}} b^{5} x^{6} \operatorname {C}\left (\sqrt {b^{2}} x\right ) + \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{5} x^{5} - 3 \, b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 15 \, \operatorname {S}\left (b x\right )}{90 \, x^{6}} \]
-1/90*(pi^3*sqrt(b^2)*b^5*x^6*fresnel_cos(sqrt(b^2)*x) + pi*b^3*x^3*cos(1/ 2*pi*b^2*x^2) - (pi^2*b^5*x^5 - 3*b*x)*sin(1/2*pi*b^2*x^2) + 15*fresnel_si n(b*x))/x^6
Time = 0.75 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\frac {\pi b^{3} \Gamma \left (- \frac {3}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4}, \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{32 x^{3} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {7}{4}\right )} \]
pi*b**3*gamma(-3/4)*gamma(3/4)*hyper((-3/4, 3/4), (1/4, 3/2, 7/4), -pi**2* b**4*x**4/16)/(32*x**3*gamma(1/4)*gamma(7/4))
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.65 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {5}{2}} {\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{6}}{192 \, x^{5}} - \frac {\operatorname {S}\left (b x\right )}{6 \, x^{6}} \]
-1/192*sqrt(1/2)*(pi*x^2)^(5/2)*(-(I - 1)*sqrt(2)*gamma(-5/2, 1/2*I*pi*b^2 *x^2) + (I + 1)*sqrt(2)*gamma(-5/2, -1/2*I*pi*b^2*x^2))*b^6/x^5 - 1/6*fres nel_sin(b*x)/x^6
\[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^7} \,d x \]