Integrand size = 8, antiderivative size = 94 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {\operatorname {FresnelC}(b x)}{6 x^6}+\frac {1}{90} b^6 \pi ^3 \operatorname {FresnelS}(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3} \]
-1/30*b*cos(1/2*b^2*Pi*x^2)/x^5+1/90*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)/x-1/6*Fr esnelC(b*x)/x^6+1/90*b^6*Pi^3*FresnelS(b*x)+1/90*b^3*Pi*sin(1/2*b^2*Pi*x^2 )/x^3
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.79 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {1}{90} \left (\frac {b \left (-3+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}-\frac {15 \operatorname {FresnelC}(b x)}{x^6}+b^6 \pi ^3 \operatorname {FresnelS}(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}\right ) \]
((b*(-3 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2])/x^5 - (15*FresnelC[b*x])/x^6 + b^6*Pi^3*FresnelS[b*x] + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/x^3)/90
Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6981, 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.
\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx\) |
\(\Big \downarrow \) 6981 |
\(\displaystyle \frac {1}{6} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelC}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle \frac {1}{6} b \left (-\frac {1}{5} \pi b^2 \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle \frac {1}{6} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle \frac {1}{6} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\pi b^2 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x)}{6 x^6}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {1}{6} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}-\pi b \operatorname {FresnelS}(b x)\right )-\frac {\sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x)}{6 x^6}\) |
-1/6*FresnelC[b*x]/x^6 + (b*(-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*Pi*FresnelS[b*x]))/3 - Sin[(b^2*Pi*x^2)/2 ]/(3*x^3)))/5))/6
3.2.24.3.1 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[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]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(-\frac {b \operatorname {hypergeom}\left (\left [-\frac {5}{4}, \frac {1}{4}\right ], \left [-\frac {1}{4}, \frac {1}{2}, \frac {5}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{5 x^{5}}\) | \(26\) |
derivativedivides | \(b^{6} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\operatorname {FresnelS}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(87\) |
default | \(b^{6} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\operatorname {FresnelS}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(87\) |
parts | \(-\frac {\operatorname {FresnelC}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 x^{5}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}-\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{5}\right )}{6}\) | \(105\) |
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {\pi ^{3} \sqrt {b^{2}} b^{5} x^{6} \operatorname {S}\left (\sqrt {b^{2}} x\right ) + \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{2} b^{5} x^{5} - 3 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 15 \, \operatorname {C}\left (b x\right )}{90 \, x^{6}} \]
1/90*(pi^3*sqrt(b^2)*b^5*x^6*fresnel_sin(sqrt(b^2)*x) + pi*b^3*x^3*sin(1/2 *pi*b^2*x^2) + (pi^2*b^5*x^5 - 3*b*x)*cos(1/2*pi*b^2*x^2) - 15*fresnel_cos (b*x))/x^6
Time = 0.69 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {b \Gamma \left (- \frac {5}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{4} \\ - \frac {1}{4}, \frac {1}{2}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 x^{5} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {5}{4}\right )} \]
b*gamma(-5/4)*gamma(1/4)*hyper((-5/4, 1/4), (-1/4, 1/2, 5/4), -pi**2*b**4* x**4/16)/(16*x**5*gamma(-1/4)*gamma(5/4))
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.65 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {5}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{6}}{192 \, x^{5}} - \frac {\operatorname {C}\left (b x\right )}{6 \, x^{6}} \]
-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_cos(b*x)/x^6
\[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^7} \,d x \]