\(\int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx\) [124]

Optimal result

Integrand size = 8, antiderivative size = 94 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {\operatorname {FresnelC}(b x)}{6 x^6}+\frac {1}{90} b^6 \pi ^3 \operatorname {FresnelS}(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3} \]

[Out]

-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*FresnelC(b*x)/x^6+1/90*b^6*Pi^3*Fresne
lS(b*x)+1/90*b^3*Pi*sin(1/2*b^2*Pi*x^2)/x^3

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3469, 3468, 3432} \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {1}{90} \pi ^3 b^6 \operatorname {FresnelS}(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{30 x^5}+\frac {\pi ^2 b^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x}+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x^3}-\frac {\operatorname {FresnelC}(b x)}{6 x^6} \]

[In]

Int[FresnelC[b*x]/x^7,x]

[Out]

-1/30*(b*Cos[(b^2*Pi*x^2)/2])/x^5 + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2])/(90*x) - FresnelC[b*x]/(6*x^6) + (b^6*Pi^3*
FresnelS[b*x])/90 + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(90*x^3)

Rule 3432

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]

Rule 3468

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] - Dist[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 3469

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] + Dist[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 6562

Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*(FresnelC[b*x]/(d*(m + 1))), x] -
Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)}{6 x^6}+\frac {1}{6} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx \\ & = -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}-\frac {\operatorname {FresnelC}(b x)}{6 x^6}-\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}-\frac {\operatorname {FresnelC}(b x)}{6 x^6}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {1}{90} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, 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 {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}+\frac {1}{90} \left (b^7 \pi ^3\right ) \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.79 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {1}{90} \left (\frac {b \left (-3+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}-\frac {15 \operatorname {FresnelC}(b x)}{x^6}+b^6 \pi ^3 \operatorname {FresnelS}(b x)+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}\right ) \]

[In]

Integrate[FresnelC[b*x]/x^7,x]

[Out]

((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*S
in[(b^2*Pi*x^2)/2])/x^3)/90

Maple [C] (verified)

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	\(-\frac {\operatorname {FresnelC}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 x^{5}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}-\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{5}\right )}{6}\)	\(105\)

[In]

int(FresnelC(b*x)/x^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {\pi ^{3} \sqrt {b^{2}} b^{5} x^{6} \operatorname {S}\left (\sqrt {b^{2}} x\right ) + \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{2} b^{5} x^{5} - 3 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 15 \, \operatorname {C}\left (b x\right )}{90 \, x^{6}} \]

[In]

integrate(fresnel_cos(b*x)/x^7,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\frac {b \Gamma \left (- \frac {5}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{4} \\ - \frac {1}{4}, \frac {1}{2}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 x^{5} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate(fresnelc(b*x)/x**7,x)

[Out]

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))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.65 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {5}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{6}}{192 \, x^{5}} - \frac {\operatorname {C}\left (b x\right )}{6 \, x^{6}} \]

[In]

integrate(fresnel_cos(b*x)/x^7,x, algorithm="maxima")

[Out]

-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*fresnel_cos(b*x)/x^6

Giac [F]

\[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{7}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)/x^7,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)/x^7, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^7} \,d x \]

[In]

int(FresnelC(b*x)/x^7,x)

[Out]

int(FresnelC(b*x)/x^7, x)