\(\int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx\) [122]

Optimal result

Integrand size = 8, antiderivative size = 69 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelC}(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x} \]

[Out]

-1/12*b*cos(1/2*b^2*Pi*x^2)/x^3-1/12*b^4*Pi^2*FresnelC(b*x)-1/4*FresnelC(b*x)/x^4+1/12*b^3*Pi*sin(1/2*b^2*Pi*x
^2)/x

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3469, 3468, 3433} \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=-\frac {1}{12} \pi ^2 b^4 \operatorname {FresnelC}(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x^3}+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x}-\frac {\operatorname {FresnelC}(b x)}{4 x^4} \]

[In]

Int[FresnelC[b*x]/x^5,x]

[Out]

-1/12*(b*Cos[(b^2*Pi*x^2)/2])/x^3 - (b^4*Pi^2*FresnelC[b*x])/12 - FresnelC[b*x]/(4*x^4) + (b^3*Pi*Sin[(b^2*Pi*
x^2)/2])/(12*x)

Rule 3433

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 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)}{4 x^4}+\frac {1}{4} b \int \frac {\cos \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 )}{12 x^3}-\frac {\operatorname {FresnelC}(b x)}{4 x^4}-\frac {1}{12} \left (b^3 \pi \right ) \int \frac {\sin \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 )}{12 x^3}-\frac {\operatorname {FresnelC}(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {1}{12} \left (b^5 \pi ^2\right ) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelC}(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelC}(b x)}{4 x^4}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x} \]

[In]

Integrate[FresnelC[b*x]/x^5,x]

[Out]

-1/12*(b*Cos[(b^2*Pi*x^2)/2])/x^3 - (b^4*Pi^2*FresnelC[b*x])/12 - FresnelC[b*x]/(4*x^4) + (b^3*Pi*Sin[(b^2*Pi*
x^2)/2])/(12*x)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(b^{4} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{4 b^{4} x^{4}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{12}\right )\)	\(64\)
default	\(b^{4} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{4 b^{4} x^{4}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{12}\right )\)	\(64\)
meijerg	\(\frac {\pi ^{2} b^{4} \left (-\frac {32 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2} x^{3} b^{3}}+\frac {32 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi x b}-\frac {32 \left (x^{4} \pi ^{2} b^{4}+3\right ) \operatorname {FresnelC}\left (b x \right )}{3 \pi ^{2} x^{4} b^{4}}\right )}{128}\)	\(79\)
parts	\(-\frac {\operatorname {FresnelC}\left (b x \right )}{4 x^{4}}+\frac {b \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}+\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{4}\)	\(82\)

[In]

int(FresnelC(b*x)/x^5,x,method=_RETURNVERBOSE)

[Out]

b^4*(-1/4*FresnelC(b*x)/b^4/x^4-1/12/b^3/x^3*cos(1/2*b^2*Pi*x^2)-1/12*Pi*(-1/b/x*sin(1/2*b^2*Pi*x^2)+Pi*Fresne
lC(b*x)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=\frac {\pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )}{12 \, x^{4}} \]

[In]

integrate(fresnel_cos(b*x)/x^5,x, algorithm="fricas")

[Out]

1/12*(pi*b^3*x^3*sin(1/2*pi*b^2*x^2) - b*x*cos(1/2*pi*b^2*x^2) - (pi^2*b^4*x^4 + 3)*fresnel_cos(b*x))/x^4

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=\frac {\pi ^{2} b^{4} C\left (b x\right ) \Gamma \left (- \frac {3}{4}\right )}{64 \Gamma \left (\frac {5}{4}\right )} - \frac {\pi b^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {3}{4}\right )}{64 x \Gamma \left (\frac {5}{4}\right )} + \frac {b \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {3}{4}\right )}{64 x^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {3 C\left (b x\right ) \Gamma \left (- \frac {3}{4}\right )}{64 x^{4} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate(fresnelc(b*x)/x**5,x)

[Out]

pi**2*b**4*fresnelc(b*x)*gamma(-3/4)/(64*gamma(5/4)) - pi*b**3*sin(pi*b**2*x**2/2)*gamma(-3/4)/(64*x*gamma(5/4
)) + b*cos(pi*b**2*x**2/2)*gamma(-3/4)/(64*x**3*gamma(5/4)) + 3*fresnelc(b*x)*gamma(-3/4)/(64*x**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 = 0.88 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {3}{2}} {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{4}}{64 \, x^{3}} - \frac {\operatorname {C}\left (b x\right )}{4 \, x^{4}} \]

[In]

integrate(fresnel_cos(b*x)/x^5,x, algorithm="maxima")

[Out]

-1/64*sqrt(1/2)*(pi*x^2)^(3/2)*((I - 1)*sqrt(2)*gamma(-3/2, 1/2*I*pi*b^2*x^2) - (I + 1)*sqrt(2)*gamma(-3/2, -1
/2*I*pi*b^2*x^2))*b^4/x^3 - 1/4*fresnel_cos(b*x)/x^4

Giac [F]

\[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{5}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)/x^5,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^5} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^5} \,d x \]

[In]

int(FresnelC(b*x)/x^5,x)

[Out]

int(FresnelC(b*x)/x^5, x)