\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx\) [155]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {2}{315} \sqrt {2} b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} b^7 \pi ^3 \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

[Out]

-1/210*b^2/x^5+1/336*b^6*Pi^2/x-1/210*b^2*cos(b^2*Pi*x^2)/x^5+67/5040*b^6*Pi^2*cos(b^2*Pi*x^2)/x-1/21*b*cos(1/
2*b^2*Pi*x^2)*FresnelC(b*x)/x^6+1/168*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/7*FresnelC(b*x)^2/x^7+1
/84*b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^4+13/2520*b^4*Pi*sin(b^2*Pi*x^2)/x^3+67/5040*b^7*Pi^3*FresnelS(
b*x*2^(1/2))*2^(1/2)+1/168*b^7*Pi^3*Unintegrable(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx \]

[In]

Int[FresnelC[b*x]^2/x^8,x]

[Out]

-1/210*b^2/x^5 + (b^6*Pi^2)/(336*x) - (b^2*Cos[b^2*Pi*x^2])/(210*x^5) + (67*b^6*Pi^2*Cos[b^2*Pi*x^2])/(5040*x)
- (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(21*x^6) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(168*x^2) - F
resnelC[b*x]^2/(7*x^7) + (b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/(72*Sqrt[2]) + (2*Sqrt[2]*b^7*Pi^3*FresnelS[Sqrt[2]*
b*x])/315 + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(84*x^4) + (13*b^4*Pi*Sin[b^2*Pi*x^2])/(2520*x^3) + (b^
7*Pi^3*Defer[Int][(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/168

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {1}{7} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx \\ & = -\frac {b^2}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {1}{42} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{21} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \\ & = -\frac {b^2}{210 x^5}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}-\frac {1}{168} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{105} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{84} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3} \, dx \\ & = -\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac {1}{336} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{252} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{315} \left (2 b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac {1}{168} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{126} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{315} \left (4 b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {2}{315} \sqrt {2} b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx \]

[In]

Integrate[FresnelC[b*x]^2/x^8,x]

[Out]

Integrate[FresnelC[b*x]^2/x^8, x]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{8}}d x\]

[In]

int(FresnelC(b*x)^2/x^8,x)

[Out]

int(FresnelC(b*x)^2/x^8,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^8,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)^2/x^8, x)

Sympy [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{8}}\, dx \]

[In]

integrate(fresnelc(b*x)**2/x**8,x)

[Out]

Integral(fresnelc(b*x)**2/x**8, x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^8,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)^2/x^8, x)

Giac [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^8,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)^2/x^8, x)

Mupad [N/A]

Not integrable

Time = 5.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^8} \,d x \]

[In]

int(FresnelC(b*x)^2/x^8,x)

[Out]

int(FresnelC(b*x)^2/x^8, x)