\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx\) [157]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {853 b^9 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{181440 \sqrt {2}}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {b^9 \pi ^4 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x},x\right )}{1728} \]

[Out]

-1/504*b^2/x^7+1/5184*b^6*Pi^2/x^3-1/504*b^2*cos(b^2*Pi*x^2)/x^7+187/181440*b^6*Pi^2*cos(b^2*Pi*x^2)/x^3-1/36*
b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^8+1/864*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^4-1/9*FresnelC(b*x)
^2/x^9+1/216*b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^6-1/1728*b^7*Pi^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^
2+19/15120*b^4*Pi*sin(b^2*Pi*x^2)/x^5-853/362880*b^8*Pi^3*sin(b^2*Pi*x^2)/x+853/362880*b^9*Pi^4*FresnelC(b*x*2
^(1/2))*2^(1/2)+1/1728*b^9*Pi^4*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x,x)

Rubi [N/A]

Not integrable

Time = 0.23 (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 = {} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx \]

[In]

Int[FresnelC[b*x]^2/x^10,x]

[Out]

-1/504*b^2/x^7 + (b^6*Pi^2)/(5184*x^3) - (b^2*Cos[b^2*Pi*x^2])/(504*x^7) + (187*b^6*Pi^2*Cos[b^2*Pi*x^2])/(181
440*x^3) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(36*x^8) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(864*
x^4) - FresnelC[b*x]^2/(9*x^9) + (853*b^9*Pi^4*FresnelC[Sqrt[2]*b*x])/(181440*Sqrt[2]) + (b^3*Pi*FresnelC[b*x]
*Sin[(b^2*Pi*x^2)/2])/(216*x^6) - (b^7*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(1728*x^2) + (19*b^4*Pi*Sin[b^2
*Pi*x^2])/(15120*x^5) - (853*b^8*Pi^3*Sin[b^2*Pi*x^2])/(362880*x) + (b^9*Pi^4*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*
FresnelC[b*x])/x, x])/1728

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {1}{9} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^9} \, dx \\ & = -\frac {b^2}{504 x^7}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {1}{72} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8} \, dx-\frac {1}{36} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx \\ & = -\frac {b^2}{504 x^7}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {1}{432} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{252} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{216} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {\left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1728}-\frac {\left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1080}-\frac {1}{630} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{864} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{3456}+\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{2592}+\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{1620}+\frac {1}{945} \left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {\left (b^9 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x} \, dx}{1728} \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {\left (b^9 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x} \, dx}{1728}+\frac {\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1728}+\frac {\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1296}+\frac {1}{810} \left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac {1}{945} \left (2 b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {67 b^9 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{25920 \sqrt {2}}+\frac {1}{945} \sqrt {2} b^9 \pi ^4 \operatorname {FresnelC}\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 )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {\left (b^9 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x} \, dx}{1728} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx \]

[In]

Integrate[FresnelC[b*x]^2/x^10,x]

[Out]

Integrate[FresnelC[b*x]^2/x^10, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{10}}d x\]

[In]

int(FresnelC(b*x)^2/x^10,x)

[Out]

int(FresnelC(b*x)^2/x^10,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^10,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)^2/x^10, x)

Sympy [N/A]

Not integrable

Time = 2.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{10}}\, dx \]

[In]

integrate(fresnelc(b*x)**2/x**10,x)

[Out]

Integral(fresnelc(b*x)**2/x**10, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^10,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)^2/x^10, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)^2/x^10,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)^2/x^10, x)

Mupad [N/A]

Not integrable

Time = 4.72 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^{10}} \,d x \]

[In]

int(FresnelC(b*x)^2/x^10,x)

[Out]

int(FresnelC(b*x)^2/x^10, x)