\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x)}{x^{10}} \, dx\) [198]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {5 b^9 \pi ^4 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} b^8 \pi ^4 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2},x\right ) \]

[Out]

-1/144*b/x^8+1/2520*b^5*Pi^2/x^4+5/2016*b^9*Pi^4*Ci(b^2*Pi*x^2)-1/144*b*cos(b^2*Pi*x^2)/x^8+67/30240*b^5*Pi^2*
cos(b^2*Pi*x^2)/x^4-1/9*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^9+1/315*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)
/x^5+1/63*b^2*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^7-1/945*b^6*Pi^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3+11
/3024*b^3*Pi*sin(b^2*Pi*x^2)/x^6-5/2016*b^7*Pi^3*sin(b^2*Pi*x^2)/x^2+1/945*b^8*Pi^4*Unintegrable(cos(1/2*b^2*P
i*x^2)*FresnelC(b*x)/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx \]

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^10,x]

[Out]

-1/144*b/x^8 + (b^5*Pi^2)/(2520*x^4) - (b*Cos[b^2*Pi*x^2])/(144*x^8) + (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(30240*x^
4) + (5*b^9*Pi^4*CosIntegral[b^2*Pi*x^2])/2016 - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(9*x^9) + (b^4*Pi^2*Cos[(
b^2*Pi*x^2)/2]*FresnelC[b*x])/(315*x^5) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(63*x^7) - (b^6*Pi^3*Fres
nelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(945*x^3) + (11*b^3*Pi*Sin[b^2*Pi*x^2])/(3024*x^6) - (5*b^7*Pi^3*Sin[b^2*Pi*x^2
])/(2016*x^2) + (b^8*Pi^4*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/945

Rubi steps \begin{align*} \text {integral}& = -\frac {b}{144 x^8}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {1}{18} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^9} \, dx-\frac {1}{9} \left (b^2 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx \\ & = -\frac {b}{144 x^8}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}+\frac {1}{36} b \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )-\frac {1}{126} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac {1}{63} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx \\ & = -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {1}{252} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac {1}{144} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )-\frac {1}{630} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx+\frac {1}{315} \left (b^6 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {\left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )}{1260}-\frac {1}{756} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{432} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {\left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx}{1890}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx \\ & = -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3780}+\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{2520}+\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1512}+\frac {1}{864} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx \\ & = -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx+\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{3780}+\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{2520}+\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1512}+\frac {1}{864} \left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right ) \\ & = -\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {5 b^9 \pi ^4 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx \]

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^10,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^10, x]

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^10,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^10,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{10}} \,d x } \]

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)/x^10,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^10, x)

Sympy [N/A]

Not integrable

Time = 65.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{10}}\, dx \]

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnelc(b*x)/x**10,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnelc(b*x)/x**10, x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{10}} \,d x } \]

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)/x^10,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^10, x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{10}} \,d x } \]

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)/x^10,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^10, x)

Mupad [N/A]

Not integrable

Time = 4.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \]

[In]

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

[Out]

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