Integrand size = 18, antiderivative size = 60 \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {x}{2 b \pi }-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi } \] Output:
1/2*x/b/Pi-cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^2/Pi+1/4*FresnelC(2^(1/2)*b *x)*2^(1/2)/b^2/Pi
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80 \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {2 b x-4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)+\sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 b^2 \pi } \] Input:
Integrate[x*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2],x]
Output:
(2*b*x - 4*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x] + Sqrt[2]*FresnelC[Sqrt[2]*b* x])/(4*b^2*Pi)
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {7015, 3839, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 7015 |
\(\displaystyle \frac {\int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 3839 |
\(\displaystyle \frac {\int \left (\frac {1}{2} \cos \left (b^2 \pi x^2\right )+\frac {1}{2}\right )dx}{\pi b}-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b}+\frac {x}{2}}{\pi b}-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\) |
Input:
Int[x*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2],x]
Output:
-((Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi)) + (x/2 + FresnelC[Sqrt[2]* b*x]/(2*Sqrt[2]*b))/(b*Pi)
Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_), x_Sy mbol] :> Int[ExpandTrigReduce[(a + b*Cos[c + d*(e + f*x)^n])^p, x], x] /; F reeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]
Int[FresnelC[(b_.)*(x_)]*(x_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-Cos[d* x^2])*(FresnelC[b*x]/(2*d)), x] + Simp[b/(2*d) Int[Cos[d*x^2]^2, x], x] / ; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]
Time = 1.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {-\frac {\operatorname {FresnelC}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b \pi }+\frac {\frac {b x}{2}+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4}}{b \pi }}{b}\) | \(52\) |
Input:
int(x*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x,method=_RETURNVERBOSE)
Output:
(-FresnelC(b*x)/b/Pi*cos(1/2*b^2*Pi*x^2)+1/b/Pi*(1/2*b*x+1/4*2^(1/2)*Fresn elC(2^(1/2)*b*x)))/b
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {2 \, b^{2} x - 4 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, \pi b^{3}} \] Input:
integrate(x*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")
Output:
1/4*(2*b^2*x - 4*b*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x) + sqrt(2)*sqrt(b^2 )*fresnel_cos(sqrt(2)*sqrt(b^2)*x))/(pi*b^3)
\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \] Input:
integrate(x*fresnelc(b*x)*sin(1/2*b**2*pi*x**2),x)
Output:
Integral(x*sin(pi*b**2*x**2/2)*fresnelc(b*x), x)
\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:
integrate(x*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="maxima")
Output:
integrate(x*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)
\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:
integrate(x*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="giac")
Output:
integrate(x*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)
Timed out. \[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:
int(x*FresnelC(b*x)*sin((Pi*b^2*x^2)/2),x)
Output:
int(x*FresnelC(b*x)*sin((Pi*b^2*x^2)/2), x)
\[ \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x \,\mathrm {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )d x \] Input:
int(x*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)
Output:
int(x*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)