Integrand size = 18, antiderivative size = 59 \[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {x}{2 b \pi }+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2 \pi }+\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi } \] Output:
-1/2*x/b/Pi+1/4*FresnelC(2^(1/2)*b*x)*2^(1/2)/b^2/Pi+FresnelS(b*x)*sin(1/2 *b^2*Pi*x^2)/b^2/Pi
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81 \[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {-2 b x+\sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )+4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^2 \pi } \] Input:
Integrate[x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
Output:
(-2*b*x + Sqrt[2]*FresnelC[Sqrt[2]*b*x] + 4*FresnelS[b*x]*Sin[(b^2*Pi*x^2) /2])/(4*b^2*Pi)
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {7014, 3838, 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.
\(\displaystyle \int x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 7014 |
\(\displaystyle \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}\) |
\(\Big \downarrow \) 3838 |
\(\displaystyle \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right )dx}{\pi b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {x}{2}-\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b}}{\pi b}\) |
Input:
Int[x*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
Output:
-((x/2 - FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b))/(b*Pi)) + (FresnelS[b*x]*Sin [(b^2*Pi*x^2)/2])/(b^2*Pi)
Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Sy mbol] :> Int[ExpandTrigReduce[(a + b*Sin[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[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_), x_Symbol] :> Simp[Sin[d*x^ 2]*(FresnelS[b*x]/(2*d)), x] - Simp[1/(Pi*b) Int[Sin[d*x^2]^2, x], x] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]
Time = 1.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b \pi }-\frac {\frac {b x}{2}-\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4}}{b \pi }}{b}\) | \(52\) |
Input:
int(x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x,method=_RETURNVERBOSE)
Output:
(FresnelS(b*x)/b/Pi*sin(1/2*b^2*Pi*x^2)-1/b/Pi*(1/2*b*x-1/4*2^(1/2)*Fresne lC(2^(1/2)*b*x)))/b
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {2 \, b^{2} x - 4 \, b \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, \pi b^{3}} \] Input:
integrate(x*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")
Output:
-1/4*(2*b^2*x - 4*b*fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2) - sqrt(2)*sqrt(b^ 2)*fresnel_cos(sqrt(2)*sqrt(b^2)*x))/(pi*b^3)
\[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \] Input:
integrate(x*cos(1/2*b**2*pi*x**2)*fresnels(b*x),x)
Output:
Integral(x*cos(pi*b**2*x**2/2)*fresnels(b*x), x)
\[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:
integrate(x*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="maxima")
Output:
integrate(x*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
\[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:
integrate(x*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="giac")
Output:
integrate(x*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
Timed out. \[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:
int(x*FresnelS(b*x)*cos((Pi*b^2*x^2)/2),x)
Output:
int(x*FresnelS(b*x)*cos((Pi*b^2*x^2)/2), x)
\[ \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )d x \] Input:
int(x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
Output:
int(x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)