Integrand size = 6, antiderivative size = 55 \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+x \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi } \] Output:
2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi+x*FresnelS(b*x)^2-1/2*FresnelS(2^ (1/2)*b*x)*2^(1/2)/b/Pi
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }+x \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi } \] Input:
Integrate[FresnelS[b*x]^2,x]
Output:
(2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b*Pi) + x*FresnelS[b*x]^2 - Fresnel S[Sqrt[2]*b*x]/(Sqrt[2]*b*Pi)
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6974, 27, 7006, 3832}
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.
\(\displaystyle \int \operatorname {FresnelS}(b x)^2 \, dx\) |
\(\Big \downarrow \) 6974 |
\(\displaystyle x \operatorname {FresnelS}(b x)^2-2 \int b x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \operatorname {FresnelS}(b x)^2-2 b \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
\(\Big \downarrow \) 7006 |
\(\displaystyle x \operatorname {FresnelS}(b x)^2-2 b \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle x \operatorname {FresnelS}(b x)^2-2 b \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
Input:
Int[FresnelS[b*x]^2,x]
Output:
x*FresnelS[b*x]^2 - 2*b*(-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi)) + FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[FresnelS[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(FresnelS[a + b*x]^2/b), x] - Simp[2 Int[(a + b*x)*Sin[(Pi/2)*(a + b*x)^2]*FresnelS[ a + b*x], x], x] /; FreeQ[{a, b}, x]
Int[FresnelS[(b_.)*(x_)]*(x_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-Cos[d* x^2])*(FresnelS[b*x]/(2*d)), x] + Simp[1/(2*b*Pi) Int[Sin[2*d*x^2], x], x ] /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]
Time = 0.69 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\operatorname {FresnelS}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {FresnelS}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{2 \pi }}{b}\) | \(49\) |
default | \(\frac {\operatorname {FresnelS}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {FresnelS}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{2 \pi }}{b}\) | \(49\) |
Input:
int(FresnelS(b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/b*(FresnelS(b*x)^2*b*x+2*FresnelS(b*x)/Pi*cos(1/2*b^2*Pi*x^2)-1/2/Pi*2^( 1/2)*FresnelS(2^(1/2)*b*x))
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 \, \pi b^{2} x \operatorname {S}\left (b x\right )^{2} + 4 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{2 \, \pi b^{2}} \] Input:
integrate(fresnel_sin(b*x)^2,x, algorithm="fricas")
Output:
1/2*(2*pi*b^2*x*fresnel_sin(b*x)^2 + 4*b*cos(1/2*pi*b^2*x^2)*fresnel_sin(b *x) - sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x))/(pi*b^2)
\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int S^{2}\left (b x\right )\, dx \] Input:
integrate(fresnels(b*x)**2,x)
Output:
Integral(fresnels(b*x)**2, x)
\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int { \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:
integrate(fresnel_sin(b*x)^2,x, algorithm="maxima")
Output:
integrate(fresnel_sin(b*x)^2, x)
\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int { \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:
integrate(fresnel_sin(b*x)^2,x, algorithm="giac")
Output:
integrate(fresnel_sin(b*x)^2, x)
Timed out. \[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int {\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \] Input:
int(FresnelS(b*x)^2,x)
Output:
int(FresnelS(b*x)^2, x)
\[ \int \operatorname {FresnelS}(b x)^2 \, dx=\int \mathrm {FresnelS}\left (b x \right )^{2}d x \] Input:
int(FresnelS(b*x)^2,x)
Output:
int(FresnelS(b*x)^2,x)