Integrand size = 8, antiderivative size = 143 \[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\frac {\cos \left (b^2 \pi x^2\right )}{4 b^2 \pi ^2}+\frac {x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b \pi }-\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^2 \pi }+\frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2+\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi }-\frac {i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi } \] Output:
1/4*cos(b^2*Pi*x^2)/b^2/Pi^2+x*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi-1/2* FresnelC(b*x)*FresnelS(b*x)/b^2/Pi+1/2*x^2*FresnelS(b*x)^2+1/8*I*x^2*hyper geom([1, 1],[3/2, 2],-1/2*I*b^2*Pi*x^2)/Pi-1/8*I*x^2*hypergeom([1, 1],[3/2 , 2],1/2*I*b^2*Pi*x^2)/Pi
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x \operatorname {FresnelS}(b x)^2 \, dx \] Input:
Integrate[x*FresnelS[b*x]^2,x]
Output:
Integrate[x*FresnelS[b*x]^2, x]
Time = 0.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6984, 7008, 3860, 3042, 3118, 7000}
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.
\(\displaystyle \int x \operatorname {FresnelS}(b x)^2 \, dx\) |
\(\Big \downarrow \) 6984 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
\(\Big \downarrow \) 7008 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int \sin \left (b^2 \pi x^2\right )dx^2}{4 \pi b}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )\) |
\(\Big \downarrow \) 7000 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {FresnelS}(b x)^2-b \left (\frac {-\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{8} i b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )+\frac {\operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b}}{\pi b^2}-\frac {x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}\right )\) |
Input:
Int[x*FresnelS[b*x]^2,x]
Output:
(x^2*FresnelS[b*x]^2)/2 - b*(-1/4*Cos[b^2*Pi*x^2]/(b^3*Pi^2) - (x*Cos[(b^2 *Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi) + ((FresnelC[b*x]*FresnelS[b*x])/(2*b) - (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2] + (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^2*Pi ))
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Int[FresnelS[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel S[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1)) Int[x^(m + 1)*Sin[(Pi/2)*b^2*x ^2]*FresnelS[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)], x_Symbol] :> Simp[FresnelC[b*x] *(FresnelS[b*x]/(2*b)), x] + (-Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-2^(-1))*I*b^2*Pi*x^2], x] + Simp[(1/8)*I*b*x^2*HypergeometricP FQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2], x]) /; FreeQ[{b, d}, x] && EqQ[d^ 2, (Pi^2/4)*b^4]
Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x ^(m - 1))*Cos[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (Simp[(m - 1)/(2*d) Int[ x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Simp[1/(2*b*Pi) Int[x^(m - 1 )*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IG tQ[m, 1]
\[\int x \operatorname {FresnelS}\left (b x \right )^{2}d x\]
Input:
int(x*FresnelS(b*x)^2,x)
Output:
int(x*FresnelS(b*x)^2,x)
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int { x \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*fresnel_sin(b*x)^2,x, algorithm="fricas")
Output:
integral(x*fresnel_sin(b*x)^2, x)
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x S^{2}\left (b x\right )\, dx \] Input:
integrate(x*fresnels(b*x)**2,x)
Output:
Integral(x*fresnels(b*x)**2, x)
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int { x \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*fresnel_sin(b*x)^2,x, algorithm="maxima")
Output:
integrate(x*fresnel_sin(b*x)^2, x)
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int { x \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:
integrate(x*fresnel_sin(b*x)^2,x, algorithm="giac")
Output:
integrate(x*fresnel_sin(b*x)^2, x)
Timed out. \[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \] Input:
int(x*FresnelS(b*x)^2,x)
Output:
int(x*FresnelS(b*x)^2, x)
\[ \int x \operatorname {FresnelS}(b x)^2 \, dx=\int x \mathrm {FresnelS}\left (b x \right )^{2}d x \] Input:
int(x*FresnelS(b*x)^2,x)
Output:
int(x*FresnelS(b*x)^2,x)