\(\int x^4 \operatorname {FresnelS}(b x)^2 \, dx\) [34]

Optimal result

Integrand size = 10, antiderivative size = 177 \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {4 x^3}{15 b^2 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2+\frac {43 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3} \] Output:

4/15*x^3/b^2/Pi^2+1/10*x^3*cos(b^2*Pi*x^2)/b^2/Pi^2-16/5*cos(1/2*b^2*Pi*x^ 
2)*FresnelS(b*x)/b^5/Pi^3+2/5*x^4*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi+1 
/5*x^5*FresnelS(b*x)^2+43/40*FresnelS(2^(1/2)*b*x)*2^(1/2)/b^5/Pi^3-8/5*x^ 
2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^3/Pi^2-11/20*x*sin(b^2*Pi*x^2)/b^4/P 
i^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.77 \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {32 b^3 \pi x^3+12 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+24 b^5 \pi ^3 x^5 \operatorname {FresnelS}(b x)^2+129 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )+48 \operatorname {FresnelS}(b x) \left (\left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-4 b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )-66 b x \sin \left (b^2 \pi x^2\right )}{120 b^5 \pi ^3} \] Input:

Integrate[x^4*FresnelS[b*x]^2,x]
 

Output:

(32*b^3*Pi*x^3 + 12*b^3*Pi*x^3*Cos[b^2*Pi*x^2] + 24*b^5*Pi^3*x^5*FresnelS[ 
b*x]^2 + 129*Sqrt[2]*FresnelS[Sqrt[2]*b*x] + 48*FresnelS[b*x]*((-8 + b^4*P 
i^2*x^4)*Cos[(b^2*Pi*x^2)/2] - 4*b^2*Pi*x^2*Sin[(b^2*Pi*x^2)/2]) - 66*b*x* 
Sin[b^2*Pi*x^2])/(120*b^5*Pi^3)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.65, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6984, 7008, 3866, 3867, 3832, 7016, 3872, 15, 3867, 3832, 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.

\(\displaystyle \int x^4 \operatorname {FresnelS}(b x)^2 \, dx\)

\(\Big \downarrow \) 6984

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx\)

\(\Big \downarrow \) 7008

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^4 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 7016

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3872

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {\int x^2dx}{2}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^3}{6}-\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \left (-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 7006

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (\frac {4 \left (-\frac {2 \left (\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {2}{5} b \left (-\frac {x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}+\frac {4 \left (\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {2 \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}\right )\)

Input:

Int[x^4*FresnelS[b*x]^2,x]
 

Output:

(x^5*FresnelS[b*x]^2)/5 - (2*b*(-((x^4*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/ 
(b^2*Pi)) + (-1/2*(x^3*Cos[b^2*Pi*x^2])/(b^2*Pi) + (3*(-1/2*FresnelS[Sqrt[ 
2]*b*x]/(Sqrt[2]*b^3*Pi) + (x*Sin[b^2*Pi*x^2])/(2*b^2*Pi)))/(2*b^2*Pi))/(2 
*b*Pi) + (4*((-2*(-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi)) + Fresne 
lS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)))/(b^2*Pi) + (x^2*FresnelS[b*x]*Sin[(b^ 
2*Pi*x^2)/2])/(b^2*Pi) - (x^3/6 + (FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^3*Pi 
) - (x*Sin[b^2*Pi*x^2])/(2*b^2*Pi))/2)/(b*Pi)))/(b^2*Pi)))/5
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 3832

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]
 

rule 3866

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^ 
(n - 1))*(e*x)^(m - n + 1)*(Cos[c + d*x^n]/(d*n)), x] + Simp[e^n*((m - n + 
1)/(d*n))   Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] 
 && IGtQ[n, 0] && LtQ[n, m + 1]
 

rule 3867

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n 
 - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ 
(d*n))   Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && 
 IGtQ[n, 0] && LtQ[n, m + 1]
 

rule 3872

Int[(x_)^(m_.)*Sin[(a_.) + ((b_.)*(x_)^(n_))/2]^2, x_Symbol] :> Simp[1/2 
Int[x^m, x], x] - Simp[1/2   Int[x^m*Cos[2*a + b*x^n], x], x] /; FreeQ[{a, 
b, m, n}, x]
 

rule 6984

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]
 

rule 7006

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]
 

rule 7008

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]
 

rule 7016

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m - 1)*Sin[d*x^2]*(FresnelS[b*x]/(2*d)), x] + (-Simp[1/(Pi*b)   Int[x^(m - 
1)*Sin[d*x^2]^2, x], x] - Simp[(m - 1)/(2*d)   Int[x^(m - 2)*Sin[d*x^2]*Fre 
snelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m 
, 1]
 

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{5} x^{5}}{5}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi ^{2}}}{\pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}-\frac {4 \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{5 \pi ^{2}}-\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{5 \pi ^{3}}}{b^{5}}\)	\(208\)
default	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{5} x^{5}}{5}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi ^{2}}}{\pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}-\frac {4 \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{5 \pi ^{2}}-\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{5 \pi ^{3}}}{b^{5}}\)	\(208\)

Input:

int(x^4*FresnelS(b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b^5*(1/5*FresnelS(b*x)^2*b^5*x^5-2*FresnelS(b*x)*(-1/5/Pi*b^4*x^4*cos(1/ 
2*b^2*Pi*x^2)+4/5/Pi*(1/Pi*b^2*x^2*sin(1/2*b^2*Pi*x^2)+2/Pi^2*cos(1/2*b^2* 
Pi*x^2)))+4/15/Pi^2*b^3*x^3-4/5/Pi^2*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^ 
(1/2)*FresnelS(2^(1/2)*b*x))-1/5/Pi^3*(-1/2*Pi*b^3*x^3*cos(b^2*Pi*x^2)+3/2 
*Pi*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2)*FresnelS(2^(1/2)*b*x))-4*2^ 
(1/2)*FresnelS(2^(1/2)*b*x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.84 \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {24 \, \pi ^{3} b^{6} x^{5} \operatorname {S}\left (b x\right )^{2} + 24 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 20 \, \pi b^{4} x^{3} + 48 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 129 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 12 \, {\left (16 \, \pi b^{3} x^{2} \operatorname {S}\left (b x\right ) + 11 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, \pi ^{3} b^{6}} \] Input:

integrate(x^4*fresnel_sin(b*x)^2,x, algorithm="fricas")
 

Output:

1/120*(24*pi^3*b^6*x^5*fresnel_sin(b*x)^2 + 24*pi*b^4*x^3*cos(1/2*pi*b^2*x 
^2)^2 + 20*pi*b^4*x^3 + 48*(pi^2*b^5*x^4 - 8*b)*cos(1/2*pi*b^2*x^2)*fresne 
l_sin(b*x) + 129*sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x) - 12*( 
16*pi*b^3*x^2*fresnel_sin(b*x) + 11*b^2*x*cos(1/2*pi*b^2*x^2))*sin(1/2*pi* 
b^2*x^2))/(pi^3*b^6)
 

Sympy [F]

\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{4} S^{2}\left (b x\right )\, dx \] Input:

integrate(x**4*fresnels(b*x)**2,x)
 

Output:

Integral(x**4*fresnels(b*x)**2, x)
 

Maxima [F]

\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{4} \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^4*fresnel_sin(b*x)^2,x, algorithm="maxima")
 

Output:

integrate(x^4*fresnel_sin(b*x)^2, x)
 

Giac [F]

\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{4} \operatorname {S}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^4*fresnel_sin(b*x)^2,x, algorithm="giac")
 

Output:

integrate(x^4*fresnel_sin(b*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int x^4\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \] Input:

int(x^4*FresnelS(b*x)^2,x)
 

Output:

int(x^4*FresnelS(b*x)^2, x)
 

Reduce [F]

\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{4} \mathrm {FresnelS}\left (b x \right )^{2}d x \] Input:

int(x^4*FresnelS(b*x)^2,x)
 

Output:

int(x^4*FresnelS(b*x)^2,x)