\(\int x^5 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x) \, dx\) [94]

Optimal result

Integrand size = 20, antiderivative size = 166 \[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {4 x}{b^5 \pi ^3}-\frac {x^5}{10 b \pi }+\frac {11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {43 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {8 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Output:

4*x/b^5/Pi^3-1/10*x^5/b/Pi+11/8*x*cos(b^2*Pi*x^2)/b^5/Pi^3-43/16*FresnelC( 
2^(1/2)*b*x)*2^(1/2)/b^6/Pi^3+4*x^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^4/ 
Pi^2-8*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^4*FresnelS(b*x)*sin(1/ 
2*b^2*Pi*x^2)/b^2/Pi+1/4*x^3*sin(b^2*Pi*x^2)/b^3/Pi^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.76 \[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {-215 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )+80 \operatorname {FresnelS}(b x) \left (4 b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (160-4 b^4 \pi ^2 x^4+55 \cos \left (b^2 \pi x^2\right )+10 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )}{80 b^6 \pi ^3} \] Input:

Integrate[x^5*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
 

Output:

(-215*Sqrt[2]*FresnelC[Sqrt[2]*b*x] + 80*FresnelS[b*x]*(4*b^2*Pi*x^2*Cos[( 
b^2*Pi*x^2)/2] + (-8 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]) + 2*b*x*(160 - 4 
*b^4*Pi^2*x^4 + 55*Cos[b^2*Pi*x^2] + 10*b^2*Pi*x^2*Sin[b^2*Pi*x^2]))/(80*b 
^6*Pi^3)
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.70, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {7016, 3872, 15, 3867, 3866, 3833, 7008, 3866, 3833, 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.

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

\(\Big \downarrow \) 7016

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

\(\Big \downarrow \) 3872

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {x^5}{10}-\frac {1}{2} \int x^4 \cos \left (b^2 \pi x^2\right )dx}{\pi b}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3867

\(\displaystyle -\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \int x^2 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3866

\(\displaystyle -\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\int \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {4 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

\(\Big \downarrow \) 7008

\(\displaystyle -\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\int x^2 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

\(\Big \downarrow \) 3866

\(\displaystyle -\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}+\frac {\frac {\int \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)dx}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

\(\Big \downarrow \) 7014

\(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

\(\Big \downarrow \) 3838

\(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right )dx}{\pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {x}{2}-\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b}}{\pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )}{\pi b^2}-\frac {\frac {1}{2} \left (\frac {3 \left (\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}-\frac {x \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}\right )+\frac {x^5}{10}}{\pi b}\)

Input:

Int[x^5*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x],x]
 

Output:

(x^4*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (4*((-1/2*(x*Cos[b^2*Pi 
*x^2])/(b^2*Pi) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^3*Pi))/(2*b*Pi) - (x^ 
2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(b^2*Pi) + (2*(-((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)))/(b^2*Pi)))/(b^2*Pi) - (x^5/10 + ((3*(-1/2*(x*Cos[b^2*Pi*x^2])/(b^2* 
Pi) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^3*Pi)))/(2*b^2*Pi) - (x^3*Sin[b^2 
*Pi*x^2])/(2*b^2*Pi))/2)/(b*Pi)
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3838

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]
 

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 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 7014

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]
 

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 = 3.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.28

method	result	size
default	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{\pi }\right )}{b^{5}}-\frac {\frac {\frac {1}{5} \pi ^{2} b^{5} x^{5}-8 b x}{2 \pi ^{3}}+\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{2 \pi }}{\pi ^{2}}-\frac {\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelC}\left (\sqrt {2}\, b x \right )}{2 \pi ^{3}}}{b^{5}}}{b}\)	\(212\)

Input:

int(x^5*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x,method=_RETURNVERBOSE)
 

Output:

(FresnelS(b*x)/b^5*(1/Pi*b^4*x^4*sin(1/2*b^2*Pi*x^2)-4/Pi*(-1/Pi*b^2*x^2*c 
os(1/2*b^2*Pi*x^2)+2/Pi^2*sin(1/2*b^2*Pi*x^2)))-1/b^5*(1/2/Pi^3*(1/5*Pi^2* 
b^5*x^5-8*b*x)+2/Pi^2*(-1/2/Pi*b*x*cos(b^2*Pi*x^2)+1/4/Pi*2^(1/2)*FresnelC 
(2^(1/2)*b*x))-1/2/Pi^3*(1/2*Pi*b^3*x^3*sin(b^2*Pi*x^2)-3/2*Pi*(-1/2/Pi*b* 
x*cos(b^2*Pi*x^2)+1/4/Pi*2^(1/2)*FresnelC(2^(1/2)*b*x))-4*2^(1/2)*FresnelC 
(2^(1/2)*b*x))))/b
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.84 \[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {8 \, \pi ^{2} b^{6} x^{5} - 320 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 220 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 210 \, b^{2} x + 215 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 40 \, {\left (\pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{80 \, \pi ^{3} b^{7}} \] Input:

integrate(x^5*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="fricas")
 

Output:

-1/80*(8*pi^2*b^6*x^5 - 320*pi*b^3*x^2*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x 
) - 220*b^2*x*cos(1/2*pi*b^2*x^2)^2 - 210*b^2*x + 215*sqrt(2)*sqrt(b^2)*fr 
esnel_cos(sqrt(2)*sqrt(b^2)*x) - 40*(pi*b^4*x^3*cos(1/2*pi*b^2*x^2) + 2*(p 
i^2*b^5*x^4 - 8*b)*fresnel_sin(b*x))*sin(1/2*pi*b^2*x^2))/(pi^3*b^7)
 

Sympy [F]

\[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \] Input:

integrate(x**5*cos(1/2*b**2*pi*x**2)*fresnels(b*x),x)
 

Output:

Integral(x**5*cos(pi*b**2*x**2/2)*fresnels(b*x), x)
 

Maxima [F]

\[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{5} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(x^5*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="maxima")
 

Output:

integrate(x^5*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
 

Giac [F]

\[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{5} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \] Input:

integrate(x^5*cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x),x, algorithm="giac")
 

Output:

integrate(x^5*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^5\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:

int(x^5*FresnelS(b*x)*cos((Pi*b^2*x^2)/2),x)
 

Output:

int(x^5*FresnelS(b*x)*cos((Pi*b^2*x^2)/2), x)
 

Reduce [F]

\[ \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{5} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )d x \] Input:

int(x^5*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)
 

Output:

int(x^5*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x),x)