\(\int x^6 \operatorname {FresnelS}(b x)^2 \, dx\) [32]

Optimal result

Integrand size = 10, antiderivative size = 239 \[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=-\frac {48 x}{7 b^6 \pi ^4}+\frac {6 x^5}{35 b^2 \pi ^2}-\frac {21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac {531 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{56 \sqrt {2} b^7 \pi ^4}-\frac {48 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b^5 \pi ^3}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2+\frac {96 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3} \]

[Out]

-48/7*x/b^6/Pi^4+6/35*x^5/b^2/Pi^2-21/8*x*cos(b^2*Pi*x^2)/b^6/Pi^4+1/14*x^5*cos(b^2*Pi*x^2)/b^2/Pi^2-48/7*x^2*
cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b^5/Pi^3+2/7*x^6*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/b/Pi+1/7*x^7*FresnelS(b*x
)^2+96/7*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^7/Pi^4-12/7*x^4*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/b^3/Pi^2-17/28*
x^3*sin(b^2*Pi*x^2)/b^4/Pi^3+531/112*FresnelC(b*x*2^(1/2))/b^7/Pi^4*2^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6565, 6589, 6597, 3472, 30, 3467, 3466, 3433, 6595, 3438} \[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\frac {531 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{56 \sqrt {2} \pi ^4 b^7}-\frac {48 x}{7 \pi ^4 b^6}+\frac {2 x^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac {6 x^5}{35 \pi ^2 b^2}+\frac {x^5 \cos \left (\pi b^2 x^2\right )}{14 \pi ^2 b^2}+\frac {96 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}-\frac {21 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^4 b^6}-\frac {48 x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac {17 x^3 \sin \left (\pi b^2 x^2\right )}{28 \pi ^3 b^4}-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2 \]

[In]

Int[x^6*FresnelS[b*x]^2,x]

[Out]

(-48*x)/(7*b^6*Pi^4) + (6*x^5)/(35*b^2*Pi^2) - (21*x*Cos[b^2*Pi*x^2])/(8*b^6*Pi^4) + (x^5*Cos[b^2*Pi*x^2])/(14
*b^2*Pi^2) + (531*FresnelC[Sqrt[2]*b*x])/(56*Sqrt[2]*b^7*Pi^4) - (48*x^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(7
*b^5*Pi^3) + (2*x^6*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(7*b*Pi) + (x^7*FresnelS[b*x]^2)/7 + (96*FresnelS[b*x]*
Sin[(b^2*Pi*x^2)/2])/(7*b^7*Pi^4) - (12*x^4*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*b^3*Pi^2) - (17*x^3*Sin[b^2*
Pi*x^2])/(28*b^4*Pi^3)

Rule 30

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

Rule 3433

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 3438

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a +
b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]

Rule 3466

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] + Dist[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 3467

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] - Dist[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 3472

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

Rule 6565

Int[FresnelS[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(FresnelS[b*x]^2/(m + 1)), x] - Dist[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 6589

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] + (Dist[(m - 1)/(2*d), Int[x^(m - 2)*Cos[d*x^2]*FresnelS[b*x], x], x] + Dist[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] && IGtQ[m, 1]

Rule 6595

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

Rule 6597

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] + (-Dist[1/(Pi*b), Int[x^(m - 1)*Sin[d*x^2]^2, x], x] - Dist[(m - 1)/(2*d), Int[x^(m - 2)*Sin[d*x^2]*
FresnelS[b*x], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2-\frac {1}{7} (2 b) \int x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2-\frac {\int x^6 \sin \left (b^2 \pi x^2\right ) \, dx}{7 \pi }-\frac {12 \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{7 b \pi } \\ & = \frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {48 \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{7 b^3 \pi ^2}-\frac {5 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{14 b^2 \pi ^2}+\frac {12 \int x^4 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{7 b^2 \pi ^2} \\ & = \frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}-\frac {48 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b^5 \pi ^3}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac {5 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac {96 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{7 b^5 \pi ^3}+\frac {15 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{28 b^4 \pi ^3}+\frac {24 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{7 b^4 \pi ^3}+\frac {6 \int x^4 \, dx}{7 b^2 \pi ^2}-\frac {6 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{7 b^2 \pi ^2} \\ & = \frac {6 x^5}{35 b^2 \pi ^2}-\frac {111 x \cos \left (b^2 \pi x^2\right )}{56 b^6 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}-\frac {48 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b^5 \pi ^3}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2+\frac {96 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac {15 \int \cos \left (b^2 \pi x^2\right ) \, dx}{56 b^6 \pi ^4}+\frac {12 \int \cos \left (b^2 \pi x^2\right ) \, dx}{7 b^6 \pi ^4}-\frac {96 \int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{7 b^6 \pi ^4}+\frac {9 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{7 b^4 \pi ^3} \\ & = \frac {6 x^5}{35 b^2 \pi ^2}-\frac {21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac {15 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{56 \sqrt {2} b^7 \pi ^4}+\frac {6 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{7 b^7 \pi ^4}-\frac {48 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b^5 \pi ^3}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2+\frac {96 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac {9 \int \cos \left (b^2 \pi x^2\right ) \, dx}{14 b^6 \pi ^4}-\frac {96 \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{7 b^6 \pi ^4} \\ & = -\frac {48 x}{7 b^6 \pi ^4}+\frac {6 x^5}{35 b^2 \pi ^2}-\frac {21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac {51 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{56 \sqrt {2} b^7 \pi ^4}+\frac {6 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{7 b^7 \pi ^4}-\frac {48 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b^5 \pi ^3}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2+\frac {96 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac {48 \int \cos \left (b^2 \pi x^2\right ) \, dx}{7 b^6 \pi ^4} \\ & = -\frac {48 x}{7 b^6 \pi ^4}+\frac {6 x^5}{35 b^2 \pi ^2}-\frac {21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac {51 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{56 \sqrt {2} b^7 \pi ^4}+\frac {30 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{7 b^7 \pi ^4}-\frac {48 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b^5 \pi ^3}+\frac {2 x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{7 b \pi }+\frac {1}{7} x^7 \operatorname {FresnelS}(b x)^2+\frac {96 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {12 x^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72 \[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\frac {2655 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )+80 b^7 \pi ^4 x^7 \operatorname {FresnelS}(b x)^2+160 \operatorname {FresnelS}(b x) \left (b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-6 \left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (5 \left (-147+4 b^4 \pi ^2 x^4\right ) \cos \left (b^2 \pi x^2\right )-2 \left (960-24 b^4 \pi ^2 x^4+85 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )\right )}{560 b^7 \pi ^4} \]

[In]

Integrate[x^6*FresnelS[b*x]^2,x]

[Out]

(2655*Sqrt[2]*FresnelC[Sqrt[2]*b*x] + 80*b^7*Pi^4*x^7*FresnelS[b*x]^2 + 160*FresnelS[b*x]*(b^2*Pi*x^2*(-24 + b
^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2] - 6*(-8 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]) + 2*b*x*(5*(-147 + 4*b^4*Pi^2*x^
4)*Cos[b^2*Pi*x^2] - 2*(960 - 24*b^4*Pi^2*x^4 + 85*b^2*Pi*x^2*Sin[b^2*Pi*x^2])))/(560*b^7*Pi^4)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{7} x^{7}}{7}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{6} x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {\frac {6 b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }-\frac {24 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{7 \pi }}{\pi }\right )+\frac {\frac {6}{35} b^{5} x^{5} \pi ^{2}-\frac {48}{7} b x}{\pi ^{4}}-\frac {6 \left (\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )\right )}{7 \pi ^{4}}-\frac {-\frac {\pi \,b^{5} x^{5} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {5 \pi \left (\frac {b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {3 \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2 \pi }\right )}{2}+\frac {12 b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {6 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{\pi }}{7 \pi ^{3}}}{b^{7}}\)	\(324\)
default	\(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{7} x^{7}}{7}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{6} x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {\frac {6 b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }-\frac {24 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{7 \pi }}{\pi }\right )+\frac {\frac {6}{35} b^{5} x^{5} \pi ^{2}-\frac {48}{7} b x}{\pi ^{4}}-\frac {6 \left (\frac {\pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {3 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )\right )}{7 \pi ^{4}}-\frac {-\frac {\pi \,b^{5} x^{5} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {5 \pi \left (\frac {b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {3 \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2 \pi }\right )}{2}+\frac {12 b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {6 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{\pi }}{7 \pi ^{3}}}{b^{7}}\)	\(324\)

[In]

int(x^6*FresnelS(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/b^7*(1/7*FresnelS(b*x)^2*b^7*x^7-2*FresnelS(b*x)*(-1/7/Pi*b^6*x^6*cos(1/2*b^2*Pi*x^2)+6/7/Pi*(1/Pi*b^4*x^4*s
in(1/2*b^2*Pi*x^2)-4/Pi*(-1/Pi*b^2*x^2*cos(1/2*b^2*Pi*x^2)+2/Pi^2*sin(1/2*b^2*Pi*x^2))))+6/7/Pi^4*(1/5*b^5*x^5
*Pi^2-8*b*x)-6/7/Pi^4*(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)*Fresn
elC(b*x*2^(1/2)))-4*2^(1/2)*FresnelC(b*x*2^(1/2)))-1/7/Pi^3*(-1/2*Pi*b^5*x^5*cos(b^2*Pi*x^2)+5/2*Pi*(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(b*x*2^(1/2))))+12/Pi*b*x*cos
(b^2*Pi*x^2)-6/Pi*2^(1/2)*FresnelC(b*x*2^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.77 \[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\frac {80 \, \pi ^{4} b^{8} x^{7} \operatorname {S}\left (b x\right )^{2} + 56 \, \pi ^{2} b^{6} x^{5} - 2370 \, b^{2} x + 20 \, {\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 160 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 2655 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 40 \, {\left (17 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 24 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{560 \, \pi ^{4} b^{8}} \]

[In]

integrate(x^6*fresnel_sin(b*x)^2,x, algorithm="fricas")

[Out]

1/560*(80*pi^4*b^8*x^7*fresnel_sin(b*x)^2 + 56*pi^2*b^6*x^5 - 2370*b^2*x + 20*(4*pi^2*b^6*x^5 - 147*b^2*x)*cos
(1/2*pi*b^2*x^2)^2 + 160*(pi^3*b^7*x^6 - 24*pi*b^3*x^2)*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x) + 2655*sqrt(2)*sq
rt(b^2)*fresnel_cos(sqrt(2)*sqrt(b^2)*x) - 40*(17*pi*b^4*x^3*cos(1/2*pi*b^2*x^2) + 24*(pi^2*b^5*x^4 - 8*b)*fre
snel_sin(b*x))*sin(1/2*pi*b^2*x^2))/(pi^4*b^8)

Sympy [F]

\[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{6} S^{2}\left (b x\right )\, dx \]

[In]

integrate(x**6*fresnels(b*x)**2,x)

[Out]

Integral(x**6*fresnels(b*x)**2, x)

Maxima [F]

\[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{6} \operatorname {S}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^6*fresnel_sin(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^6*fresnel_sin(b*x)^2, x)

Giac [F]

\[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{6} \operatorname {S}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^6*fresnel_sin(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^6*fresnel_sin(b*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^6 \operatorname {FresnelS}(b x)^2 \, dx=\int x^6\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]

[In]

int(x^6*FresnelS(b*x)^2,x)

[Out]

int(x^6*FresnelS(b*x)^2, x)