\(\int x^8 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x) \, dx\) [180]

Optimal result

Integrand size = 20, antiderivative size = 231 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \]

[Out]

105/4*x^2/b^7/Pi^4-7/12*x^6/b^3/Pi^2-55/4*x^2*cos(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^6*cos(b^2*Pi*x^2)/b^3/Pi^2-105*x*
cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^8/Pi^4+7*x^5*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^4/Pi^2+105/2*FresnelC(b*x
)^2/b^9/Pi^4-35*x^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^6/Pi^3+x^7*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^2/Pi+40
*sin(b^2*Pi*x^2)/b^9/Pi^5-5/2*x^4*sin(b^2*Pi*x^2)/b^5/Pi^3

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6590, 6598, 6576, 30, 3461, 2714, 3460, 3377, 2717, 3390} \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {105 \operatorname {FresnelC}(b x)^2}{2 \pi ^4 b^9}+\frac {105 x^2}{4 \pi ^4 b^7}-\frac {7 x^6}{12 \pi ^2 b^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]

[In]

Int[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x],x]

[Out]

(105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Cos[b^2*Pi*x^2])
/(4*b^3*Pi^2) - (105*x*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x
])/(b^4*Pi^2) + (105*FresnelC[b*x]^2)/(2*b^9*Pi^4) - (35*x^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (
x^7*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) + (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(2
*b^5*Pi^3)

Rule 30

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

Rule 2714

Int[sin[(c_.) + ((d_.)*(x_))/2]^2, x_Symbol] :> Simp[x/2, x] - Simp[Sin[2*c + d*x]/(2*d), x] /; FreeQ[{c, d},
x]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3390

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + ((f_.)*(x_))/2]^2, x_Symbol] :> Dist[1/2, Int[(c + d*x)^m, x], x] -
Dist[1/2, Int[(c + d*x)^m*Cos[2*e + f*x], x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 3460

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 6576

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[Pi*(b/(2*d)), Subst[Int[x^n, x], x, Fresne
lC[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2/4)*b^4]

Rule 6590

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*Sin[d*x^2]*(FresnelC[b*x]/(2
*d)), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*Sin[d*x^2]*FresnelC[b*x], x], x] - Dist[b/(4*d), 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 6598

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 \int x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi } \\ & = \frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {35 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx}{b^4 \pi ^2}-\frac {7 \int x^5 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\text {Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi } \\ & = \frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {105 \int x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}+\frac {35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}-\frac {3 \text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {7 \text {Subst}\left (\int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2} \\ & = \frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 x^4 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {105 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx}{b^8 \pi ^4}+\frac {105 \int x \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}+\frac {3 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac {35 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {7 \text {Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {7 \text {Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2} \\ & = -\frac {7 x^6}{12 b^3 \pi ^2}-\frac {41 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {105 \text {Subst}(\int x \, dx,x,\operatorname {FresnelC}(b x))}{b^9 \pi ^4}+\frac {3 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {35 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}+\frac {105 \text {Subst}\left (\int \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {7 \text {Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3} \\ & = \frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {73 \sin \left (b^2 \pi x^2\right )}{2 b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {7 \text {Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4} \\ & = \frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {105 x^2}{4 b^7 \pi ^4}-\frac {7 x^6}{12 b^3 \pi ^2}-\frac {55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x)^2}{2 b^9 \pi ^4}-\frac {35 x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}-\frac {5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3} \]

[In]

Integrate[x^8*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x],x]

[Out]

(105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Cos[b^2*Pi*x^2])
/(4*b^3*Pi^2) - (105*x*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x
])/(b^4*Pi^2) + (105*FresnelC[b*x]^2)/(2*b^9*Pi^4) - (35*x^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (
x^7*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) + (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(2
*b^5*Pi^3)

Maple [F]

\[\int x^{8} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelC}\left (b x \right )d x\]

[In]

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x),x)

[Out]

int(x^8*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x),x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.73 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {5 \, \pi ^{3} b^{6} x^{6} - 240 \, \pi b^{2} x^{2} - 3 \, {\left (\pi ^{3} b^{6} x^{6} - 55 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 42 \, {\left (\pi ^{3} b^{5} x^{5} - 15 \, \pi b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 315 \, \pi \operatorname {C}\left (b x\right )^{2} + 6 \, {\left (5 \, {\left (\pi ^{2} b^{4} x^{4} - 16\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{4} b^{7} x^{7} - 35 \, \pi ^{2} b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{6 \, \pi ^{5} b^{9}} \]

[In]

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="fricas")

[Out]

-1/6*(5*pi^3*b^6*x^6 - 240*pi*b^2*x^2 - 3*(pi^3*b^6*x^6 - 55*pi*b^2*x^2)*cos(1/2*pi*b^2*x^2)^2 - 42*(pi^3*b^5*
x^5 - 15*pi*b*x)*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x) - 315*pi*fresnel_cos(b*x)^2 + 6*(5*(pi^2*b^4*x^4 - 16)*c
os(1/2*pi*b^2*x^2) - (pi^4*b^7*x^7 - 35*pi^2*b^3*x^3)*fresnel_cos(b*x))*sin(1/2*pi*b^2*x^2))/(pi^5*b^9)

Sympy [A] (verification not implemented)

Time = 14.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.30 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\begin {cases} \frac {x^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi b^{2}} - \frac {5 x^{6} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{6 \pi ^{2} b^{3}} - \frac {x^{6} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{3 \pi ^{2} b^{3}} + \frac {7 x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{2} b^{4}} - \frac {5 x^{4} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{3} b^{5}} - \frac {35 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{3} b^{6}} + \frac {40 x^{2} \sin ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{4} b^{7}} + \frac {25 x^{2} \cos ^{2}{\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{4} b^{7}} - \frac {105 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{\pi ^{4} b^{8}} + \frac {80 \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )}}{\pi ^{5} b^{9}} + \frac {105 C^{2}\left (b x\right )}{2 \pi ^{4} b^{9}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**8*cos(1/2*b**2*pi*x**2)*fresnelc(b*x),x)

[Out]

Piecewise((x**7*sin(pi*b**2*x**2/2)*fresnelc(b*x)/(pi*b**2) - 5*x**6*sin(pi*b**2*x**2/2)**2/(6*pi**2*b**3) - x
**6*cos(pi*b**2*x**2/2)**2/(3*pi**2*b**3) + 7*x**5*cos(pi*b**2*x**2/2)*fresnelc(b*x)/(pi**2*b**4) - 5*x**4*sin
(pi*b**2*x**2/2)*cos(pi*b**2*x**2/2)/(pi**3*b**5) - 35*x**3*sin(pi*b**2*x**2/2)*fresnelc(b*x)/(pi**3*b**6) + 4
0*x**2*sin(pi*b**2*x**2/2)**2/(pi**4*b**7) + 25*x**2*cos(pi*b**2*x**2/2)**2/(2*pi**4*b**7) - 105*x*cos(pi*b**2
*x**2/2)*fresnelc(b*x)/(pi**4*b**8) + 80*sin(pi*b**2*x**2/2)*cos(pi*b**2*x**2/2)/(pi**5*b**9) + 105*fresnelc(b
*x)**2/(2*pi**4*b**9), Ne(b, 0)), (0, True))

Maxima [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="maxima")

[Out]

integrate(x^8*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)

Giac [F]

\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

integrate(x^8*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="giac")

[Out]

integrate(x^8*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^8\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]

[In]

int(x^8*FresnelC(b*x)*cos((Pi*b^2*x^2)/2),x)

[Out]

int(x^8*FresnelC(b*x)*cos((Pi*b^2*x^2)/2), x)