\(\int x^6 \cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x) \, dx\) [182]

Optimal result

Integrand size = 20, antiderivative size = 247 \[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {5 x^4}{8 b^3 \pi ^2}-\frac {11 \cos \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {15 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^7 \pi ^3}+\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3} \]

[Out]

-5/8*x^4/b^3/Pi^2-11/2*cos(b^2*Pi*x^2)/b^7/Pi^4+1/4*x^4*cos(b^2*Pi*x^2)/b^3/Pi^2+5*x^3*cos(1/2*b^2*Pi*x^2)*Fre
snelC(b*x)/b^4/Pi^2+15/2*FresnelC(b*x)*FresnelS(b*x)/b^7/Pi^3+15/8*I*x^2*hypergeom([1, 1],[3/2, 2],-1/2*I*b^2*
Pi*x^2)/b^5/Pi^3-15/8*I*x^2*hypergeom([1, 1],[3/2, 2],1/2*I*b^2*Pi*x^2)/b^5/Pi^3-15*x*FresnelC(b*x)*sin(1/2*b^
2*Pi*x^2)/b^6/Pi^3+x^5*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^2/Pi-7/4*x^2*sin(b^2*Pi*x^2)/b^5/Pi^3

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6590, 6598, 6582, 3460, 2718, 3461, 3390, 30, 3377} \[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^5}-\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^5}+\frac {15 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 \pi ^3 b^7}-\frac {5 x^4}{8 \pi ^2 b^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {11 \cos \left (\pi b^2 x^2\right )}{2 \pi ^4 b^7}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {7 x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^3 b^5}+\frac {5 x^3 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^4 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]

[In]

Int[x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x],x]

[Out]

(-5*x^4)/(8*b^3*Pi^2) - (11*Cos[b^2*Pi*x^2])/(2*b^7*Pi^4) + (x^4*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (5*x^3*Cos[(b
^2*Pi*x^2)/2]*FresnelC[b*x])/(b^4*Pi^2) + (15*FresnelC[b*x]*FresnelS[b*x])/(2*b^7*Pi^3) + (((15*I)/8)*x^2*Hype
rgeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*b^2*Pi*x^2])/(b^5*Pi^3) - (((15*I)/8)*x^2*HypergeometricPFQ[{1, 1}, {
3/2, 2}, (I/2)*b^2*Pi*x^2])/(b^5*Pi^3) - (15*x*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^5*FresnelC[b
*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (7*x^2*Sin[b^2*Pi*x^2])/(4*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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 6582

Int[FresnelC[(b_.)*(x_)]*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[b*Pi*FresnelC[b*x]*(FresnelS[b*x]/(4*d)), x] + (
Simp[(1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I)*d*x^2], x] - Simp[(1/8)*I*b*x^2*HypergeometricPFQ[
{1, 1}, {3/2, 2}, I*d*x^2], x]) /; FreeQ[{b, d}, 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^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {5 \int x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^5 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi } \\ & = \frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {15 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx}{b^4 \pi ^2}-\frac {5 \int x^3 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\text {Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi } \\ & = \frac {x^4 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {15 \int \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}+\frac {15 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}-\frac {\text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}-\frac {5 \text {Subst}\left (\int x \cos ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2} \\ & = \frac {x^4 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {15 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^7 \pi ^3}+\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {x^2 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac {\text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac {15 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {5 \text {Subst}\left (\int x \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {5 \text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2} \\ & = -\frac {5 x^4}{8 b^3 \pi ^2}-\frac {17 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {15 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^7 \pi ^3}+\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac {5 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3} \\ & = -\frac {5 x^4}{8 b^3 \pi ^2}-\frac {11 \cos \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^4 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {15 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^7 \pi ^3}+\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac {15 x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3} \\ \end{align*}

Mathematica [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx \]

[In]

Integrate[x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x],x]

[Out]

Integrate[x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x]

Maple [F]

\[\int x^{6} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelC}\left (b x \right )d x\]

[In]

int(x^6*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x),x)

[Out]

int(x^6*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x),x)

Fricas [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

integrate(x^6*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="fricas")

[Out]

integral(x^6*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)

Sympy [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^{6} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]

[In]

integrate(x**6*cos(1/2*b**2*pi*x**2)*fresnelc(b*x),x)

[Out]

Integral(x**6*cos(pi*b**2*x**2/2)*fresnelc(b*x), x)

Maxima [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

integrate(x^6*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="maxima")

[Out]

integrate(x^6*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)

Giac [F]

\[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{6} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]

[In]

integrate(x^6*cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x),x, algorithm="giac")

[Out]

integrate(x^6*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^6\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]

[In]

int(x^6*FresnelC(b*x)*cos((Pi*b^2*x^2)/2),x)

[Out]

int(x^6*FresnelC(b*x)*cos((Pi*b^2*x^2)/2), x)