3.141 \(\int x^6 \text{FresnelC}(b x)^2 \, dx\)

Optimal. Leaf size=239 \[ -\frac{2 x^6 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac{48 x^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac{12 x^4 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{96 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}-\frac{531 \text{FresnelC}\left (\sqrt{2} b x\right )}{56 \sqrt{2} \pi ^4 b^7}+\frac{6 x^5}{35 \pi ^2 b^2}+\frac{17 x^3 \sin \left (\pi b^2 x^2\right )}{28 \pi ^3 b^4}-\frac{x^5 \cos \left (\pi b^2 x^2\right )}{14 \pi ^2 b^2}+\frac{21 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^4 b^6}-\frac{48 x}{7 \pi ^4 b^6}+\frac{1}{7} x^7 \text{FresnelC}(b x)^2 \]

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

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Rubi [A]  time = 0.298269, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6431, 6455, 6463, 6461, 3358, 3352, 3385, 3392, 30, 3386} \[ -\frac{2 x^6 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac{48 x^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac{12 x^4 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{96 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}-\frac{531 \text{FresnelC}\left (\sqrt{2} b x\right )}{56 \sqrt{2} \pi ^4 b^7}+\frac{6 x^5}{35 \pi ^2 b^2}+\frac{17 x^3 \sin \left (\pi b^2 x^2\right )}{28 \pi ^3 b^4}-\frac{x^5 \cos \left (\pi b^2 x^2\right )}{14 \pi ^2 b^2}+\frac{21 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^4 b^6}-\frac{48 x}{7 \pi ^4 b^6}+\frac{1}{7} x^7 \text{FresnelC}(b x)^2 \]

Antiderivative was successfully verified.

[In]

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

Rule 6431

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*FresnelC[b*x]^2)/(m + 1), x] - Dist[(2*b)/
(m + 1), Int[x^(m + 1)*Cos[(Pi*b^2*x^2)/2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6455

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*b^4)/4] && IGtQ[m, 1]

Rule 6463

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*b^4)/4] && IGtQ[m, 1]

Rule 6461

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

Rule 3358

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

Rule 3352

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

Rule 3385

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 3392

Int[Cos[(a_.) + ((b_.)*(x_)^(n_))/2]^2*(x_)^(m_.), 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 30

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

Rule 3386

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]

Rubi steps

\begin{align*} \int x^6 C(b x)^2 \, dx &=\frac{1}{7} x^7 C(b x)^2-\frac{1}{7} (2 b) \int x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=\frac{1}{7} x^7 C(b x)^2-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{\int x^6 \sin \left (b^2 \pi x^2\right ) \, dx}{7 \pi }+\frac{12 \int x^5 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{7 b \pi }\\ &=-\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}-\frac{12 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^3 \pi ^2}+\frac{1}{7} x^7 C(b x)^2-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{48 \int x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, 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 \cos ^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{12 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^3 \pi ^2}+\frac{1}{7} x^7 C(b x)^2+\frac{48 x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{5 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}-\frac{96 \int x C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, 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{96 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^7 \pi ^4}-\frac{12 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^3 \pi ^2}+\frac{1}{7} x^7 C(b x)^2+\frac{48 x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\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 \cos ^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{96 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^7 \pi ^4}-\frac{12 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^3 \pi ^2}+\frac{1}{7} x^7 C(b x)^2-\frac{15 C\left (\sqrt{2} b x\right )}{56 \sqrt{2} b^7 \pi ^4}-\frac{6 \sqrt{2} C\left (\sqrt{2} b x\right )}{7 b^7 \pi ^4}+\frac{48 x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\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{96 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^7 \pi ^4}-\frac{12 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^3 \pi ^2}+\frac{1}{7} x^7 C(b x)^2-\frac{51 C\left (\sqrt{2} b x\right )}{56 \sqrt{2} b^7 \pi ^4}-\frac{6 \sqrt{2} C\left (\sqrt{2} b x\right )}{7 b^7 \pi ^4}+\frac{48 x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\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{96 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^7 \pi ^4}-\frac{12 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{7 b^3 \pi ^2}+\frac{1}{7} x^7 C(b x)^2-\frac{51 C\left (\sqrt{2} b x\right )}{56 \sqrt{2} b^7 \pi ^4}-\frac{30 \sqrt{2} C\left (\sqrt{2} b x\right )}{7 b^7 \pi ^4}+\frac{48 x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac{2 x^6 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}\\ \end{align*}

Mathematica [A]  time = 0.240461, size = 170, normalized size = 0.71 \[ \frac{80 \pi ^4 b^7 x^7 \text{FresnelC}(b x)^2-160 \text{FresnelC}(b x) \left (\pi b^2 x^2 \left (\pi ^2 b^4 x^4-24\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )+6 \left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )\right )+2 b x \left (2 \left (24 \pi ^2 b^4 x^4+85 \pi b^2 x^2 \sin \left (\pi b^2 x^2\right )-960\right )+\left (735-20 \pi ^2 b^4 x^4\right ) \cos \left (\pi b^2 x^2\right )\right )-2655 \sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )}{560 \pi ^4 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(80*b^7*Pi^4*x^7*FresnelC[b*x]^2 - 2655*Sqrt[2]*FresnelC[Sqrt[2]*b*x] - 160*FresnelC[b*x]*(6*(-8 + b^4*Pi^2*x^
4)*Cos[(b^2*Pi*x^2)/2] + b^2*Pi*x^2*(-24 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]) + 2*b*x*((735 - 20*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)

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Maple [A]  time = 0.077, size = 324, normalized size = 1.4 \begin{align*}{\frac{1}{{b}^{7}} \left ({\frac{{b}^{7}{x}^{7} \left ({\it FresnelC} \left ( bx \right ) \right ) ^{2}}{7}}-2\,{\it FresnelC} \left ( bx \right ) \left ( 1/7\,{\frac{{b}^{6}{x}^{6}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-6/7\,{\frac{1}{\pi } \left ( -{\frac{{x}^{4}{b}^{4}\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+4\,{\frac{1}{\pi } \left ({\frac{{b}^{2}{x}^{2}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+2\,{\frac{\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) } \right ) } \right ) +{\frac{6}{7\,{\pi }^{4}} \left ({\frac{{\pi }^{2}{b}^{5}{x}^{5}}{5}}-8\,bx \right ) }+{\frac{6}{7\,{\pi }^{4}} \left ({\frac{\pi \,{b}^{3}{x}^{3}\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2}}-{\frac{3\,\pi }{2} \left ( -{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) }-4\,\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) \right ) }+{\frac{1}{7\,{\pi }^{3}} \left ( -{\frac{\pi \,{b}^{5}{x}^{5}\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2}}+{\frac{5\,\pi }{2} \left ({\frac{{x}^{3}{b}^{3}\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}-{\frac{3}{2\,\pi } \left ( -{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) } \right ) }+12\,{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-6\,{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{\pi }} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^7*(1/7*b^7*x^7*FresnelC(b*x)^2-2*FresnelC(b*x)*(1/7/Pi*b^6*x^6*sin(1/2*b^2*Pi*x^2)-6/7/Pi*(-1/Pi*b^4*x^4*c
os(1/2*b^2*Pi*x^2)+4/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))))+6/7/Pi^4*(1/5*Pi^2*b^5
*x^5-8*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)*Fresne
lC(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))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6}{\rm fresnelc}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6*fresnelc(b*x)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{6}{\rm fresnelc}\left (b x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^6*fresnelc(b*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6} C^{2}\left (b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**6*fresnelc(b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6}{\rm fresnelc}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^6*fresnelc(b*x)^2, x)