3.134 \(\int x^3 \text{FresnelC}(a+b x) \, dx\)

Optimal. Leaf size=227 \[ -\frac{a^4 \text{FresnelC}(a+b x)}{4 b^4}+\frac{3 a^2 S(a+b x)}{2 \pi b^4}+\frac{a^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac{3 a^2 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^4}+\frac{3 \text{FresnelC}(a+b x)}{4 \pi ^2 b^4}+\frac{a (a+b x)^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac{(a+b x)^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi b^4}+\frac{2 a \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi ^2 b^4}-\frac{3 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4}+\frac{1}{4} x^4 \text{FresnelC}(a+b x) \]

[Out]

(2*a*Cos[(Pi*(a + b*x)^2)/2])/(b^4*Pi^2) - (3*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi^2) - (a^4*FresnelC[
a + b*x])/(4*b^4) + (3*FresnelC[a + b*x])/(4*b^4*Pi^2) + (x^4*FresnelC[a + b*x])/4 + (3*a^2*FresnelS[a + b*x])
/(2*b^4*Pi) + (a^3*Sin[(Pi*(a + b*x)^2)/2])/(b^4*Pi) - (3*a^2*(a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/(2*b^4*Pi) +
(a*(a + b*x)^2*Sin[(Pi*(a + b*x)^2)/2])/(b^4*Pi) - ((a + b*x)^3*Sin[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi)

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Rubi [A]  time = 0.18712, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6429, 3434, 3352, 3380, 2637, 3386, 3351, 3296, 2638, 3385} \[ -\frac{a^4 \text{FresnelC}(a+b x)}{4 b^4}+\frac{3 a^2 S(a+b x)}{2 \pi b^4}+\frac{a^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac{3 a^2 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 \pi b^4}+\frac{3 \text{FresnelC}(a+b x)}{4 \pi ^2 b^4}+\frac{a (a+b x)^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^4}-\frac{(a+b x)^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi b^4}+\frac{2 a \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi ^2 b^4}-\frac{3 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4}+\frac{1}{4} x^4 \text{FresnelC}(a+b x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*FresnelC[a + b*x],x]

[Out]

(2*a*Cos[(Pi*(a + b*x)^2)/2])/(b^4*Pi^2) - (3*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi^2) - (a^4*FresnelC[
a + b*x])/(4*b^4) + (3*FresnelC[a + b*x])/(4*b^4*Pi^2) + (x^4*FresnelC[a + b*x])/4 + (3*a^2*FresnelS[a + b*x])
/(2*b^4*Pi) + (a^3*Sin[(Pi*(a + b*x)^2)/2])/(b^4*Pi) - (3*a^2*(a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/(2*b^4*Pi) +
(a*(a + b*x)^2*Sin[(Pi*(a + b*x)^2)/2])/(b^4*Pi) - ((a + b*x)^3*Sin[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi)

Rule 6429

Int[FresnelC[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*FresnelC[a +
b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[(Pi*(a + b*x)^2)/2], x], x] /; FreeQ[{a
, b, c, d}, x] && IGtQ[m, 0]

Rule 3434

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :
> Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Cos[c +
d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]

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 3380

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 2637

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

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]

Rule 3351

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

Rule 3296

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 2638

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

Rubi steps

\begin{align*} \int x^3 C(a+b x) \, dx &=\frac{1}{4} x^4 C(a+b x)-\frac{1}{4} b \int x^4 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) \, dx\\ &=\frac{1}{4} x^4 C(a+b x)-\frac{\operatorname{Subst}\left (\int \left (a^4 \cos \left (\frac{\pi x^2}{2}\right )-4 a^3 x \cos \left (\frac{\pi x^2}{2}\right )+6 a^2 x^2 \cos \left (\frac{\pi x^2}{2}\right )-4 a x^3 \cos \left (\frac{\pi x^2}{2}\right )+x^4 \cos \left (\frac{\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{4 b^4}\\ &=\frac{1}{4} x^4 C(a+b x)-\frac{\operatorname{Subst}\left (\int x^4 \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4}+\frac{a \operatorname{Subst}\left (\int x^3 \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^4}+\frac{a^3 \operatorname{Subst}\left (\int x \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^4}-\frac{a^4 \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4}\\ &=-\frac{a^4 C(a+b x)}{4 b^4}+\frac{1}{4} x^4 C(a+b x)-\frac{3 a^2 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }-\frac{(a+b x)^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }+\frac{a \operatorname{Subst}\left (\int x \cos \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^4}+\frac{a^3 \operatorname{Subst}\left (\int \cos \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^4}+\frac{3 \operatorname{Subst}\left (\int x^2 \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 \pi }+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^4 \pi }\\ &=-\frac{3 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi ^2}-\frac{a^4 C(a+b x)}{4 b^4}+\frac{1}{4} x^4 C(a+b x)+\frac{3 a^2 S(a+b x)}{2 b^4 \pi }+\frac{a^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac{3 a^2 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac{a (a+b x)^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac{(a+b x)^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }+\frac{3 \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{4 b^4 \pi ^2}-\frac{a \operatorname{Subst}\left (\int \sin \left (\frac{\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{b^4 \pi }\\ &=\frac{2 a \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi ^2}-\frac{3 (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi ^2}-\frac{a^4 C(a+b x)}{4 b^4}+\frac{3 C(a+b x)}{4 b^4 \pi ^2}+\frac{1}{4} x^4 C(a+b x)+\frac{3 a^2 S(a+b x)}{2 b^4 \pi }+\frac{a^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac{3 a^2 (a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{2 b^4 \pi }+\frac{a (a+b x)^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^4 \pi }-\frac{(a+b x)^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 b^4 \pi }\\ \end{align*}

Mathematica [A]  time = 0.294206, size = 166, normalized size = 0.73 \[ \frac{\left (-\pi ^2 a^4+\pi ^2 b^4 x^4+3\right ) \text{FresnelC}(a+b x)+6 \pi a^2 S(a+b x)+\pi a^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )-\pi a^2 b x \sin \left (\frac{1}{2} \pi (a+b x)^2\right )+\pi a b^2 x^2 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )-\pi b^3 x^3 \sin \left (\frac{1}{2} \pi (a+b x)^2\right )+5 a \cos \left (\frac{1}{2} \pi (a+b x)^2\right )-3 b x \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{4 \pi ^2 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*FresnelC[a + b*x],x]

[Out]

(5*a*Cos[(Pi*(a + b*x)^2)/2] - 3*b*x*Cos[(Pi*(a + b*x)^2)/2] + (3 - a^4*Pi^2 + b^4*Pi^2*x^4)*FresnelC[a + b*x]
+ 6*a^2*Pi*FresnelS[a + b*x] + a^3*Pi*Sin[(Pi*(a + b*x)^2)/2] - a^2*b*Pi*x*Sin[(Pi*(a + b*x)^2)/2] + a*b^2*Pi
*x^2*Sin[(Pi*(a + b*x)^2)/2] - b^3*Pi*x^3*Sin[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi^2)

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Maple [A]  time = 0.05, size = 187, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{{\it FresnelC} \left ( bx+a \right ){b}^{4}{x}^{4}}{4}}-{\frac{ \left ( bx+a \right ) ^{3}}{4\,\pi }\sin \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{\frac{3}{4\,\pi } \left ( -{\frac{bx+a}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{\frac{{\it FresnelC} \left ( bx+a \right ) }{\pi }} \right ) }+{\frac{a \left ( bx+a \right ) ^{2}}{\pi }\sin \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+2\,{\frac{a\cos \left ( 1/2\,\pi \, \left ( bx+a \right ) ^{2} \right ) }{{\pi }^{2}}}-{\frac{3\,{a}^{2} \left ( bx+a \right ) }{2\,\pi }\sin \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }+{\frac{3\,{a}^{2}{\it FresnelS} \left ( bx+a \right ) }{2\,\pi }}+{\frac{{a}^{3}}{\pi }\sin \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) }-{\frac{{a}^{4}{\it FresnelC} \left ( bx+a \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*FresnelC(b*x+a),x)

[Out]

1/b^4*(1/4*FresnelC(b*x+a)*b^4*x^4-1/4/Pi*(b*x+a)^3*sin(1/2*Pi*(b*x+a)^2)+3/4/Pi*(-1/Pi*(b*x+a)*cos(1/2*Pi*(b*
x+a)^2)+1/Pi*FresnelC(b*x+a))+a/Pi*(b*x+a)^2*sin(1/2*Pi*(b*x+a)^2)+2*a/Pi^2*cos(1/2*Pi*(b*x+a)^2)-3/2*a^2/Pi*(
b*x+a)*sin(1/2*Pi*(b*x+a)^2)+3/2*a^2/Pi*FresnelS(b*x+a)+a^3/Pi*sin(1/2*Pi*(b*x+a)^2)-1/4*a^4*FresnelC(b*x+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm fresnelc}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*fresnelc(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^3*fresnelc(b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3}{\rm fresnelc}\left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*fresnelc(b*x+a),x, algorithm="fricas")

[Out]

integral(x^3*fresnelc(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} C\left (a + b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*fresnelc(b*x+a),x)

[Out]

Integral(x**3*fresnelc(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm fresnelc}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*fresnelc(b*x+a),x, algorithm="giac")

[Out]

integrate(x^3*fresnelc(b*x + a), x)