3.76 \(\int x^2 \sqrt{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=86 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{12 a^3}+\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)} \]

[Out]

(x^3*Sqrt[ArcCos[a*x]])/3 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^3) - (Sqrt[Pi/6]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(12*a^3)

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Rubi [A]  time = 0.183938, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4630, 4724, 3312, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{12 a^3}+\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[ArcCos[a*x]],x]

[Out]

(x^3*Sqrt[ArcCos[a*x]])/3 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^3) - (Sqrt[Pi/6]*FresnelC
[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(12*a^3)

Rule 4630

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCos[c*x])^n)/(m
+ 1), x] + Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 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]

Rubi steps

\begin{align*} \int x^2 \sqrt{\cos ^{-1}(a x)} \, dx &=\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}+\frac{1}{6} a \int \frac{x^3}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{x}}+\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{24 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{12 a^3}-\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{4 a^3}\\ &=\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4 a^3}-\frac{\sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{12 a^3}\\ \end{align*}

Mathematica [C]  time = 0.185464, size = 122, normalized size = 1.42 \[ \frac{\sqrt{i \cos ^{-1}(a x)} \left (9 \sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{3}{2},-i \cos ^{-1}(a x)\right )-9 i \cos ^{-1}(a x) \text{Gamma}\left (\frac{3}{2},i \cos ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{3}{2},-3 i \cos ^{-1}(a x)\right )-i \cos ^{-1}(a x) \text{Gamma}\left (\frac{3}{2},3 i \cos ^{-1}(a x)\right )\right )\right )}{72 a^3 \cos ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*Sqrt[ArcCos[a*x]],x]

[Out]

(Sqrt[I*ArcCos[a*x]]*(9*Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (-I)*ArcCos[a*x]] - (9*I)*ArcCos[a*x]*Gamma[3/2, I*ArcC
os[a*x]] + Sqrt[3]*(Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (-3*I)*ArcCos[a*x]] - I*ArcCos[a*x]*Gamma[3/2, (3*I)*ArcCos
[a*x]])))/(72*a^3*ArcCos[a*x]^(3/2))

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Maple [A]  time = 0.079, size = 96, normalized size = 1.1 \begin{align*}{\frac{1}{72\,{a}^{3}} \left ( -\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) -9\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +18\,ax\arccos \left ( ax \right ) +6\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccos(a*x)^(1/2),x)

[Out]

1/72/a^3/arccos(a*x)^(1/2)*(-3^(1/2)*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcc
os(a*x)^(1/2))-9*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))+18*a*x*arccos
(a*x)+6*arccos(a*x)*cos(3*arccos(a*x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acos(a*x)**(1/2),x)

[Out]

Integral(x**2*sqrt(acos(a*x)), x)

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Giac [B]  time = 1.27354, size = 284, normalized size = 3.3 \begin{align*} \frac{\sqrt{6} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{144 \, a^{3}{\left (i - 1\right )}} + \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{16 \, a^{3}{\left (i - 1\right )}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{8 \, a^{3}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-3 \, i \arccos \left (a x\right )\right )}}{24 \, a^{3}} - \frac{\sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{6} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{144 \, a^{3}{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{16 \, a^{3}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^(1/2),x, algorithm="giac")

[Out]

1/144*sqrt(6)*sqrt(pi)*i*erf(sqrt(6)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) + 1/16*sqrt(2)*sqrt(pi)*i*erf(sq
rt(2)*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) + 1/24*sqrt(arccos(a*x))*e^(3*i*arccos(a*x))/a^3 + 1/8*sqrt(arc
cos(a*x))*e^(i*arccos(a*x))/a^3 + 1/8*sqrt(arccos(a*x))*e^(-i*arccos(a*x))/a^3 + 1/24*sqrt(arccos(a*x))*e^(-3*
i*arccos(a*x))/a^3 - 1/144*sqrt(6)*sqrt(pi)*erf(-sqrt(6)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1)) - 1/16*sqr
t(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(arccos(a*x))/(i - 1))/(a^3*(i - 1))