3.173 \(\int x^2 \sqrt{a+b \cos ^{-1}(c x)} \, dx\)

Optimal. Leaf size=242 \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)} \]

[Out]

(x^3*Sqrt[a + b*ArcCos[c*x]])/3 - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/S
qrt[b]])/(4*c^3) - (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(1
2*c^3) - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[
b]*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)

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Rubi [A]  time = 0.702423, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4630, 4724, 3312, 3306, 3305, 3351, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*ArcCos[c*x]],x]

[Out]

(x^3*Sqrt[a + b*ArcCos[c*x]])/3 - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/S
qrt[b]])/(4*c^3) - (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(1
2*c^3) - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[
b]*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^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 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 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 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{a+b \cos ^{-1}(c x)} \, dx &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}+\frac{1}{6} (b c) \int \frac{x^3}{\sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{b \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{6 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{b \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 \sqrt{a+b x}}+\frac{\cos (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{6 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{b \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{24 c^3}-\frac{b \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{8 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{\left (b \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{8 c^3}-\frac{\left (b \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{24 c^3}-\frac{\left (b \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{8 c^3}-\frac{\left (b \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{24 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{4 c^3}-\frac{\cos \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{12 c^3}-\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{4 c^3}-\frac{\sin \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{12 c^3}\\ &=\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{\sqrt{b} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{4 c^3}-\frac{\sqrt{b} \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}-\frac{\sqrt{b} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{4 c^3}-\frac{\sqrt{b} \sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{12 c^3}\\ \end{align*}

Mathematica [C]  time = 0.516004, size = 243, normalized size = 1. \[ \frac{e^{-\frac{3 i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left (9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+\sqrt{3} \left (\sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )\right )\right )}{72 c^3 \sqrt{\frac{\left (a+b \cos ^{-1}(c x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*Sqrt[a + b*ArcCos[c*x]],x]

[Out]

(Sqrt[a + b*ArcCos[c*x]]*(9*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x
]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*
(Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a +
b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcCos[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcCos[c*x
])^2/b^2])

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Maple [A]  time = 0.134, size = 357, normalized size = 1.5 \begin{align*}{\frac{1}{72\,{c}^{3}} \left ( -\sqrt{3}\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arccos \left ( cx \right ) }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sin \left ( 3\,{\frac{a}{b}} \right ) b-\sqrt{3}\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arccos \left ( cx \right ) }\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) b-9\,\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arccos \left ( cx \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) b-9\,\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arccos \left ( cx \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) b+18\,\arccos \left ( cx \right ) \cos \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) b+18\,\cos \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) a+6\,\arccos \left ( cx \right ) \cos \left ( 3\,{\frac{a+b\arccos \left ( cx \right ) }{b}}-3\,{\frac{a}{b}} \right ) b+6\,\cos \left ( 3\,{\frac{a+b\arccos \left ( cx \right ) }{b}}-3\,{\frac{a}{b}} \right ) a \right ){\frac{1}{\sqrt{a+b\arccos \left ( cx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \arccos \left (c x\right ) + a} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*arccos(c*x) + a)*x^2, x)

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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*(a+b*arccos(c*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{a + b \operatorname{acos}{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acos(c*x))**(1/2),x)

[Out]

Integral(x**2*sqrt(a + b*acos(c*x)), x)

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Giac [B]  time = 2.00748, size = 549, normalized size = 2.27 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{a i}{b}\right )}}{16 \,{\left (\frac{b i}{\sqrt{{\left | b \right |}}} + \sqrt{{\left | b \right |}}\right )} c^{3}} - \frac{\sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{a i}{b}\right )}}{16 \,{\left (\frac{b i}{\sqrt{{\left | b \right |}}} - \sqrt{{\left | b \right |}}\right )} c^{3}} + \frac{\sqrt{\pi } \sqrt{b} \operatorname{erf}\left (-\frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (\frac{3 \, a i}{b}\right )}}{24 \,{\left (\frac{\sqrt{6} b i}{{\left | b \right |}} + \sqrt{6}\right )} c^{3}} - \frac{\sqrt{\pi } \sqrt{b} \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (-\frac{3 \, a i}{b}\right )}}{24 \,{\left (\frac{\sqrt{6} b i}{{\left | b \right |}} - \sqrt{6}\right )} c^{3}} + \frac{\sqrt{b \arccos \left (c x\right ) + a} e^{\left (3 \, i \arccos \left (c x\right )\right )}}{24 \, c^{3}} + \frac{\sqrt{b \arccos \left (c x\right ) + a} e^{\left (i \arccos \left (c x\right )\right )}}{8 \, c^{3}} + \frac{\sqrt{b \arccos \left (c x\right ) + a} e^{\left (-i \arccos \left (c x\right )\right )}}{8 \, c^{3}} + \frac{\sqrt{b \arccos \left (c x\right ) + a} e^{\left (-3 \, i \arccos \left (c x\right )\right )}}{24 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*
x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b*i/sqrt(abs(b)) + sqrt(abs(b)))*c^3) - 1/16*sqrt(2)*sqrt(pi)*b*erf(1/2*sq
rt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/
((b*i/sqrt(abs(b)) - sqrt(abs(b)))*c^3) + 1/24*sqrt(pi)*sqrt(b)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(
b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b))*e^(3*a*i/b)/((sqrt(6)*b*i/abs(b) + sqrt(6))*c^3) -
1/24*sqrt(pi)*sqrt(b)*erf(1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)*i/abs(b) - 1/2*sqrt(6)*sqrt(b*arccos(c*x
) + a)/sqrt(b))*e^(-3*a*i/b)/((sqrt(6)*b*i/abs(b) - sqrt(6))*c^3) + 1/24*sqrt(b*arccos(c*x) + a)*e^(3*i*arccos
(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*e^(i*arccos(c*x))/c^3 + 1/8*sqrt(b*arccos(c*x) + a)*e^(-i*arccos(c*x)
)/c^3 + 1/24*sqrt(b*arccos(c*x) + a)*e^(-3*i*arccos(c*x))/c^3