\(\int \frac {x^2}{(a+b \arccos (c x))^3} \, dx\) [168]

Optimal result

Integrand size = 14, antiderivative size = 197 \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^3 c^3}-\frac {9 \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3 c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{8 b^3 c^3} \]

[Out]

-x/b^2/c^2/(a+b*arccos(c*x))+3/2*x^3/b^2/(a+b*arccos(c*x))+1/8*cos(a/b)*Si((a+b*arccos(c*x))/b)/b^3/c^3+9/8*co
s(3*a/b)*Si(3*(a+b*arccos(c*x))/b)/b^3/c^3-1/8*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b^3/c^3-9/8*Ci(3*(a+b*arccos(c
*x))/b)*sin(3*a/b)/b^3/c^3+1/2*x^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^2

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4730, 4808, 4732, 4491, 3384, 3380, 3383, 4720} \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=-\frac {\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b^3 c^3}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{8 b^3 c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{8 b^3 c^3}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}+\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2} \]

[In]

Int[x^2/(a + b*ArcCos[c*x])^3,x]

[Out]

(x^2*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - x/(b^2*c^2*(a + b*ArcCos[c*x])) + (3*x^3)/(2*b^2*(a +
b*ArcCos[c*x])) - (CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(8*b^3*c^3) - (9*CosIntegral[(3*(a + b*ArcCos[
c*x]))/b]*Sin[(3*a)/b])/(8*b^3*c^3) + (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(8*b^3*c^3) + (9*Cos[(3*a)
/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/(8*b^3*c^3)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

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

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4720

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[-(b*c)^(-1), Subst[Int[x^n*Sin[-a/b + x/b], x],
x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4730

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

Rule 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(b*c^(m + 1))^(-1), Subst[Int[x^n*C
os[-a/b + x/b]^m*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4808

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Dist[f*(m/(
b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /
; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {\int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2} \, dx}{b c}+\frac {(3 c) \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2} \, dx}{2 b} \\ & = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}-\frac {9 \int \frac {x^2}{a+b \arccos (c x)} \, dx}{2 b^2}+\frac {\int \frac {1}{a+b \arccos (c x)} \, dx}{b^2 c^2} \\ & = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}+\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^3 c^3}-\frac {9 \text {Subst}\left (\int \frac {\cos ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{2 b^3 c^3} \\ & = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}-\frac {9 \text {Subst}\left (\int \left (\frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arccos (c x)\right )}{2 b^3 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^3 c^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^3 c^3} \\ & = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^3 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^3 c^3}-\frac {9 \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{8 b^3 c^3}-\frac {9 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{8 b^3 c^3} \\ & = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^3 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^3 c^3}+\frac {\left (9 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{8 b^3 c^3}+\frac {\left (9 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{8 b^3 c^3}-\frac {\left (9 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{8 b^3 c^3}-\frac {\left (9 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{8 b^3 c^3} \\ & = \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {x}{b^2 c^2 (a+b \arccos (c x))}+\frac {3 x^3}{2 b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^3 c^3}-\frac {9 \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3 c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{8 b^3 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\frac {\frac {4 b^2 x^2 \sqrt {1-c^2 x^2}}{c (a+b \arccos (c x))^2}-\frac {8 b x}{c^2 (a+b \arccos (c x))}+\frac {12 b x^3}{a+b \arccos (c x)}-\frac {\operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )}{c^3}-\frac {9 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )}{c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{c^3}}{8 b^3} \]

[In]

Integrate[x^2/(a + b*ArcCos[c*x])^3,x]

[Out]

((4*b^2*x^2*Sqrt[1 - c^2*x^2])/(c*(a + b*ArcCos[c*x])^2) - (8*b*x)/(c^2*(a + b*ArcCos[c*x])) + (12*b*x^3)/(a +
b*ArcCos[c*x]) - (CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b])/c^3 - (9*CosIntegral[3*(a/b + ArcCos[c*x])]*Sin[(3
*a)/b])/c^3 + (Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/c^3 + (9*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x]
)])/c^3)/(8*b^3)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (3 \arccos \left (c x \right )\right )}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arccos \left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b}{8}-\frac {9 \arccos \left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b}{8}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a}{8}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a}{8}+\frac {3 \cos \left (3 \arccos \left (c x \right )\right ) b}{8}}{\left (a +b \arccos \left (c x \right )\right ) b^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b -\arccos \left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a +x b c}{8 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)
default	\(\frac {\frac {\sin \left (3 \arccos \left (c x \right )\right )}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arccos \left (c x \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b}{8}-\frac {9 \arccos \left (c x \right ) \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) b}{8}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a}{8}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) a}{8}+\frac {3 \cos \left (3 \arccos \left (c x \right )\right ) b}{8}}{\left (a +b \arccos \left (c x \right )\right ) b^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b -\arccos \left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a +x b c}{8 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{3}}\)	\(290\)

[In]

int(x^2/(a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/c^3*(1/8*sin(3*arccos(c*x))/(a+b*arccos(c*x))^2/b+3/8*(3*arccos(c*x)*cos(3*a/b)*Si(3*arccos(c*x)+3*a/b)*b-3*
arccos(c*x)*sin(3*a/b)*Ci(3*arccos(c*x)+3*a/b)*b+3*cos(3*a/b)*Si(3*arccos(c*x)+3*a/b)*a-3*sin(3*a/b)*Ci(3*arcc
os(c*x)+3*a/b)*a+cos(3*arccos(c*x))*b)/(a+b*arccos(c*x))/b^3+1/8*(-c^2*x^2+1)^(1/2)/(a+b*arccos(c*x))^2/b+1/8*
(arccos(c*x)*cos(a/b)*Si(arccos(c*x)+a/b)*b-arccos(c*x)*sin(a/b)*Ci(arccos(c*x)+a/b)*b+cos(a/b)*Si(arccos(c*x)
+a/b)*a-sin(a/b)*Ci(arccos(c*x)+a/b)*a+x*b*c)/(a+b*arccos(c*x))/b^3)

Fricas [F]

\[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\int { \frac {x^{2}}{{\left (b \arccos \left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(x^2/(a+b*arccos(c*x))^3,x, algorithm="fricas")

[Out]

integral(x^2/(b^3*arccos(c*x)^3 + 3*a*b^2*arccos(c*x)^2 + 3*a^2*b*arccos(c*x) + a^3), x)

Sympy [F]

\[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}\, dx \]

[In]

integrate(x**2/(a+b*acos(c*x))**3,x)

[Out]

Integral(x**2/(a + b*acos(c*x))**3, x)

Maxima [F]

\[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\int { \frac {x^{2}}{{\left (b \arccos \left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(x^2/(a+b*arccos(c*x))^3,x, algorithm="maxima")

[Out]

1/2*(3*a*c^2*x^3 + sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x^2 - 2*a*x + (3*b*c^2*x^3 - 2*b*x)*arctan2(sqrt(c*x + 1)*
sqrt(-c*x + 1), c*x) - 2*(b^4*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*arctan2(sqrt(c*x
+ 1)*sqrt(-c*x + 1), c*x) + a^2*b^2*c^2)*integrate(1/2*(9*c^2*x^2 - 2)/(b^3*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*
x + 1), c*x) + a*b^2*c^2), x))/(b^4*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*arctan2(sqr
t(c*x + 1)*sqrt(-c*x + 1), c*x) + a^2*b^2*c^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (183) = 366\).

Time = 0.39 (sec) , antiderivative size = 1479, normalized size of antiderivative = 7.51 \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(x^2/(a+b*arccos(c*x))^3,x, algorithm="giac")

[Out]

3/2*b^2*c^3*x^3*arccos(c*x)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 3/2*a*b*c^3*x^3/
(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 9/2*b^2*arccos(c*x)^2*cos(a/b)^2*cos_integra
l(3*a/b + 3*arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 9/2*b^2*ar
ccos(c*x)^2*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) +
a^2*b^3*c^3) - 9*a*b*arccos(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^
2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 9*a*b*arccos(c*x)*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(c*x))/
(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 1/2*sqrt(-c^2*x^2 + 1)*b^2*c^2*x^2/(b^5*c^3*
arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 9/8*b^2*arccos(c*x)^2*cos_integral(3*a/b + 3*arccos(c
*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 9/2*a^2*cos(a/b)^2*cos_integra
l(3*a/b + 3*arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 1/8*b^2*ar
ccos(c*x)^2*cos_integral(a/b + arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^
3*c^3) - 27/8*b^2*arccos(c*x)^2*cos(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*
c^3*arccos(c*x) + a^2*b^3*c^3) + 9/2*a^2*cos(a/b)^3*sin_integral(3*a/b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2
+ 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 1/8*b^2*arccos(c*x)^2*cos(a/b)*sin_integral(a/b + arccos(c*x))/(b^
5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - b^2*c*x*arccos(c*x)/(b^5*c^3*arccos(c*x)^2 + 2*
a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 9/4*a*b*arccos(c*x)*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b^5*c
^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 1/4*a*b*arccos(c*x)*cos_integral(a/b + arccos(c*x)
)*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 27/4*a*b*arccos(c*x)*cos(a/b)*sin
_integral(3*a/b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 1/4*a*b*arc
cos(c*x)*cos(a/b)*sin_integral(a/b + arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c
^3) - a*b*c*x/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 9/8*a^2*cos_integral(3*a/b + 3
*arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 1/8*a^2*cos_integral(
a/b + arccos(c*x))*sin(a/b)/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) - 27/8*a^2*cos(a/b
)*sin_integral(3*a/b + 3*arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3) + 1/8*a^
2*cos(a/b)*sin_integral(a/b + arccos(c*x))/(b^5*c^3*arccos(c*x)^2 + 2*a*b^4*c^3*arccos(c*x) + a^2*b^3*c^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{(a+b \arccos (c x))^3} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3} \,d x \]

[In]

int(x^2/(a + b*acos(c*x))^3,x)

[Out]

int(x^2/(a + b*acos(c*x))^3, x)