3.41 \(\int \frac{\cos ^{-1}(a x)^4}{x^4} \, dx\)

Optimal. Leaf size=304 \[ 2 i a^3 \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-2 i a^3 \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-4 a^3 \cos ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+4 a^3 \cos ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (4,-i e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (4,i e^{i \cos ^{-1}(a x)}\right )+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^4}{3 x^3} \]

[Out]

(-2*a^2*ArcCos[a*x]^2)/x + (2*a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(3*x^2) - ArcCos[a*x]^4/(3*x^3) - (8*I)*a^3*A
rcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] - ((4*I)/3)*a^3*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (4*I)*a^3*PolyL
og[2, (-I)*E^(I*ArcCos[a*x])] + (2*I)*a^3*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (4*I)*a^3*PolyLog
[2, I*E^(I*ArcCos[a*x])] - (2*I)*a^3*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 4*a^3*ArcCos[a*x]*PolyLog
[3, (-I)*E^(I*ArcCos[a*x])] + 4*a^3*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (4*I)*a^3*PolyLog[4, (-I)*E^
(I*ArcCos[a*x])] + (4*I)*a^3*PolyLog[4, I*E^(I*ArcCos[a*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.420771, antiderivative size = 304, 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 = {4628, 4702, 4710, 4181, 2531, 6609, 2282, 6589, 2279, 2391} \[ 2 i a^3 \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-2 i a^3 \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-4 a^3 \cos ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+4 a^3 \cos ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (4,-i e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (4,i e^{i \cos ^{-1}(a x)}\right )+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^4}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^4/x^4,x]

[Out]

(-2*a^2*ArcCos[a*x]^2)/x + (2*a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(3*x^2) - ArcCos[a*x]^4/(3*x^3) - (8*I)*a^3*A
rcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] - ((4*I)/3)*a^3*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (4*I)*a^3*PolyL
og[2, (-I)*E^(I*ArcCos[a*x])] + (2*I)*a^3*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (4*I)*a^3*PolyLog
[2, I*E^(I*ArcCos[a*x])] - (2*I)*a^3*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 4*a^3*ArcCos[a*x]*PolyLog
[3, (-I)*E^(I*ArcCos[a*x])] + 4*a^3*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (4*I)*a^3*PolyLog[4, (-I)*E^
(I*ArcCos[a*x])] + (4*I)*a^3*PolyLog[4, I*E^(I*ArcCos[a*x])]

Rule 4628

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

Rule 4702

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m
+ 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4710

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

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\cos ^{-1}(a x)^4}{x^4} \, dx &=-\frac{\cos ^{-1}(a x)^4}{3 x^3}-\frac{1}{3} (4 a) \int \frac{\cos ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac{\cos ^{-1}(a x)^2}{x^2} \, dx-\frac{1}{3} \left (2 a^3\right ) \int \frac{\cos ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}+\frac{1}{3} \left (2 a^3\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )-\left (4 a^3\right ) \int \frac{\cos ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\left (2 a^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\left (2 a^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname{Subst}\left (\int x \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-4 a^3 \cos ^{-1}(a x) \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+4 a^3 \cos ^{-1}(a x) \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+\left (4 a^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )+2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-4 a^3 \cos ^{-1}(a x) \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+4 a^3 \cos ^{-1}(a x) \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{\cos ^{-1}(a x)^4}{3 x^3}-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )+2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-4 a^3 \cos ^{-1}(a x) \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+4 a^3 \cos ^{-1}(a x) \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-4 i a^3 \text{Li}_4\left (-i e^{i \cos ^{-1}(a x)}\right )+4 i a^3 \text{Li}_4\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [B]  time = 12.067, size = 1475, normalized size = 4.85 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCos[a*x]^4/x^4,x]

[Out]

a^3*(-(ArcCos[a*x]^2*(12 + ArcCos[a*x]^2))/6 + 4*(ArcCos[a*x]*(Log[1 - I*E^(I*ArcCos[a*x])] - Log[1 + I*E^(I*A
rcCos[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])])) + (2*((Pi^3*Log[Cot[
(Pi/2 - ArcCos[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcCos[a*x])*(Log[1 - E^(I*(Pi/2 - ArcCos[a*x]))] - Log[1 + E^(
I*(Pi/2 - ArcCos[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x]))
])))/4 - (3*Pi*((Pi/2 - ArcCos[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcCos[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcCos[a*x]
))]) + (2*I)*(Pi/2 - ArcCos[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*
x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcCos[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcCos[a*x]))])))/2 + 8*((I/64)
*(Pi/2 - ArcCos[a*x])^4 + (I/4)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2)^4 - ((Pi/2 - ArcCos[a*x])^3*Log[1 + E^(I*(Pi/
2 - ArcCos[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a
*x])/2))]))/8 - (Pi/2 + (-Pi/2 + ArcCos[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))] + ((3*I
)/8)*(Pi/2 - ArcCos[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-Pi/2 + ArcCos[a
*x])/2)^2 - (Pi/2 + (-Pi/2 + ArcCos[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))] + (I/2)*PolyL
og[2, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2)^2*PolyLog[
2, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))] - (3*(Pi/2 - ArcCos[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcCos[a*x]
))])/4 - (3*Pi*((I/3)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2)^3 - (Pi/2 + (-Pi/2 + ArcCos[a*x])/2)^2*Log[1 + E^((2*I)
*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))] + I*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-Pi/2 +
ArcCos[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))]/2))/2 - (3*(Pi/2 + (-Pi/2 + ArcCo
s[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - Arc
Cos[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcCos[a*x])/2))])))/3 - (-4*ArcCos[a*x]^3 + Arc
Cos[a*x]^4)/(12*(Cos[ArcCos[a*x]/2] - Sin[ArcCos[a*x]/2])^2) - (ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcC
os[a*x]/2] - Sin[ArcCos[a*x]/2])^3) + (ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] + Sin[ArcCos[a
*x]/2])^3) - (4*ArcCos[a*x]^3 + ArcCos[a*x]^4)/(12*(Cos[ArcCos[a*x]/2] + Sin[ArcCos[a*x]/2])^2) - (-12*ArcCos[
a*x]^2*Sin[ArcCos[a*x]/2] - ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] + Sin[ArcCos[a*x]/2])) -
(12*ArcCos[a*x]^2*Sin[ArcCos[a*x]/2] + ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] - Sin[ArcCos[a
*x]/2])))

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Maple [A]  time = 0.184, size = 451, normalized size = 1.5 \begin{align*}{\frac{2\,a \left ( \arccos \left ( ax \right ) \right ) ^{3}}{3\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-2\,{\frac{{a}^{2} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{4}}{3\,{x}^{3}}}-{\frac{2\,{a}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{3}}{3}\ln \left ( 1+i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) }+2\,i{a}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) -4\,{a}^{3}\arccos \left ( ax \right ){\it polylog} \left ( 3,-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) -4\,i{a}^{3}{\it polylog} \left ( 4,-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +{\frac{2\,{a}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{3}}{3}\ln \left ( 1-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) }-2\,i{a}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +4\,{a}^{3}\arccos \left ( ax \right ){\it polylog} \left ( 3,i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +4\,i{a}^{3}{\it polylog} \left ( 4,i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) -4\,{a}^{3}\arccos \left ( ax \right ) \ln \left ( 1+i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +4\,{a}^{3}\arccos \left ( ax \right ) \ln \left ( 1-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +4\,i{a}^{3}{\it dilog} \left ( 1+i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) -4\,i{a}^{3}{\it dilog} \left ( 1-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^4/x^4,x)

[Out]

2/3*a*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/x^2-2*a^2*arccos(a*x)^2/x-1/3*arccos(a*x)^4/x^3-2/3*a^3*arccos(a*x)^3*l
n(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))+2*I*a^3*arccos(a*x)^2*polylog(2,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*a^3*arccos(
a*x)*polylog(3,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*I*a^3*polylog(4,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))+2/3*a^3*arccos(
a*x)^3*ln(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-2*I*a^3*arccos(a*x)^2*polylog(2,I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*a^3*
arccos(a*x)*polylog(3,I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*I*a^3*polylog(4,I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*a^3*arcc
os(a*x)*ln(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*a^3*arccos(a*x)*ln(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*I*a^3*dilog(
1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*I*a^3*dilog(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, a x^{3} \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3}}{a^{2} x^{5} - x^{3}}\,{d x} - \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^4,x, algorithm="maxima")

[Out]

1/3*(12*a*x^3*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3/(a^2*x^5
- x^3), x) - arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^4)/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arccos \left (a x\right )^{4}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^4,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^4/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}^{4}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**4/x**4,x)

[Out]

Integral(acos(a*x)**4/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x\right )^{4}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^4,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^4/x^4, x)