\(\int \frac {\arccos (a x)^4}{x^4} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 304 \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \]

[Out]

-2*a^2*arccos(a*x)^2/x-1/3*arccos(a*x)^4/x^3-8*I*a^3*arccos(a*x)*arctan(a*x+I*(-a^2*x^2+1)^(1/2))-4/3*I*a^3*ar
ccos(a*x)^3*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+4*I*a^3*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+2*I*a^3*arccos(a
*x)^2*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*I*a^3*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-2*I*a^3*arccos(
a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*a^3*arccos(a*x)*polylog(3,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*a^
3*arccos(a*x)*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*I*a^3*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*I*a^3
*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+2/3*a*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4724, 4790, 4804, 4266, 2611, 6744, 2320, 6724, 2317, 2438} \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {2 a^2 \arccos (a x)^2}{x}-\frac {\arccos (a x)^4}{3 x^3} \]

[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 2317

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 2320

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 2438

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]

Rule 2611

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 4266

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 4724

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 4790

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/(f*(m + 1)))*Simp[(d + e*x
^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4804

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(-(c^(m
+ 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], 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] && IGtQ[n, 0] && IntegerQ[m]

Rule 6724

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 6744

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^4}{3 x^3}-\frac {1}{3} (4 a) \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac {\arccos (a x)^2}{x^2} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\arccos (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arccos (a x)\right )-\left (4 a^3\right ) \int \frac {\arccos (a x)}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )-\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+\left (4 a^3\right ) \text {Subst}(\int x \sec (x) \, dx,x,\arccos (a x)) \\ & = -\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arccos (a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )+\left (4 a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \arccos (a x)}\right ) \\ & = -\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1475\) vs. \(2(304)=608\).

Time = 12.06 (sec) , antiderivative size = 1475, normalized size of antiderivative = 4.85 \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=a^3 \left (-\frac {1}{6} \arccos (a x)^2 \left (12+\arccos (a x)^2\right )+4 \left (\arccos (a x) \left (\log \left (1-i e^{i \arccos (a x)}\right )-\log \left (1+i e^{i \arccos (a x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )\right )+\frac {2}{3} \left (\frac {1}{8} \pi ^3 \log \left (\cot \left (\frac {1}{2} \left (\frac {\pi }{2}-\arccos (a x)\right )\right )\right )+\frac {3}{4} \pi ^2 \left (\left (\frac {\pi }{2}-\arccos (a x)\right ) \left (\log \left (1-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\log \left (1+e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\operatorname {PolyLog}\left (2,e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )\right )\right )-\frac {3}{2} \pi \left (\left (\frac {\pi }{2}-\arccos (a x)\right )^2 \left (\log \left (1-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\log \left (1+e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )\right )+2 i \left (\frac {\pi }{2}-\arccos (a x)\right ) \left (\operatorname {PolyLog}\left (2,-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\operatorname {PolyLog}\left (2,e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )\right )+2 \left (-\operatorname {PolyLog}\left (3,-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )+\operatorname {PolyLog}\left (3,e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )\right )\right )+8 \left (\frac {1}{64} i \left (\frac {\pi }{2}-\arccos (a x)\right )^4+\frac {1}{4} i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )^4-\frac {1}{8} \left (\frac {\pi }{2}-\arccos (a x)\right )^3 \log \left (1+e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\frac {1}{8} \pi ^3 \left (i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )-\log \left (1+e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )\right )-\left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )^3 \log \left (1+e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )+\frac {3}{8} i \left (\frac {\pi }{2}-\arccos (a x)\right )^2 \operatorname {PolyLog}\left (2,-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )+\frac {3}{4} \pi ^2 \left (\frac {1}{2} i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )^2-\left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right ) \log \left (1+e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )\right )+\frac {3}{2} i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )^2 \operatorname {PolyLog}\left (2,-e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )-\frac {3}{4} \left (\frac {\pi }{2}-\arccos (a x)\right ) \operatorname {PolyLog}\left (3,-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\frac {3}{2} \pi \left (\frac {1}{3} i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )^3-\left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )^2 \log \left (1+e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )+i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right ) \operatorname {PolyLog}\left (2,-e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )\right )-\frac {3}{2} \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right ) \operatorname {PolyLog}\left (3,-e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,-e^{i \left (\frac {\pi }{2}-\arccos (a x)\right )}\right )-\frac {3}{4} i \operatorname {PolyLog}\left (4,-e^{2 i \left (\frac {\pi }{2}+\frac {1}{2} \left (-\frac {\pi }{2}+\arccos (a x)\right )\right )}\right )\right )\right )-\frac {-4 \arccos (a x)^3+\arccos (a x)^4}{12 \left (\cos \left (\frac {1}{2} \arccos (a x)\right )-\sin \left (\frac {1}{2} \arccos (a x)\right )\right )^2}-\frac {\arccos (a x)^4 \sin \left (\frac {1}{2} \arccos (a x)\right )}{6 \left (\cos \left (\frac {1}{2} \arccos (a x)\right )-\sin \left (\frac {1}{2} \arccos (a x)\right )\right )^3}+\frac {\arccos (a x)^4 \sin \left (\frac {1}{2} \arccos (a x)\right )}{6 \left (\cos \left (\frac {1}{2} \arccos (a x)\right )+\sin \left (\frac {1}{2} \arccos (a x)\right )\right )^3}-\frac {4 \arccos (a x)^3+\arccos (a x)^4}{12 \left (\cos \left (\frac {1}{2} \arccos (a x)\right )+\sin \left (\frac {1}{2} \arccos (a x)\right )\right )^2}-\frac {-12 \arccos (a x)^2 \sin \left (\frac {1}{2} \arccos (a x)\right )-\arccos (a x)^4 \sin \left (\frac {1}{2} \arccos (a x)\right )}{6 \left (\cos \left (\frac {1}{2} \arccos (a x)\right )+\sin \left (\frac {1}{2} \arccos (a x)\right )\right )}-\frac {12 \arccos (a x)^2 \sin \left (\frac {1}{2} \arccos (a x)\right )+\arccos (a x)^4 \sin \left (\frac {1}{2} \arccos (a x)\right )}{6 \left (\cos \left (\frac {1}{2} \arccos (a x)\right )-\sin \left (\frac {1}{2} \arccos (a x)\right )\right )}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arccos \left (a x \right )^{2} \left (-2 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +\arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}-\frac {2 \arccos \left (a x \right )^{3} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+2 i \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {polylog}\left (4, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {2 \arccos \left (a x \right )^{3} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}-2 i \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {polylog}\left (4, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\)	\(419\)
default	\(a^{3} \left (-\frac {\arccos \left (a x \right )^{2} \left (-2 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +\arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}-\frac {2 \arccos \left (a x \right )^{3} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+2 i \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {polylog}\left (4, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {2 \arccos \left (a x \right )^{3} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}-2 i \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {polylog}\left (4, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\)	\(419\)

[In]

int(arccos(a*x)^4/x^4,x,method=_RETURNVERBOSE)

[Out]

a^3*(-1/3/a^3/x^3*arccos(a*x)^2*(-2*(-a^2*x^2+1)^(1/2)*arccos(a*x)*a*x+arccos(a*x)^2+6*a^2*x^2)-2/3*arccos(a*x
)^3*ln(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))+2*I*polylog(2,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))*arccos(a*x)^2-4*arccos(a*x
)*polylog(3,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*I*polylog(4,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))+2/3*arccos(a*x)^3*ln(1
-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-2*I*polylog(2,I*(I*(-a^2*x^2+1)^(1/2)+a*x))*arccos(a*x)^2+4*arccos(a*x)*polylog
(3,I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*I*polylog(4,I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*arccos(a*x)*ln(1+I*(I*(-a^2*x^2
+1)^(1/2)+a*x))+4*arccos(a*x)*ln(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*I*dilog(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*I
*dilog(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x)))

Fricas [F]

\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{4}} \,d x } \]

[In]

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

[Out]

integral(arccos(a*x)^4/x^4, x)

Sympy [F]

\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

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

[Out]

Integral(acos(a*x)**4/x**4, x)

Maxima [F]

\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{4}} \,d x } \]

[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

Giac [F]

\[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate(arccos(a*x)^4/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^4} \,d x \]

[In]

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

[Out]

int(acos(a*x)^4/x^4, x)