\(\int \frac {\arccos (a x)^3}{x^4} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 192 \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=-\frac {a^2 \arccos (a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x^2}-\frac {\arccos (a x)^3}{3 x^3}-i a^3 \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i a^3 \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right ) \]

[Out]

-a^2*arccos(a*x)/x-1/3*arccos(a*x)^3/x^3-I*a^3*arccos(a*x)^2*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+a^3*arctanh((-a^
2*x^2+1)^(1/2))+I*a^3*arccos(a*x)*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-I*a^3*arccos(a*x)*polylog(2,I*(a*x+
I*(-a^2*x^2+1)^(1/2)))-a^3*polylog(3,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+a^3*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2))
)+1/2*a*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4724, 4790, 4804, 4266, 2611, 2320, 6724, 272, 65, 214} \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=-i a^3 \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+i a^3 \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i a^3 \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x^2}-\frac {a^2 \arccos (a x)}{x}+a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arccos (a x)^3}{3 x^3} \]

[In]

Int[ArcCos[a*x]^3/x^4,x]

[Out]

-((a^2*ArcCos[a*x])/x) + (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(2*x^2) - ArcCos[a*x]^3/(3*x^3) - I*a^3*ArcCos[a*
x]^2*ArcTan[E^(I*ArcCos[a*x])] + a^3*ArcTanh[Sqrt[1 - a^2*x^2]] + I*a^3*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCo
s[a*x])] - I*a^3*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - a^3*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + a^3*Po
lyLog[3, I*E^(I*ArcCos[a*x])]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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]

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.86 \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=a^3 \left (-i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+\operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )-\frac {\arccos (a x) \left (12 a^2 x^2+4 \arccos (a x)^2-3 \arccos (a x) \sin (2 \arccos (a x))\right )}{12 x^3} \]

[In]

Integrate[ArcCos[a*x]^3/x^4,x]

[Out]

a^3*((-I)*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + ArcTanh[Sqrt[1 - a^2*x^2]] + I*ArcCos[a*x]*PolyLog[2, (-I)
*E^(I*ArcCos[a*x])] - I*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + Pol
yLog[3, I*E^(I*ArcCos[a*x])]) - (ArcCos[a*x]*(12*a^2*x^2 + 4*ArcCos[a*x]^2 - 3*ArcCos[a*x]*Sin[2*ArcCos[a*x]])
)/(12*x^3)

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arccos \left (a x \right ) \left (-3 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +2 \arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}-\frac {\arccos \left (a x \right )^{2} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}+i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-\operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {\arccos \left (a x \right )^{2} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \arctan \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\)	\(256\)
default	\(a^{3} \left (-\frac {\arccos \left (a x \right ) \left (-3 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +2 \arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}-\frac {\arccos \left (a x \right )^{2} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}+i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-\operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {\arccos \left (a x \right )^{2} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \arctan \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\)	\(256\)

[In]

int(arccos(a*x)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{4}} \,d x } \]

[In]

integrate(arccos(a*x)^3/x^4,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^3/x^4, x)

Sympy [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

integrate(acos(a*x)**3/x**4,x)

[Out]

Integral(acos(a*x)**3/x**4, x)

Maxima [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{4}} \,d x } \]

[In]

integrate(arccos(a*x)^3/x^4,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{4}} \,d x } \]

[In]

integrate(arccos(a*x)^3/x^4,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^3/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^4} \,d x \]

[In]

int(acos(a*x)^3/x^4,x)

[Out]

int(acos(a*x)^3/x^4, x)