3.30 \(\int \frac{\cos ^{-1}(a x)^3}{x^4} \, dx\)

Optimal. Leaf size=192 \[ i a^3 \cos ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-a^3 \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+a^3 \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}+a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a^2 \cos ^{-1}(a x)}{x}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^3}{3 x^3} \]

[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])]

________________________________________________________________________________________

Rubi [A]  time = 0.303869, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4628, 4702, 4710, 4181, 2531, 2282, 6589, 266, 63, 208} \[ i a^3 \cos ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-a^3 \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+a^3 \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}+a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a^2 \cos ^{-1}(a x)}{x}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^3}{3 x^3} \]

Antiderivative was successfully verified.

[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 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 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 266

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 63

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 208

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

Rubi steps

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

Mathematica [A]  time = 0.73871, size = 165, normalized size = 0.86 \[ -\frac{\cos ^{-1}(a x) \left (12 a^2 x^2+4 \cos ^{-1}(a x)^2-3 \cos ^{-1}(a x) \sin \left (2 \cos ^{-1}(a x)\right )\right )}{12 x^3}+a^3 \left (i \cos ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-i \cos ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-\text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+\text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-i \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )\right ) \]

Antiderivative was successfully verified.

[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]  time = 0.175, size = 272, normalized size = 1.4 \begin{align*}{\frac{a \left ( \arccos \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}\arccos \left ( ax \right ) }{x}}-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{3}}{3\,{x}^{3}}}-{\frac{{a}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1+i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) }+i{a}^{3}\arccos \left ( ax \right ){\it polylog} \left ( 2,-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) -{a}^{3}{\it polylog} \left ( 3,-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +{\frac{{a}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1-i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) }-i{a}^{3}\arccos \left ( ax \right ){\it polylog} \left ( 2,i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) +{a}^{3}{\it polylog} \left ( 3,i \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \right ) -2\,i{a}^{3}\arctan \left ( i\sqrt{-{a}^{2}{x}^{2}+1}+ax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arccos \left (a x\right )^{3}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(a*x)^3/x^4, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}^{3}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**3/x**4, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x)^3/x^4, x)