3.39 \(\int \frac{\cos ^{-1}(a x)^4}{x^2} \, dx\)

Optimal. Leaf size=176 \[ 12 i a \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )-24 i a \text{PolyLog}\left (4,-i e^{i \cos ^{-1}(a x)}\right )+24 i a \text{PolyLog}\left (4,i e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right ) \]

[Out]

-(ArcCos[a*x]^4/x) - (8*I)*a*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (12*I)*a*ArcCos[a*x]^2*PolyLog[2, (-I)*
E^(I*ArcCos[a*x])] - (12*I)*a*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 24*a*ArcCos[a*x]*PolyLog[3, (-I)
*E^(I*ArcCos[a*x])] + 24*a*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (24*I)*a*PolyLog[4, (-I)*E^(I*ArcCos[
a*x])] + (24*I)*a*PolyLog[4, I*E^(I*ArcCos[a*x])]

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Rubi [A]  time = 0.204032, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4628, 4710, 4181, 2531, 6609, 2282, 6589} \[ 12 i a \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )-24 i a \text{PolyLog}\left (4,-i e^{i \cos ^{-1}(a x)}\right )+24 i a \text{PolyLog}\left (4,i e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^4/x^2,x]

[Out]

-(ArcCos[a*x]^4/x) - (8*I)*a*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (12*I)*a*ArcCos[a*x]^2*PolyLog[2, (-I)*
E^(I*ArcCos[a*x])] - (12*I)*a*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 24*a*ArcCos[a*x]*PolyLog[3, (-I)
*E^(I*ArcCos[a*x])] + 24*a*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (24*I)*a*PolyLog[4, (-I)*E^(I*ArcCos[
a*x])] + (24*I)*a*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 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]

Rubi steps

\begin{align*} \int \frac{\cos ^{-1}(a x)^4}{x^2} \, dx &=-\frac{\cos ^{-1}(a x)^4}{x}-(4 a) \int \frac{\cos ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\cos ^{-1}(a x)^4}{x}+(4 a) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-(12 a) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(12 a) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(24 i a) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(24 i a) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )+(24 a) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )-(24 a) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-(24 i a) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+(24 i a) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-24 i a \text{Li}_4\left (-i e^{i \cos ^{-1}(a x)}\right )+24 i a \text{Li}_4\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [B]  time = 1.08086, size = 549, normalized size = 3.12 \[ a \left (12 i \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{-i \cos ^{-1}(a x)}\right )+12 i \cos ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-12 i \pi \cos ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )+24 \cos ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-i \cos ^{-1}(a x)}\right )-24 \cos ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+3 i \pi \left (\pi -4 \cos ^{-1}(a x)\right ) \text{PolyLog}\left (2,i e^{-i \cos ^{-1}(a x)}\right )+3 i \pi ^2 \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-12 \pi \text{PolyLog}\left (3,i e^{-i \cos ^{-1}(a x)}\right )+12 \pi \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )-24 i \text{PolyLog}\left (4,-i e^{-i \cos ^{-1}(a x)}\right )-24 i \text{PolyLog}\left (4,-i e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^4}{a x}+i \cos ^{-1}(a x)^4-2 i \pi \cos ^{-1}(a x)^3+\frac{3}{2} i \pi ^2 \cos ^{-1}(a x)^2-\frac{1}{2} i \pi ^3 \cos ^{-1}(a x)+4 \cos ^{-1}(a x)^3 \log \left (1+i e^{-i \cos ^{-1}(a x)}\right )-4 \cos ^{-1}(a x)^3 \log \left (1+i e^{i \cos ^{-1}(a x)}\right )-6 \pi \cos ^{-1}(a x)^2 \log \left (1-i e^{-i \cos ^{-1}(a x)}\right )+6 \pi \cos ^{-1}(a x)^2 \log \left (1+i e^{i \cos ^{-1}(a x)}\right )+3 \pi ^2 \cos ^{-1}(a x) \log \left (1-i e^{-i \cos ^{-1}(a x)}\right )-3 \pi ^2 \cos ^{-1}(a x) \log \left (1+i e^{i \cos ^{-1}(a x)}\right )-\frac{1}{2} \pi ^3 \log \left (1+i e^{-i \cos ^{-1}(a x)}\right )+\frac{1}{2} \pi ^3 \log \left (1+i e^{i \cos ^{-1}(a x)}\right )+\frac{1}{2} \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (2 \cos ^{-1}(a x)+\pi \right )\right )\right )-\frac{7 i \pi ^4}{16}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCos[a*x]^4/x^2,x]

[Out]

a*(((-7*I)/16)*Pi^4 - (I/2)*Pi^3*ArcCos[a*x] + ((3*I)/2)*Pi^2*ArcCos[a*x]^2 - (2*I)*Pi*ArcCos[a*x]^3 + I*ArcCo
s[a*x]^4 - ArcCos[a*x]^4/(a*x) + 3*Pi^2*ArcCos[a*x]*Log[1 - I/E^(I*ArcCos[a*x])] - 6*Pi*ArcCos[a*x]^2*Log[1 -
I/E^(I*ArcCos[a*x])] - (Pi^3*Log[1 + I/E^(I*ArcCos[a*x])])/2 + 4*ArcCos[a*x]^3*Log[1 + I/E^(I*ArcCos[a*x])] +
(Pi^3*Log[1 + I*E^(I*ArcCos[a*x])])/2 - 3*Pi^2*ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] + 6*Pi*ArcCos[a*x]^2*L
og[1 + I*E^(I*ArcCos[a*x])] - 4*ArcCos[a*x]^3*Log[1 + I*E^(I*ArcCos[a*x])] + (Pi^3*Log[Tan[(Pi + 2*ArcCos[a*x]
)/4]])/2 + (12*I)*ArcCos[a*x]^2*PolyLog[2, (-I)/E^(I*ArcCos[a*x])] + (3*I)*Pi*(Pi - 4*ArcCos[a*x])*PolyLog[2,
I/E^(I*ArcCos[a*x])] + (3*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (12*I)*Pi*ArcCos[a*x]*PolyLog[2, (-I)*E
^(I*ArcCos[a*x])] + (12*I)*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + 24*ArcCos[a*x]*PolyLog[3, (-I)/E
^(I*ArcCos[a*x])] - 12*Pi*PolyLog[3, I/E^(I*ArcCos[a*x])] + 12*Pi*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - 24*ArcC
os[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I)/E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I
)*E^(I*ArcCos[a*x])])

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Maple [F]  time = 0.206, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{4}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^4/x^2,x)

[Out]

int(arccos(a*x)^4/x^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^2,x, algorithm="maxima")

[Out]

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

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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^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^2,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^4/x^2, 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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**4/x**2,x)

[Out]

Integral(acos(a*x)**4/x**2, 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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4/x^2,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^4/x^2, x)