3.28 \(\int \frac{\cos ^{-1}(a x)^3}{x^2} \, dx\)

Optimal. Leaf size=122 \[ 6 i a \cos ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right ) \]

[Out]

-(ArcCos[a*x]^3/x) - (6*I)*a*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + (6*I)*a*ArcCos[a*x]*PolyLog[2, (-I)*E^(
I*ArcCos[a*x])] - (6*I)*a*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - 6*a*PolyLog[3, (-I)*E^(I*ArcCos[a*x])]
+ 6*a*PolyLog[3, I*E^(I*ArcCos[a*x])]

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Rubi [A]  time = 0.174938, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4628, 4710, 4181, 2531, 2282, 6589} \[ 6 i a \cos ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^3/x^2,x]

[Out]

-(ArcCos[a*x]^3/x) - (6*I)*a*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + (6*I)*a*ArcCos[a*x]*PolyLog[2, (-I)*E^(
I*ArcCos[a*x])] - (6*I)*a*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - 6*a*PolyLog[3, (-I)*E^(I*ArcCos[a*x])]
+ 6*a*PolyLog[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 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]

Rubi steps

\begin{align*} \int \frac{\cos ^{-1}(a x)^3}{x^2} \, dx &=-\frac{\cos ^{-1}(a x)^3}{x}-(3 a) \int \frac{\cos ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\cos ^{-1}(a x)^3}{x}+(3 a) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-(6 a) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(6 a) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+6 i a \cos ^{-1}(a x) \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(6 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(6 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+6 i a \cos ^{-1}(a x) \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+6 i a \cos ^{-1}(a x) \text{Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-6 a \text{Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+6 a \text{Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.109815, size = 139, normalized size = 1.14 \[ -\frac{\cos ^{-1}(a x)^3}{x}+3 a \left (2 i \cos ^{-1}(a x) \left (\text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )\right )-2 \text{PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+\cos ^{-1}(a x)^2 \left (\log \left (1-i e^{i \cos ^{-1}(a x)}\right )-\log \left (1+i e^{i \cos ^{-1}(a x)}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCos[a*x]^3/x^2,x]

[Out]

-(ArcCos[a*x]^3/x) + 3*a*(ArcCos[a*x]^2*(Log[1 - I*E^(I*ArcCos[a*x])] - Log[1 + I*E^(I*ArcCos[a*x])]) + (2*I)*
ArcCos[a*x]*(PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])]) - 2*PolyLog[3, (-I)*E^(I*Ar
cCos[a*x])] + 2*PolyLog[3, I*E^(I*ArcCos[a*x])])

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Maple [F]  time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^3/x^2,x)

[Out]

int(arccos(a*x)^3/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 )^{3} - 3 \, a x \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}\,{d x}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^3/x^2,x, algorithm="maxima")

[Out]

-(arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 - 3*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x
+ 1)*sqrt(-a*x + 1), a*x)^2/(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 )^{3}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^3/x^2,x, algorithm="fricas")

[Out]

integral(arccos(a*x)^3/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}^{3}{\left (a x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**3/x**2,x)

[Out]

Integral(acos(a*x)**3/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 )^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^3/x^2,x, algorithm="giac")

[Out]

integrate(arccos(a*x)^3/x^2, x)