3.260 \(\int \frac{\cosh ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=460 \[ -\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{3 i a^2 \sqrt{a x-1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{3 i a^2 \sqrt{a x-1} \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{6 a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{3 a \sqrt{a x-1} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a x}} \]

[Out]

(3*a*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(2*x*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(2*x^2) - (6*a^2*
Sqrt[-1 + a*x]*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^3*ArcTan[
E^ArcCosh[a*x]])/Sqrt[1 - a*x] + ((3*I)*a^2*Sqrt[-1 + a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((
(3*I)/2)*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((3*I)*a^2*Sqrt[-1
+ a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (((3*I)/2)*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2,
I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + ((3*I)*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqr
t[1 - a*x] - ((3*I)*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((3*I)*a^2*S
qrt[-1 + a*x]*PolyLog[4, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + ((3*I)*a^2*Sqrt[-1 + a*x]*PolyLog[4, I*E^ArcCos
h[a*x]])/Sqrt[1 - a*x]

________________________________________________________________________________________

Rubi [A]  time = 1.01337, antiderivative size = 614, normalized size of antiderivative = 1.33, number of steps used = 19, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {5798, 5748, 5761, 4180, 2531, 6609, 2282, 6589, 5662, 2279, 2391} \[ -\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{6 a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]

[Out]

(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x*Sqrt[1 - a^2*x^2]) - ((1 - a*x)*(1 + a*x)*ArcCosh[a*x]^
3)/(2*x^2*Sqrt[1 - a^2*x^2]) - (6*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/Sqrt[1
- a^2*x^2] + (a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] + ((3
*I)*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] - (((3*I)/2)*a^2*Sqrt[
-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] - ((3*I)*a^2*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] + (((3*I)/2)*a^2*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] + ((3*I)*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]
*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] - ((3*I)*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Arc
Cosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] - ((3*I)*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[4,
(-I)*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] + ((3*I)*a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[4, I*E^ArcCosh[a*x]
])/Sqrt[1 - a^2*x^2]

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5748

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b
*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, 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]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{x^2} \, dx}{2 \sqrt{1-a^2 x^2}}+\frac{\left (a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{\left (a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^3 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}-\frac{\left (3 a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}+\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}-\frac{\left (3 a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{6 a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{6 a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{6 a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{6 a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [B]  time = 6.07934, size = 1051, normalized size = 2.28 \[ -\frac{i a^2 (a x+1) \left (-16 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^4+\frac{64 i (a x-1) \cosh ^{-1}(a x)^3}{a^2 x^2}-64 \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{-\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)^3+64 \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)^3-32 i \pi \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3-96 i \pi \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{-\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)^2+96 i \pi \sqrt{\frac{a x-1}{a x+1}} \log \left (1-i e^{\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)^2+192 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)^2+\frac{192 i \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^2}{a x}+24 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^2-384 \sqrt{\frac{a x-1}{a x+1}} \log \left (1-i e^{-\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)+48 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{-\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)+384 \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{-\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)-48 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \log \left (1-i e^{\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)+192 i \pi \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)+384 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)-384 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right ) \cosh ^{-1}(a x)+8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)+8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \log \left (\tan \left (\frac{1}{4} \left (2 i \cosh ^{-1}(a x)+\pi \right )\right )\right )-48 \sqrt{\frac{a x-1}{a x+1}} \left (-4 \cosh ^{-1}(a x)^2-4 i \pi \cosh ^{-1}(a x)+\pi ^2+8\right ) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+384 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )-48 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+192 i \pi \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-192 i \pi \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )+384 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (4,-i e^{-\cosh ^{-1}(a x)}\right )+384 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+7 \pi ^4 \sqrt{\frac{a x-1}{a x+1}}\right )}{128 \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]

[Out]

((-I/128)*a^2*(1 + a*x)*(7*Pi^4*Sqrt[(-1 + a*x)/(1 + a*x)] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x
] + 24*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 + ((192*I)*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2)/(a
*x) + ((64*I)*(-1 + a*x)*ArcCosh[a*x]^3)/(a^2*x^2) - (32*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3 - 16*
Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^4 - 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*
x]] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh
[a*x]*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] -
(96*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*Sqrt[(-1 + a*x)/(1 + a*x)]*
ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I*E^ArcCosh
[a*x]] + (96*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I)*Pi^3*Sqrt[(-1 +
a*x)/(1 + a*x)]*Log[1 + I*E^ArcCosh[a*x]] + 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[
a*x]] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4]] - 48*Sqrt[(-1 + a*x)/(1 +
a*x)]*(8 + Pi^2 - (4*I)*Pi*ArcCosh[a*x] - 4*ArcCosh[a*x]^2)*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a
*x)/(1 + a*x)]*PolyLog[2, I/E^ArcCosh[a*x]] + 192*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^
ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*Sqrt[(-1 + a*x)/(
1 + a*x)]*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[3, (-I)/E^
ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 384*Sqrt[(-1 + a
*x)/(1 + a*x)]*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - (192*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[3,
I*E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[4, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 +
a*x)]*PolyLog[4, (-I)*E^ArcCosh[a*x]]))/Sqrt[1 - a^2*x^2]

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{{x}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x)

[Out]

int(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )^{3}}{a^{2} x^{5} - x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3/(a^2*x^5 - x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{3}{\left (a x \right )}}{x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/x**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(acosh(a*x)**3/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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