Integrand size = 10, antiderivative size = 174 \[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \text {arccosh}(a x)}{4 x^2}+\frac {1}{2} a^4 \text {arccosh}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{4 x^4}-a^4 \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} a^4 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right ) \]
1/4*a^2*arccosh(a*x)/x^2+1/2*a^4*arccosh(a*x)^2-1/4*arccosh(a*x)^3/x^4-a^4 *arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-1/2*a^4*polylog(2, -(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-1/4*a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/ x+1/4*a*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^3+1/2*a^3*arccosh(a*x )^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x
Time = 0.51 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=\frac {a^3 x^3-a^5 x^5-a x (1+a x) \left (1-a x+2 a^2 x^2+2 a^3 x^3 \left (-1+\sqrt {\frac {-1+a x}{1+a x}}\right )\right ) \text {arccosh}(a x)^2-\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3-a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x) \left (-1+4 a^2 x^2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+2 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )}{4 x^4 \sqrt {-1+a x} \sqrt {1+a x}} \]
(a^3*x^3 - a^5*x^5 - a*x*(1 + a*x)*(1 - a*x + 2*a^2*x^2 + 2*a^3*x^3*(-1 + Sqrt[(-1 + a*x)/(1 + a*x)]))*ArcCosh[a*x]^2 - Sqrt[-1 + a*x]*Sqrt[1 + a*x] *ArcCosh[a*x]^3 - a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x ]*(-1 + 4*a^2*x^2*Log[1 + E^(-2*ArcCosh[a*x])]) + 2*a^4*x^4*Sqrt[(-1 + a*x )/(1 + a*x)]*(1 + a*x)*PolyLog[2, -E^(-2*ArcCosh[a*x])])/(4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6298, 6348, 6298, 106, 6333, 6297, 3042, 26, 4201, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {3}{4} a \int \frac {\text {arccosh}(a x)^2}{x^4 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6348 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {2}{3} a \int \frac {\text {arccosh}(a x)}{x^3}dx+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}\right )-\frac {\text {arccosh}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {2}{3} a \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)}{2 x^2}\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}\right )-\frac {\text {arccosh}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 106 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {a x-1} \sqrt {a x+1}}dx+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )-\frac {\text {arccosh}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6333 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}-2 a \int \frac {\text {arccosh}(a x)}{x}dx\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )-\frac {\text {arccosh}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}-2 a \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \text {arccosh}(a x)}{a x}d\text {arccosh}(a x)\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )-\frac {\text {arccosh}(a x)^3}{4 x^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^3}{4 x^4}+\frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}-2 a \int -i \text {arccosh}(a x) \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^3}{4 x^4}+\frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}+2 i a \int \text {arccosh}(a x) \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^3}{4 x^4}+\frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}+2 i a \left (2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)}{1+e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \text {arccosh}(a x)^2\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^3}{4 x^4}+\frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}+2 i a \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^3}{4 x^4}+\frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}+2 i a \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \log \left (1+e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^3}{4 x^4}+\frac {3}{4} a \left (\frac {2}{3} a^2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{x}+2 i a \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )\right )+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{3 x^3}-\frac {2}{3} a \left (\frac {a \sqrt {a x-1} \sqrt {a x+1}}{2 x}-\frac {\text {arccosh}(a x)}{2 x^2}\right )\right )\) |
-1/4*ArcCosh[a*x]^3/x^4 + (3*a*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] ^2)/(3*x^3) - (2*a*((a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*x) - ArcCosh[a*x]/ (2*x^2)))/3 + (2*a^2*((Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/x + (2 *I)*a*((-1/2*I)*ArcCosh[a*x]^2 + (2*I)*((ArcCosh[a*x]*Log[1 + E^(2*ArcCosh [a*x])])/2 + PolyLog[2, -E^(2*ArcCosh[a*x])]/4))))/3))/4
3.1.31.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(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] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Sim p[(d2 + e2*x)^p/(-1 + c*x)^p] Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(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] + (Simp[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] + Simp[b*c*(n/(f *(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && Eq Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.23 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(a^{4} \left (-\frac {-2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2}-a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-a^{4} x^{4}+\operatorname {arccosh}\left (a x \right )^{3}-a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{4 a^{4} x^{4}}+\operatorname {arccosh}\left (a x \right )^{2}-\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\) | \(191\) |
| default | \(a^{4} \left (-\frac {-2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2}-a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-a^{4} x^{4}+\operatorname {arccosh}\left (a x \right )^{3}-a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{4 a^{4} x^{4}}+\operatorname {arccosh}\left (a x \right )^{2}-\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\) | \(191\) |
a^4*(-1/4*(-2*a^3*x^3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a^4*x^4 *arccosh(a*x)^2-a*x*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+a^3*x^3*(a* x-1)^(1/2)*(a*x+1)^(1/2)-a^4*x^4+arccosh(a*x)^3-a^2*x^2*arccosh(a*x))/a^4/ x^4+arccosh(a*x)^2-arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)- 1/2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))
\[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{5}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{5}} \,d x } \]
-1/4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^4 + integrate(3/4*(a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a^3*x^7 - a*x^5 + (a^2*x^6 - x^4)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
Exception generated. \[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^5} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^5} \,d x \]