\(\int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [260]

Optimal result

Integrand size = 24, antiderivative size = 460 \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {6 a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \]

[Out]

3/2*a*arccosh(a*x)^2*(a*x-1)^(1/2)/x/(-a*x+1)^(1/2)-6*a^2*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))
*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+a^2*arccosh(a*x)^3*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-1)^(1/2)/(-a*x+1
)^(1/2)+3*I*a^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3/2*I*a^2*arccosh
(a*x)^2*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*I*a^2*polylog(2,I*(a*x+
(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+3/2*I*a^2*arccosh(a*x)^2*polylog(2,I*(a*x+(a*x-1)^(
1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+3*I*a^2*arccosh(a*x)*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)
^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-3*I*a^2*arccosh(a*x)*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x
-1)^(1/2)/(-a*x+1)^(1/2)-3*I*a^2*polylog(4,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+
3*I*a^2*polylog(4,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-1/2*arccosh(a*x)^3*(-a^2*x
^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5932, 5946, 4265, 2611, 6744, 2320, 6724, 5883, 5947, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {a^2 \sqrt {a x-1} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {6 a^2 \sqrt {a x-1} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {a x-1} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {a x-1} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {a x-1} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {a x-1} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {a x-1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {a x-1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {a x-1} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {a x-1} \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}+\frac {3 a \sqrt {a x-1} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}} \]

[In]

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

Rule 2317

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 2320

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 2438

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]

Rule 2611

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 4265

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5932

Int[((a_.) + ArcCosh[(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*ArcCosh[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*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/((1 + c*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, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5946

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c
*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rule 6724

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 6744

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}+\frac {1}{2} a^2 \int \frac {\text {arccosh}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx-\frac {\left (3 a \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx}{2 \sqrt {1-a x}} \\ & = \frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}+\frac {\left (a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {1-a x}}-\frac {\left (3 a^2 \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x}} \\ & = \frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {1-a x}}+\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 \sqrt {1-a x}}-\frac {\left (3 a^2 \sqrt {-1+a x}\right ) \text {Subst}(\int x \text {sech}(x) \, dx,x,\text {arccosh}(a x))}{\sqrt {1-a x}} \\ & = \frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {6 a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}}-\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}}+\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}}-\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {6 a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}}+\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {6 a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (3 i a^2 \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \\ & = \frac {3 a \sqrt {-1+a x} \text {arccosh}(a x)^2}{2 x \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 x^2}-\frac {6 a^2 \sqrt {-1+a x} \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a^2 \sqrt {-1+a x} \text {arccosh}(a x)^3 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{2 \sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {3 i a^2 \sqrt {-1+a x} \operatorname {PolyLog}\left (4,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1051\) vs. \(2(460)=920\).

Time = 4.23 (sec) , antiderivative size = 1051, normalized size of antiderivative = 2.28 \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {i a^2 (1+a x) \left (7 \pi ^4 \sqrt {\frac {-1+a x}{1+a x}}+8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+24 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2+\frac {192 i \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2}{a x}+\frac {64 i (-1+a x) \text {arccosh}(a x)^3}{a^2 x^2}-32 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3-16 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^4-384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1-i e^{-\text {arccosh}(a x)}\right )+8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \log \left (1+i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )+48 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1+i e^{-\text {arccosh}(a x)}\right )-96 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-64 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3 \log \left (1+i e^{-\text {arccosh}(a x)}\right )-48 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \log \left (1-i e^{\text {arccosh}(a x)}\right )+96 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2 \log \left (1-i e^{\text {arccosh}(a x)}\right )-8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \log \left (1+i e^{\text {arccosh}(a x)}\right )+64 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3 \log \left (1+i e^{\text {arccosh}(a x)}\right )+8 i \pi ^3 \sqrt {\frac {-1+a x}{1+a x}} \log \left (\tan \left (\frac {1}{4} (\pi +2 i \text {arccosh}(a x))\right )\right )-48 \sqrt {\frac {-1+a x}{1+a x}} \left (8+\pi ^2-4 i \pi \text {arccosh}(a x)-4 \text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )+192 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )-48 \pi ^2 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )+192 i \pi \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )-384 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )-192 i \pi \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (4,-i e^{-\text {arccosh}(a x)}\right )+384 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {PolyLog}\left (4,-i e^{\text {arccosh}(a x)}\right )\right )}{128 \sqrt {1-a^2 x^2}} \]

[In]

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

[Out]

((-1/128*I)*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 - 1
6*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)]*ArcCo
sh[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^ArcCo
sh[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^ArcCos
h[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]

Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{3}}{x^{3} \sqrt {-a^{2} x^{2}+1}}d x\]

[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)

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]

[In]

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

[Out]

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