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

Optimal. Leaf size=387 \[ \frac{9 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x) \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac{5 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac{9 \text{PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{9 \text{PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{1}{4 a c^3 \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3} \]

[Out]

1/(4*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (x*ArcCosh[a*x])/(4*c^3*(1 - a^2*x^2)) + ArcCosh[a*x]^2/(4*a*c^3*(-
1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - (9*ArcCosh[a*x]^2)/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^
3)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x]^3)/(8*c^3*(1 - a^2*x^2)) - (5*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*
x]])/(a*c^3) + (3*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) - (5*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^3
) + (9*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) + (5*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^3) - (9*A
rcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3) - (9*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(4*a*c^3) +
(9*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(4*a*c^3) + (9*PolyLog[4, -E^ArcCosh[a*x]])/(4*a*c^3) - (9*PolyLo
g[4, E^ArcCosh[a*x]])/(4*a*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.814226, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5689, 5718, 74, 5694, 4182, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac{9 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x) \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac{5 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac{9 \text{PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{9 \text{PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{1}{4 a c^3 \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(4*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (x*ArcCosh[a*x])/(4*c^3*(1 - a^2*x^2)) + ArcCosh[a*x]^2/(4*a*c^3*(-
1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - (9*ArcCosh[a*x]^2)/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^
3)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x]^3)/(8*c^3*(1 - a^2*x^2)) - (5*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*
x]])/(a*c^3) + (3*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) - (5*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^3
) + (9*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) + (5*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^3) - (9*A
rcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3) - (9*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(4*a*c^3) +
(9*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(4*a*c^3) + (9*PolyLog[4, -E^ArcCosh[a*x]])/(4*a*c^3) - (9*PolyLo
g[4, E^ArcCosh[a*x]])/(4*a*c^3)

Rule 5689

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcCosh[c*x])^n)/(2*d*(p + 1)), x] + (-Dist[(b*c*n*(-d)^p)/(2*(p + 1)), Int[x*(1 + c*x)^(p + 1/2)
*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p
+ 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p,
-1] && IntegerQ[p]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 74

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

Rule 5694

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

Rule 4182

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

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]

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{\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{(3 a) \int \frac{x \cosh ^{-1}(a x)^2}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{4 c^3}+\frac{3 \int \frac{\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{\int \frac{\cosh ^{-1}(a x)}{\left (-1+a^2 x^2\right )^2} \, dx}{2 c^3}+\frac{(9 a) \int \frac{x \cosh ^{-1}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{8 c^3}+\frac{3 \int \frac{\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{\int \frac{\cosh ^{-1}(a x)}{-1+a^2 x^2} \, dx}{4 c^3}+\frac{9 \int \frac{\cosh ^{-1}(a x)}{-1+a^2 x^2} \, dx}{4 c^3}-\frac{3 \operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}-\frac{a \int \frac{x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{4 c^3}\\ &=\frac{1}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{\operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}-\frac{9 \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}+\frac{9 \operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=\frac{1}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{\operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=\frac{1}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=\frac{1}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac{5 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{9 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ &=\frac{1}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^3}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}+\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac{5 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{9 \text{Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{9 \text{Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ \end{align*}

Mathematica [A]  time = 8.253, size = 455, normalized size = 1.18 \[ -\frac{72 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )+144 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )-144 \cosh ^{-1}(a x) \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )+8 \left (9 \cosh ^{-1}(a x)^2-20\right ) \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )+160 \text{PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )+144 \text{PolyLog}\left (4,-e^{-\cosh ^{-1}(a x)}\right )+144 \text{PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x)^4-24 \cosh ^{-1}(a x)^3 \log \left (e^{-\cosh ^{-1}(a x)}+1\right )+24 \cosh ^{-1}(a x)^3 \log \left (1-e^{\cosh ^{-1}(a x)}\right )-160 \cosh ^{-1}(a x) \log \left (1-e^{-\cosh ^{-1}(a x)}\right )+160 \cosh ^{-1}(a x) \log \left (e^{-\cosh ^{-1}(a x)}+1\right )-\frac{16 \cosh ^{-1}(a x)^2 \sinh ^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )}{\left (\frac{a x-1}{a x+1}\right )^{3/2} (a x+1)^3}-40 \cosh ^{-1}(a x)^2 \tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )+8 \tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )+40 \cosh ^{-1}(a x)^2 \coth \left (\frac{1}{2} \cosh ^{-1}(a x)\right )-8 \coth \left (\frac{1}{2} \cosh ^{-1}(a x)\right )-\cosh ^{-1}(a x)^3 \text{csch}^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+6 \cosh ^{-1}(a x)^3 \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2 \text{csch}^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x) \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^3 \text{sech}^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+6 \cosh ^{-1}(a x)^3 \text{sech}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x) \text{sech}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+3 \pi ^4}{64 a c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(3*Pi^4 - 6*ArcCosh[a*x]^4 - 8*Coth[ArcCosh[a*x]/2] + 40*ArcCosh[a*x]^2*Coth[ArcCosh[a*x]/2] - 4*ArcCosh[a*x]
*Csch[ArcCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 - Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCo
sh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^4 - 160*ArcCosh[a*x]*Log[1 - E^(-ArcCos
h[a*x])] + 160*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^3*Log[1 + E^(-ArcCosh[a*x])] + 24*Arc
Cosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] + 8*(-20 + 9*ArcCosh[a*x]^2)*PolyLog[2, -E^(-ArcCosh[a*x])] + 160*PolyLog[
2, E^(-ArcCosh[a*x])] + 72*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] + 144*ArcCosh[a*x]*PolyLog[3, -E^(-ArcCos
h[a*x])] - 144*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] + 144*PolyLog[4, -E^(-ArcCosh[a*x])] + 144*PolyLog[4, E
^ArcCosh[a*x]] - 4*ArcCosh[a*x]*Sech[ArcCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + ArcCosh[a*x
]^3*Sech[ArcCosh[a*x]/2]^4 - (16*ArcCosh[a*x]^2*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x
)^3) + 8*Tanh[ArcCosh[a*x]/2] - 40*ArcCosh[a*x]^2*Tanh[ArcCosh[a*x]/2])/(64*a*c^3)

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Maple [A]  time = 0.171, size = 710, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*a^2/(a^4*x^4-2*a^2*x^2+1)/c^3*arccosh(a*x)^3*x^3-9/8*a/(a^4*x^4-2*a^2*x^2+1)/c^3*arccosh(a*x)^2*(a*x-1)^(
1/2)*(a*x+1)^(1/2)*x^2+1/4*a^2/(a^4*x^4-2*a^2*x^2+1)/c^3*x^3*arccosh(a*x)+1/4*a/(a^4*x^4-2*a^2*x^2+1)/c^3*(a*x
+1)^(1/2)*(a*x-1)^(1/2)*x^2+5/8/(a^4*x^4-2*a^2*x^2+1)/c^3*arccosh(a*x)^3*x+11/8/a/(a^4*x^4-2*a^2*x^2+1)/c^3*ar
ccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/4/(a^4*x^4-2*a^2*x^2+1)/c^3*x*arccosh(a*x)-1/4/a/(a^4*x^4-2*a^2*x^2
+1)/c^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+5/2/a/c^3*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+5/2*polylog(2
,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-5/2/a/c^3*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-5/2*polyl
og(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/8/a/c^3*arccosh(a*x)^3*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-9/
8*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+9/4*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)
*(a*x+1)^(1/2))/a/c^3-9/4*polylog(4,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+3/8/a/c^3*arccosh(a*x)^3*ln(1+a*x+(
a*x-1)^(1/2)*(a*x+1)^(1/2))+9/8*arccosh(a*x)^2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-9/4*arccosh(a
*x)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+9/4*polylog(4,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (6 \, a^{3} x^{3} - 10 \, a x - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3}}{16 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} - \int -\frac{3 \,{\left (6 \, a^{5} x^{5} - 16 \, a^{3} x^{3} +{\left (6 \, a^{4} x^{4} - 10 \, a^{2} x^{2} - 3 \,{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x - 1\right )\right )} \sqrt{a x + 1} \sqrt{a x - 1} + 10 \, a x - 3 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2}}{16 \,{\left (a^{7} c^{3} x^{7} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{3} c^{3} x^{3} - a c^{3} x +{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt{a x + 1} \sqrt{a x - 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(6*a^3*x^3 - 10*a*x - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1
))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) - integrate(-3/16*(6*a^5*x^5
- 16*a^3*x^3 + (6*a^4*x^4 - 10*a^2*x^2 - 3*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) + 3*(a^5*x^5 - 2*a^3*x^3
+ a*x)*log(a*x - 1))*sqrt(a*x + 1)*sqrt(a*x - 1) + 10*a*x - 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x +
1) + 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a^7*c^3*x
^7 - 3*a^5*c^3*x^5 + 3*a^3*c^3*x^3 - a*c^3*x + (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt(a*x +
1)*sqrt(a*x - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\operatorname{arcosh}\left (a x\right )^{3}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-arccosh(a*x)^3/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\operatorname{acosh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(acosh(a*x)**3/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)/c**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-arccosh(a*x)^3/(a^2*c*x^2 - c)^3, x)