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

Optimal. Leaf size=260 \[ \frac{3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{3 \text{PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{3 \text{PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac{3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

[Out]

(-3*ArcCosh[a*x]^2)/(2*a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^3)/(2*c^2*(1 - a^2*x^2)) - (6*Arc
Cosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) + (ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - (3*PolyLog[2,
-E^ArcCosh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^2) + (3*PolyLog[2, E^ArcCosh
[a*x]])/(a*c^2) - (3*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^2) - (3*ArcCosh[a*x]*PolyLog[3, -E^ArcC
osh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(a*c^2) + (3*PolyLog[4, -E^ArcCosh[a*x]])/(a*
c^2) - (3*PolyLog[4, E^ArcCosh[a*x]])/(a*c^2)

________________________________________________________________________________________

Rubi [A]  time = 0.441588, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5689, 5718, 5694, 4182, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac{3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{3 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{3 \text{PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{3 \text{PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac{3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*ArcCosh[a*x]^2)/(2*a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^3)/(2*c^2*(1 - a^2*x^2)) - (6*Arc
Cosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) + (ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - (3*PolyLog[2,
-E^ArcCosh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^2) + (3*PolyLog[2, E^ArcCosh
[a*x]])/(a*c^2) - (3*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^2) - (3*ArcCosh[a*x]*PolyLog[3, -E^ArcC
osh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(a*c^2) + (3*PolyLog[4, -E^ArcCosh[a*x]])/(a*
c^2) - (3*PolyLog[4, E^ArcCosh[a*x]])/(a*c^2)

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

Mathematica [A]  time = 2.25245, size = 276, normalized size = 1.06 \[ \frac{-24 \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-48 \cosh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )+48 \cosh ^{-1}(a x) \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )-24 \left (\cosh ^{-1}(a x)^2-2\right ) \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )-48 \text{PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )-48 \text{PolyLog}\left (4,-e^{-\cosh ^{-1}(a x)}\right )-48 \text{PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^4+8 \cosh ^{-1}(a x)^3 \log \left (e^{-\cosh ^{-1}(a x)}+1\right )-8 \cosh ^{-1}(a x)^3 \log \left (1-e^{\cosh ^{-1}(a x)}\right )+48 \cosh ^{-1}(a x) \log \left (1-e^{-\cosh ^{-1}(a x)}\right )-48 \cosh ^{-1}(a x) \log \left (e^{-\cosh ^{-1}(a x)}+1\right )+12 \cosh ^{-1}(a x)^2 \tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )-12 \cosh ^{-1}(a x)^2 \coth \left (\frac{1}{2} \cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^3 \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^3 \text{sech}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-\pi ^4}{16 a c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-Pi^4 + 2*ArcCosh[a*x]^4 - 12*ArcCosh[a*x]^2*Coth[ArcCosh[a*x]/2] - 2*ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 +
48*ArcCosh[a*x]*Log[1 - E^(-ArcCosh[a*x])] - 48*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] + 8*ArcCosh[a*x]^3*Lo
g[1 + E^(-ArcCosh[a*x])] - 8*ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] - 24*(-2 + ArcCosh[a*x]^2)*PolyLog[2, -E^(
-ArcCosh[a*x])] - 48*PolyLog[2, E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] - 48*ArcCosh
[a*x]*PolyLog[3, -E^(-ArcCosh[a*x])] + 48*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] - 48*PolyLog[4, -E^(-ArcCosh
[a*x])] - 48*PolyLog[4, E^ArcCosh[a*x]] - 2*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + 12*ArcCosh[a*x]^2*Tanh[Arc
Cosh[a*x]/2])/(16*a*c^2)

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Maple [A]  time = 0.095, size = 464, normalized size = 1.8 \begin{align*} -{\frac{x \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){c}^{2}}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,a \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{2\,a{c}^{2}}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,a{c}^{2}}{\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+3\,{\frac{{\rm arccosh} \left (ax\right ){\it polylog} \left ( 3,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}-3\,{\frac{{\it polylog} \left ( 4,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{2\,a{c}^{2}}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,a{c}^{2}}{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-3\,{\frac{{\rm arccosh} \left (ax\right ){\it polylog} \left ( 3,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}+3\,{\frac{{\it polylog} \left ( 4,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}+3\,{\frac{{\rm arccosh} \left (ax\right )\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}+3\,{\frac{{\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}-3\,{\frac{{\rm arccosh} \left (ax\right )\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}}-3\,{\frac{{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a^2*x^2-1)*arccosh(a*x)^3/c^2*x-3/2/a/(a^2*x^2-1)*arccosh(a*x)^2/c^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/2/a/c
^2*arccosh(a*x)^3*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/2*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)
^(1/2))/a/c^2+3*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3*polylog(4,a*x+(a*x-1)^(1/2)*(a
*x+1)^(1/2))/a/c^2+1/2/a/c^2*arccosh(a*x)^3*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/2*arccosh(a*x)^2*polylog(2
,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3*arccosh(a*x)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+3*po
lylog(4,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+3/a/c^2*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+3*p
olylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3/a/c^2*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-3*p
olylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2

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

[Out]

-1/4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^
3/(a^3*c^2*x^2 - a*c^2) - integrate(-3/4*(2*a^3*x^3 + (2*a^2*x^2 - (a^3*x^3 - a*x)*log(a*x + 1) + (a^3*x^3 - a
*x)*log(a*x - 1))*sqrt(a*x + 1)*sqrt(a*x - 1) - 2*a*x - (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) + (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))^2/(a^5*c^2*x^5 - 2*a^3*c^2*x^3 + a*c^2*x + (
a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*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^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^3/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), 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^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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