3.252 \(\int \frac{\cosh ^{-1}(a x)^3}{(c-a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=607 \[ -\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{a x-1} \sqrt{a x+1}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - (x*ArcCosh[a*x])/(c^3*Sqrt[c -
a^2*c*x^2]) - (x*ArcCosh[a*x])/(10*c^3*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[-1 + a*x]*Sqrt[1 + a
*x]*ArcCosh[a*x]^2)/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a
*x]^2)/(5*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^3)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*Arc
Cosh[a*x]^3)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcCosh[a*x]^3)/(15*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a
*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x
^2])/(2*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x]
)])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, E^(2*ArcCosh[a*x])])/(5*a*c^3*S
qrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.07689, antiderivative size = 637, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {5713, 5691, 5688, 5715, 3716, 2190, 2531, 2282, 6589, 5716, 260, 261} \[ -\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{a x-1} \sqrt{a x+1}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(7/2),x]

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - (x*ArcCosh[a*x])/(c^3*Sqrt[c -
a^2*c*x^2]) - (x*ArcCosh[a*x])/(10*c^3*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[-1 + a*x]*Sqrt[1 + a
*x]*ArcCosh[a*x]^2)/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a
*x]^2)/(5*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (8*x*ArcCosh[a*x]^3)/(15*c^3*Sqrt[c - a^2*c*x^2]) + (x*Ar
cCosh[a*x]^3)/(5*c^3*(1 - a*x)^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) + (4*x*ArcCosh[a*x]^3)/(15*c^3*(1 - a*x)*(1
+ a*x)*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) -
(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (
Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(2*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]
*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*
PolyLog[3, E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5691

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

Rule 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5715

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

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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]

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 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 5716

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{(-1+a x)^{7/2} (1+a x)^{7/2}} \, dx}{c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\left (-1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{10 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (8 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\left (-1+a^2 x^2\right )^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{\left (-1+a^2 x^2\right )^2} \, dx}{10 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (8 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}-\frac{\left (8 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (4 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (16 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (16 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (8 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{-1+a x} \sqrt{1+a x}}{20 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^3 (1-a x) (1+a x) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.71108, size = 363, normalized size = 0.6 \[ -\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (96 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )-48 \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )+\frac{3}{1-a^2 x^2}+\frac{24 \cosh ^{-1}(a x)^2}{a^2 x^2-1}-\frac{9 \cosh ^{-1}(a x)^2}{\left (a^2 x^2-1\right )^2}-60 \log \left (\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right )+\frac{16 a x \left (\frac{a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x)^3}{(a x-1)^3}-\frac{32 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{a x-1}-\frac{12 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{(a x-1)^3 (a x+1)^2}-32 \cosh ^{-1}(a x)^3-\frac{6 a x \left (\frac{a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x)}{(a x-1)^3}+\frac{60 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)}{a x-1}+96 \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )+4 i \pi ^3\right )}{60 a c^3 \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(7/2),x]

[Out]

-(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*((4*I)*Pi^3 + 3/(1 - a^2*x^2) + (60*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcC
osh[a*x])/(-1 + a*x) - (6*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x])/(-1 + a*x)^3 - (9*ArcCosh[a*x]^2)/(-1
+ a^2*x^2)^2 + (24*ArcCosh[a*x]^2)/(-1 + a^2*x^2) - 32*ArcCosh[a*x]^3 - (32*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*Ar
cCosh[a*x]^3)/(-1 + a*x) + (16*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]^3)/(-1 + a*x)^3 - (12*a*x*Sqrt[(-
1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/((-1 + a*x)^3*(1 + a*x)^2) + 96*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])]
- 60*Log[Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)] + 96*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])] - 48*PolyLog[
3, E^(2*ArcCosh[a*x])]))/(60*a*c^3*Sqrt[c - a^2*c*x^2])

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Maple [B]  time = 0.359, size = 1319, normalized size = 2.2 \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)^(7/2),x)

[Out]

-1/60*(-c*(a^2*x^2-1))^(1/2)*(8*x^5*a^5-20*x^3*a^3-8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+15*a*x+16*(a*x+1)^(1/
2)*(a*x-1)^(1/2)*x^2*a^2-8*(a*x-1)^(1/2)*(a*x+1)^(1/2))*(24-192*arccosh(a*x)^2*x^8*a^8+840*arccosh(a*x)^2*x^6*
a^6+160*arccosh(a*x)^3*x^4*a^4-96*a^2*x^2-192*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)*x^7*a^7-192*arccosh(a*x
)^2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^7*a^7+852*arccosh(a*x)*x^6*a^6-1368*arccosh(a*x)^2*x^4*a^4+256*arccosh(a*x)^
3-264*arccosh(a*x)^2-936*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^3*x^3+372*arccosh(a*x)*a*x*(a*x-1)^(1/2)*(
a*x+1)^(1/2)+1410*a^2*x^2*arccosh(a*x)+105*a^3*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-45*(a*x+1)^(1/2)*(a*x-1)^(1/2)*
a*x+984*arccosh(a*x)^2*a^2*x^2-380*arccosh(a*x)^3*a^2*x^2-192*arccosh(a*x)*x^8*a^8+24*x^8*a^8-96*x^6*a^6+144*x
^4*a^4+756*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^5*x^5-480*arccosh(a*x)-1020*arccosh(a*x)^2*(a*x-1)^(1/2)
*(a*x+1)^(1/2)*a^3*x^3+495*arccosh(a*x)^2*a*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)+24*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^7*a
^7-84*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^5*a^5-1590*arccosh(a*x)*x^4*a^4+744*arccosh(a*x)^2*(a*x+1)^(1/2)*(a*x-1)^(
1/2)*x^5*a^5)/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x^4+287*a^2*x^2-64)/a/c^4-(a*x+1)^(1/2)*(a*x-1)^(1
/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)-1)+2*(a*x+1)^(1/2)*(a*x-1)^(1/
2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*ln(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-
c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-16/15*(a*x+1)^(1/2)*(a*x-1)^(1/2)
*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)^3+8/5*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2
)/c^4/a/(a^2*x^2-1)*arccosh(a*x)^2*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+16/5*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*
(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-16/5*(a*x+1)^(1/2
)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+8/5*(a*x+1
)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(
1/2))+16/5*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*arccosh(a*x)*polylog(2,-a*x-(a
*x-1)^(1/2)*(a*x+1)^(1/2))-16/5*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^4/a/(a^2*x^2-1)*polylog(3
,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))

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Maxima [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 )}^{\frac{7}{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)^(7/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^3/(-a^2*c*x^2 + c)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, 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)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^
4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/(-a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

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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 )}^{\frac{7}{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)^(7/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^3/(-a^2*c*x^2 + c)^(7/2), x)