3.251 \(\int \frac {\cosh ^{-1}(a x)^3}{(c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=413 \[ -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

1/3*x*arccosh(a*x)^3/c/(-a^2*c*x^2+c)^(3/2)-x*arccosh(a*x)/c^2/(-a^2*c*x^2+c)^(1/2)+2/3*x*arccosh(a*x)^3/c^2/(
-a^2*c*x^2+c)^(1/2)+1/2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2)+2/3
*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*arccosh(a*x)^2*ln(1-(a*x+(a*x-1)^(1/2
)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+1/2*ln(-a^2*x^2+1)*(a*x-1)^(1/2)*(a
*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*arccosh(a*x)*polylog(2,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(
1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+polylog(3,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*
x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.64, antiderivative size = 428, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5713, 5691, 5688, 5715, 3716, 2190, 2531, 2282, 6589, 5716, 260} \[ -\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{3 c^2 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \cosh ^{-1}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]

[Out]

-((x*ArcCosh[a*x])/(c^2*Sqrt[c - a^2*c*x^2])) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*a*c^2*(1 - a^
2*x^2)*Sqrt[c - a^2*c*x^2]) + (2*x*ArcCosh[a*x]^3)/(3*c^2*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^3)/(3*c^2*(1
- a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(3*a*c^2*Sqrt[c - a^2*
c*x^2]) - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^
2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(2*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*
x]*PolyLog[3, E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2])

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

________________________________________________________________________________________

Mathematica [C]  time = 1.00, size = 258, normalized size = 0.62 \[ \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \left (\frac {6 \cosh ^{-1}(a x)^2}{1-a^2 x^2}-24 \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )+12 \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )+12 \log \left (\sqrt {\frac {a x-1}{a x+1}} (a x+1)\right )-\frac {4 a x \left (\frac {a x-1}{a x+1}\right )^{3/2} \cosh ^{-1}(a x)^3}{(a x-1)^3}+\frac {8 a x \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{a x-1}+8 \cosh ^{-1}(a x)^3-\frac {12 a x \sqrt {\frac {a x-1}{a x+1}} \cosh ^{-1}(a x)}{a x-1}-24 \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )-i \pi ^3\right )}{12 a c^2 \sqrt {c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]

[Out]

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*((-I)*Pi^3 - (12*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x])/(-1 + a*x)
+ (6*ArcCosh[a*x]^2)/(1 - a^2*x^2) + 8*ArcCosh[a*x]^3 + (8*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/(-1
+ a*x) - (4*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]^3)/(-1 + a*x)^3 - 24*ArcCosh[a*x]^2*Log[1 - E^(2*Ar
cCosh[a*x])] + 12*Log[Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)] - 24*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])] +
12*PolyLog[3, E^(2*ArcCosh[a*x])]))/(12*a*c^2*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{3}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, choosing root of [1,0,%%%{2,[2,1,2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{-2,[0,1,0]%%%}+%%%{2,[0,0,0]%%%},0,%%%{1,[
4,2,4]%%%}+%%%{2,[4,1,4]%%%}+%%%{1,[4,0,4]%%%}+%%%{-2,[2,2,2]%%%}+%%%{-4,[2,1,2]%%%}+%%%{-2,[2,0,2]%%%}+%%%{1,
[0,2,0]%%%}+%%%{2,[0,1,0]%%%}+%%%{1,[0,0,0]%%%}] at parameters values [86,-97,-82]sym2poly/r2sym(const gen & e
,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.56, size = 955, normalized size = 2.31 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-3 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \mathrm {arccosh}\left (a x \right ) \left (6 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{3} x^{3}+6 a^{4} x^{4} \mathrm {arccosh}\left (a x \right )+6 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+6 x^{4} a^{4}+6 a^{2} x^{2} \mathrm {arccosh}\left (a x \right )^{2}-9 \,\mathrm {arccosh}\left (a x \right ) a x \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{2} x^{2} \mathrm {arccosh}\left (a x \right )-6 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -18 a^{2} x^{2}-8 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+12\right )}{6 \left (3 x^{6} a^{6}-10 x^{4} a^{4}+11 a^{2} x^{2}-4\right ) a \,c^{3}}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}-1\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{3 c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x)

[Out]

-1/6*(-c*(a^2*x^2-1))^(1/2)*(2*x^3*a^3-3*a*x-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*(a*x-1)^(1/2)*(a*x+1)^(1/
2))*arccosh(a*x)*(6*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^3*x^3+6*a^4*x^4*arccosh(a*x)+6*a^3*x^3*(a*x-1)^
(1/2)*(a*x+1)^(1/2)+6*x^4*a^4+6*a^2*x^2*arccosh(a*x)^2-9*arccosh(a*x)*a*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)-12*a^2*x
^2*arccosh(a*x)-6*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-18*a^2*x^2-8*arccosh(a*x)^2+6*arccosh(a*x)+12)/(3*a^6*x^6-10
*a^4*x^4+11*a^2*x^2-4)/a/c^3+2*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*ln(a*x+(a*
x-1)^(1/2)*(a*x+1)^(1/2))-(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*ln(a*x+(a*x-1)^
(1/2)*(a*x+1)^(1/2)-1)-(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*ln(1+a*x+(a*x-1)^(
1/2)*(a*x+1)^(1/2))-4/3*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^3+2*
(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^2*ln(1-a*x-(a*x-1)^(1/2)*(a*
x+1)^(1/2))+4*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)*polylog(2,a*x+
(a*x-1)^(1/2)*(a*x+1)^(1/2))-4*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*polylog(3,
a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+2*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2*x^2-1)*arccos
h(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+4*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/c^3/a/(a^2
*x^2-1)*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-4*(-c*(a^2*x^2-1))^(1/2)*(a*x-1)^(1/2)*(a*x+1
)^(1/2)/c^3/a/(a^2*x^2-1)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^3/(-a^2*c*x^2 + c)^(5/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3/(c - a^2*c*x^2)^(5/2),x)

[Out]

int(acosh(a*x)^3/(c - a^2*c*x^2)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(acosh(a*x)**3/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

________________________________________________________________________________________