\(\int \frac {(c-a^2 c x^2)^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx\) [414]

Optimal result

Integrand size = 24, antiderivative size = 329 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

-2/3*(-a^2*c*x^2+c)^(3/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)-2/3*c*erf(2*arccosh(a*x)^(1/2))*Pi^
(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-2/3*c*erfi(2*arccosh(a*x)^(1/2))*Pi^(1/2)*(-a^2*c*x^2
+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+2/3*c*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^
(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+2/3*c*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/
2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-16/3*c*x*(-a*x+1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5904, 5912, 5942, 5907, 3393, 3388, 2211, 2235, 2236, 5953, 5556} \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {\pi } c \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(3/2))/(3*a*ArcCosh[a*x]^(3/2)) - (16*c*x*(1 - a*x)*(1 + a*x)
*Sqrt[c - a^2*c*x^2])/(3*Sqrt[ArcCosh[a*x]]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(3
*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (2*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*S
qrt[-1 + a*x]*Sqrt[1 + a*x]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]) + (2*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*
Sqrt[1 + a*x])

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

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

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5904

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

Rule 5907

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Dist[(1/(b*c))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subst[Int[x^n*Sinh[-a/b
+ x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] &
& EqQ[e2, (-c)*d2] && IGtQ[2*p, 0]

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5942

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

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {\left (8 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x (-1+a x) (1+a x)}{\text {arccosh}(a x)^{3/2}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {\left (8 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (-1+a^2 x^2\right )}{\text {arccosh}(a x)^{3/2}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}+\frac {\left (16 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {\text {arccosh}(a x)}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (64 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{\sqrt {\text {arccosh}(a x)}} \, dx}{3 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}+\frac {\left (16 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (64 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {\left (16 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (64 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{8 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}+\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (4 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (8 c \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {16 c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 c \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {c e^{-4 \text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \left (-1-14 e^{4 \text {arccosh}(a x)}-e^{8 \text {arccosh}(a x)}+16 a^2 e^{4 \text {arccosh}(a x)} x^2+8 \text {arccosh}(a x)-8 e^{8 \text {arccosh}(a x)} \text {arccosh}(a x)+64 a e^{4 \text {arccosh}(a x)} x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)+64 a^2 e^{4 \text {arccosh}(a x)} x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)-16 e^{4 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )+16 \sqrt {2} e^{4 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+16 \sqrt {2} e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )-16 e^{4 \text {arccosh}(a x)} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )}{24 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^{3/2}} \]

[In]

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

[Out]

-1/24*(c*Sqrt[c - a^2*c*x^2]*(-1 - 14*E^(4*ArcCosh[a*x]) - E^(8*ArcCosh[a*x]) + 16*a^2*E^(4*ArcCosh[a*x])*x^2
+ 8*ArcCosh[a*x] - 8*E^(8*ArcCosh[a*x])*ArcCosh[a*x] + 64*a*E^(4*ArcCosh[a*x])*x*Sqrt[(-1 + a*x)/(1 + a*x)]*Ar
cCosh[a*x] + 64*a^2*E^(4*ArcCosh[a*x])*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] - 16*E^(4*ArcCosh[a*x])*(-A
rcCosh[a*x])^(3/2)*Gamma[1/2, -4*ArcCosh[a*x]] + 16*Sqrt[2]*E^(4*ArcCosh[a*x])*(-ArcCosh[a*x])^(3/2)*Gamma[1/2
, -2*ArcCosh[a*x]] + 16*Sqrt[2]*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/2, 2*ArcCosh[a*x]] - 16*E^(4*Arc
Cosh[a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/2, 4*ArcCosh[a*x]]))/(a*E^(4*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1
+ a*x)*ArcCosh[a*x]^(3/2))

Maple [F]

\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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