\(\int \frac {(c-a^2 c x^2)^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx\) [408]

Optimal result

Integrand size = 24, antiderivative size = 433 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

-15/32*c^2*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)
+15/32*c^2*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
)+3/8*c^2*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)-3/8*c^2*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)-1/32*c^2*erf(6^(1/2)*arccosh(
a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1/32*c^2*erfi(6^(1/2)*arccosh(
a*x)^(1/2))*6^(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*(-a^2*c*x^2+c)^(5/2)*(a*x-1)
^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5904, 5912, 5952, 5556, 3389, 2211, 2235, 2236} \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {3 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {3 \sqrt {\pi } c^2 \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15 \sqrt {\frac {\pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {\sqrt {\frac {3 \pi }{2}} c^2 \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(5/2))/(a*Sqrt[ArcCosh[a*x]]) + (3*c^2*Sqrt[Pi]*Sqrt[c - a^2*
c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*
Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c^2*Sqrt[(3*Pi)/2]*Sqrt[c - a^2*c*x^2]
*Erf[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Er
fi[2*Sqrt[ArcCosh[a*x]]])/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqr
t[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*Sqrt[(3*Pi)/2]*Sqrt[c - a^2*c*x^2]*Erfi[S
qrt[6]*Sqrt[ArcCosh[a*x]]])/(16*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 3389

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

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 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 5952

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Subst[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, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\left (12 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x (-1+a x)^2 (1+a x)^2}{\sqrt {\text {arccosh}(a x)}} \, dx}{\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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\left (12 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (-1+a^2 x^2\right )^2}{\sqrt {\text {arccosh}(a x)}} \, dx}{\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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\left (12 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\left (12 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 \sqrt {x}}-\frac {\sinh (4 x)}{8 \sqrt {x}}+\frac {\sinh (6 x)}{32 \sqrt {x}}\right ) \, dx,x,\text {arccosh}(a x)\right )}{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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (6 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \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 )}{8 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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {x}} \, dx,x,\text {arccosh}(a x)\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \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 )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c^2 \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 )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c^2 \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 )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \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 )}{16 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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{2 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}(a x)}\right )}{8 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 )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {c^2 e^{-6 \text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \left (-1+6 e^{2 \text {arccosh}(a x)}+e^{4 \text {arccosh}(a x)}+52 e^{6 \text {arccosh}(a x)}+e^{8 \text {arccosh}(a x)}+6 e^{10 \text {arccosh}(a x)}-e^{12 \text {arccosh}(a x)}-64 a^2 e^{6 \text {arccosh}(a x)} x^2-16 e^{6 \text {arccosh}(a x)} \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+16 e^{6 \text {arccosh}(a x)} \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\sqrt {6} e^{6 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-6 \text {arccosh}(a x)\right )-12 e^{6 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-\sqrt {2} e^{6 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )-\sqrt {2} e^{6 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )-12 e^{6 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )+\sqrt {6} e^{6 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},6 \text {arccosh}(a x)\right )\right )}{32 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

(c^2*Sqrt[c - a^2*c*x^2]*(-1 + 6*E^(2*ArcCosh[a*x]) + E^(4*ArcCosh[a*x]) + 52*E^(6*ArcCosh[a*x]) + E^(8*ArcCos
h[a*x]) + 6*E^(10*ArcCosh[a*x]) - E^(12*ArcCosh[a*x]) - 64*a^2*E^(6*ArcCosh[a*x])*x^2 - 16*E^(6*ArcCosh[a*x])*
Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 16*E^(6*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[
a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[6]*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -6*ArcCosh[
a*x]] - 12*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - Sqrt[2]*E^(6*ArcCosh[a*x])*Sqr
t[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] - Sqrt[2]*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 2*ArcC
osh[a*x]] - 12*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]] + Sqrt[6]*E^(6*ArcCosh[a*x])*S
qrt[ArcCosh[a*x]]*Gamma[1/2, 6*ArcCosh[a*x]]))/(32*a*E^(6*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*S
qrt[ArcCosh[a*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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