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

Optimal. Leaf size=329 \[ -\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \left (c-a^2 c x^2\right )^{3/2}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.745105, antiderivative size = 337, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {5713, 5697, 5776, 5701, 3312, 3307, 2180, 2204, 2205, 5781, 5448} \[ -\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 c (a x+1)^{3/2} (1-a x)^2 \sqrt{c-a^2 c x^2}}{3 a \sqrt{a x-1} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*(1 - a*x)^2*(1 + a*x)^(3/2)*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[-1 + a*x]*ArcCosh[a*x]^(3/2)) - (16*c*x*(1 - a
^2*x^2)*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*Sqrt[-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 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 5697

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

Rule 5776

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

Rule 5701

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

Rule 3312

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 3307

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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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 5781

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

Rule 5448

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]

Rubi steps

\begin{align*} \int \frac{\left (c-a^2 c x^2\right )^{3/2}}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int \frac{(-1+a x)^{3/2} (1+a x)^{3/2}}{\cosh ^{-1}(a x)^{5/2}} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(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 )}{\cosh ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(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{\cosh ^{-1}(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{\cosh ^{-1}(a x)}} \, dx}{3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (16 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int \left (-\frac{1}{8 \sqrt{x}}+\frac{\cosh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{\left (8 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (4 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{\left (8 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(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 ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{2 c (1-a x)^2 (1+a x)^{3/2} \sqrt{c-a^2 c x^2}}{3 a \sqrt{-1+a x} \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(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{\cosh ^{-1}(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{\cosh ^{-1}(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{\cosh ^{-1}(a x)}\right )}{3 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.580683, size = 317, normalized size = 0.96 \[ -\frac{c \sqrt{c-a^2 c x^2} e^{-4 \cosh ^{-1}(a x)} \left (-16 e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-4 \cosh ^{-1}(a x)\right )+16 \sqrt{2} e^{4 \cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 \cosh ^{-1}(a x)\right )+16 \sqrt{2} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},2 \cosh ^{-1}(a x)\right )-16 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},4 \cosh ^{-1}(a x)\right )+16 a^2 x^2 e^{4 \cosh ^{-1}(a x)}+64 a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+64 a x \sqrt{\frac{a x-1}{a x+1}} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-14 e^{4 \cosh ^{-1}(a x)}-e^{8 \cosh ^{-1}(a x)}-8 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+8 \cosh ^{-1}(a x)-1\right )}{24 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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*A
rcCosh[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)]*ArcCosh
[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])*(-ArcCos
h[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*ArcCosh[
a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/2, 4*ArcCosh[a*x]]))/(24*a*E^(4*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 +
a*x)*ArcCosh[a*x]^(3/2))

________________________________________________________________________________________

Maple [F]  time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}{\operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x