3.379 \(\int (c-a^2 c x^2)^{3/2} \sqrt{\cosh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=351 \[ -\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]

[Out]

(3*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/8 + (x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]])/4 - (c*Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sq
rt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[A
rcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]
])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/
(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

________________________________________________________________________________________

Rubi [A]  time = 0.723019, antiderivative size = 363, normalized size of antiderivative = 1.03, number of steps used = 25, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5713, 5685, 5683, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205, 5780} \[ -\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/8 + (c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a
*x]])/4 - (c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c -
a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*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*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[
2*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*S
qrt[ArcCosh[a*x]]])/(16*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 5685

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

Rule 5683

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

Rule 5676

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

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3308

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 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 5780

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

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \sqrt{\cosh ^{-1}(a x)} \, dx &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \sqrt{\cosh ^{-1}(a x)} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)} \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (-1+a^2 x^2\right )}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (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 )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (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 )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (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 )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (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 )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (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 )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 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 )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (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 )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (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 )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (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 )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (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 )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 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 )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 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 )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 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 )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 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 )}{32 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{16 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.239761, size = 154, normalized size = 0.44 \[ -\frac{c \sqrt{c-a^2 c x^2} \left (-\sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-4 \cosh ^{-1}(a x)\right )+8 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 \cosh ^{-1}(a x)\right )+\sqrt{\cosh ^{-1}(a x)} \left (8 \sqrt{2} \text{Gamma}\left (\frac{3}{2},2 \cosh ^{-1}(a x)\right )-\text{Gamma}\left (\frac{3}{2},4 \cosh ^{-1}(a x)\right )+32 \cosh ^{-1}(a x)^{3/2}\right )\right )}{128 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*Sqrt[c - a^2*c*x^2]*(-(Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -4*ArcCosh[a*x]]) + 8*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Ga
mma[3/2, -2*ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(32*ArcCosh[a*x]^(3/2) + 8*Sqrt[2]*Gamma[3/2, 2*ArcCosh[a*x]] -
Gamma[3/2, 4*ArcCosh[a*x]])))/(128*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])

________________________________________________________________________________________

Maple [F]  time = 0.354, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{{\rm arccosh} \left (ax\right )}\, 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)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \sqrt{\operatorname{arcosh}\left (a x\right )}\,{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)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^(1/2),x, algorithm="giac")

[Out]

sage0*x