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

Optimal. Leaf size=368 \[ -\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a^3 \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.778797, antiderivative size = 376, normalized size of antiderivative = 1.02, 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} \[ -\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a^3 \sqrt{a^2-x^2} \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{4 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}+\frac{1}{4} x (a-x) (a+x) \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*x*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/8 + ((a - x)*x*(a + x)*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/4 - (a
^3*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/(4*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erf[2*
Sqrt[ArcCosh[x/a]]])/(256*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[Sqrt[2]*Sqrt[Arc
Cosh[x/a]]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erfi[2*Sqrt[ArcCosh[x/a]]])/(25
6*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(16*Sqrt[-
1 + x/a]*Sqrt[1 + x/a])

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

Mathematica [A]  time = 0.270036, size = 165, normalized size = 0.45 \[ -\frac{a^4 \sqrt{a^2-x^2} \left (-\sqrt{-\cosh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-4 \cosh ^{-1}\left (\frac{x}{a}\right )\right )+8 \sqrt{2} \sqrt{-\cosh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 \cosh ^{-1}\left (\frac{x}{a}\right )\right )+\sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )} \left (8 \sqrt{2} \text{Gamma}\left (\frac{3}{2},2 \cosh ^{-1}\left (\frac{x}{a}\right )\right )-\text{Gamma}\left (\frac{3}{2},4 \cosh ^{-1}\left (\frac{x}{a}\right )\right )+32 \cosh ^{-1}\left (\frac{x}{a}\right )^{3/2}\right )\right )}{128 \sqrt{\frac{x-a}{a+x}} (a+x) \sqrt{\cosh ^{-1}\left (\frac{x}{a}\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^4*Sqrt[a^2 - x^2]*(-(Sqrt[-ArcCosh[x/a]]*Gamma[3/2, -4*ArcCosh[x/a]]) + 8*Sqrt[2]*Sqrt[-ArcCosh[x/a]]*Gamm
a[3/2, -2*ArcCosh[x/a]] + Sqrt[ArcCosh[x/a]]*(32*ArcCosh[x/a]^(3/2) + 8*Sqrt[2]*Gamma[3/2, 2*ArcCosh[x/a]] - G
amma[3/2, 4*ArcCosh[x/a]])))/(128*Sqrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])

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Maple [F]  time = 0.336, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}-{x}^{2} \right ) ^{{\frac{3}{2}}}\sqrt{{\rm arccosh} \left ({\frac{x}{a}}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2-x^2)^(3/2)*arccosh(x/a)^(1/2),x, algorithm="maxima")

[Out]

integrate((a^2 - x^2)^(3/2)*sqrt(arccosh(x/a)), 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-x^2)^(3/2)*arccosh(x/a)^(1/2),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-x**2)**(3/2)*acosh(x/a)**(1/2),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-x^2)^(3/2)*arccosh(x/a)^(1/2),x, algorithm="giac")

[Out]

sage0*x