\(\int \sqrt {a^2-x^2} \text {arccosh}(\frac {x}{a})^{3/2} \, dx\) [398]

Optimal result

Integrand size = 24, antiderivative size = 316 \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \]

[Out]

1/2*x*arccosh(x/a)^(3/2)*(a^2-x^2)^(1/2)-1/5*a*arccosh(x/a)^(5/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)
+3/128*a*erf(2^(1/2)*arccosh(x/a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+3/128*a
*erfi(2^(1/2)*arccosh(x/a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+3/16*a*(a^2-x^
2)^(1/2)*arccosh(x/a)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)-3/8*x^2*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/2)/a/(-1+x/a)
^(1/2)/(1+x/a)^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5895, 5893, 5884, 5953, 3393, 3388, 2211, 2235, 2236} \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \]

[In]

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

[Out]

(3*a*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*x^2*Sqrt[a^2 - x^2]*Sqrt[ArcCo
sh[x/a]])/(8*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (x*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/2 - (a*Sqrt[a^2 - x^2]*A
rcCosh[x/a]^(5/2))/(5*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh
[x/a]]])/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])
/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a])

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 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])
), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5893

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

Rule 5895

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

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 {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {\sqrt {a^2-x^2} \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx}{2 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 \sqrt {a^2-x^2}\right ) \int x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx}{4 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = -\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 \sqrt {a^2-x^2}\right ) \int \frac {x^2}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}} \, dx}{16 a^2 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = -\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {x}} \, dx,x,\text {arccosh}\left (\frac {x}{a}\right )\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = -\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-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}\left (\frac {x}{a}\right )\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = \frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arccosh}\left (\frac {x}{a}\right )\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = \frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arccosh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = \frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ & = \frac {3 a \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46 \[ \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {a^2 \sqrt {a^2-x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-8 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \left (16 \text {arccosh}\left (\frac {x}{a}\right )^2+15 \cosh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )-20 \text {arccosh}\left (\frac {x}{a}\right ) \sinh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )\right )\right )}{640 \sqrt {\frac {-a+x}{a+x}} (a+x)} \]

[In]

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

[Out]

(a^2*Sqrt[a^2 - x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[
x/a]]] - 8*Sqrt[ArcCosh[x/a]]*(16*ArcCosh[x/a]^2 + 15*Cosh[2*ArcCosh[x/a]] - 20*ArcCosh[x/a]*Sinh[2*ArcCosh[x/
a]])))/(640*Sqrt[(-a + x)/(a + x)]*(a + x))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate(acosh(x/a)**(3/2)*(a**2-x**2)**(1/2),x)

[Out]

Integral(sqrt(-(-a + x)*(a + x))*acosh(x/a)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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