3.4.97 \(\int (a^2-x^2)^{3/2} \cosh ^{-1}(\frac {x}{a})^{3/2} \, dx\) [397]

Optimal. Leaf size=525 \[ \frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \]

[Out]

1/4*x*(a^2-x^2)^(3/2)*arccosh(x/a)^(3/2)+3/8*a^2*x*arccosh(x/a)^(3/2)*(a^2-x^2)^(1/2)-3/20*a^3*arccosh(x/a)^(5
/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+3/128*a^3*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^3*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)-3/2048*a^3*erf(2*arccosh(x/a)^(1/2))*Pi^(1/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(
1/2)/(1+x/a)^(1/2)-3/2048*a^3*erfi(2*arccosh(x/a)^(1/2))*Pi^(1/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)
+3/32*(a^2-x^2)^(5/2)*arccosh(x/a)^(1/2)/a/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+27/256*a^3*(a^2-x^2)^(1/2)*arccosh(x/a
)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)-9/32*a*x^2*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2
)

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 525, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5897, 5895, 5893, 5884, 5953, 3393, 3388, 2211, 2235, 2236, 5912, 5914, 5907} \begin {gather*} \frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \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 )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \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 )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(27*a^3*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/(256*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (9*a*x^2*Sqrt[a^2 - x^2]*Sqrt
[ArcCosh[x/a]])/(32*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*(a^2 - x^2)^(5/2)*Sqrt[ArcCosh[x/a]])/(32*a*Sqrt[-1 + x
/a]*Sqrt[1 + x/a]) + (3*a^2*x*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/8 + (x*(a^2 - x^2)^(3/2)*ArcCosh[x/a]^(3/2))
/4 - (3*a^3*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(5/2))/(20*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*a^3*Sqrt[Pi]*Sqrt[a^2 -
x^2]*Erf[2*Sqrt[ArcCosh[x/a]]])/(2048*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[S
qrt[2]*Sqrt[ArcCosh[x/a]]])/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erfi[2*Sqrt[Ar
cCosh[x/a]]])/(2048*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCos
h[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 5897

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

Rule 5907

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

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 5914

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

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*} \int \left (a^2-x^2\right )^{3/2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, 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} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \int 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}} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {\left (9 a \sqrt {a^2-x^2}\right ) \int 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 {\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}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx}{8 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 \sqrt {a^2-x^2}\right ) \int \frac {x^2}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \, dx}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {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 {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^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 {9 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{512 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 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 (9 a^3 \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,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{1024 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{1024 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 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 {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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 )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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 )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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 {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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}}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 219, normalized size = 0.42 \begin {gather*} \frac {a^4 \sqrt {a^2-x^2} \left (-384 \cosh ^{-1}\left (\frac {x}{a}\right )^3-480 \cosh ^{-1}\left (\frac {x}{a}\right ) \cosh \left (2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+60 \sqrt {2 \pi } \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )+60 \sqrt {2 \pi } \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )-5 \sqrt {-\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},-4 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+5 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},4 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+640 \cosh ^{-1}\left (\frac {x}{a}\right )^2 \sinh \left (2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )\right )}{2560 \sqrt {\frac {-a+x}{a+x}} (a+x) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^4*Sqrt[a^2 - x^2]*(-384*ArcCosh[x/a]^3 - 480*ArcCosh[x/a]*Cosh[2*ArcCosh[x/a]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh
[x/a]]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[x/a]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]] - 5
*Sqrt[-ArcCosh[x/a]]*Gamma[5/2, -4*ArcCosh[x/a]] + 5*Sqrt[ArcCosh[x/a]]*Gamma[5/2, 4*ArcCosh[x/a]] + 640*ArcCo
sh[x/a]^2*Sinh[2*ArcCosh[x/a]]))/(2560*Sqrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}\,{\left (a^2-x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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