\(\int \sqrt {a^2-x^2} \sqrt {\text {arccosh}(\frac {x}{a})} \, dx\) [393]

Optimal result

Integrand size = 24, antiderivative size = 211 \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \]

[Out]

-1/3*a*arccosh(x/a)^(3/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+1/32*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)-1/32*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)+1/2*x*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5895, 5893, 5887, 5556, 12, 3389, 2211, 2235, 2236} \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \]

[In]

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

[Out]

(x*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/2 - (a*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/(3*Sqrt[-1 + x/a]*Sqrt[1 + x
/a]) + (a*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*S
qrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a])

Rule 12

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

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 3389

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 5556

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 5887

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

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.57 \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=-\frac {a^2 \sqrt {a^2-x^2} \left (16 \text {arccosh}\left (\frac {x}{a}\right )^2+3 \sqrt {2} \sqrt {-\text {arccosh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \text {arccosh}\left (\frac {x}{a}\right )\right )+3 \sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 \text {arccosh}\left (\frac {x}{a}\right )\right )\right )}{48 \sqrt {\frac {-a+x}{a+x}} (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}} \]

[In]

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

[Out]

-1/48*(a^2*Sqrt[a^2 - x^2]*(16*ArcCosh[x/a]^2 + 3*Sqrt[2]*Sqrt[-ArcCosh[x/a]]*Gamma[3/2, -2*ArcCosh[x/a]] + 3*
Sqrt[2]*Sqrt[ArcCosh[x/a]]*Gamma[3/2, 2*ArcCosh[x/a]]))/(Sqrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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