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

Optimal. Leaf size=211 \[ \frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \]

[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)

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Rubi [A]  time = 0.39, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \]

Antiderivative was successfully verified.

[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 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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \, dx &=\frac {\sqrt {a^2-x^2} \int \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \, dx}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\cosh ^{-1}\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 {\cosh ^{-1}\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 {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\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 ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\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 {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\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 ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}\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 {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\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 ) \operatorname {Subst}\left (\int \frac {\sinh (2 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 {1}{2} x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\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 ) \operatorname {Subst}\left (\int \frac {e^{-2 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 {\left (a \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 )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\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 ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\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 ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\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 {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\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 {\cosh ^{-1}\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 {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 121, normalized size = 0.57 \[ -\frac {a^2 \sqrt {a^2-x^2} \left (16 \cosh ^{-1}\left (\frac {x}{a}\right )^2+3 \sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+3 \sqrt {2} \sqrt {-\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )\right )}{48 \sqrt {\frac {x-a}{a+x}} (a+x) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \]

Warning: Unable to verify antiderivative.

[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]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} - x^{2}} \sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.69, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2}-x^{2}}\, \sqrt {\mathrm {arccosh}\left (\frac {x}{a}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} - x^{2}} \sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}\,\sqrt {a^2-x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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