Integrand size = 24, antiderivative size = 211 \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \]
-1/3*a*arccosh(x/a)^(3/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+1/3 2*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)
Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.57 \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=-\frac {a^2 \sqrt {a^2-x^2} \left (16 \text {arccosh}\left (\frac {x}{a}\right )^2+3 \sqrt {2} \sqrt {-\text {arccosh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \text {arccosh}\left (\frac {x}{a}\right )\right )+3 \sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 \text {arccosh}\left (\frac {x}{a}\right )\right )\right )}{48 \sqrt {\frac {-a+x}{a+x}} (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}} \]
-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]])
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.88, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6310, 6302, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6308}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle -\frac {\sqrt {a^2-x^2} \int \frac {x}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle -\frac {a \sqrt {a^2-x^2} \int \frac {x \sqrt {\frac {\frac {x}{a}-1}{\frac {x}{a}+1}} \left (\frac {x}{a}+1\right )}{a \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )}{4 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sinh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )}{4 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sinh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \int -\frac {i \sin \left (2 i \text {arccosh}\left (\frac {x}{a}\right )\right )}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {i a \sqrt {a^2-x^2} \int \frac {\sin \left (2 i \text {arccosh}\left (\frac {x}{a}\right )\right )}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {i a \sqrt {a^2-x^2} \left (\frac {1}{2} i \int \frac {e^{2 \text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )-\frac {1}{2} i \int \frac {e^{-2 \text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {i a \sqrt {a^2-x^2} \left (i \int e^{2 \text {arccosh}\left (\frac {x}{a}\right )}d\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-i \int e^{-2 \text {arccosh}\left (\frac {x}{a}\right )}d\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {i a \sqrt {a^2-x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-i \int e^{-2 \text {arccosh}\left (\frac {x}{a}\right )}d\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {i a \sqrt {a^2-x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {i a \sqrt {a^2-x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )\right )}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\) |
(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]) + ((I/8)*a*Sqrt[a^2 - x^2]*((-1/2 *I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqr t[2]*Sqrt[ArcCosh[x/a]]]))/(Sqrt[-1 + x/a]*Sqrt[1 + x/a])
3.4.93.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(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*ArcCosh[ 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]
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] + (-Simp[( 1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(a + b*ArcC osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq rt[d + e*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, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n , 0]
\[\int \sqrt {a^{2}-x^{2}}\, \sqrt {\operatorname {arccosh}\left (\frac {x}{a}\right )}d x\]
Exception generated. \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}\, dx \]
\[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\int { \sqrt {a^{2} - x^{2}} \sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )} \,d x } \]
\[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\int { \sqrt {a^{2} - x^{2}} \sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )} \,d x } \]
Timed out. \[ \int \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \, dx=\int \sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}\,\sqrt {a^2-x^2} \,d x \]