Integrand size = 22, antiderivative size = 176 \[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1+\frac {x^2}{a^2}}}+\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1+\frac {x^2}{a^2}}} \]
1/3*a*arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2)/(1+x^2/a^2)^(1/2)+1/32*a*erf(2^(1 /2)*arcsinh(x/a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2+x^2)^(1/2)/(1+x^2/a^2)^(1/2) -1/32*a*erfi(2^(1/2)*arcsinh(x/a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2+x^2)^(1/2)/ (1+x^2/a^2)^(1/2)+1/2*x*(a^2+x^2)^(1/2)*arcsinh(x/a)^(1/2)
Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\frac {a \sqrt {a^2+x^2} \left (16 \text {arcsinh}\left (\frac {x}{a}\right )^2-3 \sqrt {2} \sqrt {-\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \text {arcsinh}\left (\frac {x}{a}\right )\right )-3 \sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 \text {arcsinh}\left (\frac {x}{a}\right )\right )\right )}{48 \sqrt {1+\frac {x^2}{a^2}} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \]
(a*Sqrt[a^2 + x^2]*(16*ArcSinh[x/a]^2 - 3*Sqrt[2]*Sqrt[-ArcSinh[x/a]]*Gamm a[3/2, -2*ArcSinh[x/a]] - 3*Sqrt[2]*Sqrt[ArcSinh[x/a]]*Gamma[3/2, 2*ArcSin h[x/a]]))/(48*Sqrt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])
Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6200, 6195, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6198}
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.
| \(\displaystyle \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {\sqrt {a^2+x^2} \int \frac {x}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle -\frac {a \sqrt {a^2+x^2} \int \frac {x \sqrt {\frac {x^2}{a^2}+1}}{a \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )}{4 \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}-\frac {a \sqrt {a^2+x^2} \int \frac {\sinh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )}{4 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}-\frac {a \sqrt {a^2+x^2} \int \frac {\sinh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}-\frac {a \sqrt {a^2+x^2} \int -\frac {i \sin \left (2 i \text {arcsinh}\left (\frac {x}{a}\right )\right )}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {i a \sqrt {a^2+x^2} \int \frac {\sin \left (2 i \text {arcsinh}\left (\frac {x}{a}\right )\right )}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {i a \sqrt {a^2+x^2} \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {i a \sqrt {a^2+x^2} \left (i \int e^{2 \text {arcsinh}\left (\frac {x}{a}\right )}d\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-i \int e^{-2 \text {arcsinh}\left (\frac {x}{a}\right )}d\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {i a \sqrt {a^2+x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )-i \int e^{-2 \text {arcsinh}\left (\frac {x}{a}\right )}d\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {\sqrt {a^2+x^2} \int \frac {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {i a \sqrt {a^2+x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {i a \sqrt {a^2+x^2} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{8 \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\) |
(x*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a] ^(3/2))/(3*Sqrt[1 + x^2/a^2]) + ((I/8)*a*Sqrt[a^2 + x^2]*((-1/2*I)*Sqrt[Pi /2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[A rcSinh[x/a]]]))/Sqrt[1 + x^2/a^2]
3.5.86.3.1 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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
\[\int \sqrt {a^{2}+x^{2}}\, \sqrt {\operatorname {arcsinh}\left (\frac {x}{a}\right )}d x\]
Exception generated. \[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int \sqrt {a^{2} + x^{2}} \sqrt {\operatorname {asinh}{\left (\frac {x}{a} \right )}}\, dx \]
\[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int { \sqrt {a^{2} + x^{2}} \sqrt {\operatorname {arsinh}\left (\frac {x}{a}\right )} \,d x } \]
\[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int { \sqrt {a^{2} + x^{2}} \sqrt {\operatorname {arsinh}\left (\frac {x}{a}\right )} \,d x } \]
Timed out. \[ \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int \sqrt {\mathrm {asinh}\left (\frac {x}{a}\right )}\,\sqrt {a^2+x^2} \,d x \]