Integrand size = 23, antiderivative size = 185 \[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {3 \sqrt {x+\sqrt {1+x^2}}}{8 \left (1+x^2+x \sqrt {1+x^2}\right )^2}+\frac {\left (x+\sqrt {1+x^2}\right )^{5/2}}{8 \left (1+x^2+x \sqrt {1+x^2}\right )^2}+\frac {3 \arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )}{4 \sqrt {2}}+\frac {3 \text {arctanh}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )}{4 \sqrt {2}} \]
-3/8*(x+(x^2+1)^(1/2))^(1/2)/(1+x^2+x*(x^2+1)^(1/2))^2+1/8*(x+(x^2+1)^(1/2 ))^(5/2)/(1+x^2+x*(x^2+1)^(1/2))^2+3/8*2^(1/2)*arctan((-1/2*2^(1/2)+1/2*x* 2^(1/2)+1/2*(x^2+1)^(1/2)*2^(1/2))/(x+(x^2+1)^(1/2))^(1/2))+3/8*2^(1/2)*ar ctanh((1/2*2^(1/2)+1/2*x*2^(1/2)+1/2*(x^2+1)^(1/2)*2^(1/2))/(x+(x^2+1)^(1/ 2))^(1/2))
Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{8} \left (\frac {2 \sqrt {x+\sqrt {1+x^2}} \left (-1+x^2+x \sqrt {1+x^2}\right )}{\left (1+x^2+x \sqrt {1+x^2}\right )^2}+3 \sqrt {2} \arctan \left (\frac {-1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right )+3 \sqrt {2} \text {arctanh}\left (\frac {1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right )\right ) \]
((2*Sqrt[x + Sqrt[1 + x^2]]*(-1 + x^2 + x*Sqrt[1 + x^2]))/(1 + x^2 + x*Sqr t[1 + x^2])^2 + 3*Sqrt[2]*ArcTan[(-1 + x + Sqrt[1 + x^2])/(Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]])] + 3*Sqrt[2]*ArcTanh[(1 + x + Sqrt[1 + x^2])/(Sqrt[2]*Sq rt[x + Sqrt[1 + x^2]])])/8
Time = 0.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2547, 252, 253, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (x^2+1\right )^2 \sqrt {\sqrt {x^2+1}+x}} \, dx\) |
| \(\Big \downarrow \) 2547 |
| \(\displaystyle 8 \int \frac {\left (x+\sqrt {x^2+1}\right )^{3/2}}{\left (\left (x+\sqrt {x^2+1}\right )^2+1\right )^3}d\left (x+\sqrt {x^2+1}\right )\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle 8 \left (\frac {1}{8} \int \frac {1}{\sqrt {x+\sqrt {x^2+1}} \left (\left (x+\sqrt {x^2+1}\right )^2+1\right )^2}d\left (x+\sqrt {x^2+1}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\sqrt {x+\sqrt {x^2+1}} \left (\left (x+\sqrt {x^2+1}\right )^2+1\right )}d\left (x+\sqrt {x^2+1}\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \int \frac {1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {1}{2} \int \frac {x+\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {1}{2} \int \frac {1}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x-\sqrt {x^2+1}-1}d\left (1-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-\sqrt {x^2+1}-1}d\left (\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {-x-\sqrt {x^2+1}+1}{\left (x+\sqrt {x^2+1}\right )^2+1}d\sqrt {x+\sqrt {x^2+1}}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {x+\sqrt {x^2+1}}}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1\right )}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x+\sqrt {x^2+1}}}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1\right )}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {x+\sqrt {x^2+1}}}{x+\sqrt {x^2+1}-\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}{x+\sqrt {x^2+1}+\sqrt {2} \sqrt {x+\sqrt {x^2+1}}+1}d\sqrt {x+\sqrt {x^2+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 8 \left (\frac {1}{8} \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{2 \sqrt {2}}\right )\right )+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )}\right )-\frac {\sqrt {\sqrt {x^2+1}+x}}{4 \left (\left (\sqrt {x^2+1}+x\right )^2+1\right )^2}\right )\) |
8*(-1/4*Sqrt[x + Sqrt[1 + x^2]]/(1 + (x + Sqrt[1 + x^2])^2)^2 + (Sqrt[x + Sqrt[1 + x^2]]/(2*(1 + (x + Sqrt[1 + x^2])^2)) + (3*((-(ArcTan[1 - Sqrt[2] *Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2])/2 + (-1/2*Log[1 + x + Sqrt[1 + x^2] - Sqrt[2]*Sqrt[x + Sqr t[1 + x^2]]]/Sqrt[2] + Log[1 + x + Sqrt[1 + x^2] + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/(2*Sqrt[2]))/2))/2)/8)
3.24.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_ .)*(x_)^2])^(n_.), x_Symbol] :> Simp[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && IntegerQ[2*m] && (Intege rQ[m] || GtQ[i/c, 0])
\[\int \frac {1}{\left (x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}d x\]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {3 \, \sqrt {2} {\left (-\left (i + 1\right ) \, x^{2} - i - 1\right )} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + 3 \, \sqrt {2} {\left (\left (i - 1\right ) \, x^{2} + i - 1\right )} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + 3 \, \sqrt {2} {\left (-\left (i - 1\right ) \, x^{2} - i + 1\right )} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + 3 \, \sqrt {2} {\left (\left (i + 1\right ) \, x^{2} + i + 1\right )} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + 4 \, {\left (3 \, x^{2} - 3 \, \sqrt {x^{2} + 1} x + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{16 \, {\left (x^{2} + 1\right )}} \]
-1/16*(3*sqrt(2)*(-(I + 1)*x^2 - I - 1)*log((I + 1)*sqrt(2) + 2*sqrt(x + s qrt(x^2 + 1))) + 3*sqrt(2)*((I - 1)*x^2 + I - 1)*log(-(I - 1)*sqrt(2) + 2* sqrt(x + sqrt(x^2 + 1))) + 3*sqrt(2)*(-(I - 1)*x^2 - I + 1)*log((I - 1)*sq rt(2) + 2*sqrt(x + sqrt(x^2 + 1))) + 3*sqrt(2)*((I + 1)*x^2 + I + 1)*log(- (I + 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1))) + 4*(3*x^2 - 3*sqrt(x^2 + 1)* x + 1)*sqrt(x + sqrt(x^2 + 1)))/(x^2 + 1)
\[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )^{2}}\, dx \]
\[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
\[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{{\left (x^{2} + 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{{\left (x^2+1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]