Integrand size = 43, antiderivative size = 213 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {5 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}{2 \left (-1+\sqrt {1+x}\right ) \sqrt {1+\sqrt {1+x}}}-\frac {\left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}}{2 \left (-1+\sqrt {1+x}\right ) \sqrt {1+\sqrt {1+x}}}-\frac {1}{4} \sqrt {17+25 \sqrt {2}} \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {-1+\sqrt {2}}}\right )+\text {arctanh}\left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-\frac {1}{4} \sqrt {-17+25 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right ) \]
5/2*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)/(-1+(1+x)^(1/2))/(1+(1+x)^(1/2))^(1/2) -1/2*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)/(-1+(1+x)^(1/2))/(1+(1+x)^(1/2))^(1/2 )-1/4*(17+25*2^(1/2))^(1/2)*arctan((1+(1+(1+x)^(1/2))^(1/2))^(1/2)/(2^(1/2 )-1)^(1/2))+arctanh((1+(1+(1+x)^(1/2))^(1/2))^(1/2))-1/4*(-17+25*2^(1/2))^ (1/2)*arctanh((1+(1+(1+x)^(1/2))^(1/2))^(1/2)/(1+2^(1/2))^(1/2))
Time = 0.81 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{1-\sqrt {1+x}}+\frac {\left (3-\sqrt {1+x}\right ) \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}}{2 x}-\frac {1}{4} \sqrt {17+25 \sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+\text {arctanh}\left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-\frac {1}{4} \sqrt {-17+25 \sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \]
Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/(1 - Sqrt[1 + x]) + ((3 - Sqrt[1 + x])*Sqr t[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/(2*x) - (Sqrt[17 + 25* Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]])/4 + Ar cTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] - (Sqrt[-17 + 25*Sqrt[2]]*ArcTanh[S qrt[-1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]])/4
Time = 2.19 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.55, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {7267, 2003, 7267, 2003, 2353, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {x+1} \sqrt {\sqrt {x+1}+1}}{x^2 \sqrt {\sqrt {\sqrt {x+1}+1}+1}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {(x+1) \sqrt {\sqrt {x+1}+1}}{x^2 \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle 2 \int \frac {x+1}{\left (1-\sqrt {x+1}\right )^2 \left (\sqrt {x+1}+1\right )^{3/2} \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 4 \int \frac {x^2}{(1-x)^2 (x+1) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle 4 \int \frac {\left (1-\sqrt {\sqrt {x+1}+1}\right )^2 \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}}{(1-x)^2 (x+1)}d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 2353 |
| \(\displaystyle 4 \int \left (\frac {\left (3-2 \sqrt {\sqrt {x+1}+1}\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}}{2 (x-1)^2}+\frac {\left (2 \sqrt {\sqrt {x+1}+1}-1\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}}{4 (x-1)}-\frac {\left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}}{2 \sqrt {x+1}}+\frac {\left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}}{4 (x+1)}\right )d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \left (\frac {1}{4} \sqrt {\frac {1}{2} \left (17 \sqrt {2}-23\right )} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {\sqrt {2}-1}}\right )-\frac {3}{16} \sqrt {13 \sqrt {2}-7} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {\sqrt {2}-1}}\right )+\frac {1}{4} \text {arctanh}\left (\sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (23+17 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {1+\sqrt {2}}}\right )+\frac {3}{16} \sqrt {7+13 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {1+\sqrt {2}}}\right )-\frac {\left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}}{4 \sqrt {x+1}}+\frac {\left (2-\sqrt {\sqrt {x+1}+1}\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}}{8 (1-x)}+\frac {1}{4} \sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )\) |
4*(Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/4 + ((2 - Sqrt[1 + Sqrt[1 + x]])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/(8*(1 - x)) - (1 + Sqrt[1 + Sqrt[1 + x]])^(3/2) /(4*Sqrt[1 + x]) - (3*Sqrt[-7 + 13*Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[1 + Sqrt[ 1 + x]]]/Sqrt[-1 + Sqrt[2]]])/16 + (Sqrt[(-23 + 17*Sqrt[2])/2]*ArcTan[Sqrt [1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[-1 + Sqrt[2]]])/4 + ArcTanh[Sqrt[1 + Sqrt [1 + Sqrt[1 + x]]]]/4 + (3*Sqrt[7 + 13*Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[1 + Sqrt[2]]])/16 - (Sqrt[(23 + 17*Sqrt[2])/2]*ArcTanh[ Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[1 + Sqrt[2]]])/4)
3.26.41.3.1 Defintions of rubi rules used
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.17 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(-\frac {1}{2 \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-1\right )}-\frac {\ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-1\right )}{2}+\frac {\frac {\left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{2}-\frac {3 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}{2}}{\left (1+\sqrt {1+\sqrt {1+x}}\right )^{2}-2 \sqrt {1+\sqrt {1+x}}-3}-\frac {\sqrt {2}\, \left (8+\sqrt {2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\left (-8+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {\sqrt {2}-1}}-\frac {1}{2 \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}+1\right )}+\frac {\ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}+1\right )}{2}\) | \(215\) |
| default | \(-\frac {1}{2 \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-1\right )}-\frac {\ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-1\right )}{2}+\frac {\frac {\left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{2}-\frac {3 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}{2}}{\left (1+\sqrt {1+\sqrt {1+x}}\right )^{2}-2 \sqrt {1+\sqrt {1+x}}-3}-\frac {\sqrt {2}\, \left (8+\sqrt {2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\left (-8+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {\sqrt {2}-1}}-\frac {1}{2 \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}+1\right )}+\frac {\ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}+1\right )}{2}\) | \(215\) |
int((1+x)^(1/2)*(1+(1+x)^(1/2))^(1/2)/x^2/(1+(1+(1+x)^(1/2))^(1/2))^(1/2), x,method=_RETURNVERBOSE)
-1/2/((1+(1+(1+x)^(1/2))^(1/2))^(1/2)-1)-1/2*ln((1+(1+(1+x)^(1/2))^(1/2))^ (1/2)-1)+2*(1/4*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)-3/4*(1+(1+(1+x)^(1/2))^(1/ 2))^(1/2))/((1+(1+(1+x)^(1/2))^(1/2))^2-2*(1+(1+x)^(1/2))^(1/2)-3)-1/8*2^( 1/2)*(8+2^(1/2))/(1+2^(1/2))^(1/2)*arctanh((1+(1+(1+x)^(1/2))^(1/2))^(1/2) /(1+2^(1/2))^(1/2))+1/8*(-8+2^(1/2))*2^(1/2)/(2^(1/2)-1)^(1/2)*arctan((1+( 1+(1+x)^(1/2))^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))-1/2/((1+(1+(1+x)^(1/2))^(1/ 2))^(1/2)+1)+1/2*ln((1+(1+(1+x)^(1/2))^(1/2))^(1/2)+1)
Time = 0.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=-\frac {x \sqrt {25 \, \sqrt {2} - 17} \log \left (\sqrt {25 \, \sqrt {2} - 17} {\left (3 \, \sqrt {2} + 7\right )} + 31 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) - x \sqrt {25 \, \sqrt {2} - 17} \log \left (-\sqrt {25 \, \sqrt {2} - 17} {\left (3 \, \sqrt {2} + 7\right )} + 31 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) - x \sqrt {-25 \, \sqrt {2} - 17} \log \left ({\left (3 \, \sqrt {2} - 7\right )} \sqrt {-25 \, \sqrt {2} - 17} + 31 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) + x \sqrt {-25 \, \sqrt {2} - 17} \log \left (-{\left (3 \, \sqrt {2} - 7\right )} \sqrt {-25 \, \sqrt {2} - 17} + 31 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right ) - 4 \, x \log \left (\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} + 1\right ) + 4 \, x \log \left (\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} - 1\right ) + 4 \, {\left (\sqrt {\sqrt {x + 1} + 1} {\left (\sqrt {x + 1} - 3\right )} + 2 \, \sqrt {x + 1} + 2\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}{8 \, x} \]
integrate((1+x)^(1/2)*(1+(1+x)^(1/2))^(1/2)/x^2/(1+(1+(1+x)^(1/2))^(1/2))^ (1/2),x, algorithm="fricas")
-1/8*(x*sqrt(25*sqrt(2) - 17)*log(sqrt(25*sqrt(2) - 17)*(3*sqrt(2) + 7) + 31*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - x*sqrt(25*sqrt(2) - 17)*log(-sqrt(25 *sqrt(2) - 17)*(3*sqrt(2) + 7) + 31*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - x*s qrt(-25*sqrt(2) - 17)*log((3*sqrt(2) - 7)*sqrt(-25*sqrt(2) - 17) + 31*sqrt (sqrt(sqrt(x + 1) + 1) + 1)) + x*sqrt(-25*sqrt(2) - 17)*log(-(3*sqrt(2) - 7)*sqrt(-25*sqrt(2) - 17) + 31*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - 4*x*log( sqrt(sqrt(sqrt(x + 1) + 1) + 1) + 1) + 4*x*log(sqrt(sqrt(sqrt(x + 1) + 1) + 1) - 1) + 4*(sqrt(sqrt(x + 1) + 1)*(sqrt(x + 1) - 3) + 2*sqrt(x + 1) + 2 )*sqrt(sqrt(sqrt(x + 1) + 1) + 1))/x
Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {\sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}{x^{2} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \]
integrate((1+x)^(1/2)*(1+(1+x)^(1/2))^(1/2)/x^2/(1+(1+(1+x)^(1/2))^(1/2))^ (1/2),x, algorithm="maxima")
Result contains complex when optimal does not.
Time = 8.39 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.92 \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {3}{4} \, {\left (\sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {\sqrt {\sqrt {\frac {1}{2} \, \sqrt {3} + 1} + 1}}{\sqrt {\sqrt {2} - 1}}\right ) - i \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {\frac {1}{2} \, \sqrt {3} + 1} + 1}}{\sqrt {-\sqrt {2} - 1}}\right )\right )} \mathrm {sgn}\left (4 \, x + 1\right ) - \frac {\frac {{\left (7 i \, \sqrt {\sqrt {2} + 1} {\left | \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) \right |} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) - 3 i \, \sqrt {2 \, \sqrt {2} + 2}\right )} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}{\sqrt {-\frac {\sqrt {2} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) + \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )}}}\right )}{{\left | \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) \right |}} + \frac {{\left (7 \, \sqrt {\sqrt {2} - 1} {\left | \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) \right |} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) + 3 \, \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}{\sqrt {\frac {\sqrt {2} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) - \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )}}}\right )}{{\left | \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right ) \right |}} - \frac {2 \, \log \left (\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} + 1\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )} + \frac {2 \, \log \left ({\left | \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} - 1 \right |}\right )}{\mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )} + \frac {2 \, {\left ({\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} - 5 \, \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )}}{{\left ({\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{3} - 3 \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{2} + \sqrt {\sqrt {x + 1} + 1} + 2\right )} \mathrm {sgn}\left (4 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - 8 \, \sqrt {x + 1} - 7\right )}}{4 \, \mathrm {sgn}\left (4 \, x + 1\right )} \]
integrate((1+x)^(1/2)*(1+(1+x)^(1/2))^(1/2)/x^2/(1+(1+(1+x)^(1/2))^(1/2))^ (1/2),x, algorithm="giac")
3/4*(sqrt(2*sqrt(2) - 2)*arctan(sqrt(sqrt(1/2*sqrt(3) + 1) + 1)/sqrt(sqrt( 2) - 1)) - I*sqrt(2*sqrt(2) + 2)*arctan(sqrt(sqrt(1/2*sqrt(3) + 1) + 1)/sq rt(-sqrt(2) - 1)))*sgn(4*x + 1) - 1/4*((7*I*sqrt(sqrt(2) + 1)*abs(sgn(4*(s qrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7))*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt (x + 1) - 7) - 3*I*sqrt(2*sqrt(2) + 2))*arctan(sqrt(sqrt(sqrt(x + 1) + 1) + 1)/sqrt(-(sqrt(2)*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7) + sgn(4 *(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7))/sgn(4*(sqrt(x + 1) + 1)^2 - 8*s qrt(x + 1) - 7)))/abs(sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7)) + (7 *sqrt(sqrt(2) - 1)*abs(sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7))*sgn (4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7) + 3*sqrt(2*sqrt(2) - 2))*arcta n(sqrt(sqrt(sqrt(x + 1) + 1) + 1)/sqrt((sqrt(2)*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7) - sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7))/sgn (4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7)))/abs(sgn(4*(sqrt(x + 1) + 1)^ 2 - 8*sqrt(x + 1) - 7)) - 2*log(sqrt(sqrt(sqrt(x + 1) + 1) + 1) + 1)/sgn(4 *(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7) + 2*log(abs(sqrt(sqrt(sqrt(x + 1 ) + 1) + 1) - 1))/sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7) + 2*((sqr t(sqrt(x + 1) + 1) + 1)^(5/2) - 5*sqrt(sqrt(sqrt(x + 1) + 1) + 1))/(((sqrt (sqrt(x + 1) + 1) + 1)^3 - 3*(sqrt(sqrt(x + 1) + 1) + 1)^2 + sqrt(sqrt(x + 1) + 1) + 2)*sgn(4*(sqrt(x + 1) + 1)^2 - 8*sqrt(x + 1) - 7)))/sgn(4*x + 1 )
Timed out. \[ \int \frac {\sqrt {1+x} \sqrt {1+\sqrt {1+x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}}{x^2\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \,d x \]