Integrand size = 27, antiderivative size = 160 \[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=-4 \sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}+\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\sqrt {1-\sqrt {\frac {1+x^2}{x^2}}}}}{\sqrt {-1+\sqrt {2}}}\right )+2 \text {arctanh}\left (\sqrt {1+\sqrt {1-\sqrt {\frac {1+x^2}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1-\sqrt {\frac {1+x^2}{x^2}}}}}{\sqrt {1+\sqrt {2}}}\right ) \]
-4*(1+(1-(1+1/x^2)^(1/2))^(1/2))^(1/2)+(2^(1/2)-1)^(1/2)*arctan((1+(1-((x^ 2+1)/x^2)^(1/2))^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))+2*arctanh((1+(1-((x^2+1)/ x^2)^(1/2))^(1/2))^(1/2))+(1+2^(1/2))^(1/2)*arctanh((1+(1-((x^2+1)/x^2)^(1 /2))^(1/2))^(1/2)/(1+2^(1/2))^(1/2))
Time = 0.81 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=-4 \sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}+\sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}\right )+2 \text {arctanh}\left (\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}\right ) \]
-4*Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]] + 2*ArcTanh[Sqrt[1 + Sqr t[1 - Sqrt[1 + x^(-2)]]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*S qrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]]
Time = 1.21 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.80, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {7282, 7267, 25, 7267, 2003, 2351, 25, 481, 25, 561, 25, 654, 1480, 217, 219, 1610, 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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{x} \, dx\) |
| \(\Big \downarrow \) 7282 |
| \(\displaystyle -\frac {1}{2} \int \sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} x^2d\frac {1}{x^2}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -\int -\frac {\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} \sqrt {1+\frac {1}{x^2}}}{1-\frac {1}{x^4}}d\sqrt {1+\frac {1}{x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1} \sqrt {\frac {1}{x^2}+1}}{1-\frac {1}{x^4}}d\sqrt {\frac {1}{x^2}+1}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -2 \int \frac {\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} \left (1-\frac {1}{x^4}\right ) x^2}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle -2 \int \frac {\left (1-\sqrt {1-\sqrt {1+\frac {1}{x^2}}}\right ) \left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2} x^2}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -2 \left (\int -\frac {\left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2}}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+\int \frac {\left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2} x^2}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\int \frac {\left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2} x^2}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}-\int \frac {\left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2}}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}\right )\) |
| \(\Big \downarrow \) 481 |
| \(\displaystyle -2 \left (\int -\frac {2 \sqrt {1-\sqrt {1+\frac {1}{x^2}}}+3}{\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} \left (2-\frac {1}{x^4}\right )}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+\int \frac {\left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2} x^2}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (-\int \frac {2 \sqrt {1-\sqrt {1+\frac {1}{x^2}}}+3}{\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} \left (2-\frac {1}{x^4}\right )}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+\int \frac {\left (\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1\right )^{3/2} x^2}{2-\frac {1}{x^4}}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle -2 \left (-\int \frac {2 \sqrt {1-\sqrt {1+\frac {1}{x^2}}}+3}{\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} \left (2-\frac {1}{x^4}\right )}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+2 \int -\frac {1}{\left (1-\frac {1}{x^4}\right ) \left (1+\frac {2}{x^4}-\frac {1}{x^8}\right ) x^8}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (-\int \frac {2 \sqrt {1-\sqrt {1+\frac {1}{x^2}}}+3}{\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1} \left (2-\frac {1}{x^4}\right )}d\sqrt {1-\sqrt {1+\frac {1}{x^2}}}-2 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \left (1+\frac {2}{x^4}-\frac {1}{x^8}\right ) x^8}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle -2 \left (-2 \int \frac {1+\frac {2}{x^4}}{1+\frac {2}{x^4}-\frac {1}{x^8}}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}-2 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \left (1+\frac {2}{x^4}-\frac {1}{x^8}\right ) x^8}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -2 \left (-2 \left (\frac {1}{4} \left (4-3 \sqrt {2}\right ) \int \frac {1}{-\sqrt {2}+1-\frac {1}{x^4}}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}+\frac {1}{4} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {2}+1-\frac {1}{x^4}}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}\right )-2 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \left (1+\frac {2}{x^4}-\frac {1}{x^8}\right ) x^8}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 \left (-2 \left (\frac {1}{4} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {2}+1-\frac {1}{x^4}}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}-\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}\right )-2 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \left (1+\frac {2}{x^4}-\frac {1}{x^8}\right ) x^8}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 \left (-2 \int \frac {1}{\left (1-\frac {1}{x^4}\right ) \left (1+\frac {2}{x^4}-\frac {1}{x^8}\right ) x^8}d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}-2 \left (\frac {\left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}\right )+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle -2 \left (-2 \int \left (\frac {1+\frac {3}{x^4}}{2 \left (-1-\frac {2}{x^4}+\frac {1}{x^8}\right )}-\frac {1}{2 \left (\frac {1}{x^4}-1\right )}\right )d\sqrt {\sqrt {1-\sqrt {1+\frac {1}{x^2}}}+1}-2 \left (\frac {\left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}\right )+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (-2 \left (\frac {\left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\left (4-3 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}\right )+2 \left (-\frac {1}{4} \sqrt {5 \sqrt {2}-7} \arctan \left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {\sqrt {2}-1}}\right )-\frac {1}{2} \text {arctanh}\left (\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )+\frac {1}{4} \sqrt {7+5 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{\sqrt {1+\sqrt {2}}}\right )\right )+2 \sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}\right )\) |
-2*(2*Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]] - 2*(-1/4*((4 - 3*Sqrt[2])*ArcT an[Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]/Sqrt[-1 + Sqrt[2]]])/Sqrt[-1 + Sqr t[2]] + ((4 + 3*Sqrt[2])*ArcTanh[Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]/Sqrt [1 + Sqrt[2]]])/(4*Sqrt[1 + Sqrt[2]])) + 2*(-1/4*(Sqrt[-7 + 5*Sqrt[2]]*Arc Tan[Sqrt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]/Sqrt[-1 + Sqrt[2]]]) - ArcTanh[Sq rt[1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]]/2 + (Sqrt[7 + 5*Sqrt[2]]*ArcTanh[Sqrt[ 1 + Sqrt[1 - Sqrt[1 + x^(-2)]]]/Sqrt[1 + Sqrt[2]]])/4))
3.22.71.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d*((c + d*x)^(n - 1)/(b*(n - 1))), x] + Simp[1/b Int[(c + d*x)^(n - 2)*(Simp[b *c^2 - a*d^2 + 2*b*c*d*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 *a*c, 0] && IntegerQ[q] && IntegerQ[m]
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_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
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]]
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
\[\int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^{2}}}}}}{x}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1197 vs. \(2 (120) = 240\).
Time = 100.64 (sec) , antiderivative size = 1197, normalized size of antiderivative = 7.48 \[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=\text {Too large to display} \]
-1/4*sqrt(sqrt(2) + 1)*log(4*(101*sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(sqrt(2) + 1)*sqrt(-sqrt((x^2 + 1)/x^ 2) + 1) + 2*(404*x^2 + sqrt(2)*(300*x^2 + 49) - 4*(75*sqrt(2)*x^2 + 101*x^ 2)*sqrt((x^2 + 1)/x^2) + 52)*sqrt(sqrt(2) + 1) - 4*(150*sqrt(2)*x^2 + 202* x^2 + (101*sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(-sqrt((x^2 + 1)/x^2) + 1) - 2*(75*sqrt(2)*x^2 + 101*x^2)*sq rt((x^2 + 1)/x^2))*sqrt(sqrt(-sqrt((x^2 + 1)/x^2) + 1) + 1)) + 1/4*sqrt(sq rt(2) + 1)*log(-4*(101*sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2) *sqrt((x^2 + 1)/x^2))*sqrt(sqrt(2) + 1)*sqrt(-sqrt((x^2 + 1)/x^2) + 1) - 2 *(404*x^2 + sqrt(2)*(300*x^2 + 49) - 4*(75*sqrt(2)*x^2 + 101*x^2)*sqrt((x^ 2 + 1)/x^2) + 52)*sqrt(sqrt(2) + 1) - 4*(150*sqrt(2)*x^2 + 202*x^2 + (101* sqrt(2)*x^2 + 150*x^2 - (101*sqrt(2)*x^2 + 150*x^2)*sqrt((x^2 + 1)/x^2))*s qrt(-sqrt((x^2 + 1)/x^2) + 1) - 2*(75*sqrt(2)*x^2 + 101*x^2)*sqrt((x^2 + 1 )/x^2))*sqrt(sqrt(-sqrt((x^2 + 1)/x^2) + 1) + 1)) - 1/8*sqrt(-4*sqrt(2) + 4)*log(4*(75*sqrt(2)*x^2 - 101*x^2)*sqrt(-4*sqrt(2) + 4)*sqrt((x^2 + 1)/x^ 2) + (404*x^2 - sqrt(2)*(300*x^2 + 49) + 52)*sqrt(-4*sqrt(2) + 4) + 4*(150 *sqrt(2)*x^2 - 202*x^2 + (101*sqrt(2)*x^2 - 150*x^2 - (101*sqrt(2)*x^2 - 1 50*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(-sqrt((x^2 + 1)/x^2) + 1) - 2*(75*sqrt(2 )*x^2 - 101*x^2)*sqrt((x^2 + 1)/x^2))*sqrt(sqrt(-sqrt((x^2 + 1)/x^2) + 1) + 1) + 2*((101*sqrt(2)*x^2 - 150*x^2)*sqrt(-4*sqrt(2) + 4)*sqrt((x^2 + ...
\[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=\int \frac {\sqrt {\sqrt {1 - \sqrt {1 + \frac {1}{x^{2}}}} + 1}}{x}\, dx \]
\[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=\int { \frac {\sqrt {\sqrt {-\sqrt {\frac {1}{x^{2}} + 1} + 1} + 1}}{x} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument ValueError ind ex.cc ind
Timed out. \[ \int \frac {\sqrt {1+\sqrt {1-\sqrt {1+\frac {1}{x^2}}}}}{x} \, dx=\int \frac {\sqrt {\sqrt {1-\sqrt {\frac {1}{x^2}+1}}+1}}{x} \,d x \]