Integrand size = 27, antiderivative size = 196 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=2 \sqrt {x+\sqrt {1+x}}-3 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )-4 \text {RootSum}\left [25-40 \text {$\#$1}+6 \text {$\#$1}^2-8 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {5 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-10+3 \text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
Time = 0.14 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=2 \sqrt {x+\sqrt {1+x}}-3 \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [-1+6 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{3-3 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
2*Sqrt[x + Sqrt[1 + x]] - 3*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x ]]] + 4*RootSum[-1 + 6*#1 - 3*#1^2 + 2*#1^3 + #1^4 & , (2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] - 2*Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x] ] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1^2)/(3 - 3*# 1 + 3*#1^2 + 2*#1^3) & ]
Time = 0.57 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {7267, 25, 1352, 27, 2143, 27, 1092, 219, 1365, 1154, 217, 219}
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.
| \(\displaystyle \int \frac {\sqrt {x+\sqrt {x+1}}}{x-\sqrt {x+1}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {\sqrt {x+1} \sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}-x}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\sqrt {x+1} \sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}-x}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 1352 |
| \(\displaystyle 2 \left (\sqrt {x+\sqrt {x+1}}-\int \frac {3 (x+1)+\sqrt {x+1}+1}{2 \left (\sqrt {x+1}-x\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\sqrt {x+\sqrt {x+1}}-\frac {1}{2} \int \frac {3 (x+1)+\sqrt {x+1}+1}{\left (\sqrt {x+1}-x\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \int \frac {1}{\sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}+\int -\frac {4 \left (\sqrt {x+1}+1\right )}{\left (\sqrt {x+1}-x\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \int \frac {1}{\sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}-4 \int \frac {\sqrt {x+1}+1}{\left (\sqrt {x+1}-x\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (6 \int \frac {1}{3-x}d\frac {2 \sqrt {x+1}+1}{\sqrt {x+\sqrt {x+1}}}-4 \int \frac {\sqrt {x+1}+1}{\left (\sqrt {x+1}-x\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-4 \int \frac {\sqrt {x+1}+1}{\left (\sqrt {x+1}-x\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-4 \left (\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \frac {1}{\left (-2 \sqrt {x+1}-\sqrt {5}+1\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}+\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\left (-2 \sqrt {x+1}+\sqrt {5}+1\right ) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-4 \left (-\frac {2}{5} \left (5-3 \sqrt {5}\right ) \int \frac {1}{-x+16 \left (1-\sqrt {5}\right )-1}d\frac {-2 \left (2-\sqrt {5}\right ) \sqrt {x+1}+\sqrt {5}+3}{\sqrt {x+\sqrt {x+1}}}-\frac {2}{5} \left (5+3 \sqrt {5}\right ) \int \frac {1}{-x+16 \left (1+\sqrt {5}\right )-1}d\frac {-2 \left (2+\sqrt {5}\right ) \sqrt {x+1}-\sqrt {5}+3}{\sqrt {x+\sqrt {x+1}}}\right )\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-4 \left (\frac {\left (5-3 \sqrt {5}\right ) \arctan \left (\frac {-2 \left (2-\sqrt {5}\right ) \sqrt {x+1}+\sqrt {5}+3}{4 \sqrt {\sqrt {5}-1} \sqrt {x+\sqrt {x+1}}}\right )}{10 \sqrt {\sqrt {5}-1}}-\frac {2}{5} \left (5+3 \sqrt {5}\right ) \int \frac {1}{-x+16 \left (1+\sqrt {5}\right )-1}d\frac {-2 \left (2+\sqrt {5}\right ) \sqrt {x+1}-\sqrt {5}+3}{\sqrt {x+\sqrt {x+1}}}\right )\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {1}{2} \left (3 \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-4 \left (\frac {\left (5-3 \sqrt {5}\right ) \arctan \left (\frac {-2 \left (2-\sqrt {5}\right ) \sqrt {x+1}+\sqrt {5}+3}{4 \sqrt {\sqrt {5}-1} \sqrt {x+\sqrt {x+1}}}\right )}{10 \sqrt {\sqrt {5}-1}}-\frac {\left (5+3 \sqrt {5}\right ) \text {arctanh}\left (\frac {-2 \left (2+\sqrt {5}\right ) \sqrt {x+1}-\sqrt {5}+3}{4 \sqrt {1+\sqrt {5}} \sqrt {x+\sqrt {x+1}}}\right )}{10 \sqrt {1+\sqrt {5}}}\right )\right )+\sqrt {x+\sqrt {x+1}}\right )\) |
2*(Sqrt[x + Sqrt[1 + x]] + (3*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt [1 + x]])] - 4*(((5 - 3*Sqrt[5])*ArcTan[(3 + Sqrt[5] - 2*(2 - Sqrt[5])*Sqr t[1 + x])/(4*Sqrt[-1 + Sqrt[5]]*Sqrt[x + Sqrt[1 + x]])])/(10*Sqrt[-1 + Sqr t[5]]) - ((5 + 3*Sqrt[5])*ArcTanh[(3 - Sqrt[5] - 2*(2 + Sqrt[5])*Sqrt[1 + x])/(4*Sqrt[1 + Sqrt[5]]*Sqrt[x + Sqrt[1 + x]])])/(10*Sqrt[1 + Sqrt[5]]))) /2)
3.25.24.3.1 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[((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[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a* (h*e - 2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1 ))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4* d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 ] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
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 = 0.13 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.06
| method | result | size |
| derivativedivides | \(\frac {2 \left (\sqrt {5}-1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}{2}+\frac {\left (2-\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}\right )}{4}+\frac {\left (-\sqrt {5}+1\right ) \arctan \left (\frac {-2 \sqrt {5}+2+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}-1}\, \sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}\right )}{2 \sqrt {\sqrt {5}-1}}\right )}{5}+\frac {2 \left (\sqrt {5}+1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}{2}+\frac {\left (2+\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}\right )}{4}-\frac {\sqrt {\sqrt {5}+1}\, \operatorname {arctanh}\left (\frac {2 \sqrt {5}+2+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}+1}\, \sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}\right )}{2}\right )}{5}\) | \(403\) |
| default | \(\frac {2 \left (\sqrt {5}-1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}{2}+\frac {\left (2-\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}\right )}{4}+\frac {\left (-\sqrt {5}+1\right ) \arctan \left (\frac {-2 \sqrt {5}+2+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}-1}\, \sqrt {\left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2-\sqrt {5}\right ) \left (\sqrt {1+x}+\frac {\sqrt {5}}{2}-\frac {1}{2}\right )-\sqrt {5}+1}}\right )}{2 \sqrt {\sqrt {5}-1}}\right )}{5}+\frac {2 \left (\sqrt {5}+1\right ) \sqrt {5}\, \left (\frac {\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}{2}+\frac {\left (2+\sqrt {5}\right ) \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}\right )}{4}-\frac {\sqrt {\sqrt {5}+1}\, \operatorname {arctanh}\left (\frac {2 \sqrt {5}+2+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}{2 \sqrt {\sqrt {5}+1}\, \sqrt {\left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )^{2}+\left (2+\sqrt {5}\right ) \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )+\sqrt {5}+1}}\right )}{2}\right )}{5}\) | \(403\) |
2/5*(5^(1/2)-1)*5^(1/2)*(1/2*(((1+x)^(1/2)+1/2*5^(1/2)-1/2)^2+(2-5^(1/2))* ((1+x)^(1/2)+1/2*5^(1/2)-1/2)-5^(1/2)+1)^(1/2)+1/4*(2-5^(1/2))*ln(1/2+(1+x )^(1/2)+(((1+x)^(1/2)+1/2*5^(1/2)-1/2)^2+(2-5^(1/2))*((1+x)^(1/2)+1/2*5^(1 /2)-1/2)-5^(1/2)+1)^(1/2))+1/2*(-5^(1/2)+1)/(5^(1/2)-1)^(1/2)*arctan(1/2*( -2*5^(1/2)+2+(2-5^(1/2))*((1+x)^(1/2)+1/2*5^(1/2)-1/2))/(5^(1/2)-1)^(1/2)/ (((1+x)^(1/2)+1/2*5^(1/2)-1/2)^2+(2-5^(1/2))*((1+x)^(1/2)+1/2*5^(1/2)-1/2) -5^(1/2)+1)^(1/2)))+2/5*(5^(1/2)+1)*5^(1/2)*(1/2*(((1+x)^(1/2)-1/2*5^(1/2) -1/2)^2+(2+5^(1/2))*((1+x)^(1/2)-1/2*5^(1/2)-1/2)+5^(1/2)+1)^(1/2)+1/4*(2+ 5^(1/2))*ln(1/2+(1+x)^(1/2)+(((1+x)^(1/2)-1/2*5^(1/2)-1/2)^2+(2+5^(1/2))*( (1+x)^(1/2)-1/2*5^(1/2)-1/2)+5^(1/2)+1)^(1/2))-1/2*(5^(1/2)+1)^(1/2)*arcta nh(1/2*(2*5^(1/2)+2+(2+5^(1/2))*((1+x)^(1/2)-1/2*5^(1/2)-1/2))/(5^(1/2)+1) ^(1/2)/(((1+x)^(1/2)-1/2*5^(1/2)-1/2)^2+(2+5^(1/2))*((1+x)^(1/2)-1/2*5^(1/ 2)-1/2)+5^(1/2)+1)^(1/2)))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 9.19 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.64 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=-\frac {1}{5} \, \sqrt {5} \sqrt {2} \sqrt {\sqrt {5} + 2} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (3 \, x^{2} + \sqrt {5} {\left (x^{2} + 3 \, x - 1\right )} + 4 \, {\left (2 \, x + \sqrt {5} - 1\right )} \sqrt {x + 1} + x + 5\right )} \sqrt {\sqrt {5} + 2} + 4 \, {\left ({\left (\sqrt {5} x + x + 2\right )} \sqrt {x + 1} + \sqrt {5} {\left (x + 1\right )} + 3 \, x + 1\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x^{2} - x - 1}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {2} \sqrt {\sqrt {5} + 2} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (3 \, x^{2} + \sqrt {5} {\left (x^{2} + 3 \, x - 1\right )} + 4 \, {\left (2 \, x + \sqrt {5} - 1\right )} \sqrt {x + 1} + x + 5\right )} \sqrt {\sqrt {5} + 2} - 4 \, {\left ({\left (\sqrt {5} x + x + 2\right )} \sqrt {x + 1} + \sqrt {5} {\left (x + 1\right )} + 3 \, x + 1\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x^{2} - x - 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-32 \, \sqrt {5} + 64} \log \left (-\frac {4 \, {\left (2 \, x - \sqrt {5} - 1\right )} \sqrt {x + 1} \sqrt {-32 \, \sqrt {5} + 64} + 16 \, {\left ({\left (\sqrt {5} x - x - 2\right )} \sqrt {x + 1} + \sqrt {5} {\left (x + 1\right )} - 3 \, x - 1\right )} \sqrt {x + \sqrt {x + 1}} + {\left (3 \, x^{2} - \sqrt {5} {\left (x^{2} + 3 \, x - 1\right )} + x + 5\right )} \sqrt {-32 \, \sqrt {5} + 64}}{x^{2} - x - 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-32 \, \sqrt {5} + 64} \log \left (\frac {4 \, {\left (2 \, x - \sqrt {5} - 1\right )} \sqrt {x + 1} \sqrt {-32 \, \sqrt {5} + 64} - 16 \, {\left ({\left (\sqrt {5} x - x - 2\right )} \sqrt {x + 1} + \sqrt {5} {\left (x + 1\right )} - 3 \, x - 1\right )} \sqrt {x + \sqrt {x + 1}} + {\left (3 \, x^{2} - \sqrt {5} {\left (x^{2} + 3 \, x - 1\right )} + x + 5\right )} \sqrt {-32 \, \sqrt {5} + 64}}{x^{2} - x - 1}\right ) + 2 \, \sqrt {x + \sqrt {x + 1}} + \frac {3}{2} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \]
-1/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5) + 2)*log(4*(sqrt(2)*(3*x^2 + sqrt(5)*(x^ 2 + 3*x - 1) + 4*(2*x + sqrt(5) - 1)*sqrt(x + 1) + x + 5)*sqrt(sqrt(5) + 2 ) + 4*((sqrt(5)*x + x + 2)*sqrt(x + 1) + sqrt(5)*(x + 1) + 3*x + 1)*sqrt(x + sqrt(x + 1)))/(x^2 - x - 1)) + 1/5*sqrt(5)*sqrt(2)*sqrt(sqrt(5) + 2)*lo g(-4*(sqrt(2)*(3*x^2 + sqrt(5)*(x^2 + 3*x - 1) + 4*(2*x + sqrt(5) - 1)*sqr t(x + 1) + x + 5)*sqrt(sqrt(5) + 2) - 4*((sqrt(5)*x + x + 2)*sqrt(x + 1) + sqrt(5)*(x + 1) + 3*x + 1)*sqrt(x + sqrt(x + 1)))/(x^2 - x - 1)) - 1/20*s qrt(5)*sqrt(-32*sqrt(5) + 64)*log(-(4*(2*x - sqrt(5) - 1)*sqrt(x + 1)*sqrt (-32*sqrt(5) + 64) + 16*((sqrt(5)*x - x - 2)*sqrt(x + 1) + sqrt(5)*(x + 1) - 3*x - 1)*sqrt(x + sqrt(x + 1)) + (3*x^2 - sqrt(5)*(x^2 + 3*x - 1) + x + 5)*sqrt(-32*sqrt(5) + 64))/(x^2 - x - 1)) + 1/20*sqrt(5)*sqrt(-32*sqrt(5) + 64)*log((4*(2*x - sqrt(5) - 1)*sqrt(x + 1)*sqrt(-32*sqrt(5) + 64) - 16* ((sqrt(5)*x - x - 2)*sqrt(x + 1) + sqrt(5)*(x + 1) - 3*x - 1)*sqrt(x + sqr t(x + 1)) + (3*x^2 - sqrt(5)*(x^2 + 3*x - 1) + x + 5)*sqrt(-32*sqrt(5) + 6 4))/(x^2 - x - 1)) + 2*sqrt(x + sqrt(x + 1)) + 3/2*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) + 8*x + 8*sqrt(x + 1) + 5)
Not integrable
Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\int \frac {\sqrt {x + \sqrt {x + 1}}}{x - \sqrt {x + 1}}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\int { \frac {\sqrt {x + \sqrt {x + 1}}}{x - \sqrt {x + 1}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.70 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=-8 \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{5}} {\left (\arctan \left (2\right ) + \arctan \left (\frac {1}{2} \, \sqrt {\sqrt {5} + 1} {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )} - \frac {1}{2} \, \sqrt {\sqrt {5} - 1}\right )\right )} + 4 \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{5}} \log \left ({\left | 4 \, \sqrt {5} \sqrt {10 \, \sqrt {5} + 20} + 20 \, \sqrt {5} + 40 \, \sqrt {x + \sqrt {x + 1}} - 40 \, \sqrt {x + 1} - 20 \, \sqrt {10 \, \sqrt {5} + 20} + 20 \right |}\right ) - 4 \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{5}} \log \left ({\left | -4 \, \sqrt {5} \sqrt {10 \, \sqrt {5} + 20} + 20 \, \sqrt {5} + 40 \, \sqrt {x + \sqrt {x + 1}} - 40 \, \sqrt {x + 1} + 20 \, \sqrt {10 \, \sqrt {5} + 20} + 20 \right |}\right ) + 2 \, \sqrt {x + \sqrt {x + 1}} - 3 \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
-8*sqrt(1/10*sqrt(5) - 1/5)*(arctan(2) + arctan(1/2*sqrt(sqrt(5) + 1)*(sqr t(x + sqrt(x + 1)) - sqrt(x + 1)) - 1/2*sqrt(sqrt(5) - 1))) + 4*sqrt(1/10* sqrt(5) + 1/5)*log(abs(4*sqrt(5)*sqrt(10*sqrt(5) + 20) + 20*sqrt(5) + 40*s qrt(x + sqrt(x + 1)) - 40*sqrt(x + 1) - 20*sqrt(10*sqrt(5) + 20) + 20)) - 4*sqrt(1/10*sqrt(5) + 1/5)*log(abs(-4*sqrt(5)*sqrt(10*sqrt(5) + 20) + 20*s qrt(5) + 40*sqrt(x + sqrt(x + 1)) - 40*sqrt(x + 1) + 20*sqrt(10*sqrt(5) + 20) + 20)) + 2*sqrt(x + sqrt(x + 1)) - 3*log(-2*sqrt(x + sqrt(x + 1)) + 2* sqrt(x + 1) + 1)
Not integrable
Time = 6.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx=\int \frac {\sqrt {x+\sqrt {x+1}}}{x-\sqrt {x+1}} \,d x \]