Integrand size = 40, antiderivative size = 266 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=-4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\text {RootSum}\left [-1-8 \text {$\#$1}^2-8 \text {$\#$1}^3+14 \text {$\#$1}^4+32 \text {$\#$1}^5+24 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-1+\text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [-1+8 \text {$\#$1}^2+8 \text {$\#$1}^3+18 \text {$\#$1}^4+32 \text {$\#$1}^5+24 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (-1+\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{1+4 \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=-4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+4 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
-4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4 *#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*# 1^2 + #1^4) & ]
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.
\(\displaystyle \int \frac {\sqrt {x^2+1} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{1-x^2} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt {x^2+1} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{2 (1-x)}+\frac {\sqrt {x^2+1} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{2 (x+1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {x^2+1} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{1-x}dx+\frac {1}{2} \int \frac {\sqrt {x^2+1} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x+1}dx\) |
3.28.79.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Not integrable
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.12
\[\int \frac {\sqrt {x^{2}+1}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{2}+1}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.35 (sec) , antiderivative size = 914, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=\frac {1}{2} \, \sqrt {2 \, \sqrt {\sqrt {2} + 1} + 2} \log \left (\sqrt {2} \sqrt {2 \, \sqrt {\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {\sqrt {2} + 1} + 2} \log \left (-\sqrt {2} \sqrt {2 \, \sqrt {\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {\sqrt {2} + 1} + 2} \log \left (\sqrt {2} \sqrt {-2 \, \sqrt {\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {\sqrt {2} + 1} + 2} \log \left (-\sqrt {2} \sqrt {-2 \, \sqrt {\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {\sqrt {2} - 1} + 2} \log \left (\sqrt {2} \sqrt {2 \, \sqrt {\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {\sqrt {2} - 1} + 2} \log \left (-\sqrt {2} \sqrt {2 \, \sqrt {\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {\sqrt {2} - 1} + 2} \log \left (\sqrt {2} \sqrt {-2 \, \sqrt {\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {\sqrt {2} - 1} + 2} \log \left (-\sqrt {2} \sqrt {-2 \, \sqrt {\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {-\sqrt {2} + 1} + 2} \log \left (\sqrt {2} \sqrt {2 \, \sqrt {-\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {-\sqrt {2} + 1} + 2} \log \left (-\sqrt {2} \sqrt {2 \, \sqrt {-\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {-\sqrt {2} + 1} + 2} \log \left (\sqrt {2} \sqrt {-2 \, \sqrt {-\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {-\sqrt {2} + 1} + 2} \log \left (-\sqrt {2} \sqrt {-2 \, \sqrt {-\sqrt {2} + 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {-\sqrt {2} - 1} + 2} \log \left (\sqrt {2} \sqrt {2 \, \sqrt {-\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {-\sqrt {2} - 1} + 2} \log \left (-\sqrt {2} \sqrt {2 \, \sqrt {-\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {-\sqrt {2} - 1} + 2} \log \left (\sqrt {2} \sqrt {-2 \, \sqrt {-\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {-\sqrt {2} - 1} + 2} \log \left (-\sqrt {2} \sqrt {-2 \, \sqrt {-\sqrt {2} - 1} + 2} + 2 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - 4 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 2 \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) - 2 \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \]
1/2*sqrt(2*sqrt(sqrt(2) + 1) + 2)*log(sqrt(2)*sqrt(2*sqrt(sqrt(2) + 1) + 2 ) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(sqrt(2) + 1) + 2)*log(-sqrt(2)*sqrt(2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(sqrt(2) + 1) + 2)*log(sqrt(2)*sqrt(-2*sqrt(s qrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt (sqrt(2) + 1) + 2)*log(-sqrt(2)*sqrt(-2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sq rt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(sqrt(2) - 1) + 2)*log(sqrt(2 )*sqrt(2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1 /2*sqrt(2*sqrt(sqrt(2) - 1) + 2)*log(-sqrt(2)*sqrt(2*sqrt(sqrt(2) - 1) + 2 ) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(sqrt(2) - 1) + 2)*log(sqrt(2)*sqrt(-2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(sqrt(2) - 1) + 2)*log(-sqrt(2)*sqrt(-2*sqrt (sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqr t(-sqrt(2) + 1) + 2)*log(sqrt(2)*sqrt(2*sqrt(-sqrt(2) + 1) + 2) + 2*sqrt(s qrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(2*sqrt(-sqrt(2) + 1) + 2)*log(-sqr t(2)*sqrt(2*sqrt(-sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(-sqrt(2) + 1) + 2)*log(sqrt(2)*sqrt(-2*sqrt(-sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(-sqrt(2 ) + 1) + 2)*log(-sqrt(2)*sqrt(-2*sqrt(-sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(2*sqrt(-sqrt(2) - 1) + 2)*log(sqrt(2)*...
Not integrable
Time = 1.68 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=- \int \frac {\sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} - 1}\, dx \]
Not integrable
Time = 0.61 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=\int { -\frac {\sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} - 1} \,d x } \]
Not integrable
Time = 125.38 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=\int { -\frac {\sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} - 1} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx=-\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x^2+1}}{x^2-1} \,d x \]