Integrand size = 40, antiderivative size = 140 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \]
-1/2*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) & ]/2
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right ) \sqrt {x^2+1}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{2 (1-x) \sqrt {x^2+1}}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{2 (x+1) \sqrt {x^2+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(1-x) \sqrt {x^2+1}}dx+\frac {1}{2} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1) \sqrt {x^2+1}}dx\) |
3.20.90.3.1 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 = 0.23
\[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right ) \sqrt {x^{2}+1}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.37 (sec) , antiderivative size = 785, normalized size of antiderivative = 5.61 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=\frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {\sqrt {\sqrt {2} + 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {\sqrt {\sqrt {2} + 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {-\sqrt {\sqrt {2} + 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {-\sqrt {\sqrt {2} + 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {\sqrt {\sqrt {2} - 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {\sqrt {\sqrt {2} - 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {-\sqrt {\sqrt {2} - 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {-\sqrt {\sqrt {2} - 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {\sqrt {-\sqrt {2} + 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {\sqrt {-\sqrt {2} + 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {-\sqrt {-\sqrt {2} + 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-\sqrt {2} + 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {-\sqrt {-\sqrt {2} + 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {\sqrt {-\sqrt {2} - 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {\sqrt {-\sqrt {2} - 1} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \sqrt {-\sqrt {-\sqrt {2} - 1} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-\sqrt {2} - 1} + 1} \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \sqrt {-\sqrt {-\sqrt {2} - 1} + 1}\right ) \]
1/4*sqrt(2)*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(sqrt(2) + 1) + 1)) - 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(sqrt(2) + 1) + 1)) + 1/4*sqrt(2)*sqrt(-sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(-sqrt(sqrt(2) + 1) + 1)) - 1/4*sqrt(2)*sqrt(-sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(-sqrt(sqrt(2) + 1) + 1) ) - 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1 )) + 1) + sqrt(sqrt(sqrt(2) - 1) + 1)) + 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) - 1 ) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(sqrt(2) - 1) + 1) ) - 1/4*sqrt(2)*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(-sqrt(sqrt(2) - 1) + 1)) + 1/4*sqrt(2)*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(-sqrt(sqrt(2) - 1) + 1)) - 1/4*sqrt(2)*sqrt(sqrt(-sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^ 2 + 1)) + 1) + sqrt(sqrt(-sqrt(2) + 1) + 1)) + 1/4*sqrt(2)*sqrt(sqrt(-sqrt (2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(-sqrt(2) + 1) + 1)) - 1/4*sqrt(2)*sqrt(-sqrt(-sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sq rt(x^2 + 1)) + 1) + sqrt(-sqrt(-sqrt(2) + 1) + 1)) + 1/4*sqrt(2)*sqrt(-sqr t(-sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(-sqrt(-s qrt(2) + 1) + 1)) + 1/4*sqrt(2)*sqrt(sqrt(-sqrt(2) - 1) + 1)*log(sqrt(sqrt (x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(-sqrt(2) - 1) + 1)) - 1/4*sqrt(2)*...
Not integrable
Time = 3.47 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=- \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx \]
Not integrable
Time = 0.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x^{2} + 1} {\left (x^{2} - 1\right )}} \,d x } \]
Not integrable
Time = 126.62 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x^{2} + 1} {\left (x^{2} - 1\right )}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx=-\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\left (x^2-1\right )\,\sqrt {x^2+1}} \,d x \]