Integrand size = 51, antiderivative size = 323 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=-\frac {x \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{-1+x^2}+\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]+\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \] Output:
Unintegrable
Leaf count is larger than twice the leaf count of optimal. \(682\) vs. \(2(323)=646\).
Time = 0.59 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.11 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {1}{8} \left (-\frac {8 x \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{-1+x^2}+4 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\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 )+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+4 \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 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\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 )-9 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \] Input:
Integrate[((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^ 2]]])/(1 - x^2)^2,x]
Output:
((-8*x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(-1 + x^ 2) + 4*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (3*Log[Sqrt[1 + Sqrt[x + Sq rt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[ 1 + x^2]]] - #1] + 7*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Lo g[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqr t[1 + x^2]]] - #1]*#1^6)/(2*#1^3 - 3*#1^5 + #1^7) & ] + 4*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*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + S qrt[1 + x^2]]] - #1] - 9*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ])/8
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.
\(\displaystyle \int \frac {\left (x^2+1\right ) \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right )^2}-\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{1-x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(1-x)^2}dx+\frac {1}{2} \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1)^2}dx\) |
Input:
Int[((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/ (1 - x^2)^2,x]
Output:
$Aborted
Not integrable
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.13
\[\int \frac {\left (x^{2}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}d x\]
Input:
int((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^ 2+1)^2,x)
Output:
int((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^ 2+1)^2,x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.96 (sec) , antiderivative size = 6671, normalized size of antiderivative = 20.65 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2 )/(-x^2+1)^2,x, algorithm="fricas")
Output:
Too large to include
Not integrable
Time = 31.52 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.15 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right ) \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx \] Input:
integrate((x**2+1)*(x+(x**2+1)**(1/2))**(1/2)*(1+(x+(x**2+1)**(1/2))**(1/2 ))**(1/2)/(-x**2+1)**2,x)
Output:
Integral(sqrt(x + sqrt(x**2 + 1))*(x**2 + 1)*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/((x - 1)**2*(x + 1)**2), x)
Not integrable
Time = 0.70 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.13 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int { \frac {{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2 )/(-x^2+1)^2,x, algorithm="maxima")
Output:
integrate((x^2 + 1)*sqrt(x + sqrt(x^2 + 1))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 - 1)^2, x)
Timed out. \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2 )/(-x^2+1)^2,x, algorithm="giac")
Output:
Timed out
Not integrable
Time = 10.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.13 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\left (x^2+1\right )\,\sqrt {x+\sqrt {x^2+1}}}{{\left (x^2-1\right )}^2} \,d x \] Input:
int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)*(x + (x^2 + 1)^(1/2 ))^(1/2))/(x^2 - 1)^2,x)
Output:
int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)*(x + (x^2 + 1)^(1/2 ))^(1/2))/(x^2 - 1)^2, x)
Not integrable
Time = 200.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.13 \[ \int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\left (x^{2}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}d x \] Input:
int((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^ 2+1)^2,x)
Output:
int((x^2+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^ 2+1)^2,x)