Integrand size = 53, antiderivative size = 639 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\left (75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-16-18 x-16 x^2-6 x^3+32 x^4+24 x^5\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (8-120 x+24 x^2-1170 x^3-32 x^4+240 x^5\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-6-32 x-18 x^2+32 x^3+24 x^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{105 \left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{5/2}}-\text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-15 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-15 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
Time = 0.01 (sec) , antiderivative size = 849, normalized size of antiderivative = 1.33 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (75+24 x-735 x^2+8 x^3-1050 x^4-32 x^5+240 x^6+2 \left (-8-9 x-8 x^2-3 x^3+16 x^4+12 x^5\right ) \sqrt {x+\sqrt {1+x^2}}+2 \sqrt {1+x^2} \left (4-60 x+12 x^2-585 x^3-16 x^4+120 x^5+\left (-3-16 x-9 x^2+16 x^3+12 x^4\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{105 \left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{5/2}}-\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 {5 \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+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 ]-\frac {1}{4} \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 ]+\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}-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+2 \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 ]-\frac {1}{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 )-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 ] \]
Integrate[((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]
(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(75 + 24*x - 735*x^2 + 8*x^3 - 1050*x^4 - 32*x^5 + 240*x^6 + 2*(-8 - 9*x - 8*x^2 - 3*x^3 + 16*x^4 + 12*x^5)*Sqrt[ x + Sqrt[1 + x^2]] + 2*Sqrt[1 + x^2]*(4 - 60*x + 12*x^2 - 585*x^3 - 16*x^4 + 120*x^5 + (-3 - 16*x - 9*x^2 + 16*x^3 + 12*x^4)*Sqrt[x + Sqrt[1 + x^2]] )))/(105*(-1 + x^2)*(x + Sqrt[1 + x^2])^(5/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 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 - 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*#1^3 - 3*#1^5 + #1^7) & ]/4 + R ootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sq rt[1 + x^2]]] - #1]*#1) - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1 ^3 + 2*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[Sq rt[1 + Sqrt[x + Sqrt[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) & ]/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.
| \(\displaystyle \int \frac {\left (x^2+1\right )^2 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{-x-1}+\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{x-1}+\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{(x-1)^2}+\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{(x+1)^2}+\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}dx+\int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{-x-1}dx+\int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x-1)^2}dx+\int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x-1}dx+\int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1)^2}dx\) |
3.32.14.3.1 Defintions of rubi rules used
Not integrable
Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.07
\[\int \frac {\left (x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.24 (sec) , antiderivative size = 7099, normalized size of antiderivative = 11.11 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Too large to display} \]
integrate((x^2+1)^2*(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")
Timed out. \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \]
integrate((x**2+1)**2*(x+(x**2+1)**(1/2))**(1/2)*(1+(x+(x**2+1)**(1/2))**( 1/2))**(1/2)/(-x**2+1)**2,x)
Not integrable
Time = 0.90 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.07 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2}} \,d x } \]
integrate((x^2+1)^2*(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")
integrate((x^2 + 1)^2*sqrt(x + sqrt(x^2 + 1))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 - 1)^2, x)
Timed out. \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \]
integrate((x^2+1)^2*(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")
Not integrable
Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.07 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}}{{\left (x^2-1\right )}^2} \,d x \]
int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^2*(x + (x^2 + 1)^(1 /2))^(1/2))/(x^2 - 1)^2,x)