Integrand size = 53, antiderivative size = 639 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\left (-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (2-48 x+2 x^2-16 x^3-4 x^4+64 x^5\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-16+4 x-48 x^2-4 x^3+64 x^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{24 \left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{8} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+9 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
Time = 1.16 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{24} \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-8-9 x-248 x^2-3 x^3-208 x^4+12 x^5+128 x^6+2 \left (1-24 x+x^2-8 x^3-2 x^4+32 x^5\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-3-96 x-9 x^2-272 x^3+12 x^4+128 x^5+4 \left (-4+x-12 x^2-x^3+16 x^4\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (-1+x^2\right ) \left (x+\sqrt {1+x^2}\right )^{7/2}}-3 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+24 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]-6 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-24 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]-6 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]
Integrate[((1 + x^2)^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^2)^2*Sqr t[x + Sqrt[1 + x^2]]),x]
((Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-8 - 9*x - 248*x^2 - 3*x^3 - 208*x^4 + 12*x^5 + 128*x^6 + 2*(1 - 24*x + x^2 - 8*x^3 - 2*x^4 + 32*x^5)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-3 - 96*x - 9*x^2 - 272*x^3 + 12*x^4 + 128 *x^5 + 4*(-4 + x - 12*x^2 - x^3 + 16*x^4)*Sqrt[x + Sqrt[1 + x^2]])))/((-1 + x^2)*(x + Sqrt[1 + x^2])^(7/2)) - 3*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x ^2]]]] + 24*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 )/(-2*#1 + #1^3) & ] - 6*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt [1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2 ]]] - #1]*#1^2)/(2*#1^3 - 3*#1^5 + #1^7) & ] - 24*RootSum[2 - 8*#1^2 + 8*# 1^4 - 4*#1^6 + #1^8 & , (-(Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1) + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3)/(2 - 2*#1^2 + #1^4) & ] - 6*RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[ x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]* #1^2)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ])/24
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.
\(\displaystyle \int \frac {\left (x^2+1\right )^2 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right )^2 \sqrt {\sqrt {x^2+1}+x}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{(-x-1) \sqrt {\sqrt {x^2+1}+x}}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{(x-1) \sqrt {\sqrt {x^2+1}+x}}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{(x-1)^2 \sqrt {\sqrt {x^2+1}+x}}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{(x+1)^2 \sqrt {\sqrt {x^2+1}+x}}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {\sqrt {x^2+1}+x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x+\sqrt {x^2+1}}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(-x-1) \sqrt {x+\sqrt {x^2+1}}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x-1)^2 \sqrt {x+\sqrt {x^2+1}}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x-1) \sqrt {x+\sqrt {x^2+1}}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1)^2 \sqrt {x+\sqrt {x^2+1}}}dx\) |
3.32.11.3.1 Defintions of rubi rules used
Not integrable
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.07
\[\int \frac {\left (x^{2}+1\right )^{2} \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.27 (sec) , antiderivative size = 6983, normalized size of antiderivative = 10.93 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Too large to display} \]
integrate((x^2+1)^2*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2/(x+(x^2+1 )^(1/2))^(1/2),x, algorithm="fricas")
Timed out. \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Timed out} \]
integrate((x**2+1)**2*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**2+1)**2/( x+(x**2+1)**(1/2))**(1/2),x)
Not integrable
Time = 0.87 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.07 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{2} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
integrate((x^2+1)^2*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2/(x+(x^2+1 )^(1/2))^(1/2),x, algorithm="maxima")
integrate((x^2 + 1)^2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/((x^2 - 1)^2*sqrt( x + sqrt(x^2 + 1))), x)
Timed out. \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Timed out} \]
integrate((x^2+1)^2*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2/(x+(x^2+1 )^(1/2))^(1/2),x, algorithm="giac")
Not integrable
Time = 8.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.07 \[ \int \frac {\left (1+x^2\right )^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^2}{{\left (x^2-1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]
int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^2)/((x^2 - 1)^2*(x + (x^2 + 1)^(1/2))^(1/2)),x)