Integrand size = 40, antiderivative size = 603 \[ \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\frac {\left (-803792+690 x-6024144 x^2+10470 x^3-1112160 x^4+1128 x^5+5861376 x^6-12288 x^7+143360 x^8\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{40320 \left (-1+x^2\right ) \sqrt {1+x^2} \left (4 x+8 x^3\right )+40320 \left (-1+x^2\right ) \left (1+8 x^2+8 x^4\right )}-\frac {2299}{128} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]
Time = 1.18 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.41 \[ \int \frac {\left (1+x^2\right )^{5/2} \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 (2 \left (-401896+345 x-3012072 x^2+5235 x^3-556080 x^4+564 x^5+2930688 x^6-6144 x^7+71680 x^8\right )+\left (-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7+\left (935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{40320 \left (-1+x^2\right ) \left (1+8 x^2+8 x^4+4 \sqrt {1+x^2} \left (x+2 x^3\right )\right )}-\frac {2299}{128} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+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+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {14 \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+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{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 )-\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}+4 \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 {-16 \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+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(2*(-401896 + 345*x - 3012072*x^2 + 523 5*x^3 - 556080*x^4 + 564*x^5 + 2930688*x^6 - 6144*x^7 + 71680*x^8) + (-184 + 1525*x - 6328*x^2 - 8025*x^3 - 1680*x^4 - 3740*x^5 + 8192*x^6 + 10240*x ^7)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-282 - 3314848*x + 5298*x^2 - 3989088*x^3 + 7272*x^4 + 5789696*x^5 - 12288*x^6 + 143360*x^7 + (935 - 24 16*x - 2315*x^2 - 5776*x^3 - 8860*x^4 + 8192*x^5 + 10240*x^6)*Sqrt[x + Sqr t[1 + x^2]])))/(40320*(-1 + x^2)*(1 + 8*x^2 + 8*x^4 + 4*Sqrt[1 + x^2]*(x + 2*x^3))) - (2299*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128 + 4*Root Sum[-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 + Log[Sqrt[1 + S qrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum [-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 - 4 *RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[x + S qrt[1 + x^2]]] - #1] - 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 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sq rt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x ^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#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.
\(\displaystyle \int \frac {\left (x^2+1\right )^{5/2} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (x^2+1\right )^{5/2}}{2 \left (1-x^2\right )}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (x^2+1\right )^{5/2}}{4 (1-x)^2}+\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (x^2+1\right )^{5/2}}{4 (x+1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \int \frac {\left (x^2+1\right )^{5/2} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(1-x)^2}dx+\frac {1}{4} \int \frac {\left (x^2+1\right )^{5/2} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{1-x}dx+\frac {1}{4} \int \frac {\left (x^2+1\right )^{5/2} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1)^2}dx+\frac {1}{4} \int \frac {\left (x^2+1\right )^{5/2} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x+1}dx\) |
3.32.6.3.1 Defintions of rubi rules used
Not integrable
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05
\[\int \frac {\left (x^{2}+1\right )^{\frac {5}{2}} \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.19 (sec) , antiderivative size = 6743, normalized size of antiderivative = 11.18 \[ \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \]
Not integrable
Time = 0.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{\frac {5}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} - 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\text {Timed out} \]
Not integrable
Time = 8.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^{5/2}}{{\left (x^2-1\right )}^2} \,d x \]