Integrand size = 44, antiderivative size = 399 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\frac {\left (-1-2 x-x^2-2 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (1+18 x+x^2+26 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-1-x-2 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (5+x+26 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{16 \sqrt {1+x^2} \left (2 x+2 x^3\right )+16 \left (1+3 x^2+2 x^4\right )}+\frac {1}{64} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\frac {1}{64} \left (\frac {4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-1-2 x-x^2-2 x^3+\left (1+18 x+x^2+26 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-1-x-2 x^2+\left (5+x+26 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+32 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+\text {$\#$1}^3}\&\right ]-\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {24 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-26 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+19 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]
((4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-1 - 2*x - x^2 - 2*x^3 + (1 + 18*x + x^2 + 26*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-1 - x - 2*x^2 + (5 + x + 26*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/((1 + x^2)*(1 + 2*x^2 + 2*x*Sq rt[1 + x^2])) + 32*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sq rt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2 ]]] - #1]*#1^2)/(-#1 + #1^3) & ] - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (24*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 26*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x ^2]]] - #1]*#1^4 + 19*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(- #1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/64
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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (x^2+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{2 \left (-x^2-1\right )}-\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{4 (-x+i)^2}-\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{4 (x+i)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{4} \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(i-x)^2}dx+\frac {1}{4} i \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{i-x}dx-\frac {1}{4} \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+i)^2}dx+\frac {1}{4} i \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x+i}dx\) |
3.30.100.3.1 Defintions of rubi rules used
Not integrable
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.09
\[\int \frac {\sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{2}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.98 (sec) , antiderivative size = 3329, normalized size of antiderivative = 8.34 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\text {Too large to display} \]
integrate((x+(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")
1/64*(sqrt(2)*(x^2 + 1)*sqrt(sqrt(2)*sqrt(-1572864*(117/4096*I*sqrt(2) - 1 /2*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1572864*(-1 17/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37 /2048)^2 - 1/16*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271 /2097152) - 74)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 105271 /2097152) + 222) + 4329/2*I*sqrt(2) - 37888*sqrt(2645/524288*I*sqrt(2) - 1 05271/2097152) - 59480) + 512*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 512*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37)*log(1/4*(209715 2*(267626275*sqrt(2)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 1 05271/2097152) - 74) - 461246050066*sqrt(2))*(-117/4096*I*sqrt(2) - 1/2*sq rt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 967303076388012 032*sqrt(2)*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) - 105271/ 2097152) + 37/2048)^2 + 5*(112250595573760*sqrt(2)*(117/4096*I*sqrt(2) - 1 /2*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 + 1980434435* sqrt(2)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 105271/2097152 ) - 74) + 859779825444*sqrt(2))*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I* sqrt(2) - 105271/2097152) - 74) - sqrt(-1572864*(117/4096*I*sqrt(2) - 1/2* sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1572864*(-117/ 4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37/20 48)^2 - 1/16*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271...
Not integrable
Time = 3.62 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x^{2} + 1\right )^{2}}\, dx \]
Not integrable
Time = 0.60 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int { \frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{2}} \,d x } \]
integrate((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")
Timed out. \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\text {Timed out} \]
integrate((x+(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 = 36, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x+\sqrt {x^2+1}}}{{\left (x^2+1\right )}^2} \,d x \]