Integrand size = 44, antiderivative size = 402 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\left (-1+2 x-x^2+2 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-3-2 x+5 x^2-2 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (1-x+2 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-1+5 x-2 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{16 \sqrt {1+x^2} \left (2 x+2 x^3\right )+16 \left (1+3 x^2+2 x^4\right )}-\frac {1}{64} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-24 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{64} \left (\frac {4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-1+2 x-x^2+2 x^3-\left (3+2 x-5 x^2+2 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (1-x+2 x^2+\left (-1+5 x-2 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+32 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {36 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+136 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]
((4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-1 + 2*x - x^2 + 2*x^3 - (3 + 2*x - 5*x^2 + 2*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(1 - x + 2*x^2 + ( -1 + 5*x - 2*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/((1 + x^2)*(1 + 2*x^2 + 2*x*S qrt[1 + x^2])) + 32*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[S qrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (36*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 136*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 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.
\(\displaystyle \int \frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (x^2+1\right )^2 \sqrt {\sqrt {x^2+1}+x}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{2 \left (-x^2-1\right ) \sqrt {\sqrt {x^2+1}+x}}-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{4 (-x+i)^2 \sqrt {\sqrt {x^2+1}+x}}-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{4 (x+i)^2 \sqrt {\sqrt {x^2+1}+x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{4} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(i-x)^2 \sqrt {x+\sqrt {x^2+1}}}dx+\frac {1}{4} i \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(i-x) \sqrt {x+\sqrt {x^2+1}}}dx-\frac {1}{4} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+i)^2 \sqrt {x+\sqrt {x^2+1}}}dx+\frac {1}{4} i \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+i) \sqrt {x+\sqrt {x^2+1}}}dx\) |
3.31.4.3.1 Defintions of rubi rules used
Not integrable
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.08
\[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.96 (sec) , antiderivative size = 3327, normalized size of antiderivative = 8.28 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Too large to display} \]
integrate((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")
1/64*(sqrt(2)*(x^2 + 1)*sqrt(sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1 /2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-4 37/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2 048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/209 7152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/20 97152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 333 7/2097152) - 93816) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 51 2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37)*log(1/4*(10485760*(50 643935151*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337 /2097152) + 74) + 35783410727342*sqrt(2))*(437/4096*I*sqrt(2) - 1/2*sqrt(- 49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 + 37521625686833364992 0*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/20 97152) - 37/2048)^2 + (531040149448949760*sqrt(2)*(-437/4096*I*sqrt(2) - 1 /2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 936912800293 5*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152 ) + 74) - 33052532426664649*sqrt(2))*(-437*I*sqrt(2) + 2048*sqrt(-49125/52 4288*I*sqrt(2) + 3337/2097152) + 74) - sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*( -437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37 /2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 333...
Not integrable
Time = 9.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )^{2}}\, dx \]
Not integrable
Time = 0.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
integrate((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")
Timed out. \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Timed out} \]
integrate((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 = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{{\left (x^2+1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]