Integrand size = 44, antiderivative size = 399 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\frac {\left (-1-2 x-x^2-2 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (1+18 x+x^2+26 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-1-x-2 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (5+x+26 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{16 \sqrt {1+x^2} \left (2 x+2 x^3\right )+16 \left (1+3 x^2+2 x^4\right )}+\frac {1}{64} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \] Output:
Unintegrable
Time = 0.76 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\frac {1}{64} \left (\frac {4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-1-2 x-x^2-2 x^3+\left (1+18 x+x^2+26 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-1-x-2 x^2+\left (5+x+26 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+32 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+\text {$\#$1}^3}\&\right ]-\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {24 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-26 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+19 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \] Input:
Integrate[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2)^2,x]
Output:
((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 |
\(\displaystyle -\frac {1}{4} \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(i-x)^2}dx+\frac {1}{4} i \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{i-x}dx-\frac {1}{4} \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+i)^2}dx+\frac {1}{4} i \int \frac {\sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{x+i}dx\) |
Input:
Int[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^2)^ 2,x]
Output:
$Aborted
Not integrable
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.09
\[\int \frac {\sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{2}}d x\]
Input:
int((x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2,x)
Output:
int((x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2,x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.87 (sec) , antiderivative size = 3133, normalized size of antiderivative = 7.85 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\text {Too large to display} \] Input:
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")
Output:
Too large to include
Not integrable
Time = 3.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x^{2} + 1\right )^{2}}\, dx \] Input:
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)
Output:
Integral(sqrt(x + sqrt(x**2 + 1))*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2 + 1)**2, x)
Not integrable
Time = 0.50 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int { \frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{2}} \,d x } \] Input:
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")
Output:
integrate(sqrt(x + sqrt(x^2 + 1))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^2, x)
Timed out. \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\text {Timed out} \] Input:
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")
Output:
Timed out
Not integrable
Time = 10.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x+\sqrt {x^2+1}}}{{\left (x^2+1\right )}^2} \,d x \] Input:
int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x + (x^2 + 1)^(1/2))^(1/2))/ (x^2 + 1)^2,x)
Output:
int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x + (x^2 + 1)^(1/2))^(1/2))/ (x^2 + 1)^2, x)
Not integrable
Time = 0.63 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx=\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{\sqrt {x^{2}+1}\, x^{4}+2 \sqrt {x^{2}+1}\, x^{2}+\sqrt {x^{2}+1}+x^{5}-x^{4}+2 x^{3}-2 x^{2}+x -1}d x +\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, x}{\sqrt {x^{2}+1}\, x^{4}+2 \sqrt {x^{2}+1}\, x^{2}+\sqrt {x^{2}+1}+x^{5}-x^{4}+2 x^{3}-2 x^{2}+x -1}d x -\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {\sqrt {x^{2}+1}+x}}{\sqrt {x^{2}+1}\, x^{4}+2 \sqrt {x^{2}+1}\, x^{2}+\sqrt {x^{2}+1}+x^{5}-x^{4}+2 x^{3}-2 x^{2}+x -1}d x \right ) \] Input:
int((x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2,x)
Output:
int((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)*sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1))/(sqrt(x**2 + 1)*x**4 + 2*sqrt(x**2 + 1)*x**2 + sqrt(x**2 + 1) + x** 5 - x**4 + 2*x**3 - 2*x**2 + x - 1),x) + int((sqrt(sqrt(sqrt(x**2 + 1) + x ) + 1)*sqrt(sqrt(x**2 + 1) + x)*x)/(sqrt(x**2 + 1)*x**4 + 2*sqrt(x**2 + 1) *x**2 + sqrt(x**2 + 1) + x**5 - x**4 + 2*x**3 - 2*x**2 + x - 1),x) - int(( sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)*sqrt(sqrt(x**2 + 1) + x))/(sqrt(x**2 + 1)*x**4 + 2*sqrt(x**2 + 1)*x**2 + sqrt(x**2 + 1) + x**5 - x**4 + 2*x**3 - 2*x**2 + x - 1),x)