Integrand size = 34, antiderivative size = 245 \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
1/2*x*(x^2+(x^4+1)^(1/2))^(1/2)-1/2*arctan(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^( 1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)+(-2+2*2^(1/2))^(1/2)*arctan((2+2*2^(1/ 2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))-arctanh(2^(1/ 2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)+(2+2*2^(1/2) )^(1/2)*arctanh((-2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x ^4+1)^(1/2)))
Time = 0.87 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]
(x*Sqrt[x^2 + Sqrt[1 + x^4]])/2 - ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[ 2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*ArcTan[( Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4]))/(x*Sqrt[x^2 + Sqrt[1 + x ^4]])] - Sqrt[2]*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2*(1 + Sqrt[2])]*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4] )/(Sqrt[2*(1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^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.
\(\displaystyle \int \frac {\sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}}{x^2+1} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {i \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}}{2 (-x+i)}+\frac {i \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2}}{2 (x+i)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} i \int \frac {\sqrt {x^4+1} \sqrt {x^2+\sqrt {x^4+1}}}{i-x}dx+\frac {1}{2} i \int \frac {\sqrt {x^4+1} \sqrt {x^2+\sqrt {x^4+1}}}{x+i}dx\) |
3.28.1.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \frac {\sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{2}+1}d x\]
Timed out. \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {x^{4} + 1}}{x^{2} + 1}\, dx \]
\[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 1} \,d x } \]
\[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 1} \,d x } \]
Timed out. \[ \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2}}{x^2+1} \,d x \]