Integrand size = 25, antiderivative size = 171 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
(2^(1/2)-1)^(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)-(1+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 = 0.56 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]
Sqrt[-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[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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^2+1} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {i \sqrt {\sqrt {x^4+1}+x^2}}{2 (-x+i)}+\frac {i \sqrt {\sqrt {x^4+1}+x^2}}{2 (x+i)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {x^4+1}}}{i-x}dx+\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {x^4+1}}}{x+i}dx\) |
3.23.67.3.1 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]
\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{2}+1}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (133) = 266\).
Time = 0.96 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 1} \log \left (-\frac {\sqrt {2} x^{2} - 2 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} x \sqrt {-\sqrt {2} + 1} - {\left (x^{3} - \sqrt {2} x + x\right )} \sqrt {-\sqrt {2} + 1}\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {-\sqrt {2} + 1} \log \left (-\frac {\sqrt {2} x^{2} - 2 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} x \sqrt {-\sqrt {2} + 1} - {\left (x^{3} - \sqrt {2} x + x\right )} \sqrt {-\sqrt {2} + 1}\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) \]
1/4*sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqr t(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1) - 1/4*sqrt(sqrt(2) + 1)*log(( sqrt(2)*x^2 + 2*x^2 + (x^3 + sqrt(2)*x - sqrt(x^4 + 1)*x + x)*sqrt(x^2 + s qrt(x^4 + 1))*sqrt(sqrt(2) + 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1)) + 1/4*sqrt(sqrt(2) + 1)*log((sqrt(2)*x^2 + 2*x^2 - (x^3 + sqrt(2)*x - sqrt (x^4 + 1)*x + x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) + 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1)) - 1/4*sqrt(-sqrt(2) + 1)*log(-(sqrt(2)*x^2 - 2*x^2 + sqrt(x^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*x*sqrt(-sqrt(2) + 1) - (x ^3 - sqrt(2)*x + x)*sqrt(-sqrt(2) + 1)) + sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^ 2 + 1)) + 1/4*sqrt(-sqrt(2) + 1)*log(-(sqrt(2)*x^2 - 2*x^2 - sqrt(x^2 + sq rt(x^4 + 1))*(sqrt(x^4 + 1)*x*sqrt(-sqrt(2) + 1) - (x^3 - sqrt(2)*x + x)*s qrt(-sqrt(2) + 1)) + sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^2 + 1))
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 1}\, dx \]
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 1} \,d x } \]
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 1} \,d x } \]
Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{x^2+1} \,d x \]