Integrand size = 32, antiderivative size = 262 \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (5+x^2\right )}{2 \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+4 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2 \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
1/2*x*(x^2+5)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2)+4*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)-(14+10*2^(1/2))^(1/2)*ar ctan((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)) )+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)-(-14+10*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 = 1.55 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (5+x^2\right )}{2 \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+4 \sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]
(x*(5 + x^2))/(2*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + 4*Sqrt[2]*ArcTan[( -1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2* (7 + 5*Sqrt[2])]*ArcTan[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4])) /(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt [2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - Sqrt[2*(-7 + 5*Sqrt[2])]*ArcTa nh[(-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.
\(\displaystyle \int \frac {\left (x^2-1\right )^2}{\left (x^2+1\right )^2 \sqrt {\sqrt {x^4+1}+x^2}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4}{\left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\frac {4}{\left (x^2+1\right )^2 \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{\sqrt {\sqrt {x^4+1}+x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{\sqrt {x^2+\sqrt {x^4+1}}}dx-\int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {x^4+1}}}dx-\int \frac {1}{(x+i)^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-i \int \frac {1}{(x+i) \sqrt {x^2+\sqrt {x^4+1}}}dx\) |
3.28.68.3.1 Defintions of rubi rules used
\[\int \frac {\left (x^{2}-1\right )^{2}}{\left (x^{2}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (205) = 410\).
Time = 6.97 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.51 \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {16 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \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 ) + 2 \, {\left (x^{2} + 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} + {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} + 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} + 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - 2 \, {\left (x^{2} + 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} - {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} + 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} + 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - 2 \, {\left (x^{2} + 1\right )} \sqrt {-10 \, \sqrt {2} - 14} \log \left (-\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x - 4 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14} + {\left (4 \, x^{3} - \sqrt {2} {\left (3 \, x^{3} + 7 \, x\right )} + 10 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + 2 \, {\left (x^{2} + 1\right )} \sqrt {-10 \, \sqrt {2} - 14} \log \left (-\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x - 4 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14} + {\left (4 \, x^{3} - \sqrt {2} {\left (3 \, x^{3} + 7 \, x\right )} + 10 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + 4 \, {\left (x^{5} + 5 \, x^{3} - \sqrt {x^{4} + 1} {\left (x^{3} + 5 \, x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{2} + 1\right )}} \]
-1/8*(16*sqrt(2)*(x^2 + 1)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1 ))*sqrt(x^2 + sqrt(x^4 + 1))/x) - sqrt(2)*(x^2 + 1)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1) + 2*(x^2 + 1)*sqrt(10*sqrt(2) - 14)*log((2*sqrt(2)*x^2 + 4*x^2 + (4*x^3 + sqrt(2)*(3*x^3 + 7*x) - sqrt(x^4 + 1)*(3*sqrt(2)*x + 4*x) + 10* x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(10*sqrt(2) - 14) + 2*sqrt(x^4 + 1)*(sqrt (2) + 1))/(x^2 + 1)) - 2*(x^2 + 1)*sqrt(10*sqrt(2) - 14)*log((2*sqrt(2)*x^ 2 + 4*x^2 - (4*x^3 + sqrt(2)*(3*x^3 + 7*x) - sqrt(x^4 + 1)*(3*sqrt(2)*x + 4*x) + 10*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(10*sqrt(2) - 14) + 2*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1)) - 2*(x^2 + 1)*sqrt(-10*sqrt(2) - 14)*log(-( 2*sqrt(2)*x^2 - 4*x^2 + sqrt(x^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(3*sqrt(2 )*x - 4*x)*sqrt(-10*sqrt(2) - 14) + (4*x^3 - sqrt(2)*(3*x^3 + 7*x) + 10*x) *sqrt(-10*sqrt(2) - 14)) + 2*sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^2 + 1)) + 2*( x^2 + 1)*sqrt(-10*sqrt(2) - 14)*log(-(2*sqrt(2)*x^2 - 4*x^2 - sqrt(x^2 + s qrt(x^4 + 1))*(sqrt(x^4 + 1)*(3*sqrt(2)*x - 4*x)*sqrt(-10*sqrt(2) - 14) + (4*x^3 - sqrt(2)*(3*x^3 + 7*x) + 10*x)*sqrt(-10*sqrt(2) - 14)) + 2*sqrt(x^ 4 + 1)*(sqrt(2) - 1))/(x^2 + 1)) + 4*(x^5 + 5*x^3 - sqrt(x^4 + 1)*(x^3 + 5 *x))*sqrt(x^2 + sqrt(x^4 + 1)))/(x^2 + 1)
\[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{\left (x^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \]
\[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{2}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {{\left (x^{2} - 1\right )}^{2}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {{\left (x^2-1\right )}^2}{{\left (x^2+1\right )}^2\,\sqrt {\sqrt {x^4+1}+x^2}} \,d x \]