Integrand size = 32, antiderivative size = 362 \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (-3+x^4\right )}{2 \left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {\frac {1}{2} \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \] Output:
1/2*x*(x^4-3)/(x^4-1)/(x^2+(x^4+1)^(1/2))^(1/2)+1/4*(14+10*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)))-1/4*(14+10*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)))+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)-1/4*(-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)))-1 /4*(-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)))
Leaf count is larger than twice the leaf count of optimal. \(50637\) vs. \(2(362)=724\).
Time = 99.20 (sec) , antiderivative size = 50637, normalized size of antiderivative = 139.88 \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\text {Result too large to show} \] Input:
Integrate[(1 + x^4)^2/((-1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Output:
Result too large to show
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.
| \(\displaystyle \int \frac {\left (x^4+1\right )^2}{\left (x^4-1\right )^2 \sqrt {\sqrt {x^4+1}+x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{\left (-x^2-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{2 \left (x^2-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{4 (x-1)^2 \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{4 (x+1)^2 \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{\left (x^2+1\right )^2 \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{\sqrt {\sqrt {x^4+1}+x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} \int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x^2+\sqrt {x^4+1}}}dx+\frac {1}{4} \int \frac {1}{(x-1)^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} \int \frac {1}{(x+i)^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} i \int \frac {1}{(x+i) \sqrt {x^2+\sqrt {x^4+1}}}dx+\frac {1}{4} \int \frac {1}{(x+1)^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} \int \frac {1}{(x+1) \sqrt {x^2+\sqrt {x^4+1}}}dx\) |
Input:
Int[(1 + x^4)^2/((-1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Output:
$Aborted
\[\int \frac {\left (x^{4}+1\right )^{2}}{\left (x^{4}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}d x\]
Input:
int((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x)
Output:
int((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x)
Time = 5.11 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.29 \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {2 \, {\left (x^{4} - 1\right )} \sqrt {\frac {5}{2} \, \sqrt {2} + \frac {7}{2}} \arctan \left (\frac {2 \, {\left (10 \, x^{7} + 14 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{7} + 5 \, x^{3}\right )} - {\left (4 \, x^{5} - \sqrt {2} {\left (x^{5} + 3 \, x\right )} + 4 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {5}{2} \, \sqrt {2} + \frac {7}{2}}}{7 \, x^{8} + 10 \, x^{4} - 1}\right ) + \sqrt {2} {\left (x^{4} - 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - {\left (x^{4} - 1\right )} \sqrt {\frac {5}{2} \, \sqrt {2} - \frac {7}{2}} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (24 \, x^{4} + \sqrt {2} {\left (17 \, x^{4} + 3\right )} + 2 \, \sqrt {x^{4} + 1} {\left (7 \, \sqrt {2} x^{2} + 10 \, x^{2}\right )} + 4\right )} \sqrt {\frac {5}{2} \, \sqrt {2} - \frac {7}{2}}}{x^{4} - 1}\right ) + {\left (x^{4} - 1\right )} \sqrt {\frac {5}{2} \, \sqrt {2} - \frac {7}{2}} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (24 \, x^{4} + \sqrt {2} {\left (17 \, x^{4} + 3\right )} + 2 \, \sqrt {x^{4} + 1} {\left (7 \, \sqrt {2} x^{2} + 10 \, x^{2}\right )} + 4\right )} \sqrt {\frac {5}{2} \, \sqrt {2} - \frac {7}{2}}}{x^{4} - 1}\right ) - 4 \, {\left (x^{7} - 3 \, x^{3} - {\left (x^{5} - 3 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{4} - 1\right )}} \] Input:
integrate((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fric as")
Output:
1/8*(2*(x^4 - 1)*sqrt(5/2*sqrt(2) + 7/2)*arctan(2*(10*x^7 + 14*x^3 - 2*sqr t(2)*(3*x^7 + 5*x^3) - (4*x^5 - sqrt(2)*(x^5 + 3*x) + 4*x)*sqrt(x^4 + 1))* sqrt(x^2 + sqrt(x^4 + 1))*sqrt(5/2*sqrt(2) + 7/2)/(7*x^8 + 10*x^4 - 1)) + sqrt(2)*(x^4 - 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) - (x^4 - 1)*sqrt(5/2*sq rt(2) - 7/2)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x + x))* sqrt(x^2 + sqrt(x^4 + 1)) + (24*x^4 + sqrt(2)*(17*x^4 + 3) + 2*sqrt(x^4 + 1)*(7*sqrt(2)*x^2 + 10*x^2) + 4)*sqrt(5/2*sqrt(2) - 7/2))/(x^4 - 1)) + (x^ 4 - 1)*sqrt(5/2*sqrt(2) - 7/2)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1) *(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) - (24*x^4 + sqrt(2)*(17*x^4 + 3) + 2*sqrt(x^4 + 1)*(7*sqrt(2)*x^2 + 10*x^2) + 4)*sqrt(5/2*sqrt(2) - 7/2) )/(x^4 - 1)) - 4*(x^7 - 3*x^3 - (x^5 - 3*x)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt (x^4 + 1)))/(x^4 - 1)
\[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {\left (x^{4} + 1\right )^{2}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \] Input:
integrate((x**4+1)**2/(x**4-1)**2/(x**2+(x**4+1)**(1/2))**(1/2),x)
Output:
Integral((x**4 + 1)**2/((x - 1)**2*(x + 1)**2*(x**2 + 1)**2*sqrt(x**2 + sq rt(x**4 + 1))), x)
\[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{2}}{{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}} \,d x } \] Input:
integrate((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxi ma")
Output:
integrate((x^4 + 1)^2/((x^4 - 1)^2*sqrt(x^2 + sqrt(x^4 + 1))), x)
\[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{2}}{{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}} \,d x } \] Input:
integrate((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac ")
Output:
integrate((x^4 + 1)^2/((x^4 - 1)^2*sqrt(x^2 + sqrt(x^4 + 1))), x)
Timed out. \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\int \frac {{\left (x^4+1\right )}^2}{{\left (x^4-1\right )}^2\,\sqrt {\sqrt {x^4+1}+x^2}} \,d x \] Input:
int((x^4 + 1)^2/((x^4 - 1)^2*((x^4 + 1)^(1/2) + x^2)^(1/2)),x)
Output:
int((x^4 + 1)^2/((x^4 - 1)^2*((x^4 + 1)^(1/2) + x^2)^(1/2)), x)
\[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {3 \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{5}-7 \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x -12 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{14}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}+12 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{14}}{x^{12}-x^{8}-x^{4}+1}d x \right )-20 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{10}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}+20 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{10}}{x^{12}-x^{8}-x^{4}+1}d x \right )-24 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{6}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}+24 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{6}}{x^{12}-x^{8}-x^{4}+1}d x \right )-16 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}+16 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}}{x^{12}-x^{8}-x^{4}+1}d x \right )+38 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{8}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}-38 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{8}}{x^{12}-x^{8}-x^{4}+1}d x \right )+2 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{12}-x^{8}-x^{4}+1}d x \right ) x^{4}-2 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{12}-x^{8}-x^{4}+1}d x \right )}{9 x^{4}-9} \] Input:
int((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x)
Output:
(3*sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x**5 - 7*sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x - 12*int((sqrt(sqrt(x**4 + 1) + x**2)*x**14)/(x* *12 - x**8 - x**4 + 1),x)*x**4 + 12*int((sqrt(sqrt(x**4 + 1) + x**2)*x**14 )/(x**12 - x**8 - x**4 + 1),x) - 20*int((sqrt(sqrt(x**4 + 1) + x**2)*x**10 )/(x**12 - x**8 - x**4 + 1),x)*x**4 + 20*int((sqrt(sqrt(x**4 + 1) + x**2)* x**10)/(x**12 - x**8 - x**4 + 1),x) - 24*int((sqrt(sqrt(x**4 + 1) + x**2)* x**6)/(x**12 - x**8 - x**4 + 1),x)*x**4 + 24*int((sqrt(sqrt(x**4 + 1) + x* *2)*x**6)/(x**12 - x**8 - x**4 + 1),x) - 16*int((sqrt(sqrt(x**4 + 1) + x** 2)*x**2)/(x**12 - x**8 - x**4 + 1),x)*x**4 + 16*int((sqrt(sqrt(x**4 + 1) + x**2)*x**2)/(x**12 - x**8 - x**4 + 1),x) + 38*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x**8)/(x**12 - x**8 - x**4 + 1),x)*x**4 - 38*int((sqr t(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x**8)/(x**12 - x**8 - x**4 + 1),x) + 2*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1))/(x**12 - x**8 - x**4 + 1),x)*x**4 - 2*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1))/(x**12 - x**8 - x**4 + 1),x))/(9*(x**4 - 1))