\(\int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} (1+x^8)} \, dx\) [1310]

Optimal result

Integrand size = 26, antiderivative size = 94 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^4+6 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.36 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (4 \left (\arctan \left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )+\text {RootSum}\left [2-4 \text {$\#$1}^4+6 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (-1+x^2\right )}} \] Input:

Integrate[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]
 

Output:

(Sqrt[x]*(-1 + x^2)^(1/4)*(4*(ArcTan[Sqrt[x]/(-1 + x^2)^(1/4)] + ArcTanh[S 
qrt[x]/(-1 + x^2)^(1/4)]) + RootSum[2 - 4*#1^4 + 6*#1^8 - 4*#1^12 + #1^16 
& , (-Log[Sqrt[x]] + Log[(-1 + x^2)^(1/4) - Sqrt[x]*#1])/#1 & ]))/(4*(x^2* 
(-1 + x^2))^(1/4))
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^8-1}{\sqrt [4]{x^4-x^2} \left (x^8+1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2-1} \int -\frac {1-x^8}{\sqrt {x} \sqrt [4]{x^2-1} \left (x^8+1\right )}dx}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {1-x^8}{\sqrt {x} \sqrt [4]{x^2-1} \left (x^8+1\right )}dx}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 2019

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (-x^6-x^4-x^2-1\right )}{\sqrt {x} \left (x^8+1\right )}dx}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int -\frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \frac {\left (x^2-1\right )^{3/4} \left (x^6+x^4+x^2+1\right )}{x^8+1}d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{x^2-1} \int \left (\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-i \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (i \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [8]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [4]{-1} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{5/8} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{3/4} \sqrt {x}+\sqrt [16]{-1}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} \sqrt {x}\right )}+\frac {\left (\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \left (x^2-1\right )^{3/4}}{16 \left ((-1)^{7/8} \sqrt {x}+\sqrt [16]{-1}\right )}\right )d\sqrt {x}}{\sqrt [4]{x^4-x^2}}\)

Input:

Int[(-1 + x^8)/((-x^2 + x^4)^(1/4)*(1 + x^8)),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

\[\int \frac {x^{8}-1}{\left (x^{4}-x^{2}\right )^{\frac {1}{4}} \left (x^{8}+1\right )}d x\]

Input:

int((x^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x)
 

Output:

int((x^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\text {Timed out} \] Input:

integrate((x^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 112.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.38 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{8} + 1\right )}\, dx \] Input:

integrate((x**8-1)/(x**4-x**2)**(1/4)/(x**8+1),x)
 

Output:

Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)/((x**2*(x - 1)*(x + 1))**(1 
/4)*(x**8 + 1)), x)
 

Maxima [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((x^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x, algorithm="maxima")
 

Output:

integrate((x^8 - 1)/((x^8 + 1)*(x^4 - x^2)^(1/4)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((x^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: 
Bad Argument ValueDone
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int \frac {x^8-1}{\left (x^8+1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,d x \] Input:

int((x^8 - 1)/((x^8 + 1)*(x^4 - x^2)^(1/4)),x)
                                                                                    
                                                                                    
 

Output:

int((x^8 - 1)/((x^8 + 1)*(x^4 - x^2)^(1/4)), x)
 

Reduce [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.66 \[ \int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} \left (1+x^8\right )} \, dx=\int \frac {\sqrt {x}\, \left (x^{2}-1\right )^{\frac {3}{4}} x^{7}}{x^{10}-x^{8}+x^{2}-1}d x -\left (\int \frac {\sqrt {x}\, \left (x^{2}-1\right )^{\frac {3}{4}}}{x^{11}-x^{9}+x^{3}-x}d x \right ) \] Input:

int((x^8-1)/(x^4-x^2)^(1/4)/(x^8+1),x)
 

Output:

int((sqrt(x)*(x**2 - 1)**(3/4)*x**7)/(x**10 - x**8 + x**2 - 1),x) - int((s 
qrt(x)*(x**2 - 1)**(3/4))/(x**11 - x**9 + x**3 - x),x)