3.14.9 \(\int \frac {-1+x^8}{\sqrt [4]{-x^2+x^4} (1+x^8)} \, dx\) [1309]

3.14.9.1 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
 

3.14.9.2 Mathematica [A] (verified)

Time = 0.38 (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))
 

3.14.9.3 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
 

3.14.9.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2019

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px 
, Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && 
 EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7276

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]
 

3.14.9.4 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)
 

3.14.9.5 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
 

3.14.9.6 Sympy [N/A]

Not integrable

Time = 104.33 (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)
 

3.14.9.7 Maxima [N/A]

Not integrable

Time = 0.30 (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)
 

3.14.9.8 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 ValueInvalid _EXT in replace_ext Error: Bad Argument ValueDon 
e
 

3.14.9.9 Mupad [N/A]

Not integrable

Time = 5.89 (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)