Integrand size = 20, antiderivative size = 171 \[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=-\frac {x}{8 \sqrt {-1+x^4}}+\frac {1}{32} \arctan \left (\frac {-\frac {1}{2}-x^2+\frac {x^4}{2}}{x \sqrt {-1+x^4}}\right )+\frac {\arctan \left (\frac {\frac {1}{2^{3/4}}+\frac {x^2}{\sqrt [4]{2}}-\frac {x^4}{2^{3/4}}}{x \sqrt {-1+x^4}}\right )}{8\ 2^{3/4}}-\frac {1}{32} \text {arctanh}\left (\frac {-\frac {1}{2}+x^2+\frac {x^4}{2}}{x \sqrt {-1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{-1+\sqrt {2} x^2+x^4}\right )}{8\ 2^{3/4}} \]
-1/8*x/(x^4-1)^(1/2)+1/32*arctan((-1/2-x^2+1/2*x^4)/x/(x^4-1)^(1/2))+1/16* arctan((1/2*2^(1/4)+1/2*x^2*2^(3/4)-1/2*x^4*2^(1/4))/x/(x^4-1)^(1/2))*2^(1 /4)-1/32*arctanh((-1/2+x^2+1/2*x^4)/x/(x^4-1)^(1/2))+1/16*arctanh(2^(3/4)* x*(x^4-1)^(1/2)/(-1+2^(1/2)*x^2+x^4))*2^(1/4)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=\frac {1}{32} \left (-\frac {4 x}{\sqrt {-1+x^4}}-(1-i) \arctan \left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )+(1+i) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^4}}{x}\right )-2 \sqrt [4]{2} \arctan \left (\frac {2^{3/4} x \sqrt {-1+x^4}}{1+\sqrt {2} x^2-x^4}\right )+2 \sqrt [4]{2} \text {arctanh}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{-1+\sqrt {2} x^2+x^4}\right )\right ) \]
((-4*x)/Sqrt[-1 + x^4] - (1 - I)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] + (1 + I)*ArcTan[((1/2 + I/2)*Sqrt[-1 + x^4])/x] - 2*2^(1/4)*ArcTan[(2^(3/4)*x*S qrt[-1 + x^4])/(1 + Sqrt[2]*x^2 - x^4)] + 2*2^(1/4)*ArcTanh[(2^(3/4)*x*Sqr t[-1 + x^4])/(-1 + Sqrt[2]*x^2 + x^4)])/32
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.30 (sec) , antiderivative size = 848, normalized size of antiderivative = 4.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {x^8}{\sqrt {x^4-1} \left (x^{16}-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {1}{4 \left (x^4+1\right ) \sqrt {x^4-1}}+\frac {1}{2 \left (x^8+1\right ) \sqrt {x^4-1}}+\frac {1}{8 \left (x^2-1\right ) \sqrt {x^4-1}}-\frac {1}{8 \left (x^2+1\right ) \sqrt {x^4-1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {x^2-1}}{\sqrt {x^4-1}}\right )|\frac {1}{2}\right ) x}{8 \sqrt {2} \sqrt {x^2}}-\frac {x}{4 \sqrt {x^4-1}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )+\frac {\sqrt {x^2-1} \sqrt {x^2+1} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}-\frac {(1+i) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{\left ((8+8 i)+8 \sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{8 \left (1+(-1)^{3/4}\right ) \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \left ((2-2 i)+2 \sqrt {2}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (-\sqrt [4]{-1},\arcsin (x),-1\right )}{8 \sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\sqrt [4]{-1},\arcsin (x),-1\right )}{8 \sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (-(-1)^{3/4},\arcsin (x),-1\right )}{8 \sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left ((-1)^{3/4},\arcsin (x),-1\right )}{8 \sqrt {x^4-1}}\) |
-1/4*x/Sqrt[-1 + x^4] - (1/32 - I/32)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] - (1/32 - I/32)*ArcTanh[((1 + I)*x)/Sqrt[-1 + x^4]] + (Sqrt[-1 + x^2]*Sqrt[ 1 + x^2]*EllipticE[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*Sq rt[-1 + x^4]) + (x*EllipticE[ArcSin[(Sqrt[2]*Sqrt[-1 + x^2])/Sqrt[-1 + x^4 ]], 1/2])/(8*Sqrt[2]*Sqrt[x^2]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[A rcSin[x], -1])/(8*(1 - (-1)^(1/4))*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(8*(1 + (-1)^(1/4))*Sqrt[-1 + x^4]) - (S qrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(8*(1 + (-1)^(3/4))*S qrt[-1 + x^4]) - ((1 + I)*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[x], -1])/(((8 + 8*I) + 8*Sqrt[2])*Sqrt[-1 + x^4]) - (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(4*Sqrt[2]*Sqrt[- 1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqr t[-1 + x^2]], 1/2])/(8*Sqrt[2]*(1 - (-1)^(1/4))*Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/ (8*Sqrt[2]*(1 + (-1)^(1/4))*Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2 ]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*(1 + (-1) ^(3/4))*Sqrt[-1 + x^4]) + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[( Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(4*((2 - 2*I) + 2*Sqrt[2])*Sqrt[-1 + x^4 ]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-(-1)^(1/4), ArcSin[x], -1])/ (8*Sqrt[-1 + x^4]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(-1)^(1/4)...
3.23.65.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.35 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {x}{8 \sqrt {x^{4}-1}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-8 i \textit {\_Z}^{6}+8 \textit {\_Z}^{4}+32 i \textit {\_Z}^{2}+16\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+2 i \textit {\_R}^{4}-4 \textit {\_R}^{2}+8 i\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}-i+\sqrt {x^{4}-1}}{x}\right )}{\textit {\_R} \left (-\textit {\_R}^{6}+6 i \textit {\_R}^{4}-4 \textit {\_R}^{2}-8 i\right )}\right )}{16}+\left (-\frac {1}{32}-\frac {i}{32}\right ) \ln \left (\frac {\left (1-i\right ) \sqrt {x^{4}-1}-2 i x}{x^{2}+i}\right )+\left (\frac {1}{32}+\frac {i}{32}\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}}{x}\right )+\left (-\frac {1}{32}-\frac {i}{32}\right ) \ln \left (2\right )\) | \(163\) |
| default | \(\frac {2 \left (x^{4}-1\right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-8 i \textit {\_Z}^{6}+8 \textit {\_Z}^{4}+32 i \textit {\_Z}^{2}+16\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+2 i \textit {\_R}^{4}-4 \textit {\_R}^{2}+8 i\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}-i+\sqrt {x^{4}-1}}{x}\right )}{\textit {\_R} \left (-\textit {\_R}^{6}+6 i \textit {\_R}^{4}-4 \textit {\_R}^{2}-8 i\right )}\right )+\left (1+i\right ) \left (x^{4}-1\right ) \ln \left (\frac {\left (1-i\right ) \sqrt {x^{4}-1}-2 i x}{x^{2}+i}\right )+\left (1+i\right ) \left (-x^{4}+1\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}}{x}\right )+4 x \sqrt {x^{4}-1}+\left (1+i\right ) \ln \left (2\right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (-1+x \right )}{32 \left (1+x \right ) \left (i-x \right ) \left (i+x \right ) \left (-1+x \right )}\) | \(217\) |
| pseudoelliptic | \(\frac {2 \left (x^{4}-1\right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-8 i \textit {\_Z}^{6}+8 \textit {\_Z}^{4}+32 i \textit {\_Z}^{2}+16\right )}{\sum }\frac {\left (-\textit {\_R}^{6}+2 i \textit {\_R}^{4}-4 \textit {\_R}^{2}+8 i\right ) \ln \left (\frac {-\textit {\_R} x -x^{2}-i+\sqrt {x^{4}-1}}{x}\right )}{\textit {\_R} \left (-\textit {\_R}^{6}+6 i \textit {\_R}^{4}-4 \textit {\_R}^{2}-8 i\right )}\right )+\left (1+i\right ) \left (x^{4}-1\right ) \ln \left (\frac {\left (1-i\right ) \sqrt {x^{4}-1}-2 i x}{x^{2}+i}\right )+\left (1+i\right ) \left (-x^{4}+1\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}}{x}\right )+4 x \sqrt {x^{4}-1}+\left (1+i\right ) \ln \left (2\right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (-1+x \right )}{32 \left (1+x \right ) \left (i-x \right ) \left (i+x \right ) \left (-1+x \right )}\) | \(217\) |
| elliptic | \(\frac {\left (-\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {2^{\frac {3}{4}} \sqrt {x^{4}-1}}{2 x}+\frac {\sqrt {2}}{2}}{\frac {x^{4}-1}{2 x^{2}}+\frac {2^{\frac {3}{4}} \sqrt {x^{4}-1}}{2 x}+\frac {\sqrt {2}}{2}}\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {x^{4}-1}}{x}+1\right )+2 \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {x^{4}-1}}{x}-1\right )\right )}{32}-\frac {\sqrt {2}\, x}{8 \sqrt {x^{4}-1}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {-\frac {\sqrt {x^{4}-1}}{x}+\frac {x^{4}-1}{2 x^{2}}+1}{\frac {\sqrt {x^{4}-1}}{x}+\frac {x^{4}-1}{2 x^{2}}+1}\right )+2 \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )+2 \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )\right )}{64}\right ) \sqrt {2}}{2}\) | \(219\) |
-1/8*x/(x^4-1)^(1/2)-1/16*sum((-_R^6+2*I*_R^4-4*_R^2+8*I)*ln((-_R*x-x^2-I+ (x^4-1)^(1/2))/x)/_R/(-_R^6+6*I*_R^4-4*_R^2-8*I),_R=RootOf(_Z^8-8*I*_Z^6+8 *_Z^4+32*I*_Z^2+16))-(1/32+1/32*I)*ln(((1-I)*(x^4-1)^(1/2)-2*I*x)/(x^2+I)) +(1/32+1/32*I)*arctan((1/2+1/2*I)*(x^4-1)^(1/2)/x)-(1/32+1/32*I)*ln(2)
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.32 \[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=\frac {2^{\frac {1}{4}} {\left (-\left (i - 1\right ) \, x^{4} + i - 1\right )} \log \left (-\frac {2 \, {\left (2 \cdot 2^{\frac {3}{4}} {\left (\left (i + 1\right ) \, x^{6} - \left (i + 1\right ) \, x^{2}\right )} + 4 \, {\left (x^{5} + i \, \sqrt {2} x^{3} - x\right )} \sqrt {x^{4} - 1} - 2^{\frac {1}{4}} {\left (\left (i - 1\right ) \, x^{8} - \left (4 i - 4\right ) \, x^{4} + i - 1\right )}\right )}}{x^{8} + 1}\right ) + 2^{\frac {1}{4}} {\left (\left (i + 1\right ) \, x^{4} - i - 1\right )} \log \left (-\frac {2 \, {\left (2 \cdot 2^{\frac {3}{4}} {\left (-\left (i - 1\right ) \, x^{6} + \left (i - 1\right ) \, x^{2}\right )} + 4 \, {\left (x^{5} - i \, \sqrt {2} x^{3} - x\right )} \sqrt {x^{4} - 1} - 2^{\frac {1}{4}} {\left (-\left (i + 1\right ) \, x^{8} + \left (4 i + 4\right ) \, x^{4} - i - 1\right )}\right )}}{x^{8} + 1}\right ) + 2^{\frac {1}{4}} {\left (-\left (i + 1\right ) \, x^{4} + i + 1\right )} \log \left (-\frac {2 \, {\left (2 \cdot 2^{\frac {3}{4}} {\left (\left (i - 1\right ) \, x^{6} - \left (i - 1\right ) \, x^{2}\right )} + 4 \, {\left (x^{5} - i \, \sqrt {2} x^{3} - x\right )} \sqrt {x^{4} - 1} - 2^{\frac {1}{4}} {\left (\left (i + 1\right ) \, x^{8} - \left (4 i + 4\right ) \, x^{4} + i + 1\right )}\right )}}{x^{8} + 1}\right ) + 2^{\frac {1}{4}} {\left (\left (i - 1\right ) \, x^{4} - i + 1\right )} \log \left (-\frac {2 \, {\left (2 \cdot 2^{\frac {3}{4}} {\left (-\left (i + 1\right ) \, x^{6} + \left (i + 1\right ) \, x^{2}\right )} + 4 \, {\left (x^{5} + i \, \sqrt {2} x^{3} - x\right )} \sqrt {x^{4} - 1} - 2^{\frac {1}{4}} {\left (-\left (i - 1\right ) \, x^{8} + \left (4 i - 4\right ) \, x^{4} - i + 1\right )}\right )}}{x^{8} + 1}\right ) + 4 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {x^{4} - 1} x}{x^{2} + 1}\right ) + 2 \, {\left (x^{4} - 1\right )} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} - 1} x - 1}{x^{4} + 1}\right ) - 8 \, \sqrt {x^{4} - 1} x}{64 \, {\left (x^{4} - 1\right )}} \]
1/64*(2^(1/4)*(-(I - 1)*x^4 + I - 1)*log(-2*(2*2^(3/4)*((I + 1)*x^6 - (I + 1)*x^2) + 4*(x^5 + I*sqrt(2)*x^3 - x)*sqrt(x^4 - 1) - 2^(1/4)*((I - 1)*x^ 8 - (4*I - 4)*x^4 + I - 1))/(x^8 + 1)) + 2^(1/4)*((I + 1)*x^4 - I - 1)*log (-2*(2*2^(3/4)*(-(I - 1)*x^6 + (I - 1)*x^2) + 4*(x^5 - I*sqrt(2)*x^3 - x)* sqrt(x^4 - 1) - 2^(1/4)*(-(I + 1)*x^8 + (4*I + 4)*x^4 - I - 1))/(x^8 + 1)) + 2^(1/4)*(-(I + 1)*x^4 + I + 1)*log(-2*(2*2^(3/4)*((I - 1)*x^6 - (I - 1) *x^2) + 4*(x^5 - I*sqrt(2)*x^3 - x)*sqrt(x^4 - 1) - 2^(1/4)*((I + 1)*x^8 - (4*I + 4)*x^4 + I + 1))/(x^8 + 1)) + 2^(1/4)*((I - 1)*x^4 - I + 1)*log(-2 *(2*2^(3/4)*(-(I + 1)*x^6 + (I + 1)*x^2) + 4*(x^5 + I*sqrt(2)*x^3 - x)*sqr t(x^4 - 1) - 2^(1/4)*(-(I - 1)*x^8 + (4*I - 4)*x^4 - I + 1))/(x^8 + 1)) + 4*(x^4 - 1)*arctan(sqrt(x^4 - 1)*x/(x^2 + 1)) + 2*(x^4 - 1)*log((x^4 + 2*x ^2 - 2*sqrt(x^4 - 1)*x - 1)/(x^4 + 1)) - 8*sqrt(x^4 - 1)*x)/(x^4 - 1)
\[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=\int \frac {x^{8}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right ) \left (x^{8} + 1\right )}\, dx \]
Integral(x**8/(sqrt((x - 1)*(x + 1)*(x**2 + 1))*(x - 1)*(x + 1)*(x**2 + 1) *(x**4 + 1)*(x**8 + 1)), x)
\[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=\int { \frac {x^{8}}{{\left (x^{16} - 1\right )} \sqrt {x^{4} - 1}} \,d x } \]
\[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=\int { \frac {x^{8}}{{\left (x^{16} - 1\right )} \sqrt {x^{4} - 1}} \,d x } \]
Timed out. \[ \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx=\int \frac {x^8}{\sqrt {x^4-1}\,\left (x^{16}-1\right )} \,d x \]