Integrand size = 34, antiderivative size = 217 \[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \] Output:
arctan(x/(x^4-1)^(1/4))-1/20*(-10+10*5^(1/2))^(1/2)*arctan(1/2*(-2+2*5^(1/ 2))^(1/2)*x/(x^4-1)^(1/4))-1/20*(10+10*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1 /2))^(1/2)*x/(x^4-1)^(1/4))+arctanh(x/(x^4-1)^(1/4))-1/20*(-10+10*5^(1/2)) ^(1/2)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))-1/20*(10+10*5^(1/ 2))^(1/2)*arctanh(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4-1)^(1/4))
Time = 0.58 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=\frac {1}{20} \left (20 \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )+20 \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )\right ) \] Input:
Integrate[(-1 - 2*x^4 + 2*x^8)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]
Output:
(20*ArcTan[x/(-1 + x^4)^(1/4)] - Sqrt[10*(-1 + Sqrt[5])]*ArcTan[(Sqrt[(-1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[10*(1 + Sqrt[5])]*ArcTan[(Sqrt[( 1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] + 20*ArcTanh[x/(-1 + x^4)^(1/4)] - Sq rt[10*(-1 + Sqrt[5])]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[10*(1 + Sqrt[5])]*ArcTanh[(Sqrt[(1 + Sqrt[5])/2]*x)/(-1 + x^4)^(1/ 4)])/20
Time = 0.88 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {7279, 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.
| \(\displaystyle \int \frac {2 x^8-2 x^4-1}{\sqrt [4]{x^4-1} \left (x^8-x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2}{\sqrt [4]{x^4-1}}+\frac {1}{\left (x^8-x^4-1\right ) \sqrt [4]{x^4-1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}+\text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {5}}\) |
Input:
Int[(-1 - 2*x^4 + 2*x^8)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]
Output:
ArcTan[x/(-1 + x^4)^(1/4)] - (((3 - Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt [5]))^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[5]) - (((3 + Sqrt[5])/2)^(1/4)*A rcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[5]) + ArcTanh [x/(-1 + x^4)^(1/4)] - (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5])) ^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[5]) - (((3 + Sqrt[5])/2)^(1/4)*ArcTan h[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*Sqrt[5])
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.02 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {2+2 \sqrt {5}}\, \left (-5+\sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 \left (x^{4}-1\right )^{\frac {1}{4}}}{\sqrt {-2+2 \sqrt {5}}\, x}\right )}{40}-\frac {\sqrt {-2+2 \sqrt {5}}\, \left (5+\sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 \left (x^{4}-1\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {5}}\, x}\right )}{40}-\frac {\sqrt {2+2 \sqrt {5}}\, \left (-5+\sqrt {5}\right ) \arctan \left (\frac {2 \left (x^{4}-1\right )^{\frac {1}{4}}}{\sqrt {-2+2 \sqrt {5}}\, x}\right )}{40}+\frac {\sqrt {-2+2 \sqrt {5}}\, \left (5+\sqrt {5}\right ) \arctan \left (\frac {2 \left (x^{4}-1\right )^{\frac {1}{4}}}{\sqrt {2+2 \sqrt {5}}\, x}\right )}{40}+\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )}{2}-\arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )-\frac {\ln \left (\frac {-x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{2}\) | \(202\) |
Input:
int((2*x^8-2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x,method=_RETURNVERBOSE)
Output:
1/40*(2+2*5^(1/2))^(1/2)*(-5+5^(1/2))*arctanh(2/(-2+2*5^(1/2))^(1/2)/x*(x^ 4-1)^(1/4))-1/40*(-2+2*5^(1/2))^(1/2)*(5+5^(1/2))*arctanh(2/(2+2*5^(1/2))^ (1/2)/x*(x^4-1)^(1/4))-1/40*(2+2*5^(1/2))^(1/2)*(-5+5^(1/2))*arctan(2/(-2+ 2*5^(1/2))^(1/2)/x*(x^4-1)^(1/4))+1/40*(-2+2*5^(1/2))^(1/2)*(5+5^(1/2))*ar ctan(2/(2+2*5^(1/2))^(1/2)/x*(x^4-1)^(1/4))+1/2*ln(((x^4-1)^(1/4)+x)/x)-ar ctan((x^4-1)^(1/4)/x)-1/2*ln((-x+(x^4-1)^(1/4))/x)
Time = 0.08 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.27 \[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \arctan \left (\frac {\sqrt {5} x \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}}}{{\left (x^{4} - 1\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {5} - 5\right )} \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \log \left (\frac {\sqrt {5} x \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \log \left (-\frac {\sqrt {5} x \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \log \left (\frac {\sqrt {5} x \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \log \left (-\frac {\sqrt {5} x \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \arctan \left (\frac {x}{{\left (x^{4} - 1\right )}^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \] Input:
integrate((2*x^8-2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="fricas")
Output:
-1/2*sqrt(1/10*sqrt(5) - 1/10)*arctan(sqrt(5)*x*sqrt(1/10*sqrt(5) - 1/10)/ (x^4 - 1)^(1/4)) - 1/2*sqrt(1/10*sqrt(5) + 1/10)*arctan(1/2*(x^4 - 1)^(1/4 )*(sqrt(5) - 5)*sqrt(1/10*sqrt(5) + 1/10)/x) - 1/4*sqrt(1/10*sqrt(5) + 1/1 0)*log((sqrt(5)*x*sqrt(1/10*sqrt(5) + 1/10) + (x^4 - 1)^(1/4))/x) + 1/4*sq rt(1/10*sqrt(5) + 1/10)*log(-(sqrt(5)*x*sqrt(1/10*sqrt(5) + 1/10) - (x^4 - 1)^(1/4))/x) - 1/4*sqrt(1/10*sqrt(5) - 1/10)*log((sqrt(5)*x*sqrt(1/10*sqr t(5) - 1/10) + (x^4 - 1)^(1/4))/x) + 1/4*sqrt(1/10*sqrt(5) - 1/10)*log(-(s qrt(5)*x*sqrt(1/10*sqrt(5) - 1/10) - (x^4 - 1)^(1/4))/x) + arctan(x/(x^4 - 1)^(1/4)) + 1/2*log((x + (x^4 - 1)^(1/4))/x) - 1/2*log(-(x - (x^4 - 1)^(1 /4))/x)
Timed out. \[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((2*x**8-2*x**4-1)/(x**4-1)**(1/4)/(x**8-x**4-1),x)
Output:
Timed out
\[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, x^{4} - 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((2*x^8-2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="maxima")
Output:
integrate((2*x^8 - 2*x^4 - 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(1/4)), x)
\[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 2 \, x^{4} - 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((2*x^8-2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="giac")
Output:
integrate((2*x^8 - 2*x^4 - 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(1/4)), x)
Timed out. \[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=\int \frac {-2\,x^8+2\,x^4+1}{{\left (x^4-1\right )}^{1/4}\,\left (-x^8+x^4+1\right )} \,d x \] Input:
int((2*x^4 - 2*x^8 + 1)/((x^4 - 1)^(1/4)*(x^4 - x^8 + 1)),x)
Output:
int((2*x^4 - 2*x^8 + 1)/((x^4 - 1)^(1/4)*(x^4 - x^8 + 1)), x)
\[ \int \frac {-1-2 x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx=2 \left (\int \frac {x^{8}}{\left (x^{4}-1\right )^{\frac {1}{4}} x^{8}-\left (x^{4}-1\right )^{\frac {1}{4}} x^{4}-\left (x^{4}-1\right )^{\frac {1}{4}}}d x \right )-2 \left (\int \frac {x^{4}}{\left (x^{4}-1\right )^{\frac {1}{4}} x^{8}-\left (x^{4}-1\right )^{\frac {1}{4}} x^{4}-\left (x^{4}-1\right )^{\frac {1}{4}}}d x \right )-\left (\int \frac {1}{\left (x^{4}-1\right )^{\frac {1}{4}} x^{8}-\left (x^{4}-1\right )^{\frac {1}{4}} x^{4}-\left (x^{4}-1\right )^{\frac {1}{4}}}d x \right ) \] Input:
int((2*x^8-2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x)
Output:
2*int(x**8/((x**4 - 1)**(1/4)*x**8 - (x**4 - 1)**(1/4)*x**4 - (x**4 - 1)** (1/4)),x) - 2*int(x**4/((x**4 - 1)**(1/4)*x**8 - (x**4 - 1)**(1/4)*x**4 - (x**4 - 1)**(1/4)),x) - int(1/((x**4 - 1)**(1/4)*x**8 - (x**4 - 1)**(1/4)* x**4 - (x**4 - 1)**(1/4)),x)