Integrand size = 32, antiderivative size = 85 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {3}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {3}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+2 \text {$\#$1}^4}\&\right ] \]
ArcTan[x/(-1 + x^4)^(1/4)] + ArcTanh[x/(-1 + x^4)^(1/4)] + (3*RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1^3) + Log[(-1 + x^4)^(1/4) - x*#1]*#1^3)/(-1 + 2*#1^4) & ])/4
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 341, normalized size of antiderivative = 4.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
\(\displaystyle \int \frac {2 x^8+x^4-1}{\sqrt [4]{x^4-1} \left (x^8-x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {2}{\sqrt [4]{x^4-1}}-\frac {3 \left (1-x^4\right )}{\sqrt [4]{x^4-1} \left (x^8-x^4+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {i \sqrt {3} \arctan \left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x^4-1}}\right )}{2 \left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4}}-\frac {1}{2} i \sqrt {3} \left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt [4]{x^4-1}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {i \sqrt {3} \text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x^4-1}}\right )}{2 \left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4}}-\frac {1}{2} i \sqrt {3} \left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt [4]{x^4-1}}\right )\) |
ArcTan[x/(-1 + x^4)^(1/4)] + ((I/2)*Sqrt[3]*ArcTan[x/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(-1 + x^4)^(1/4))])/(-((I - Sqrt[3])/(I + Sqrt[3])))^(3 /4) - (I/2)*Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*ArcTan[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x)/(-1 + x^4)^(1/4)] + ArcTanh[x/(-1 + x^4 )^(1/4)] + ((I/2)*Sqrt[3]*ArcTanh[x/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4 )*(-1 + x^4)^(1/4))])/(-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4) - (I/2)*Sqrt[ 3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*ArcTanh[((-((I - Sqrt[3])/(I + S qrt[3])))^(1/4)*x)/(-1 + x^4)^(1/4)]
3.12.48.3.1 Defintions of rubi rules used
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 = 7.98 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )}{2}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\frac {9 \left (x^{4}-1\right )^{\frac {1}{4}} \textit {\_R}^{3} x -3 \textit {\_R}^{2} x^{2}+\sqrt {x^{4}-1}}{x^{2}}\right )\right )}{4}-\arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )+\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )}{2}\) | \(99\) |
trager | \(\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+2 x^{4}-1\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{8}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}} x^{7}+6 \left (x^{4}-1\right )^{\frac {3}{4}} x^{5}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{4}-1}\, x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+6 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{8}-x^{4}+1}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{8}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}} x^{7}+6 \left (x^{4}-1\right )^{\frac {3}{4}} x^{5}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{4}-1}\, x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+6 \left (x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{8}-x^{4}+1}\right )}{4}\) | \(471\) |
-1/2*ln(((x^4-1)^(1/4)-x)/x)+3/4*sum(_R*ln((9*(x^4-1)^(1/4)*_R^3*x-3*_R^2* x^2+(x^4-1)^(1/2))/x^2),_R=RootOf(9*_Z^4+1))-arctan((x^4-1)^(1/4)/x)+1/2*l n(((x^4-1)^(1/4)+x)/x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.36 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=-\frac {1}{4} \, \left (-9\right )^{\frac {1}{4}} \log \left (-\frac {\left (-9\right )^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x + 3 i \, x^{2} - 3 \, \sqrt {x^{4} - 1}}{x^{2}}\right ) + \frac {1}{4} \, \left (-9\right )^{\frac {1}{4}} \log \left (\frac {\left (-9\right )^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x - 3 i \, x^{2} + 3 \, \sqrt {x^{4} - 1}}{x^{2}}\right ) - \frac {1}{4} i \, \left (-9\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (-9\right )^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x + 3 i \, x^{2} + 3 \, \sqrt {x^{4} - 1}}{x^{2}}\right ) + \frac {1}{4} i \, \left (-9\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (-9\right )^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x + 3 i \, x^{2} + 3 \, \sqrt {x^{4} - 1}}{x^{2}}\right ) - \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]
-1/4*(-9)^(1/4)*log(-((-9)^(3/4)*(x^4 - 1)^(1/4)*x + 3*I*x^2 - 3*sqrt(x^4 - 1))/x^2) + 1/4*(-9)^(1/4)*log(((-9)^(3/4)*(x^4 - 1)^(1/4)*x - 3*I*x^2 + 3*sqrt(x^4 - 1))/x^2) - 1/4*I*(-9)^(1/4)*log((I*(-9)^(3/4)*(x^4 - 1)^(1/4) *x + 3*I*x^2 + 3*sqrt(x^4 - 1))/x^2) + 1/4*I*(-9)^(1/4)*log((-I*(-9)^(3/4) *(x^4 - 1)^(1/4)*x + 3*I*x^2 + 3*sqrt(x^4 - 1))/x^2) - arctan((x^4 - 1)^(1 /4)/x) + 1/2*log((x + (x^4 - 1)^(1/4))/x) - 1/2*log(-(x - (x^4 - 1)^(1/4)) /x)
Not integrable
Time = 14.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.42 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=\int \frac {\left (x^{4} + 1\right ) \left (2 x^{4} - 1\right )}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - x^{4} + 1\right )}\, dx \]
Integral((x**4 + 1)*(2*x**4 - 1)/(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x** 8 - x**4 + 1)), x)
Not integrable
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.38 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} + x^{4} - 1}{{\left (x^{8} - x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.38 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=\int { \frac {2 \, x^{8} + x^{4} - 1}{{\left (x^{8} - x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 6.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.38 \[ \int \frac {-1+x^4+2 x^8}{\sqrt [4]{-1+x^4} \left (1-x^4+x^8\right )} \, dx=\int \frac {2\,x^8+x^4-1}{{\left (x^4-1\right )}^{1/4}\,\left (x^8-x^4+1\right )} \,d x \]