Integrand size = 27, antiderivative size = 136 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt {2}} \]
1/2*arctan(x/(x^6+x^2)^(1/4))-1/4*arctan(2^(1/2)*x*(x^6+x^2)^(1/4)/(-x^2+( x^6+x^2)^(1/2)))*2^(1/2)-1/2*arctanh(x/(x^6+x^2)^(1/4))+1/4*arctanh((1/2*2 ^(1/2)*x^2+1/2*(x^6+x^2)^(1/2)*2^(1/2))/x/(x^6+x^2)^(1/4))*2^(1/2)
Time = 0.00 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x-\sqrt {1+x^4}}\right )-2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x+\sqrt {1+x^4}}\right )\right )}{4 \sqrt {x} \sqrt [4]{1+x^4}} \]
((x^2 + x^6)^(1/4)*(2*ArcTan[Sqrt[x]/(1 + x^4)^(1/4)] + Sqrt[2]*ArcTan[(Sq rt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(x - Sqrt[1 + x^4])] - 2*ArcTanh[Sqrt[x]/(1 + x^4)^(1/4)] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(x + Sq rt[1 + x^4])]))/(4*Sqrt[x]*(1 + x^4)^(1/4))
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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \sqrt [4]{x^6+x^2}}{x^8+x^4+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^6+x^2} \int -\frac {\sqrt {x} \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{x^6+x^2} \int \frac {\sqrt {x} \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}dx}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \frac {x \left (1-x^4\right ) \sqrt [4]{x^4+1}}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {2 \sqrt [4]{x^6+x^2} \int \left (\frac {\sqrt [4]{x^4+1} \left (1-\sqrt {x}\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}+1\right ) \sqrt [4]{x^4+1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {(2 x-1) \sqrt [4]{x^4+1}}{4 \left (x^2-x+1\right )}+\frac {x \left (1-2 x^2\right ) \sqrt [4]{x^4+1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^4+1}}\) |
3.20.45.3.1 Defintions of rubi rules used
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]
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
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 = 4.55 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.40
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\ln \left (\frac {x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2}+\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{-\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) \sqrt {2}}{8}+\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{4}\) | \(190\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) x}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) x}\right )}{4}+\frac {\ln \left (-\frac {-x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-2 \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-x^{3}-x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{6}+x^{2}}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}\) | \(420\) |
1/4*ln(((x^2*(x^4+1))^(1/4)-x)/x)-1/4*ln((x+(x^2*(x^4+1))^(1/4))/x)-1/2*ar ctan((x^2*(x^4+1))^(1/4)/x)+1/8*ln(((x^2*(x^4+1))^(1/4)*2^(1/2)*x+x^2+(x^2 *(x^4+1))^(1/2))/(-(x^2*(x^4+1))^(1/4)*2^(1/2)*x+x^2+(x^2*(x^4+1))^(1/2))) *2^(1/2)+1/4*arctan(((x^2*(x^4+1))^(1/4)*2^(1/2)+x)/x)*2^(1/2)+1/4*arctan( ((x^2*(x^4+1))^(1/4)*2^(1/2)-x)/x)*2^(1/2)
Result contains complex when optimal does not.
Time = 7.25 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.99 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + x}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x^{2}} x + x - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - x^{3} + x}\right ) \]
(1/16*I - 1/16)*sqrt(2)*log((4*I*(x^6 + x^2)^(1/4)*x^2 - (2*I - 2)*sqrt(2) *sqrt(x^6 + x^2)*x + sqrt(2)*((I + 1)*x^5 - (I + 1)*x^3 + (I + 1)*x) - 4*( x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) - (1/16*I - 1/16)*sqrt(2)*log((4*I*(x^6 + x^2)^(1/4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*(-(I + 1 )*x^5 + (I + 1)*x^3 - (I + 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) - (1/16*I + 1/16)*sqrt(2)*log((-4*I*(x^6 + x^2)^(1/4)*x^2 + (2*I + 2)*sqrt( 2)*sqrt(x^6 + x^2)*x + sqrt(2)*(-(I - 1)*x^5 + (I - 1)*x^3 - (I - 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) + (1/16*I + 1/16)*sqrt(2)*log((-4*I* (x^6 + x^2)^(1/4)*x^2 - (2*I + 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*((I - 1)*x^5 - (I - 1)*x^3 + (I - 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x) ) + 1/4*arctan(2*((x^6 + x^2)^(1/4)*x^2 + (x^6 + x^2)^(3/4))/(x^5 - x^3 + x)) + 1/4*log(-(x^5 + x^3 - 2*(x^6 + x^2)^(1/4)*x^2 + 2*sqrt(x^6 + x^2)*x + x - 2*(x^6 + x^2)^(3/4))/(x^5 - x^3 + x))
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Integral((x**2*(x**4 + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)/((x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1)), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]
\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8+x^4+1} \,d x \]