Integrand size = 29, antiderivative size = 121 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-3+2 \text {$\#$1}^4}\&\right ]+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.26 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (\text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-3+2 \text {$\#$1}^4}\&\right ]+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ]\right )}{4 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \]
(x^(3/2)*(-1 + x^2)^(3/4)*(RootSum[3 - 3*#1^4 + #1^8 & , (-(Log[Sqrt[x]]*# 1) + Log[(-1 + x^2)^(1/4) - Sqrt[x]*#1]*#1)/(-3 + 2*#1^4) & ] + RootSum[1 - #1^4 + #1^8 & , (-(Log[Sqrt[x]]*#1) + Log[(-1 + x^2)^(1/4) - Sqrt[x]*#1] *#1)/(-1 + 2*#1^4) & ]))/(4*(x^2*(-1 + x^2))^(3/4))
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.
| \(\displaystyle \int \frac {\left (x^4+1\right ) \sqrt [4]{x^4-x^2}}{x^8+x^4+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^2} \int \frac {\sqrt {x} \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}dx}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \left (\frac {\sqrt [4]{x^2-1} \left (-\sqrt {x}-1\right )}{8 \left (x+\sqrt {x}+1\right )}+\frac {\left (\sqrt {x}-1\right ) \sqrt [4]{x^2-1}}{8 \left (x-\sqrt {x}+1\right )}+\frac {\sqrt [4]{x^2-1}}{4 \left (x^2-x+1\right )}+\frac {x \sqrt [4]{x^2-1}}{2 \left (x^4-x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \sqrt [4]{x^4-x^2} \int \frac {x \sqrt [4]{x^2-1} \left (x^4+1\right )}{x^8+x^4+1}d\sqrt {x}}{\sqrt {x} \sqrt [4]{x^2-1}}\) |
3.18.94.3.1 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 = 113.55 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.81
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4}+3\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{4}-3}\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{4}-1}\right )}{4}\) | \(98\) |
| trager | \(\text {Expression too large to display}\) | \(11367\) |
1/4*sum(_R*ln((-_R*x+(x^4-x^2)^(1/4))/x)/(2*_R^4-3),_R=RootOf(_Z^8-3*_Z^4+ 3))+1/4*sum(_R*ln((-_R*x+(x^4-x^2)^(1/4))/x)/(2*_R^4-1),_R=RootOf(_Z^8-_Z^ 4+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.08 (sec) , antiderivative size = 3561, normalized size of antiderivative = 29.43 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\text {Too large to display} \]
-1/48*sqrt(6)*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 3))*log(-(24*(x^4 - x^2)^(3/4 )*(2*x^2 + sqrt(3)*(8*I*x^2 + 5*I) - 11) + 12*(x^4 - x^2)^(1/4)*(sqrt(3)*s qrt(2)*(8*I*x^4 + 5*I*x^2) + sqrt(2)*(2*x^4 - 11*x^2))*sqrt(I*sqrt(3) + 3) - (4*sqrt(6)*sqrt(x^4 - x^2)*(24*x^3 - sqrt(3)*(2*I*x^3 - 11*I*x) + 15*x) + sqrt(6)*(sqrt(3)*sqrt(2)*(-7*I*x^5 + 19*I*x^3 - 3*I*x) + sqrt(2)*(35*x^ 5 + 17*x^3 - 13*x))*sqrt(I*sqrt(3) + 3))*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 3) ))/(x^5 + x^3 + x)) + 1/48*sqrt(6)*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 3))*log( -(24*(x^4 - x^2)^(3/4)*(2*x^2 + sqrt(3)*(8*I*x^2 + 5*I) - 11) + 12*(x^4 - x^2)^(1/4)*(sqrt(3)*sqrt(2)*(8*I*x^4 + 5*I*x^2) + sqrt(2)*(2*x^4 - 11*x^2) )*sqrt(I*sqrt(3) + 3) + (4*sqrt(6)*sqrt(x^4 - x^2)*(24*x^3 + sqrt(3)*(-2*I *x^3 + 11*I*x) + 15*x) - sqrt(6)*(sqrt(3)*sqrt(2)*(7*I*x^5 - 19*I*x^3 + 3* I*x) - sqrt(2)*(35*x^5 + 17*x^3 - 13*x))*sqrt(I*sqrt(3) + 3))*sqrt(-sqrt(2 )*sqrt(I*sqrt(3) + 3)))/(x^5 + x^3 + x)) - 1/48*sqrt(6)*sqrt(sqrt(2)*sqrt( I*sqrt(3) + 3))*log(-(24*(x^4 - x^2)^(3/4)*(2*x^2 + sqrt(3)*(8*I*x^2 + 5*I ) - 11) + 12*(x^4 - x^2)^(1/4)*(sqrt(3)*sqrt(2)*(-8*I*x^4 - 5*I*x^2) - sqr t(2)*(2*x^4 - 11*x^2))*sqrt(I*sqrt(3) + 3) - (4*sqrt(6)*sqrt(x^4 - x^2)*(2 4*x^3 - sqrt(3)*(2*I*x^3 - 11*I*x) + 15*x) + sqrt(6)*(sqrt(3)*sqrt(2)*(7*I *x^5 - 19*I*x^3 + 3*I*x) - sqrt(2)*(35*x^5 + 17*x^3 - 13*x))*sqrt(I*sqrt(3 ) + 3))*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 3)))/(x^5 + x^3 + x)) + 1/48*sqrt(6) *sqrt(sqrt(2)*sqrt(I*sqrt(3) + 3))*log(-(24*(x^4 - x^2)^(3/4)*(2*x^2 + ...
Not integrable
Time = 3.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.34 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Integral((x**2*(x - 1)*(x + 1))**(1/4)*(x**4 + 1)/((x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1)), x)
Not integrable
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.24 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]
Exception generated. \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:proot error [1,0,0,0,1,0,0,0,1]proo t error [1,0,0,0,-1,0,0,0,1]proot error [1,0,-10,0,1]proot error [1,0,-10, 0,1]proot
Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.24 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^4}}{1+x^4+x^8} \, dx=\int \frac {\left (x^4+1\right )\,{\left (x^4-x^2\right )}^{1/4}}{x^8+x^4+1} \,d x \]