Integrand size = 23, antiderivative size = 280 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {x^3}{3}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} x}{1+x^2}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} x}{1+x^2}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \] Output:
-1/3*x^3+1/4*arctan((1/2*6^(1/2)-1/2*2^(1/2)-2*x)/(1/2*6^(1/2)+1/2*2^(1/2) ))/(3/2*2^(1/2)+1/2*6^(1/2))-1/4*arctan((1/2*6^(1/2)+1/2*2^(1/2)-2*x)/(1/2 *6^(1/2)-1/2*2^(1/2)))/(3/2*2^(1/2)-1/2*6^(1/2))-1/4*arctan((1/2*6^(1/2)-1 /2*2^(1/2)+2*x)/(1/2*6^(1/2)+1/2*2^(1/2)))/(3/2*2^(1/2)+1/2*6^(1/2))+1/4*a rctan((1/2*6^(1/2)+1/2*2^(1/2)+2*x)/(1/2*6^(1/2)-1/2*2^(1/2)))/(3/2*2^(1/2 )-1/2*6^(1/2))-1/4*arctanh((1/2*6^(1/2)-1/2*2^(1/2))*x/(x^2+1))/(3/2*2^(1/ 2)-1/2*6^(1/2))+1/4*arctanh((1/2*6^(1/2)+1/2*2^(1/2))*x/(x^2+1))/(3/2*2^(1 /2)+1/2*6^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.17 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {x^3}{3}+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(x^6*(1 - x^4))/(1 - x^4 + x^8),x]
Output:
-1/3*x^3 + RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1 + 2*#1^5) & ]/4
Time = 0.60 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.48, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1826, 27, 1709, 1407, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {x^6 \left (1-x^4\right )}{x^8-x^4+1} \, dx\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle -\frac {1}{3} \int -\frac {3 x^2}{x^8-x^4+1}dx-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x^2}{x^8-x^4+1}dx-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 1709 |
| \(\displaystyle \frac {\int \frac {1}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {\int \frac {1}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {2-\sqrt {3}}-x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\int \frac {\sqrt {2+\sqrt {3}}-x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {x^3}{3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {x^3}{3}\) |
Input:
Int[(x^6*(1 - x^4))/(1 - x^4 + x^8),x]
Output:
-1/3*x^3 - ((Sqrt[(2 - Sqrt[3])/(2 + Sqrt[3])]*ArcTan[(-Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]] - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/2)/(2*Sqrt[ 2 - Sqrt[3]]) + (Sqrt[(2 - Sqrt[3])/(2 + Sqrt[3])]*ArcTan[(Sqrt[2 - Sqrt[3 ]] + 2*x)/Sqrt[2 + Sqrt[3]]] + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/2)/(2*Sq rt[2 - Sqrt[3]]))/(2*Sqrt[3]) + ((Sqrt[(2 + Sqrt[3])/(2 - Sqrt[3])]*ArcTan [(-Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] - Log[1 - Sqrt[2 + Sqrt[3]] *x + x^2]/2)/(2*Sqrt[2 + Sqrt[3]]) + (Sqrt[(2 + Sqrt[3])/(2 - Sqrt[3])]*Ar cTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] + Log[1 + Sqrt[2 + Sqrt[ 3]]*x + x^2]/2)/(2*Sqrt[2 + Sqrt[3]]))/(2*Sqrt[3])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> W ith[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r) Int[x^( m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Simp[1/(2*c*r) Int[x^(m - n/2)/( q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, 3* (n/2)] && NegQ[b^2 - 4*a*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x] , x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.16
| method | result | size |
| default | \(-\frac {x^{3}}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(46\) |
| risch | \(-\frac {x^{3}}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(46\) |
Input:
int(x^6*(-x^4+1)/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/3*x^3+1/4*sum(_R^2/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.35 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx =\text {Too large to display} \] Input:
integrate(x^6*(-x^4+1)/(x^8-x^4+1),x, algorithm="fricas")
Output:
-1/3*x^3 + 1/4*sqrt(1/3)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(3*sqrt(1/3) *sqrt(3/2*sqrt(-1/3) + 1/2)*(sqrt(-1/3) + 1)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1 /2)) + 2*x) - 1/4*sqrt(1/3)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt( 1/3)*sqrt(3/2*sqrt(-1/3) + 1/2)*(sqrt(-1/3) + 1)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2)) + 2*x) - 1/4*sqrt(1/3)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*log(3*s qrt(1/3)*(sqrt(-1/3) - 1)*sqrt(-3/2*sqrt(-1/3) + 1/2)*sqrt(-sqrt(-3/2*sqrt (-1/3) + 1/2)) + 2*x) + 1/4*sqrt(1/3)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*l og(-3*sqrt(1/3)*(sqrt(-1/3) - 1)*sqrt(-3/2*sqrt(-1/3) + 1/2)*sqrt(-sqrt(-3 /2*sqrt(-1/3) + 1/2)) + 2*x) - 1/4*sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(1/4)* log(3*sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(3/4)*(sqrt(-1/3) + 1) + 2*x) + 1/4 *sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(1/4)*log(-3*sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(3/4)*(sqrt(-1/3) + 1) + 2*x) + 1/4*sqrt(1/3)*(-3/2*sqrt(-1/3) + 1/2 )^(1/4)*log(3*sqrt(1/3)*(sqrt(-1/3) - 1)*(-3/2*sqrt(-1/3) + 1/2)^(3/4) + 2 *x) - 1/4*sqrt(1/3)*(-3/2*sqrt(-1/3) + 1/2)^(1/4)*log(-3*sqrt(1/3)*(sqrt(- 1/3) - 1)*(-3/2*sqrt(-1/3) + 1/2)^(3/4) + 2*x)
Time = 1.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=- \frac {x^{3}}{3} - \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (442368 t^{7} + 192 t^{3} + x \right )} \right )\right )} \] Input:
integrate(x**6*(-x**4+1)/(x**8-x**4+1),x)
Output:
-x**3/3 - RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(442368 *_t**7 + 192*_t**3 + x)))
\[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=\int { -\frac {{\left (x^{4} - 1\right )} x^{6}}{x^{8} - x^{4} + 1} \,d x } \] Input:
integrate(x^6*(-x^4+1)/(x^8-x^4+1),x, algorithm="maxima")
Output:
-1/3*x^3 + integrate(x^2/(x^8 - x^4 + 1), x)
Time = 0.15 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {1}{3} \, x^{3} + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \] Input:
integrate(x^6*(-x^4+1)/(x^8-x^4+1),x, algorithm="giac")
Output:
-1/3*x^3 + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sq rt(6) + sqrt(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) + sqr t(2))/(sqrt(6) + sqrt(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt (6) + sqrt(2))/(sqrt(6) - sqrt(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4 *x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) - 1/48*(sqrt(6) - 3*sqrt(2))* log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log( x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 - 1/ 2*x*(sqrt(6) - sqrt(2)) + 1)
Time = 0.15 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.04 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {x^3}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \] Input:
int(-(x^6*(x^4 - 1))/(x^8 - x^4 + 1),x)
Output:
(3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*(3^(1/2)*1i + 1)) - (3^(1/2 )*x*(8 - 3^(1/2)*8i)^(1/4))/(2*(3^(1/2)*1i + 1)))*(8 - 3^(1/2)*8i)^(1/4))/ 12 - (3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4))/(2*(3^(1/2)*1i + 1)) + (3^(1 /2)*x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*(3^(1/2)*1i + 1)))*(8 - 3^(1/2)*8i)^(1 /4)*1i)/12 - x^3/3 - (2^(3/4)*3^(1/2)*atan((2^(3/4)*x*(3^(1/2)*1i + 1)^(1/ 4))/(2*(3^(1/2)*1i - 1)) - (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4)*1i)/( 2*(3^(1/2)*1i - 1)))*(3^(1/2)*1i + 1)^(1/4)*1i)/12 + (2^(3/4)*3^(1/2)*atan ((2^(3/4)*x*(3^(1/2)*1i + 1)^(1/4)*1i)/(2*(3^(1/2)*1i - 1)) + (2^(3/4)*3^( 1/2)*x*(3^(1/2)*1i + 1)^(1/4))/(2*(3^(1/2)*1i - 1)))*(3^(1/2)*1i + 1)^(1/4 ))/12
Time = 0.16 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.47 \[ \int \frac {x^6 \left (1-x^4\right )}{1-x^4+x^8} \, dx=-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{6}-\frac {\sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{4}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{6}+\frac {\sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{4}-\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}+\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{12}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{12}+\frac {\sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{8}+\frac {\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{48}-\frac {\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{48}-\frac {\sqrt {2}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{16}+\frac {\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{16}-\frac {x^{3}}{3} \] Input:
int(x^6*(-x^4+1)/(x^8-x^4+1),x)
Output:
( - 8*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) - 12*sqrt( - sqrt(3) + 2)*atan((sqrt(6) + sqrt(2) - 4*x) /(2*sqrt( - sqrt(3) + 2))) + 8*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) + 12*sqrt( - sqrt(3) + 2)*atan( (sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) - 2*sqrt(6)*atan((2*sq rt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2))) + 6*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2))) + 2*sqrt(6)*atan((2*sqrt( - sqrt (3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) - 6*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) + 4*sqrt( - sqrt(3) + 2)*sqrt(3)*log( - sq rt( - sqrt(3) + 2)*x + x**2 + 1) - 4*sqrt( - sqrt(3) + 2)*sqrt(3)*log(sqrt ( - sqrt(3) + 2)*x + x**2 + 1) + 6*sqrt( - sqrt(3) + 2)*log( - sqrt( - sqr t(3) + 2)*x + x**2 + 1) - 6*sqrt( - sqrt(3) + 2)*log(sqrt( - sqrt(3) + 2)* x + x**2 + 1) + sqrt(6)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2) - s qrt(6)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2) - 3*sqrt(2)*log(( - sqr t(6)*x - sqrt(2)*x + 2*x**2 + 2)/2) + 3*sqrt(2)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2) - 16*x**3)/48