Integrand size = 46, antiderivative size = 107 \[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2-2 x^2+\sqrt [3]{-x+x^5}}\right )+\log \left (1+x^2+\sqrt [3]{-x+x^5}\right )-\frac {1}{2} \log \left (1+2 x^2+x^4+\left (-1-x^2\right ) \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \]
3^(1/2)*arctan(3^(1/2)*(x^5-x)^(1/3)/(-2-2*x^2+(x^5-x)^(1/3)))+ln(1+x^2+(x ^5-x)^(1/3))-1/2*ln(1+2*x^2+x^4+(-x^2-1)*(x^5-x)^(1/3)+(x^5-x)^(2/3))
\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx \]
Integrate[((-1 - 2*x + x^2)*(-1 + 2*x + x^2))/((1 - x + 2*x^2 + x^3 + x^4) *(-x + x^5)^(1/3)),x]
Integrate[((-1 - 2*x + x^2)*(-1 + 2*x + x^2))/((1 - x + 2*x^2 + x^3 + x^4) *(-x + x^5)^(1/3)), x]
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.
| \(\displaystyle \int \frac {\left (x^2-2 x-1\right ) \left (x^2+2 x-1\right )}{\left (x^4+x^3+2 x^2-x+1\right ) \sqrt [3]{x^5-x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {\left (-x^2-2 x+1\right ) \left (-x^2+2 x+1\right )}{\sqrt [3]{x} \sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}dx}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {\sqrt [3]{x} \left (-x^2-2 x+1\right ) \left (-x^2+2 x+1\right )}{\sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {\sqrt [3]{x} \left (x^4-6 x^2+1\right )}{\sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \left (\frac {\left (-x^2-8 x+1\right ) x^{4/3}}{\sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}+\frac {\sqrt [3]{x}}{\sqrt [3]{x^4-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^5-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \left (\int \frac {x^{4/3}}{\sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}d\sqrt [3]{x}-8 \int \frac {x^{7/3}}{\sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}d\sqrt [3]{x}-\int \frac {x^{10/3}}{\sqrt [3]{x^4-1} \left (x^4+x^3+2 x^2-x+1\right )}d\sqrt [3]{x}+\frac {x^{2/3} \sqrt [3]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^4\right )}{2 \sqrt [3]{x^4-1}}\right )}{\sqrt [3]{x^5-x}}\) |
3.16.60.3.1 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]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.64 (sec) , antiderivative size = 782, normalized size of antiderivative = 7.31
RootOf(_Z^2+_Z+1)*ln((-215628532170574272*RootOf(_Z^2+_Z+1)^2*x^4+89845221 7377392800*RootOf(_Z^2+_Z+1)^2*x^3+812973192552633838*RootOf(_Z^2+_Z+1)*x^ 4-2253715023870923763*(x^5-x)^(1/3)*RootOf(_Z^2+_Z+1)*x^2-4312570643411485 44*RootOf(_Z^2+_Z+1)^2*x^2-326661081770322853*RootOf(_Z^2+_Z+1)*x^3-412140 106595081815*x^4+2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(2/3)-83209 0150298700567*(x^5-x)^(1/3)*x^2-898452217377392800*RootOf(_Z^2+_Z+1)^2*x+1 625946385105267676*RootOf(_Z^2+_Z+1)*x^2-130149507345815310*x^3+8320901502 98700567*(x^5-x)^(2/3)-2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3) -215628532170574272*RootOf(_Z^2+_Z+1)^2+326661081770322853*RootOf(_Z^2+_Z+ 1)*x-824280213190163630*x^2-832090150298700567*(x^5-x)^(1/3)+8129731925526 33838*RootOf(_Z^2+_Z+1)+130149507345815310*x-412140106595081815)/(x^4+x^3+ 2*x^2-x+1))-ln(-(215628532170574272*RootOf(_Z^2+_Z+1)^2*x^4-89845221737739 2800*RootOf(_Z^2+_Z+1)^2*x^3+1244230256893782382*RootOf(_Z^2+_Z+1)*x^4-225 3715023870923763*(x^5-x)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+431257064341148544*Ro otOf(_Z^2+_Z+1)^2*x^2-2123565516525108453*RootOf(_Z^2+_Z+1)*x^3+1440741831 318289925*x^4+2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(2/3)-14216248 73572223196*(x^5-x)^(1/3)*x^2+898452217377392800*RootOf(_Z^2+_Z+1)^2*x+248 8460513787564764*RootOf(_Z^2+_Z+1)*x^2-1094963791801900343*x^3+14216248735 72223196*(x^5-x)^(2/3)-2253715023870923763*RootOf(_Z^2+_Z+1)*(x^5-x)^(1/3) +215628532170574272*RootOf(_Z^2+_Z+1)^2+2123565516525108453*RootOf(_Z^2...
Time = 1.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\sqrt {3} \arctan \left (\frac {541310 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + \sqrt {3} {\left (311575 \, x^{4} + 193471 \, x^{3} + 623150 \, x^{2} - 193471 \, x + 311575\right )} + 777518 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}}}{3 \, {\left (166375 \, x^{4} - 493039 \, x^{3} + 332750 \, x^{2} + 493039 \, x + 166375\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + x^{3} + 2 \, x^{2} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} - x + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{3} + 2 \, x^{2} - x + 1}\right ) \]
-sqrt(3)*arctan(1/3*(541310*sqrt(3)*(x^5 - x)^(1/3)*(x^2 + 1) + sqrt(3)*(3 11575*x^4 + 193471*x^3 + 623150*x^2 - 193471*x + 311575) + 777518*sqrt(3)* (x^5 - x)^(2/3))/(166375*x^4 - 493039*x^3 + 332750*x^2 + 493039*x + 166375 )) + 1/2*log((x^4 + x^3 + 2*x^2 + 3*(x^5 - x)^(1/3)*(x^2 + 1) - x + 3*(x^5 - x)^(2/3) + 1)/(x^4 + x^3 + 2*x^2 - x + 1))
\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int \frac {\left (x^{2} - 2 x - 1\right ) \left (x^{2} + 2 x - 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + x^{3} + 2 x^{2} - x + 1\right )}\, dx \]
Integral((x**2 - 2*x - 1)*(x**2 + 2*x - 1)/((x*(x - 1)*(x + 1)*(x**2 + 1)) **(1/3)*(x**4 + x**3 + 2*x**2 - x + 1)), x)
\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int { \frac {{\left (x^{2} + 2 \, x - 1\right )} {\left (x^{2} - 2 \, x - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} + x^{3} + 2 \, x^{2} - x + 1\right )}} \,d x } \]
\[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=\int { \frac {{\left (x^{2} + 2 \, x - 1\right )} {\left (x^{2} - 2 \, x - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{3}} {\left (x^{4} + x^{3} + 2 \, x^{2} - x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (-1-2 x+x^2\right ) \left (-1+2 x+x^2\right )}{\left (1-x+2 x^2+x^3+x^4\right ) \sqrt [3]{-x+x^5}} \, dx=-\int \frac {\left (x^2+2\,x-1\right )\,\left (-x^2+2\,x+1\right )}{{\left (x^5-x\right )}^{1/3}\,\left (x^4+x^3+2\,x^2-x+1\right )} \,d x \]