Integrand size = 29, antiderivative size = 429 \[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^4}}\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{4 \sqrt [3]{2}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{x^2+x^4}\right )-\frac {1}{6} \log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{4 \sqrt [3]{2}}+\frac {1}{12} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}-\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{8 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}} \]
-1/6*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4+x^2)^(1/3)))-1/2*arctan(3^(1/2)*x /(x+2*(x^4+x^2)^(1/3)))*3^(1/2)-1/8*3^(1/2)*arctan(3^(1/2)*x/(-x+2^(2/3)*( x^4+x^2)^(1/3)))*2^(2/3)-1/24*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^4+x^2 )^(1/3)))*2^(2/3)+1/2*ln(-x+(x^4+x^2)^(1/3))-1/6*ln(x+(x^4+x^2)^(1/3))+1/2 4*ln(-2*x+2^(2/3)*(x^4+x^2)^(1/3))*2^(2/3)-1/8*ln(2*x+2^(2/3)*(x^4+x^2)^(1 /3))*2^(2/3)+1/12*ln(x^2-x*(x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))-1/4*ln(x^2+x*( x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))+1/16*ln(-2*x^2+2^(2/3)*x*(x^4+x^2)^(1/3)-2 ^(1/3)*(x^4+x^2)^(2/3))*2^(2/3)-1/48*ln(2*x^2+2^(2/3)*x*(x^4+x^2)^(1/3)+2^ (1/3)*(x^4+x^2)^(2/3))*2^(2/3)
Time = 3.64 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.10 \[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )-24 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x^2}}\right )+6\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2^{2/3} \sqrt [3]{1+x^2}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+24 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )-8 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+2\ 2^{2/3} \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-6\ 2^{2/3} \log \left (2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )+4 \log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-12 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )+3\ 2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}-\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )-2^{2/3} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{x^2+x^4}} \]
(x^(2/3)*(1 + x^2)^(1/3)*(8*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2* (1 + x^2)^(1/3))] - 24*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x^2)^(1/3))] + 6*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2^(2/ 3)*(1 + x^2)^(1/3))] - 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))] + 24*Log[-x^(1/3) + (1 + x^2)^(1/3)] - 8*Log[ x^(1/3) + (1 + x^2)^(1/3)] + 2*2^(2/3)*Log[-2*x^(1/3) + 2^(2/3)*(1 + x^2)^ (1/3)] - 6*2^(2/3)*Log[2*x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3)] + 4*Log[x^(2/3 ) - x^(1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)] - 12*Log[x^(2/3) + x^(1/3)* (1 + x^2)^(1/3) + (1 + x^2)^(2/3)] + 3*2^(2/3)*Log[-2*x^(2/3) + 2^(2/3)*x^ (1/3)*(1 + x^2)^(1/3) - 2^(1/3)*(1 + x^2)^(2/3)] - 2^(2/3)*Log[2*x^(2/3) + 2^(2/3)*x^(1/3)*(1 + x^2)^(1/3) + 2^(1/3)*(1 + x^2)^(2/3)]))/(48*(x^2 + x ^4)^(1/3))
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.
| \(\displaystyle \int \frac {x^6-x^3+1}{\sqrt [3]{x^4+x^2} \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^2+1} \int -\frac {x^6-x^3+1}{x^{2/3} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^2+1} \int \frac {x^6-x^3+1}{x^{2/3} \sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \frac {x^6-x^3+1}{\sqrt [3]{x^2+1} \left (1-x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \left (\frac {2-x^3}{\sqrt [3]{x^2+1} \left (1-x^6\right )}-\frac {1}{\sqrt [3]{x^2+1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \left (-\frac {1}{6} \int \frac {1}{\left (-\sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (1-\sqrt [3]{x}\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left (\sqrt [9]{-1} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (\sqrt [9]{-1} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left (-(-1)^{2/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (1-(-1)^{2/9} \sqrt [3]{x}\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left (-(-1)^{4/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (1-(-1)^{4/9} \sqrt [3]{x}\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left ((-1)^{5/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left ((-1)^{5/9} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left (-(-1)^{2/3} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{x}\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left ((-1)^{7/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left ((-1)^{7/9} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\frac {1}{6} \int \frac {1}{\left (-(-1)^{8/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \int \frac {1}{\left (1-(-1)^{8/9} \sqrt [3]{x}\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}-\sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )\right )}{\sqrt [3]{x^4+x^2}}\) |
3.31.25.3.1 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_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 321.80 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-x}{x}\right )}{2}+\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}-\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{6}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{2}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x}{x}\right )}{6}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )}{8}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{16}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right )}{8}+\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )}{24}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{48}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right )}{24}\) | \(388\) |
| trager | \(\text {Expression too large to display}\) | \(10360\) |
1/2*ln(((x^2*(x^2+1))^(1/3)-x)/x)+1/12*ln(((x^2*(x^2+1))^(2/3)-(x^2*(x^2+1 ))^(1/3)*x+x^2)/x^2)-1/6*3^(1/2)*arctan(1/3*(-2*(x^2*(x^2+1))^(1/3)+x)*3^( 1/2)/x)-1/4*ln(((x^2*(x^2+1))^(2/3)+(x^2*(x^2+1))^(1/3)*x+x^2)/x^2)+1/2*3^ (1/2)*arctan(1/3*(2*(x^2*(x^2+1))^(1/3)+x)*3^(1/2)/x)-1/6*ln(((x^2*(x^2+1) )^(1/3)+x)/x)-1/8*2^(2/3)*ln((2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x)+1/16*2^(2/ 3)*ln((2^(2/3)*x^2-2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2) -1/8*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(-2^(2/3)*(x^2*(x^2+1))^(1/3)+x)/x )+1/24*2^(2/3)*ln((-2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x)-1/48*2^(2/3)*ln((2^( 2/3)*x^2+2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2)+1/24*3^(1 /2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*(x^2*(x^2+1))^(1/3)+x)/x)
Exception generated. \[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
\[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^{6} - x^{3} + 1}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral((x**6 - x**3 + 1)/((x**2*(x**2 + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)), x)
\[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} - x^{3} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} - x^{3} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^6-x^3+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \]