Integrand size = 41, antiderivative size = 191 \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^7}}\right )+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^7}}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^7}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^7}\right )}{2^{2/3}}+\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^7}+\left (x+x^7\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^7}+\sqrt [3]{2} \left (x+x^7\right )^{2/3}\right )}{2\ 2^{2/3}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^7+x)^(1/3)))+1/2*3^(1/2)*arctan(3^(1 /2)*x/(x+2^(2/3)*(x^7+x)^(1/3)))*2^(1/3)-1/2*ln(-x+(x^7+x)^(1/3))+1/2*ln(- 2*x+2^(2/3)*(x^7+x)^(1/3))*2^(1/3)+1/4*ln(x^2+x*(x^7+x)^(1/3)+(x^7+x)^(2/3 ))-1/4*ln(2*x^2+2^(2/3)*x*(x^7+x)^(1/3)+2^(1/3)*(x^7+x)^(2/3))*2^(1/3)
\[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx \]
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.
| \(\displaystyle \int \frac {\left (2 x^6-1\right ) \sqrt [3]{x^7+x}}{\left (x^6-2 x^2+1\right ) \left (x^6-x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 2461 |
| \(\displaystyle \int \left (\frac {\sqrt [3]{x^7+x} \left (2 x^6-1\right )}{\left (x^2-1\right ) \left (x^6-x^2+1\right )}+\frac {\left (-x^2-2\right ) \sqrt [3]{x^7+x} \left (2 x^6-1\right )}{\left (x^4+x^2-1\right ) \left (x^6-x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt [3]{x^7+x} \left (1-2 x^6\right )}{\left (1-x^2\right ) \left (x^6-x^2+1\right )}+\frac {\left (x^2+2\right ) \left (2 x^6-1\right ) \sqrt [3]{x^7+x}}{\left (-x^4-x^2+1\right ) \left (x^6-x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2 x^6-1\right ) \sqrt [3]{x^7+x}}{\left (1-x^2\right ) \left (-x^4-x^2+1\right ) \left (x^6-x^2+1\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^7+x} \int -\frac {\sqrt [3]{x} \left (1-2 x^6\right ) \sqrt [3]{x^6+1}}{\left (1-x^2\right ) \left (-x^4-x^2+1\right ) \left (x^6-x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^6+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x^7+x} \int \frac {\sqrt [3]{x} \left (1-2 x^6\right ) \sqrt [3]{x^6+1}}{\left (1-x^2\right ) \left (-x^4-x^2+1\right ) \left (x^6-x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^6+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7+x} \int \frac {x \left (1-2 x^6\right ) \sqrt [3]{x^6+1}}{\left (1-x^2\right ) \left (-x^4-x^2+1\right ) \left (x^6-x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^6+1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7+x} \int \left (\frac {\sqrt [3]{x^6+1} \left (\sqrt [3]{x}-2\right )}{6 \left (x^{2/3}-\sqrt [3]{x}+1\right )}-\frac {\sqrt [3]{x} \sqrt [3]{x^6+1}}{3 \left (x^{2/3}-1\right )}+\frac {\left (\sqrt [3]{x}+2\right ) \sqrt [3]{x^6+1}}{6 \left (x^{2/3}+\sqrt [3]{x}+1\right )}-\frac {x \left (2 x^2+1\right ) \sqrt [3]{x^6+1}}{x^4+x^2-1}+\frac {x \left (3 x^4-1\right ) \sqrt [3]{x^6+1}}{x^6-x^2+1}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^6+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^7+x} \left (-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [3]{x^3+1}}{\sqrt [3]{x}-1}d\sqrt [3]{x},\sqrt [3]{x},x^{2/3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [3]{x} \sqrt [3]{x^3+1}}{x^3-x+1}d\sqrt [3]{x},\sqrt [3]{x},x^{2/3}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {x^{7/3} \sqrt [3]{x^3+1}}{x^3-x+1}d\sqrt [3]{x},\sqrt [3]{x},x^{2/3}\right )+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{2 \sqrt [3]{x}-i \sqrt {3}-1}d\sqrt [3]{x}+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{2 \sqrt [3]{x}-i \sqrt {3}+1}d\sqrt [3]{x}+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{2 \sqrt [3]{x}+i \sqrt {3}-1}d\sqrt [3]{x}+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{2 \sqrt [3]{x}+i \sqrt {3}+1}d\sqrt [3]{x}+\frac {\left (1-i \sqrt {3}\right ) \left (1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1-\sqrt {5}}-\sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{6 \sqrt {10}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1-\sqrt {5}}-\sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{6 \sqrt {10} \sqrt [3]{1+\sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{-\sqrt [6]{2} \sqrt [3]{x}-\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\int \frac {\sqrt [3]{x^6+1}}{-\sqrt [6]{2} \sqrt [3]{x}-\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10} \sqrt [3]{-1+\sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1+\sqrt {5}}-\sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1+\sqrt {5}}-\sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \sqrt {10} \sqrt [3]{-1+\sqrt {5}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{-\sqrt [6]{2} \sqrt [3]{x}-(-1)^{5/6} \sqrt [6]{1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}-\frac {\sqrt [3]{-\frac {1}{1+\sqrt {5}}} \int \frac {\sqrt [3]{x^6+1}}{-\sqrt [6]{2} \sqrt [3]{x}-(-1)^{5/6} \sqrt [6]{1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}+\sqrt [6]{-1-\sqrt {5}}}d\sqrt [3]{x}}{6 \sqrt {10}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}+\sqrt [6]{-1-\sqrt {5}}}d\sqrt [3]{x}}{6 \sqrt {10} \sqrt [3]{1+\sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}-\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}-\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10} \sqrt [3]{-1+\sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}+\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}+\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10} \sqrt [3]{-1+\sqrt {5}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}-(-1)^{5/6} \sqrt [6]{1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}-\frac {\sqrt [3]{-\frac {1}{1+\sqrt {5}}} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [3]{-1} \sqrt [6]{2} \sqrt [3]{x}-(-1)^{5/6} \sqrt [6]{1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1-\sqrt {5}}-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{6 \sqrt {10}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1-\sqrt {5}}-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{6 \sqrt {10} \sqrt [3]{1+\sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}-\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\int \frac {\sqrt [3]{x^6+1}}{-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}-\sqrt [6]{-1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10} \sqrt [3]{-1+\sqrt {5}}}+\frac {\left (-1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1+\sqrt {5}}-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \sqrt {10}}+\frac {\int \frac {\sqrt [3]{x^6+1}}{\sqrt [6]{-1+\sqrt {5}}-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \sqrt {10} \sqrt [3]{-1+\sqrt {5}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt {5}\right )^{2/3} \int \frac {\sqrt [3]{x^6+1}}{-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}-(-1)^{5/6} \sqrt [6]{1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}-\frac {\sqrt [3]{-\frac {1}{1+\sqrt {5}}} \int \frac {\sqrt [3]{x^6+1}}{-(-1)^{2/3} \sqrt [6]{2} \sqrt [3]{x}-(-1)^{5/6} \sqrt [6]{1+\sqrt {5}}}d\sqrt [3]{x}}{3 \sqrt {10}}\right )}{\sqrt [3]{x} \sqrt [3]{x^6+1}}\) |
3.24.88.3.1 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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
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_)), 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 = 6.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{7}+x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{7}+x \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{7}+x \right )^{\frac {1}{3}} x +\left (x^{7}+x \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {1}{3}}}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{7}+x \right )^{\frac {1}{3}}\right )}{3 x}\right )}{2}+\frac {\ln \left (\frac {x^{2}+x \left (x^{7}+x \right )^{\frac {1}{3}}+\left (x^{7}+x \right )^{\frac {2}{3}}}{x^{2}}\right )}{4}-\frac {\ln \left (\frac {-x +\left (x^{7}+x \right )^{\frac {1}{3}}}{x}\right )}{2}\) | \(160\) |
1/4*(-2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^7+x)^(1/3)))*3^(1/2)+2*ln((-2^( 1/3)*x+(x^7+x)^(1/3))/x)-ln((2^(2/3)*x^2+2^(1/3)*(x^7+x)^(1/3)*x+(x^7+x)^( 2/3))/x^2))*2^(1/3)+1/2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^7+x)^(1/3)))+ 1/4*ln((x^2+x*(x^7+x)^(1/3)+(x^7+x)^(2/3))/x^2)-1/2*ln((-x+(x^7+x)^(1/3))/ x)
Exception generated. \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Timed out. \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\int { \frac {{\left (x^{7} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - 1\right )}}{{\left (x^{6} - x^{2} + 1\right )} {\left (x^{6} - 2 \, x^{2} + 1\right )}} \,d x } \]
\[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\int { \frac {{\left (x^{7} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - 1\right )}}{{\left (x^{6} - x^{2} + 1\right )} {\left (x^{6} - 2 \, x^{2} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+2 x^6\right ) \sqrt [3]{x+x^7}}{\left (1-2 x^2+x^6\right ) \left (1-x^2+x^6\right )} \, dx=\int \frac {\left (2\,x^6-1\right )\,{\left (x^7+x\right )}^{1/3}}{\left (x^6-x^2+1\right )\,\left (x^6-2\,x^2+1\right )} \,d x \]