Integrand size = 37, antiderivative size = 209 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3+x^4}}\right )+\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{x+x^3+x^4}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^3+x^4}+\left (x+x^3+x^4\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^3+x^4}+\sqrt [3]{2} \left (x+x^3+x^4\right )^{2/3}\right )}{2^{2/3}} \]
-3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^4+x^3+x)^(1/3)))+2^(1/3)*3^(1/2)*arctan( 3^(1/2)*x/(x+2^(2/3)*(x^4+x^3+x)^(1/3)))-ln(-x+(x^4+x^3+x)^(1/3))+2^(1/3)* ln(-2*x+2^(2/3)*(x^4+x^3+x)^(1/3))+1/2*ln(x^2+x*(x^4+x^3+x)^(1/3)+(x^4+x^3 +x)^(2/3))-1/2*ln(2*x^2+2^(2/3)*x*(x^4+x^3+x)^(1/3)+2^(1/3)*(x^4+x^3+x)^(2 /3))*2^(1/3)
Time = 3.07 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\frac {\sqrt [3]{x+x^3+x^4} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2+x^3}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{1+x^2+x^3}}\right )-2 \log \left (-x^{2/3}+\sqrt [3]{1+x^2+x^3}\right )+2 \sqrt [3]{2} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{1+x^2+x^3}\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2+x^3}+\left (1+x^2+x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{1+x^2+x^3}+\sqrt [3]{2} \left (1+x^2+x^3\right )^{2/3}\right )\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \]
((x + x^3 + x^4)^(1/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*( 1 + x^2 + x^3)^(1/3))] + 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/ 3) + 2^(2/3)*(1 + x^2 + x^3)^(1/3))] - 2*Log[-x^(2/3) + (1 + x^2 + x^3)^(1 /3)] + 2*2^(1/3)*Log[-2*x^(2/3) + 2^(2/3)*(1 + x^2 + x^3)^(1/3)] + Log[x^( 4/3) + x^(2/3)*(1 + x^2 + x^3)^(1/3) + (1 + x^2 + x^3)^(2/3)] - 2^(1/3)*Lo g[2*x^(4/3) + 2^(2/3)*x^(2/3)*(1 + x^2 + x^3)^(1/3) + 2^(1/3)*(1 + x^2 + x ^3)^(2/3)]))/(2*x^(1/3)*(1 + x^2 + x^3)^(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.
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt [3]{x^4+x^3+x}}{\left (x^3+1\right ) \left (x^3-x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x^4+x^3+x} \int -\frac {\sqrt [3]{x} \left (2-x^3\right ) \sqrt [3]{x^3+x^2+1}}{\left (x^3+1\right ) \left (x^3-x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^3+x^2+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [3]{x^4+x^3+x} \int \frac {\sqrt [3]{x} \left (2-x^3\right ) \sqrt [3]{x^3+x^2+1}}{\left (x^3+1\right ) \left (x^3-x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^3+x^2+1}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {3 \sqrt [3]{x^4+x^3+x} \int \frac {x \left (2-x^3\right ) \sqrt [3]{x^3+x^2+1}}{\left (x^3+1\right ) \left (x^3-x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^3+x^2+1}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {3 \sqrt [3]{x^4+x^3+x} \int \left (\frac {\sqrt [3]{x^3+x^2+1} \left (2-\sqrt [3]{x}\right )}{3 \left (x^{2/3}-\sqrt [3]{x}+1\right )}+\frac {\sqrt [3]{x^3+x^2+1}}{3 \left (\sqrt [3]{x}+1\right )}+\frac {(2 x-1) \sqrt [3]{x^3+x^2+1}}{x^2-x+1}+\frac {(2-3 x) x \sqrt [3]{x^3+x^2+1}}{x^3-x^2+1}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^3+x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \sqrt [3]{x^4+x^3+x} \left (\frac {1}{3} \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^3+x^2+1}}{2 \sqrt [3]{x}-i \sqrt {3}-1}d\sqrt [3]{x}-\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^3+x^2+1}}{2 \sqrt [3]{x}+i \sqrt {3}-1}d\sqrt [3]{x}-\frac {2 \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{-2} \sqrt [3]{x}+\sqrt [3]{1-i \sqrt {3}}}d\sqrt [3]{x}}{3 \left (1-i \sqrt {3}\right )^{2/3}}-\frac {2 \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{-2} \sqrt [3]{x}+\sqrt [3]{1+i \sqrt {3}}}d\sqrt [3]{x}}{3 \left (1+i \sqrt {3}\right )^{2/3}}-\frac {2 \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1-i \sqrt {3}\right )^{2/3}}-\frac {2 \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1+i \sqrt {3}\right )^{2/3}}-\frac {2 \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1-i \sqrt {3}\right )^{2/3}}-\frac {2 \int \frac {\sqrt [3]{x^3+x^2+1}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{x}}d\sqrt [3]{x}}{3 \left (1+i \sqrt {3}\right )^{2/3}}+2 \int \frac {x \sqrt [3]{x^3+x^2+1}}{x^3-x^2+1}d\sqrt [3]{x}-3 \int \frac {x^2 \sqrt [3]{x^3+x^2+1}}{x^3-x^2+1}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{x^3+x^2+1}}\) |
3.26.6.3.1 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 = 1.38 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(2^{\frac {1}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}-2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )-\ln \left (\frac {-x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(201\) |
2^(1/3)*ln((-2^(1/3)*x+(x*(x^3+x^2+1))^(1/3))/x)-1/2*2^(1/3)*ln((2^(2/3)*x ^2+2^(1/3)*(x*(x^3+x^2+1))^(1/3)*x+(x*(x^3+x^2+1))^(2/3))/x^2)-2^(1/3)*3^( 1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*(x*(x^3+x^2+1))^(1/3)+x)/x)-ln((-x+(x*(x^ 3+x^2+1))^(1/3))/x)+1/2*ln(((x*(x^3+x^2+1))^(2/3)+(x*(x^3+x^2+1))^(1/3)*x+ x^2)/x^2)+3^(1/2)*arctan(1/3*(2*(x*(x^3+x^2+1))^(1/3)+x)*3^(1/2)/x)
Exception generated. \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{3} + x^{2} + 1\right )} \left (x^{3} - 2\right )}{\left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{3} - x^{2} + 1\right )}\, dx \]
Integral((x*(x**3 + x**2 + 1))**(1/3)*(x**3 - 2)/((x + 1)*(x**2 - x + 1)*( x**3 - x**2 + 1)), x)
\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} - x^{2} + 1\right )} {\left (x^{3} + 1\right )}} \,d x } \]
\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} - x^{2} + 1\right )} {\left (x^{3} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int \frac {\left (x^3-2\right )\,{\left (x^4+x^3+x\right )}^{1/3}}{\left (x^3+1\right )\,\left (x^3-x^2+1\right )} \,d x \]