Integrand size = 22, antiderivative size = 356 \[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \left (x^2+x^4\right )^{2/3}}\right )}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x^2+x^4}\right )-\frac {1}{6} \log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {1}{12} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}-\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}} \]
-1/6*arctan(3^(1/2)*x^2/(x^2+2*(x^4+x^2)^(2/3)))*3^(1/2)+1/24*arctan(3^(1/ 2)*x^2/(x^2+2^(1/3)*(x^4+x^2)^(2/3)))*2^(2/3)*3^(1/2)-1/6*ln(-x+(x^4+x^2)^ (1/3))-1/6*ln(x+(x^4+x^2)^(1/3))+1/24*ln(-2*x+2^(2/3)*(x^4+x^2)^(1/3))*2^( 2/3)+1/24*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))+1/12*ln(x^2+x*(x^4+x^2)^(1/3)+(x^4+x^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)-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 = 0.81 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \left (1+x^2\right )^{2/3}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}}\right )+8 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+8 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )-2\ 2^{2/3} \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-2\ 2^{2/3} \log \left (2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-4 \log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-4 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )+2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}-\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )+2^{2/3} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{x^2+x^4}} \]
-1/48*(x^(2/3)*(1 + x^2)^(1/3)*(8*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3 ) + 2*(1 + x^2)^(2/3))] - 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2 /3) + 2^(1/3)*(1 + x^2)^(2/3))] + 8*Log[-x^(1/3) + (1 + x^2)^(1/3)] + 8*Lo g[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)] - 2*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)] - 4*Log[x^(2/3) + x^(1/3) *(1 + x^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)] + 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)]))/(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.
| \(\displaystyle \int \frac {x^3}{\sqrt [3]{x^4+x^2} \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^2+1} \int -\frac {x^{7/3}}{\sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^2+1} \int \frac {x^{7/3}}{\sqrt [3]{x^2+1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \frac {x^3}{\sqrt [3]{x^2+1} \left (1-x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \left (\frac {\sqrt [3]{x}-2}{18 \left (x^{2/3}-\sqrt [3]{x}+1\right ) \sqrt [3]{x^2+1}}-\frac {\sqrt [3]{x}}{9 \left (x^{2/3}-1\right ) \sqrt [3]{x^2+1}}+\frac {\sqrt [3]{x}+2}{18 \left (x^{2/3}+\sqrt [3]{x}+1\right ) \sqrt [3]{x^2+1}}+\frac {x-2}{6 \sqrt [3]{x^2+1} \left (x^2-x+1\right )}+\frac {x+2}{6 \sqrt [3]{x^2+1} \left (x^2+x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \left (\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (2 \sqrt [3]{x}-i \sqrt {3}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (2 \sqrt [3]{x}-i \sqrt {3}+1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (2 \sqrt [3]{x}+i \sqrt {3}-1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {1}{18} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (2 \sqrt [3]{x}+i \sqrt {3}+1\right ) \sqrt [3]{x^2+1}}d\sqrt [3]{x}+\frac {\left (-\sqrt {3}+i\right ) x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{12 \left (\sqrt {3}+i\right )}+\frac {\left (\sqrt {3}+i\right ) x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{12 \left (-\sqrt {3}+i\right )}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{12 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-x^{4/3}-x^{2/3}+x^2+1\right )}{72 \sqrt [3]{2}}-\frac {\log \left (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+1\right )}{24 \sqrt [3]{2}}\right )}{\sqrt [3]{x^4+x^2}}\) |
3.30.53.3.1 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 = 26.89 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.61
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {2}{3}} x^{2}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{24}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}+2 \,2^{\frac {1}{3}} x^{2}+x^{2} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x^{2}}\right )}{48}-\frac {2^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}+x^{2}\right )}{3 x^{2}}\right )}{24}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}-x^{2}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {\left (\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+1\right ) \left (x^{2}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}\right )}{x^{2}}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}+x^{2}\right ) \sqrt {3}}{3 x^{2}}\right )}{6}\) | \(218\) |
| trager | \(\text {Expression too large to display}\) | \(2991\) |
1/24*2^(2/3)*ln((-2^(2/3)*x^2+(x^2*(x^2+1))^(2/3))/x^2)-1/48*2^(2/3)*ln(1/ x^2*(2^(2/3)*(x^2*(x^2+1))^(2/3)+2*2^(1/3)*x^2+x^2*(x^2*(x^2+1))^(1/3)+(x^ 2*(x^2+1))^(1/3)))-1/24*2^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2^(1/3)*(x^2*( x^2+1))^(2/3)+x^2)/x^2)-1/6*ln(((x^2*(x^2+1))^(2/3)-x^2)/x^2)+1/12*ln(((x^ 2*(x^2+1))^(1/3)+1)*(x^2+(x^2*(x^2+1))^(1/3))/x^2)+1/6*3^(1/2)*arctan(1/3* (2*(x^2*(x^2+1))^(2/3)+x^2)*3^(1/2)/x^2)
Time = 1.62 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.17 \[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {1}{72} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (24 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{8} + 2 \, x^{6} - 6 \, x^{4} + 2 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + \sqrt {6} 2^{\frac {1}{3}} {\left (x^{12} - 42 \, x^{10} - 417 \, x^{8} - 812 \, x^{6} - 417 \, x^{4} - 42 \, x^{2} + 1\right )} - 12 \, \sqrt {6} {\left (x^{10} + 33 \, x^{8} + 110 \, x^{6} + 110 \, x^{4} + 33 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{12} + 102 \, x^{10} + 447 \, x^{8} + 628 \, x^{6} + 447 \, x^{4} + 102 \, x^{2} + 1\right )}}\right ) - \frac {1}{144} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{4} + 4 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{8} + 32 \, x^{6} + 78 \, x^{4} + 32 \, x^{2} + 1\right )} + 6 \, {\left (x^{6} + 11 \, x^{4} + 11 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + \frac {1}{72} \cdot 2^{\frac {2}{3}} \log \left (-\frac {2^{\frac {2}{3}} {\left (x^{4} - 2 \, x^{2} + 1\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + 12 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) - \frac {1}{12} \, \log \left (\frac {x^{4} + x^{2} - 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{2} + 1}\right ) \]
-1/72*sqrt(6)*2^(1/6)*arctan(1/6*2^(1/6)*(24*sqrt(6)*2^(2/3)*(x^8 + 2*x^6 - 6*x^4 + 2*x^2 + 1)*(x^4 + x^2)^(2/3) + sqrt(6)*2^(1/3)*(x^12 - 42*x^10 - 417*x^8 - 812*x^6 - 417*x^4 - 42*x^2 + 1) - 12*sqrt(6)*(x^10 + 33*x^8 + 1 10*x^6 + 110*x^4 + 33*x^2 + 1)*(x^4 + x^2)^(1/3))/(x^12 + 102*x^10 + 447*x ^8 + 628*x^6 + 447*x^4 + 102*x^2 + 1)) - 1/144*2^(2/3)*log((12*2^(2/3)*(x^ 4 + 4*x^2 + 1)*(x^4 + x^2)^(2/3) + 2^(1/3)*(x^8 + 32*x^6 + 78*x^4 + 32*x^2 + 1) + 6*(x^6 + 11*x^4 + 11*x^2 + 1)*(x^4 + x^2)^(1/3))/(x^8 - 4*x^6 + 6* x^4 - 4*x^2 + 1)) + 1/72*2^(2/3)*log(-(2^(2/3)*(x^4 - 2*x^2 + 1) - 6*2^(1/ 3)*(x^4 + x^2)^(1/3)*(x^2 + 1) + 12*(x^4 + x^2)^(2/3))/(x^4 - 2*x^2 + 1)) - 1/6*sqrt(3)*arctan(1/3*(sqrt(3)*(x^2 + 1) + 2*sqrt(3)*(x^4 + x^2)^(1/3)) /(x^2 + 1)) - 1/12*log((x^4 + x^2 - 3*(x^4 + x^2)^(1/3)*(x^2 + 1) + 3*(x^4 + x^2)^(2/3) + 1)/(x^4 + x^2 + 1))
\[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^{3}}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
\[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^3}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \]