Integrand size = 22, antiderivative size = 198 \[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^5}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x+x^5}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{x+x^5}\right )-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x+x^5}\right )}{6 \sqrt [3]{2}}-\frac {1}{6} \log \left (x^2+x \sqrt [3]{x+x^5}+\left (x+x^5\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x+x^5}-\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{12 \sqrt [3]{2}} \]
-1/3*arctan(3^(1/2)*x/(x+2*(x^5+x)^(1/3)))*3^(1/2)-1/12*3^(1/2)*arctan(3^( 1/2)*x/(-x+2^(2/3)*(x^5+x)^(1/3)))*2^(2/3)+1/3*ln(-x+(x^5+x)^(1/3))-1/12*l n(2*x+2^(2/3)*(x^5+x)^(1/3))*2^(2/3)-1/6*ln(x^2+x*(x^5+x)^(1/3)+(x^5+x)^(2 /3))+1/24*ln(-2*x^2+2^(2/3)*x*(x^5+x)^(1/3)-2^(1/3)*(x^5+x)^(2/3))*2^(2/3)
Time = 15.81 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=\frac {1}{24} \left (-8 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^5}}\right )+2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{x+x^5}}\right )+8 \log \left (-x+\sqrt [3]{x+x^5}\right )-2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{x+x^5}\right )-4 \log \left (x^2+x \sqrt [3]{x+x^5}+\left (x+x^5\right )^{2/3}\right )+2^{2/3} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{x+x^5}-\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )\right ) \]
(-8*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^5)^(1/3))] + 2*2^(2/3)*Sqrt[3 ]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(x + x^5)^(1/3))] + 8*Log[-x + (x + x^5) ^(1/3)] - 2*2^(2/3)*Log[2*x + 2^(2/3)*(x + x^5)^(1/3)] - 4*Log[x^2 + x*(x + x^5)^(1/3) + (x + x^5)^(2/3)] + 2^(2/3)*Log[-2*x^2 + 2^(2/3)*x*(x + x^5) ^(1/3) - 2^(1/3)*(x + x^5)^(2/3)])/24
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.
| \(\displaystyle \int \frac {x^6-1}{\sqrt [3]{x^5+x} \left (x^6+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \int -\frac {1-x^6}{\sqrt [3]{x} \sqrt [3]{x^4+1} \left (x^6+1\right )}dx}{\sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {1-x^6}{\sqrt [3]{x} \sqrt [3]{x^4+1} \left (x^6+1\right )}dx}{\sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {\sqrt [3]{x} \left (1-x^6\right )}{\sqrt [3]{x^4+1} \left (x^6+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \frac {1-x^3}{\sqrt [3]{x^2+1} \left (x^3+1\right )}dx^{2/3}}{2 \sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \int \left (\frac {2}{\sqrt [3]{x^2+1} \left (x^3+1\right )}-\frac {1}{\sqrt [3]{x^2+1}}\right )dx^{2/3}}{2 \sqrt [3]{x^5+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^4+1} \left (-\frac {2}{9} \int \frac {1}{\left (-x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left (\sqrt [9]{-1} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left (-(-1)^{2/9} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left (\sqrt [3]{-1} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left (-(-1)^{4/9} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left ((-1)^{5/9} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left (-(-1)^{2/3} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left ((-1)^{7/9} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-\frac {2}{9} \int \frac {1}{\left (-(-1)^{8/9} x^{2/3}-1\right ) \sqrt [3]{x^2+1}}dx^{2/3}-x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )\right )}{2 \sqrt [3]{x^5+x}}\) |
3.25.40.3.1 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_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), 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 = 19.66 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.91
| method | result | size |
| pseudoelliptic | \(-\frac {\ln \left (\frac {{\left (\left (x^{4}+1\right ) x \right )}^{\frac {2}{3}}+x {\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}+x^{2}}{x^{2}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +{\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}}{x}\right )}{12}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} {\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}} x +{\left (\left (x^{4}+1\right ) x \right )}^{\frac {2}{3}}}{x^{2}}\right )}{24}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} {\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}+x \right )}{3 x}\right )}{12}+\frac {\ln \left (\frac {{\left (\left (x^{4}+1\right ) x \right )}^{\frac {1}{3}}-x}{x}\right )}{3}\) | \(180\) |
-1/6*ln((((x^4+1)*x)^(2/3)+x*((x^4+1)*x)^(1/3)+x^2)/x^2)+1/3*3^(1/2)*arcta n(1/3*(2*((x^4+1)*x)^(1/3)+x)*3^(1/2)/x)-1/12*2^(2/3)*ln((2^(1/3)*x+((x^4+ 1)*x)^(1/3))/x)+1/24*2^(2/3)*ln((2^(2/3)*x^2-2^(1/3)*((x^4+1)*x)^(1/3)*x+( (x^4+1)*x)^(2/3))/x^2)-1/12*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(-2^(2/3)*( (x^4+1)*x)^(1/3)+x)/x)+1/3*ln((((x^4+1)*x)^(1/3)-x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (150) = 300\).
Time = 2.72 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.01 \[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=\frac {1}{36} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{8} - 14 \, x^{6} + 6 \, x^{4} - 14 \, x^{2} + 1\right )} {\left (x^{5} + x\right )}^{\frac {2}{3}} - 24 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{9} + x^{7} + x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} + \sqrt {6} 2^{\frac {1}{3}} {\left (x^{12} + 24 \, x^{10} - 57 \, x^{8} + 56 \, x^{6} - 57 \, x^{4} + 24 \, x^{2} + 1\right )}\right )}}{6 \, {\left (x^{12} - 48 \, x^{10} + 15 \, x^{8} - 88 \, x^{6} + 15 \, x^{4} - 48 \, x^{2} + 1\right )}}\right ) - \frac {1}{72} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{5} - x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{8} - 14 \, x^{6} + 6 \, x^{4} - 14 \, x^{2} + 1\right )} - 6 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{2} + 1\right )}}{x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1}\right ) + \frac {1}{36} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 2 \, x^{2} + 1\right )} - 3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} + x\right )}^{\frac {2}{3}} + 6 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{5} + x\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{4} - x^{2} + 3 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{4} - x^{2} + 1}\right ) \]
1/36*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(-1) ^(2/3)*(x^8 - 14*x^6 + 6*x^4 - 14*x^2 + 1)*(x^5 + x)^(2/3) - 24*sqrt(6)*(- 1)^(1/3)*(x^9 + x^7 + x^3 + x)*(x^5 + x)^(1/3) + sqrt(6)*2^(1/3)*(x^12 + 2 4*x^10 - 57*x^8 + 56*x^6 - 57*x^4 + 24*x^2 + 1))/(x^12 - 48*x^10 + 15*x^8 - 88*x^6 + 15*x^4 - 48*x^2 + 1)) - 1/72*2^(2/3)*(-1)^(1/3)*log((12*2^(1/3) *(-1)^(2/3)*(x^5 - x^3 + x)*(x^5 + x)^(1/3) - 2^(2/3)*(-1)^(1/3)*(x^8 - 14 *x^6 + 6*x^4 - 14*x^2 + 1) - 6*(x^5 + x)^(2/3)*(x^4 - 4*x^2 + 1))/(x^8 + 4 *x^6 + 6*x^4 + 4*x^2 + 1)) + 1/36*2^(2/3)*(-1)^(1/3)*log(-(2^(1/3)*(-1)^(2 /3)*(x^4 + 2*x^2 + 1) - 3*2^(2/3)*(-1)^(1/3)*(x^5 + x)^(2/3) + 6*(x^5 + x) ^(1/3)*x)/(x^4 + 2*x^2 + 1)) + 1/3*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt( 3)*(x^5 + x)^(1/3))/x) + 1/6*log((x^4 - x^2 + 3*(x^5 + x)^(1/3)*x - 3*(x^5 + x)^(2/3) + 1)/(x^4 - x^2 + 1))
\[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt [3]{x \left (x^{4} + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Integral((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)/((x*(x**4 + 1))**(1 /3)*(x**2 + 1)*(x**4 - x**2 + 1)), x)
\[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {-1+x^6}{\sqrt [3]{x+x^5} \left (1+x^6\right )} \, dx=\int \frac {x^6-1}{\left (x^6+1\right )\,{\left (x^5+x\right )}^{1/3}} \,d x \]