\(\int \frac {(-2+x^3) \sqrt [3]{x+x^3+x^4}}{(1+x^3) (1-x^2+x^3)} \, dx\) [2506]

Optimal result

Integrand size = 37, antiderivative size = 209 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3+x^4}}\right )+\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{x+x^3+x^4}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^3+x^4}+\left (x+x^3+x^4\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^3+x^4}+\sqrt [3]{2} \left (x+x^3+x^4\right )^{2/3}\right )}{2^{2/3}} \]

[Out]

-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)

Rubi [F]

\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx \]

[In]

Int[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/((1 + x^3)*(1 - x^2 + x^3)),x]

[Out]

(3*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 + x
^3)^(1/3)) - (3*(1 - I*Sqrt[3])*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3), x], x, x^
(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) - (3*(1 + I*Sqrt[3])*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(
1 + x^6 + x^9)^(1/3), x], x, x^(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) - ((x + x^3 + x^4)^(1/3)*Defer[Subst]
[Defer[Int][(1 + x^6 + x^9)^(1/3)/(1 + x), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((1 + I*Sqrt[3])
*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3)/(-1 - I*Sqrt[3] + 2*x), x], x, x^(1/3)])/
(x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((1 - I*Sqrt[3])*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^
9)^(1/3)/(-1 + I*Sqrt[3] + 2*x), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((1 - I*Sqrt[3])^(1/3)*(1
+ I*Sqrt[3])*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3)/((1 - I*Sqrt[3])^(1/3) + (-2)
^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((1 - I*Sqrt[3])*(1 + I*Sqrt[3])^(1/3)*(x + x^
3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3)/((1 + I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x], x, x^
(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((1 - I*Sqrt[3])^(1/3)*(1 + I*Sqrt[3])*(x + x^3 + x^4)^(1/3)*Defer
[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3)/((1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 +
x^2 + x^3)^(1/3)) + ((1 - I*Sqrt[3])*(1 + I*Sqrt[3])^(1/3)*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 +
x^6 + x^9)^(1/3)/((1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((
1 - I*Sqrt[3])^(1/3)*(1 + I*Sqrt[3])*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3)/((1 -
I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) + ((1 - I*Sqrt[3
])*(1 + I*Sqrt[3])^(1/3)*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6 + x^9)^(1/3)/((1 + I*Sqrt[3])^
(1/3) - (-1)^(2/3)*2^(1/3)*x), x], x, x^(1/3)])/(2*x^(1/3)*(1 + x^2 + x^3)^(1/3)) - (6*(x + x^3 + x^4)^(1/3)*D
efer[Subst][Defer[Int][(x^3*(1 + x^6 + x^9)^(1/3))/(1 - x^6 + x^9), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 + x^3)^
(1/3)) + (9*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(x^6*(1 + x^6 + x^9)^(1/3))/(1 - x^6 + x^9), x], x,
x^(1/3)])/(x^(1/3)*(1 + x^2 + x^3)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+x^3+x^4} \int \frac {\sqrt [3]{x} \left (-2+x^3\right ) \sqrt [3]{1+x^2+x^3}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\sqrt [3]{x+x^3+x^4} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{1+x}+\frac {(-1-x) \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{1-x+x^2}+\frac {\sqrt [3]{x} (-2+3 x) \sqrt [3]{1+x^2+x^3}}{1-x^2+x^3}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\sqrt [3]{x+x^3+x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{1+x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\sqrt [3]{x+x^3+x^4} \int \frac {(-1-x) \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{1-x+x^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\sqrt [3]{x+x^3+x^4} \int \frac {\sqrt [3]{x} (-2+3 x) \sqrt [3]{1+x^2+x^3}}{1-x^2+x^3} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\sqrt [3]{x+x^3+x^4} \int \left (\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{-1-i \sqrt {3}+2 x}+\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{-1+i \sqrt {3}+2 x}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1+x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \left (-2+3 x^3\right ) \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (\sqrt [3]{1+x^6+x^9}-\frac {\sqrt [3]{1+x^6+x^9}}{1+x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9}+\frac {3 x^6 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}{-1-i \sqrt {3}+2 x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{1+x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (9 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [3]{1+x^6+x^9}}{3 (1+x)}+\frac {(2-x) \sqrt [3]{1+x^6+x^9}}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (9 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{2} \sqrt [3]{1+x^6+x^9}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+x^6+x^9}}{2 \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{2} \sqrt [3]{1+x^6+x^9}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+x^6+x^9}}{2 \left (-1-i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = -\frac {\sqrt [3]{x+x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\sqrt [3]{x+x^3+x^4} \text {Subst}\left (\int \frac {(2-x) \sqrt [3]{1+x^6+x^9}}{1-x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (9 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = -\frac {\sqrt [3]{x+x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\sqrt [3]{x+x^3+x^4} \text {Subst}\left (\int \left (\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{1+x^6+x^9}}{-1-i \sqrt {3}+2 x}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{1+x^6+x^9}}{-1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (9 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = -\frac {\sqrt [3]{x+x^3+x^4} \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (9 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{1+x^6+x^9}}{1-x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1-i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}+\frac {\left (3 \left (-1+i \sqrt {3}\right ) \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \sqrt [3]{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.07 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\frac {\sqrt [3]{x+x^3+x^4} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2+x^3}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{1+x^2+x^3}}\right )-2 \log \left (-x^{2/3}+\sqrt [3]{1+x^2+x^3}\right )+2 \sqrt [3]{2} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{1+x^2+x^3}\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2+x^3}+\left (1+x^2+x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{1+x^2+x^3}+\sqrt [3]{2} \left (1+x^2+x^3\right )^{2/3}\right )\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \]

[In]

Integrate[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/((1 + x^3)*(1 - x^2 + x^3)),x]

[Out]

((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)*S
qrt[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)*Log[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))

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(2^{\frac {1}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}-2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )-\ln \left (\frac {-x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\)	\(201\)

[In]

int((x^3-2)*(x^4+x^3+x)^(1/3)/(x^3+1)/(x^3-x^2+1),x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((x^3-2)*(x^4+x^3+x)^(1/3)/(x^3+1)/(x^3-x^2+1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F]

\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{3} + x^{2} + 1\right )} \left (x^{3} - 2\right )}{\left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{3} - x^{2} + 1\right )}\, dx \]

[In]

integrate((x**3-2)*(x**4+x**3+x)**(1/3)/(x**3+1)/(x**3-x**2+1),x)

[Out]

Integral((x*(x**3 + x**2 + 1))**(1/3)*(x**3 - 2)/((x + 1)*(x**2 - x + 1)*(x**3 - x**2 + 1)), x)

Maxima [F]

\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} - x^{2} + 1\right )} {\left (x^{3} + 1\right )}} \,d x } \]

[In]

integrate((x^3-2)*(x^4+x^3+x)^(1/3)/(x^3+1)/(x^3-x^2+1),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3 + x)^(1/3)*(x^3 - 2)/((x^3 - x^2 + 1)*(x^3 + 1)), x)

Giac [F]

\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} - x^{2} + 1\right )} {\left (x^{3} + 1\right )}} \,d x } \]

[In]

integrate((x^3-2)*(x^4+x^3+x)^(1/3)/(x^3+1)/(x^3-x^2+1),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 + x)^(1/3)*(x^3 - 2)/((x^3 - x^2 + 1)*(x^3 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{\left (1+x^3\right ) \left (1-x^2+x^3\right )} \, dx=\int \frac {\left (x^3-2\right )\,{\left (x^4+x^3+x\right )}^{1/3}}{\left (x^3+1\right )\,\left (x^3-x^2+1\right )} \,d x \]

[In]

int(((x^3 - 2)*(x + x^3 + x^4)^(1/3))/((x^3 + 1)*(x^3 - x^2 + 1)),x)

[Out]

int(((x^3 - 2)*(x + x^3 + x^4)^(1/3))/((x^3 + 1)*(x^3 - x^2 + 1)), x)