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

Optimal. Leaf size=209 \[ -\log \left (\sqrt [3]{x^4+x^3+x}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^4+x^3+x}-2 x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^3+x}+x}\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^4+x^3+x} x+\left (x^4+x^3+x\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^3+x} x+\sqrt [3]{2} \left (x^4+x^3+x\right )^{2/3}\right )}{2^{2/3}} \]

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Rubi [F]  time = 6.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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} \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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [F]  time = 2.60, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.46, size = 209, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3+x^4}}\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3 + x^4)^(1/3))]) + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)
*(x + x^3 + x^4)^(1/3))] - Log[-x + (x + x^3 + x^4)^(1/3)] + 2^(1/3)*Log[-2*x + 2^(2/3)*(x + x^3 + x^4)^(1/3)]
+ Log[x^2 + x*(x + x^3 + x^4)^(1/3) + (x + x^3 + x^4)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(x + x^3 + x^4)^(1/3)
+ 2^(1/3)*(x + x^3 + x^4)^(2/3)]/2^(2/3)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-2\right ) \left (x^{4}+x^{3}+x \right )^{\frac {1}{3}}}{\left (x^{3}+1\right ) \left (x^{3}-x^{2}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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