\(\int \frac {-3-x^4+3 x^6}{(1-x^4+x^6) \sqrt [3]{1-x^3-x^4+x^6}} \, dx\) [1737]

Optimal result

Integrand size = 44, antiderivative size = 117 \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{1-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3-x^4+x^6}+\left (1-x^3-x^4+x^6\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^6-x^4-x^3+1)^(1/3)))-ln(x+(x^6-x^4-x^3+1)^(1/3))+1/2*ln(x^2-x*(x^6-x^4-x^3+
1)^(1/3)+(x^6-x^4-x^3+1)^(2/3))

Rubi [F]

\[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx \]

[In]

Int[(-3 - x^4 + 3*x^6)/((1 - x^4 + x^6)*(1 - x^3 - x^4 + x^6)^(1/3)),x]

[Out]

3*Defer[Int][(1 - x^3 - x^4 + x^6)^(-1/3), x] - 6*Defer[Int][1/((1 - x^4 + x^6)*(1 - x^3 - x^4 + x^6)^(1/3)),
x] + 2*Defer[Int][x^4/((1 - x^4 + x^6)*(1 - x^3 - x^4 + x^6)^(1/3)), x]

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{1-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3-x^4+x^6}+\left (1-x^3-x^4+x^6\right )^{2/3}\right ) \]

[In]

Integrate[(-3 - x^4 + 3*x^6)/((1 - x^4 + x^6)*(1 - x^3 - x^4 + x^6)^(1/3)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3 - x^4 + x^6)^(1/3))] - Log[x + (1 - x^3 - x^4 + x^6)^(1/3)] + Log[x
^2 - x*(1 - x^3 - x^4 + x^6)^(1/3) + (1 - x^3 - x^4 + x^6)^(2/3)]/2

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {x^{2}-x \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}}+\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x +\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )\)	\(111\)
trager	\(-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x^{6}-x^{4}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}+1}{x^{6}-x^{4}+1}\right )\)	\(340\)

[In]

int((3*x^6-x^4-3)/(x^6-x^4+1)/(x^6-x^4-x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/2*ln((x^2-x*(x^6-x^4-x^3+1)^(1/3)+(x^6-x^4-x^3+1)^(2/3))/x^2)-3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^6-x^4-x^3+1
)^(1/3))/x)-ln((x+(x^6-x^4-x^3+1)^(1/3))/x)

Fricas [A] (verification not implemented)

none

Time = 1.98 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.34 \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} + 1\right )}}{3 \, {\left (x^{6} - x^{4} - 2 \, x^{3} + 1\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} - x^{4} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{4} + 1}\right ) \]

[In]

integrate((3*x^6-x^4-3)/(x^6-x^4+1)/(x^6-x^4-x^3+1)^(1/3),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan(1/3*(2*sqrt(3)*(x^6 - x^4 - x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^6 - x^4 - x^3 + 1)^(2/3)*x + sqr
t(3)*(x^6 - x^4 + 1))/(x^6 - x^4 - 2*x^3 + 1)) - 1/2*log((x^6 - x^4 + 3*(x^6 - x^4 - x^3 + 1)^(1/3)*x^2 + 3*(x
^6 - x^4 - x^3 + 1)^(2/3)*x + 1)/(x^6 - x^4 + 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\text {Timed out} \]

[In]

integrate((3*x**6-x**4-3)/(x**6-x**4+1)/(x**6-x**4-x**3+1)**(1/3),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int { \frac {3 \, x^{6} - x^{4} - 3}{{\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{6} - x^{4} + 1\right )}} \,d x } \]

[In]

integrate((3*x^6-x^4-3)/(x^6-x^4+1)/(x^6-x^4-x^3+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x^6 - x^4 - 3)/((x^6 - x^4 - x^3 + 1)^(1/3)*(x^6 - x^4 + 1)), x)

Giac [F]

\[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int { \frac {3 \, x^{6} - x^{4} - 3}{{\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{6} - x^{4} + 1\right )}} \,d x } \]

[In]

integrate((3*x^6-x^4-3)/(x^6-x^4+1)/(x^6-x^4-x^3+1)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x^6 - x^4 - 3)/((x^6 - x^4 - x^3 + 1)^(1/3)*(x^6 - x^4 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx=\int -\frac {-3\,x^6+x^4+3}{\left (x^6-x^4+1\right )\,{\left (x^6-x^4-x^3+1\right )}^{1/3}} \,d x \]

[In]

int(-(x^4 - 3*x^6 + 3)/((x^6 - x^4 + 1)*(x^6 - x^4 - x^3 + 1)^(1/3)),x)

[Out]

int(-(x^4 - 3*x^6 + 3)/((x^6 - x^4 + 1)*(x^6 - x^4 - x^3 + 1)^(1/3)), x)