\(\int \frac {-1+x}{(-4-2 x+x^2) \sqrt [3]{-2-2 x+x^2}} \, dx\) [1806]

Optimal result

Integrand size = 26, antiderivative size = 123 \[ \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-2-2 x+x^2}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2+2^{2/3} \sqrt [3]{-2-2 x+x^2}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2+2^{2/3} \sqrt [3]{-2-2 x+x^2}+\sqrt [3]{2} \left (-2-2 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]

[Out]

1/4*3^(1/2)*arctan(1/3*3^(1/2)+1/3*2^(2/3)*(x^2-2*x-2)^(1/3)*3^(1/2))*2^(2/3)+1/4*ln(-2+2^(2/3)*(x^2-2*x-2)^(1
/3))*2^(2/3)-1/8*ln(2+2^(2/3)*(x^2-2*x-2)^(1/3)+2^(1/3)*(x^2-2*x-2)^(2/3))*2^(2/3)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{-2-2 x+x^2}}{\sqrt {3}}\right )+2 \log \left (-2+2^{2/3} \sqrt [3]{-2-2 x+x^2}\right )-\log \left (2+2^{2/3} \sqrt [3]{-2-2 x+x^2}+\sqrt [3]{2} \left (-2-2 x+x^2\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 6.82 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2}-2 x -2\right )^{\frac {1}{3}}+1\right )}{3}\right )+2 \ln \left (\left (x^{2}-2 x -2\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\ln \left (\left (x^{2}-2 x -2\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (x^{2}-2 x -2\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )\right )}{8}\)	\(84\)
trager	\(\text {Expression too large to display}\)	\(1005\)

[In]

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

[Out]

1/8*2^(2/3)*(2*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*(x^2-2*x-2)^(1/3)+1))+2*ln((x^2-2*x-2)^(1/3)-2^(1/3))-ln((x
^2-2*x-2)^(2/3)+2^(1/3)*(x^2-2*x-2)^(1/3)+2^(2/3)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx=\frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} + 2 \, \sqrt {2} {\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 2\right )}^{\frac {2}{3}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}}\right ) \]

[In]

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

[Out]

1/4*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6) + 2*sqrt(2)*(x^2 - 2*x - 2)^(1/3))) - 1/8*2^(2/3)*log(
2^(2/3) + 2^(1/3)*(x^2 - 2*x - 2)^(1/3) + (x^2 - 2*x - 2)^(2/3)) + 1/4*2^(2/3)*log(-2^(1/3) + (x^2 - 2*x - 2)^
(1/3))

Sympy [F]

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

[In]

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

[Out]

Integral((x - 1)/((x**2 - 2*x - 4)*(x**2 - 2*x - 2)**(1/3)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx=\frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left ({\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left ({\left | {\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

1/4*sqrt(3)*2^(2/3)*arctan(1/3*sqrt(3)*(2*(1/2*x^2 - x - 1)^(1/3) + 1)) - 1/8*2^(2/3)*log((1/2*x^2 - x - 1)^(2
/3) + (1/2*x^2 - x - 1)^(1/3) + 1) + 1/4*2^(2/3)*log(abs((1/2*x^2 - x - 1)^(1/3) - 1))

Mupad [B] (verification not implemented)

Time = 6.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx=\frac {2^{2/3}\,\ln \left (\frac {9\,{\left (x^2-2\,x-2\right )}^{1/3}}{4}-\frac {9\,2^{1/3}}{4}\right )}{4}+\frac {2^{2/3}\,\ln \left (\frac {9\,{\left (x^2-2\,x-2\right )}^{1/3}}{4}-\frac {9\,2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{8}-\frac {2^{2/3}\,\ln \left (\frac {9\,{\left (x^2-2\,x-2\right )}^{1/3}}{4}-\frac {9\,2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{8} \]

[In]

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

[Out]

(2^(2/3)*log((9*(x^2 - 2*x - 2)^(1/3))/4 - (9*2^(1/3))/4))/4 + (2^(2/3)*log((9*(x^2 - 2*x - 2)^(1/3))/4 - (9*2
^(1/3)*(3^(1/2)*1i - 1)^2)/16)*(3^(1/2)*1i - 1))/8 - (2^(2/3)*log((9*(x^2 - 2*x - 2)^(1/3))/4 - (9*2^(1/3)*(3^
(1/2)*1i + 1)^2)/16)*(3^(1/2)*1i + 1))/8