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

Optimal. Leaf size=123 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^2-2 x-2}-2\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \left (x^2-2 x-2\right )^{2/3}+2^{2/3} \sqrt [3]{x^2-2 x-2}+2\right )}{4 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-2 x-2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \]

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Rubi [F]  time = 0.01, 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 {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 78, normalized size = 0.63 \begin {gather*} \frac {-\log \left (x^2-2 x-4\right )+3 \log \left (\sqrt [3]{2}-\sqrt [3]{x^2-2 x-2}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-2 x-2}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

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

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IntegrateAlgebraic [A]  time = 0.26, size = 123, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 93, normalized size = 0.76 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.30, size = 84, normalized size = 0.68 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [C]  time = 9.09, size = 1042, normalized size = 8.47

method	result	size
trager	\(\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}+3 \left (x^{2}-2 x -2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +6 \left (x^{2}-2 x -2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) x -2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )-2 \RootOf \left (\textit {\_Z}^{3}-4\right )}{x^{2}-2 x -4}\right )-\frac {\ln \left (-\frac {-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}+6 \left (x^{2}-2 x -2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +12 \left (x^{2}-2 x -2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+3 \left (x^{2}-2 x -2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \left (x^{2}-2 x -2\right )^{\frac {2}{3}}-8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )+2 \RootOf \left (\textit {\_Z}^{3}-4\right )}{x^{2}-2 x -4}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{4}-\ln \left (-\frac {-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{2}+6 \left (x^{2}-2 x -2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x +12 \left (x^{2}-2 x -2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+3 \left (x^{2}-2 x -2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \left (x^{2}-2 x -2\right )^{\frac {2}{3}}-8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )+2 \RootOf \left (\textit {\_Z}^{3}-4\right )}{x^{2}-2 x -4}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+16 \textit {\_Z}^{2}\right )\)	\(1042\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*ln((2*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2
)^2*RootOf(_Z^3-4)^2*x^2+RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^3*x^2+3*(x^2-2*x-
2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^2-4*RootOf(RootOf(_Z^3-4)^2+4*_Z*
RootOf(_Z^3-4)+16*_Z^2)^2*RootOf(_Z^3-4)^2*x-2*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^
3-4)^3*x+6*(x^2-2*x-2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)+2*RootOf(Root
Of(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*x^2+RootOf(_Z^3-4)*x^2-4*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)
+16*_Z^2)*x-2*RootOf(_Z^3-4)*x-4*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)-2*RootOf(_Z^3-4))/(x^2-2
*x-4))-1/4*ln(-(-4*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+RootOf(RootOf(_
Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^3*x^2+6*(x^2-2*x-2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+4*_Z*Ro
otOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^2+8*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)^2*RootOf(_Z^3-4)
^2*x-2*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^3*x+12*(x^2-2*x-2)^(1/3)*RootOf(Roo
tOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)+3*(x^2-2*x-2)^(1/3)*RootOf(_Z^3-4)^2+6*(x^2-2*x-2)^(
2/3)-8*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)+2*RootOf(_Z^3-4))/(x^2-2*x-4))*RootOf(_Z^3-4)-ln(-
(-4*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+RootOf(RootOf(_Z^3-4)^2+4*_Z*R
ootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^3*x^2+6*(x^2-2*x-2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16
*_Z^2)*RootOf(_Z^3-4)^2+8*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)^2*RootOf(_Z^3-4)^2*x-2*RootOf(R
ootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)^3*x+12*(x^2-2*x-2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+4
*_Z*RootOf(_Z^3-4)+16*_Z^2)*RootOf(_Z^3-4)+3*(x^2-2*x-2)^(1/3)*RootOf(_Z^3-4)^2+6*(x^2-2*x-2)^(2/3)-8*RootOf(R
ootOf(_Z^3-4)^2+4*_Z*RootOf(_Z^3-4)+16*_Z^2)+2*RootOf(_Z^3-4))/(x^2-2*x-4))*RootOf(RootOf(_Z^3-4)^2+4*_Z*RootO
f(_Z^3-4)+16*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x - 4\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.19, size = 109, normalized size = 0.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\left (x^{2} - 2 x - 4\right ) \sqrt [3]{x^{2} - 2 x - 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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