3.13.12 \(\int \frac {-3+x}{\sqrt [3]{-1+x^2} (2+x+x^2)} \, dx\) [1212]

Optimal. Leaf size=89 \[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{2+2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (-1-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]

[Out]

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

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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 {-3+x}{\sqrt [3]{-1+x^2} \left (2+x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 89, normalized size = 1.00 \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{2+2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (-1-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.79, size = 229, normalized size = 2.57

method	result	size
trager	\(\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{2}-1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-x +1}{x^{2}+x +2}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{2}-1\right )^{\frac {2}{3}}-x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x -1}{x^{2}+x +2}\right )\)	\(229\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(-(RootOf(_Z^2+_Z+1)^2*x^2+3*RootOf(_Z^2+_Z+1)^2*x+3*RootOf(_Z^2+_Z+1)*(x^2-1)^(1/3)*x-RootOf(_Z^2+_Z+1)*x^2
+3*RootOf(_Z^2+_Z+1)*(x^2-1)^(1/3)-2*RootOf(_Z^2+_Z+1)*x+3*(x^2-1)^(2/3)-RootOf(_Z^2+_Z+1)-x+1)/(x^2+x+2))+Roo
tOf(_Z^2+_Z+1)*ln(-(RootOf(_Z^2+_Z+1)^2*x^2+3*RootOf(_Z^2+_Z+1)^2*x+3*RootOf(_Z^2+_Z+1)*(x^2-1)^(1/3)*x+3*Root
Of(_Z^2+_Z+1)*(x^2-1)^(1/3)-RootOf(_Z^2+_Z+1)*x+3*(x^2-1)^(2/3)-x^2+RootOf(_Z^2+_Z+1)-2*x-1)/(x^2+x+2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.73, size = 95, normalized size = 1.07 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + \sqrt {3} {\left (x - 1\right )} - 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, x^{2} + 17 \, x + 7}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} - 3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + x + 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 2}{x^{2} + x + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(-(4*sqrt(3)*(x^2 - 1)^(1/3)*(x + 1) + sqrt(3)*(x - 1) - 2*sqrt(3)*(x^2 - 1)^(2/3))/(8*x^2 + 17
*x + 7)) + 1/2*log((x^2 - 3*(x^2 - 1)^(1/3)*(x + 1) + x + 3*(x^2 - 1)^(2/3) + 2)/(x^2 + x + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x-3}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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