3.11.60 \(\int \frac {-3+x^2}{\sqrt [3]{-1+x^2} (-1+x^2+x^3)} \, dx\) [1060]

Optimal. Leaf size=80 \[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{-2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]

[Out]

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

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Rubi [F]
time = 0.26, 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^2}{\sqrt [3]{-1+x^2} \left (-1+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 80, normalized size = 1.00 \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{-2 x+\sqrt [3]{-1+x^2}}\right )+\log \left (x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(-2*x + (-1 + x^2)^(1/3))] + Log[x + (-1 + x^2)^(1/3)] - Log[x^2 - 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 = 1.33, size = 230, normalized size = 2.88

method	result	size
trager	\(\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}-1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 x^{3}+2 x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{3}+x^{2}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x \left (x^{2}-1\right )^{\frac {2}{3}}+\left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{3}+x^{2}-1}\right )\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 0.78, size = 100, normalized size = 1.25 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{3} + 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 8 \, x^{2} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} + 3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} x^{2} + x^{2} + 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} x - 1}{x^{3} + x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

integrate((x^2 - 3)/((x^3 + x^2 - 1)*(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^2-3}{{\left (x^2-1\right )}^{1/3}\,\left (x^3+x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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