3.13.60 \(\int \frac {(-1+x) (3+x)}{(-1+x^2)^{2/3} (2-x+x^2)} \, dx\)

Optimal. Leaf size=92 \[ -\log \left (\left (x^2-1\right )^{2/3}+x+1\right )+\frac {1}{2} \log \left (x^2+\left (x^2-1\right )^{4/3}+(-x-1) \left (x^2-1\right )^{2/3}+2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (x^2-1\right )^{2/3}}{\left (x^2-1\right )^{2/3}-2 x-2}\right ) \]

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Rubi [F]  time = 0.33, 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) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3^(3/4)*Sqrt[2 + Sqrt[3]]*(1 + (-1 + x^2)^(1/3))*Sqrt[(1 - (-1 + x^2)^(1/3) + (-1 + x^2)^(2/3))/(1 + Sqrt[3]
+ (-1 + x^2)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))], -7
- 4*Sqrt[3]])/(x*Sqrt[(1 + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))^2]) - Defer[Int][(5 - 3*x)/((-1
+ x^2)^(2/3)*(2 - x + x^2)), x]

Rubi steps

\begin {align*} \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx &=\int \left (\frac {1}{\left (-1+x^2\right )^{2/3}}-\frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )}\right ) \, dx\\ &=\int \frac {1}{\left (-1+x^2\right )^{2/3}} \, dx-\int \frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx\\ &=\frac {\left (3 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}-\int \frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx\\ &=\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}-\int \frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.15, size = 92, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (-1+x^2\right )^{2/3}}{-2-2 x+\left (-1+x^2\right )^{2/3}}\right )-\log \left (1+x+\left (-1+x^2\right )^{2/3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(-1-x) \left (-1+x^2\right )^{2/3}+\left (-1+x^2\right )^{4/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.01, size = 98, 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((-1+x)*(3+x)/(x^2-1)^(2/3)/(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x + 3\right )} {\left (x - 1\right )}}{{\left (x^{2} - x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.60, size = 315, normalized size = 3.42

method	result	size
trager	\(-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 x^{2}-3 \left (x^{2}-1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-4 x +2}{x^{2}-x +2}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+7 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{2}-1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -2}{x^{2}-x +2}\right )\)	\(315\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-(RootOf(_Z^2-_Z+1)^2*x^2-3*RootOf(_Z^2-_Z+1)*(x^2-1)^(1/3)*x-3*RootOf(_Z^2-_Z+1)^2*x-3*RootOf(_Z^2-_Z+1)*
x^2+3*(x^2-1)^(2/3)+3*RootOf(_Z^2-_Z+1)*(x^2-1)^(1/3)+3*x*(x^2-1)^(1/3)+8*RootOf(_Z^2-_Z+1)*x+2*x^2-3*(x^2-1)^
(1/3)-RootOf(_Z^2-_Z+1)-4*x+2)/(x^2-x+2))+RootOf(_Z^2-_Z+1)*ln(-(RootOf(_Z^2-_Z+1)^2*x^2-3*RootOf(_Z^2-_Z+1)*(
x^2-1)^(1/3)*x-3*RootOf(_Z^2-_Z+1)^2*x-2*RootOf(_Z^2-_Z+1)*x^2+3*(x^2-1)^(2/3)+3*RootOf(_Z^2-_Z+1)*(x^2-1)^(1/
3)+3*x*(x^2-1)^(1/3)+7*RootOf(_Z^2-_Z+1)*x-3*(x^2-1)^(1/3)+RootOf(_Z^2-_Z+1)-2*x-2)/(x^2-x+2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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