\(\int \frac {x (-3+x^2)}{(-1+x^2)^{2/3} (1-x^2+x^3)} \, dx\) [958]

Optimal result

Integrand size = 28, antiderivative size = 73 \[ \int \frac {x \left (-3+x^2\right )}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^2}}\right )+\log \left (-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

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

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (-3+x^2\right )}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^2}}\right )+\log \left (-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]

[In]

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

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 3.15

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {x \left (-3+x^2\right )}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \log \left (-\frac {x - {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{2} - 1\right )}^{\frac {1}{3}} x + {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (-3+x^2\right )}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx=\int \frac {x\,\left (x^2-3\right )}{{\left (x^2-1\right )}^{2/3}\,\left (x^3-x^2+1\right )} \,d x \]

[In]

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

[Out]

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