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

Optimal. Leaf size=27 \[ -\frac{3 \left (1-x^2\right )}{2 \left (-(x+1) \left (1-x^2\right )\right )^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0298237, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2067, 2064, 37} \[ -\frac{3 (1-x) (x+1)}{2 \left (x^3+x^2-x-1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(-1 + x^2))^(-2/3),x]

[Out]

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

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2064

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*
x)^(2*p)), Int[(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0]
&&  !IntegerQ[p]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\left ((1+x) \left (-1+x^2\right )\right )^{2/3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (-\frac{16}{27}-\frac{4 x}{3}+x^3\right )^{2/3}} \, dx,x,\frac{1}{3}+x\right )\\ &=\frac{\left (32 \sqrt [3]{2} (-1-x)^{4/3} (-1+x)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{16}{9}-\frac{8 x}{3}\right )^{4/3} \left (-\frac{16}{9}+\frac{4 x}{3}\right )^{2/3}} \, dx,x,\frac{1}{3}+x\right )}{9 \left (-1-x+x^2+x^3\right )^{2/3}}\\ &=-\frac{3 (1-x) (1+x)}{2 \left (-1-x+x^2+x^3\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0098332, size = 23, normalized size = 0.85 \[ \frac{3 (x-1) (x+1)}{2 \left ((x-1) (x+1)^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 20, normalized size = 0.7 \begin{align*}{\frac{ \left ( 3\,x-3 \right ) \left ( 1+x \right ) }{2} \left ( \left ( 1+x \right ) \left ({x}^{2}-1 \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.39376, size = 53, normalized size = 1.96 \begin{align*} \frac{3 \,{\left (x^{3} + x^{2} - x - 1\right )}^{\frac{1}{3}}}{2 \,{\left (x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^3 + x^2 - x - 1)^(1/3)/(x + 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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