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

Optimal. Leaf size=20 \[ \frac{3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0056524, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {651} \[ \frac{3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.030237, size = 23, normalized size = 1.15 \[ \frac{3 (x-1) \sqrt [3]{x+1}}{2 \left (x^2-1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*}{\frac{3\,x-3}{2}\sqrt [3]{1+x} \left ({x}^{2}-1 \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68169, size = 47, normalized size = 2.35 \begin{align*} \frac{3 \,{\left (x^{2} - 1\right )}^{\frac{1}{3}}}{2 \,{\left (x + 1\right )}^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^2 - 1)^(1/3)/(x + 1)^(2/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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