3.1121 \(\int \frac{1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x}{\sqrt{1-x} \sqrt{x+1}} \]

[Out]

x/(Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0016086, antiderivative size = 18, 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 = {39} \[ \frac{x}{\sqrt{1-x} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(Sqrt[1 - x]*Sqrt[1 + x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0027818, size = 13, normalized size = 0.72 \[ \frac{x}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/Sqrt[1 - x^2]

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{x{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01873, size = 15, normalized size = 0.83 \begin{align*} \frac{x}{\sqrt{-x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/sqrt(-x^2 + 1)

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Fricas [A]  time = 1.74749, size = 53, normalized size = 2.94 \begin{align*} -\frac{\sqrt{x + 1} x \sqrt{-x + 1}}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x + 1)*x*sqrt(-x + 1)/(x^2 - 1)

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Sympy [A]  time = 2.71814, size = 65, normalized size = 3.61 \begin{align*} \begin{cases} \frac{1}{\sqrt{-1 + \frac{2}{x + 1}}} - \frac{1}{\sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{x - 1} + \frac{i \sqrt{1 - \frac{2}{x + 1}}}{x - 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((1/sqrt(-1 + 2/(x + 1)) - 1/(sqrt(-1 + 2/(x + 1))*(x + 1)), 2/Abs(x + 1) > 1), (-I*sqrt(1 - 2/(x + 1
))*(x + 1)/(x - 1) + I*sqrt(1 - 2/(x + 1))/(x - 1), True))

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Giac [B]  time = 1.06418, size = 84, normalized size = 4.67 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{4 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1} \sqrt{-x + 1}}{2 \,{\left (x - 1\right )}} - \frac{\sqrt{x + 1}}{4 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/2*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 1/4*sqrt(x + 1)/(sqrt(2) - s
qrt(-x + 1))