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 = 0.0122861, 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 \[ \frac{x}{\sqrt{1-x} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

x/(Sqrt[1 - x]*Sqrt[1 + x])

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Rubi in Sympy [A]  time = 2.63629, size = 14, normalized size = 0.78 \[ \frac{x}{\sqrt{- x + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(3/2)/(1+x)**(3/2),x)

[Out]

x/(sqrt(-x + 1)*sqrt(x + 1))

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Mathematica [A]  time = 0.00991435, 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 = 0.003, size = 15, normalized size = 0.8 \[{x{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

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 = 1.33924, size = 15, normalized size = 0.83 \[ \frac{x}{\sqrt{-x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/sqrt(-x^2 + 1)

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Fricas [A]  time = 0.20276, size = 54, normalized size = 3. \[ -\frac{\sqrt{x + 1} x \sqrt{-x + 1} - x}{x^{2} + \sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.5958, size = 66, normalized size = 3.67 \[ \begin{cases} \frac{1}{\sqrt{-1 + \frac{2}{x + 1}}} - \frac{1}{\sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{x - 1} + \frac{i \sqrt{1 - \frac{2}{x + 1}}}{x - 1} & \text{otherwise} \end{cases} \]

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(1/(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/XCAS [A]  time = 0.212366, size = 84, normalized size = 4.67 \[ \frac{\sqrt{2} - \sqrt{-x + 1}}{4 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1} \sqrt{-x + 1}}{2 \,{\left (x - 1\right )}} - \frac{\sqrt{x + 1}}{4 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(-x + 1)^(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) - sqrt(-x + 1))