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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.43, size = 22, normalized size = 1.22 \[ -\frac {\sqrt {x + 1} x \sqrt {-x + 1}}{x^{2} - 1} \]

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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giac [B]  time = 0.72, size = 62, normalized size = 3.44 \[ \frac {\sqrt {2} - \sqrt {-x + 1}}{4 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1} \sqrt {-x + 1}}{2 \, {\left (x - 1\right )}} - \frac {\sqrt {x + 1}}{4 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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))

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {x}{\sqrt {-x +1}\, \sqrt {x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 11, normalized size = 0.61 \[ \frac {x}{\sqrt {-x^{2} + 1}} \]

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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mupad [B]  time = 0.31, size = 14, normalized size = 0.78 \[ \frac {x}{\sqrt {1-x}\,\sqrt {x+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/((1 - x)^(1/2)*(x + 1)^(1/2))

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sympy [A]  time = 1.86, size = 65, normalized size = 3.61 \[ \begin {cases} \frac {1}{\sqrt {-1 + \frac {2}{x + 1}}} - \frac {1}{\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{x - 1} + \frac {i \sqrt {1 - \frac {2}{x + 1}}}{x - 1} & \text {otherwise} \end {cases} \]

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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