\(\int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\) [1121]

Optimal result

Integrand size = 17, antiderivative size = 18 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\frac {x}{\sqrt {1-x} \sqrt {1+x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {39} \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\frac {x}{\sqrt {1-x} \sqrt {x+1}} \]

[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*} \text {integral}& = \frac {x}{\sqrt {1-x} \sqrt {1+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\frac {x}{\sqrt {1-x^2}} \]

[In]

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

[Out]

x/Sqrt[1 - x^2]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
gosper	\(\frac {x}{\sqrt {1-x}\, \sqrt {1+x}}\)	\(15\)
default	\(\frac {1}{\sqrt {1-x}\, \sqrt {1+x}}-\frac {\sqrt {1-x}}{\sqrt {1+x}}\)	\(29\)
risch	\(\frac {x \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {1-x}\, \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1+x}}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=-\frac {\sqrt {x + 1} x \sqrt {-x + 1}}{x^{2} - 1} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.50 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\begin {cases} - \frac {\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{x - 1} + \frac {\sqrt {-1 + \frac {2}{x + 1}}}{x - 1} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i}{\sqrt {1 - \frac {2}{x + 1}}} + \frac {i}{\sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\frac {x}{\sqrt {-x^{2} + 1}} \]

[In]

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

[Out]

x/sqrt(-x^2 + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (14) = 28\).

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.44 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{4 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1} \sqrt {-x + 1}}{2 \, {\left (x - 1\right )}} - \frac {\sqrt {x + 1}}{4 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx=\frac {x}{\sqrt {1-x}\,\sqrt {x+1}} \]

[In]

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

[Out]

x/((1 - x)^(1/2)*(x + 1)^(1/2))