\(\int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\) [1120]

Optimal result

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

[Out]

-(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 = {37} \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=-\frac {\sqrt {1-x}}{\sqrt {x+1}} \]

[In]

Int[1/(Sqrt[1 - x]*(1 + x)^(3/2)),x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

Rule 37

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=-\frac {\sqrt {1-x}}{\sqrt {1+x}} \]

[In]

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

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
gosper	\(-\frac {\sqrt {1-x}}{\sqrt {1+x}}\)	\(15\)
default	\(-\frac {\sqrt {1-x}}{\sqrt {1+x}}\)	\(15\)
risch	\(\frac {\left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}\)	\(38\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=-\frac {x + \sqrt {x + 1} \sqrt {-x + 1} + 1}{x + 1} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=\begin {cases} - \sqrt {-1 + \frac {2}{x + 1}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- i \sqrt {1 - \frac {2}{x + 1}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x + 1} \]

[In]

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

[Out]

-sqrt(-x^2 + 1)/(x + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (14) = 28\).

Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{2 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx=-\frac {\sqrt {1-x}}{\sqrt {x+1}} \]

[In]

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

[Out]

-(1 - x)^(1/2)/(x + 1)^(1/2)