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

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

[Out]

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

________________________________________________________________________________________

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} \begin {gather*} \frac {x}{\sqrt {1-x} \sqrt {x+1}} \end {gather*}

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 = 0.03, size = 13, normalized size = 0.72 \begin {gather*} \frac {x}{\sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/Sqrt[1 - x^2]

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.01, size = 64, normalized size = 3.56 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {x \sqrt {\frac {1-x}{1+x}}}{-1+x},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I}{\sqrt {1-\frac {2}{1+x}}}+\frac {I}{\left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{-x Sqrt[(1 - x) / (1 + x)] / (-1 + x), 1 / Abs[1 + x] > 1 / 2}}, -I / Sqrt[1 - 2 / (1 + x)] + I /
((1 + x) Sqrt[1 - 2 / (1 + x)])]

________________________________________________________________________________________

Maple [A]
time = 0.15, size = 29, normalized size = 1.61

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	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, x}{\sqrt {1-x}\, \sqrt {1+x}\, \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 11, normalized size = 0.61 \begin {gather*} \frac {x}{\sqrt {-x^{2} + 1}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 22, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {x + 1} x \sqrt {-x + 1}}{x^{2} - 1} \end {gather*}

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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.97, size = 63, normalized size = 3.50 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (14) = 28\).
time = 0.00, size = 87, normalized size = 4.83 \begin {gather*} 2 \left (\frac {\sqrt {-x+1}}{4 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{16 \sqrt {-x+1}}-\frac {\sqrt {-x+1} \sqrt {x+1}}{4 \left (x+1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.31, size = 14, normalized size = 0.78 \begin {gather*} \frac {x}{\sqrt {1-x}\,\sqrt {x+1}} \end {gather*}

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))

________________________________________________________________________________________