3.748 \(\int \frac{x}{\sqrt{\frac{1-x}{1+x}} (1+x)} \, dx\)

Optimal. Leaf size=20 \[ -\sqrt{\frac{1-x}{x+1}} (x+1) \]

[Out]

-(Sqrt[(1 - x)/(1 + x)]*(1 + x))

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Rubi [A]  time = 0.055983, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1962, 12, 383} \[ -\sqrt{\frac{1-x}{x+1}} (x+1) \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[(1 - x)/(1 + x)]*(1 + x)),x]

[Out]

-(Sqrt[(1 - x)/(1 + x)]*(1 + x))

Rule 1962

Int[(u_)^(r_.)*(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Wi
th[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*
x^q)^((m + 1)/n - 1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^((m + 1)/n + 1),
x], x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && Frac
tionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 383

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

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{\frac{1-x}{1+x}} (1+x)} \, dx &=-\left (4 \operatorname{Subst}\left (\int \frac{1-x^2}{2 \left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-x}{1+x}}\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1-x^2}{\left (1+x^2\right )^2} \, dx,x,\sqrt{\frac{1-x}{1+x}}\right )\right )\\ &=-\sqrt{\frac{1-x}{1+x}} (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0089171, size = 19, normalized size = 0.95 \[ \frac{x-1}{\sqrt{\frac{1-x}{x+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x)/Sqrt[(1 - x)/(1 + x)]

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Maple [A]  time = 0.003, size = 17, normalized size = 0.9 \begin{align*}{(x-1){\frac{1}{\sqrt{-{\frac{x-1}{1+x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971723, size = 36, normalized size = 1.8 \begin{align*} \frac{2 \, \sqrt{-\frac{x - 1}{x + 1}}}{\frac{x - 1}{x + 1} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68215, size = 45, normalized size = 2.25 \begin{align*} -{\left (x + 1\right )} \sqrt{-\frac{x - 1}{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*sqrt(-(x - 1)/(x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \frac{x - 1}{x + 1}} \left (x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15837, size = 39, normalized size = 1.95 \begin{align*} -\frac{2}{\sqrt{-\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{-\frac{x - 1}{x + 1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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