3.8.48 \(\int \frac {x}{\sqrt {\frac {1-x}{1+x}} (1+x)} \, dx\) [748]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {1984, 12, 391} \begin {gather*} -\sqrt {\frac {1-x}{x+1}} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[(1 - x)/(1 + x)]*(1 + x))

Rule 12

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

Rule 391

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]

Rule 1984

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)/(b*e - d*x^q)^((m + 1)/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r,
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]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {\frac {1-x}{1+x}} (1+x)} \, dx &=-\left (4 \text {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 \text {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*}

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Mathematica [A]
time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} \frac {-1+x}{\sqrt {\frac {1-x}{1+x}}} \end {gather*}

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 = 0.44, size = 36, normalized size = 1.80

method	result	size
gosper	\(\frac {-1+x}{\sqrt {-\frac {-1+x}{1+x}}}\)	\(17\)
risch	\(\frac {-1+x}{\sqrt {-\frac {-1+x}{1+x}}}\)	\(17\)
trager	\(\left (-1-x \right ) \sqrt {-\frac {-1+x}{1+x}}\)	\(19\)
default	\(\frac {\left (-1+x \right ) \sqrt {-x^{2}+1}}{\sqrt {-\frac {-1+x}{1+x}}\, \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 27, normalized size = 1.35 \begin {gather*} \frac {2 \, \sqrt {-\frac {x - 1}{x + 1}}}{\frac {x - 1}{x + 1} - 1} \end {gather*}

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 = 0.32, size = 17, normalized size = 0.85 \begin {gather*} -{\left (x + 1\right )} \sqrt {-\frac {x - 1}{x + 1}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- \frac {x - 1}{x + 1}} \left (x + 1\right )}\, dx \end {gather*}

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 = 6.11, size = 17, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{\mathrm {sgn}\left (x + 1\right )} \end {gather*}

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]

-sqrt(-x^2 + 1)/sgn(x + 1)

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Mupad [B]
time = 0.06, size = 17, normalized size = 0.85 \begin {gather*} -\sqrt {-\frac {x-1}{x+1}}\,\left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-(x - 1)/(x + 1))^(1/2)*(x + 1)

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