3.744 \(\int \sqrt{\frac{-a+x}{a+x}} \, dx\)

Optimal. Leaf size=41 \[ \sqrt{-\frac{a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt{-\frac{a-x}{a+x}}\right ) \]

[Out]

Sqrt[-((a - x)/(a + x))]*(a + x) - 2*a*ArcTanh[Sqrt[-((a - x)/(a + x))]]

________________________________________________________________________________________

Rubi [A]  time = 0.0192864, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1959, 288, 206} \[ \sqrt{-\frac{a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt{-\frac{a-x}{a+x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-a + x)/(a + x)],x]

[Out]

Sqrt[-((a - x)/(a + x))]*(a + x) - 2*a*ArcTanh[Sqrt[-((a - x)/(a + x))]]

Rule 1959

Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Denominator[p]
}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n - 1))/(b*e - d*x^q)^(1/n + 1),
x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n
]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0749607, size = 78, normalized size = 1.9 \[ \frac{\sqrt{\frac{x-a}{a+x}} \left (\sqrt{x-a} (a+x)-2 a^{3/2} \sqrt{\frac{a+x}{a}} \sinh ^{-1}\left (\frac{\sqrt{x-a}}{\sqrt{2} \sqrt{a}}\right )\right )}{\sqrt{x-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-a + x)/(a + x)],x]

[Out]

(Sqrt[(-a + x)/(a + x)]*(Sqrt[-a + x]*(a + x) - 2*a^(3/2)*Sqrt[(a + x)/a]*ArcSinh[Sqrt[-a + x]/(Sqrt[2]*Sqrt[a
])]))/Sqrt[-a + x]

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 60, normalized size = 1.5 \begin{align*} -{(a+x)\sqrt{{\frac{-a+x}{a+x}}} \left ( a\ln \left ( x+\sqrt{-{a}^{2}+{x}^{2}} \right ) -\sqrt{-{a}^{2}+{x}^{2}} \right ){\frac{1}{\sqrt{ \left ( a+x \right ) \left ( -a+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.988935, size = 95, normalized size = 2.32 \begin{align*} a{\left (\frac{2 \, \sqrt{-\frac{a - x}{a + x}}}{\frac{a - x}{a + x} + 1} - \log \left (\sqrt{-\frac{a - x}{a + x}} + 1\right ) + \log \left (\sqrt{-\frac{a - x}{a + x}} - 1\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.80178, size = 142, normalized size = 3.46 \begin{align*} -a \log \left (\sqrt{-\frac{a - x}{a + x}} + 1\right ) + a \log \left (\sqrt{-\frac{a - x}{a + x}} - 1\right ) +{\left (a + x\right )} \sqrt{-\frac{a - x}{a + x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((-a + x)/(a + x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.15938, size = 54, normalized size = 1.32 \begin{align*} a \log \left ({\left | -x + \sqrt{-a^{2} + x^{2}} \right |}\right ) \mathrm{sgn}\left (a + x\right ) + \sqrt{-a^{2} + x^{2}} \mathrm{sgn}\left (a + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(abs(-x + sqrt(-a^2 + x^2)))*sgn(a + x) + sqrt(-a^2 + x^2)*sgn(a + x)