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

Optimal. Leaf size=42 \[ 2 a \tan ^{-1}\left (\sqrt{\frac{a+x}{a-x}}\right )-(a-x) \sqrt{\frac{a+x}{a-x}} \]

[Out]

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

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Rubi [A]  time = 0.0161238, antiderivative size = 42, 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, 203} \[ 2 a \tan ^{-1}\left (\sqrt{\frac{a+x}{a-x}}\right )-(a-x) \sqrt{\frac{a+x}{a-x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(a + x)/(a - x)],x]

[Out]

-((a - x)*Sqrt[(a + x)/(a - x)]) + 2*a*ArcTan[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 )\\ &=-(a-x) \sqrt{\frac{a+x}{a-x}}+(2 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{a+x}{a-x}}\right )\\ &=-(a-x) \sqrt{\frac{a+x}{a-x}}+2 a \tan ^{-1}\left (\sqrt{\frac{a+x}{a-x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0423336, size = 78, normalized size = 1.86 \[ \frac{\sqrt{\frac{a+x}{a-x}} \left (-2 a^{3/2} \sqrt{a-x} \sqrt{\frac{a+x}{a}} \sin ^{-1}\left (\frac{\sqrt{a-x}}{\sqrt{2} \sqrt{a}}\right )-a^2+x^2\right )}{a+x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(a + x)/(a - x)],x]

[Out]

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

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Maple [A]  time = 0.014, size = 64, normalized size = 1.5 \begin{align*} -{(-a+x)\sqrt{-{\frac{a+x}{-a+x}}} \left ( a\arctan \left ({x{\frac{1}{\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*arctan(x/(a^2-x^2)^(1/2))-(a^2-x^2)^(1/2))/(-(a+x)*(-a+x))^(1/2)

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Maxima [A]  time = 1.46206, size = 66, normalized size = 1.57 \begin{align*} -2 \, a{\left (\frac{\sqrt{\frac{a + x}{a - x}}}{\frac{a + x}{a - x} + 1} - \arctan \left (\sqrt{\frac{a + x}{a - x}}\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]

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

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Fricas [A]  time = 1.65947, size = 90, normalized size = 2.14 \begin{align*} 2 \, a \arctan \left (\sqrt{\frac{a + x}{a - x}}\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]

2*a*arctan(sqrt((a + x)/(a - x))) - (a - x)*sqrt((a + x)/(a - x))

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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)

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Giac [A]  time = 1.14373, size = 49, normalized size = 1.17 \begin{align*} a \arcsin \left (\frac{x}{a}\right ) \mathrm{sgn}\left (a - x\right ) \mathrm{sgn}\left (a\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*arcsin(x/a)*sgn(a - x)*sgn(a) - sqrt(a^2 - x^2)*sgn(a - x)