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

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]

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

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Rubi [A]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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])

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 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
]

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*}

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Mathematica [A]  time = 0.05, size = 83, normalized size = 1.98 \[ \frac {\sqrt {x-a} \sqrt {\frac {a+x}{a-x}} \left (2 a^{3/2} \sqrt {\frac {a+x}{a}} \sinh ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {2} \sqrt {a}}\right )+\sqrt {x-a} (a+x)\right )}{a+x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(a + x)/(a - x)],x]

[Out]

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

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fricas [A]  time = 0.42, size = 38, normalized size = 0.90 \[ 2 \, a \arctan \left (\sqrt {\frac {a + x}{a - x}}\right ) - {\left (a - x\right )} \sqrt {\frac {a + x}{a - x}} \]

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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giac [A]  time = 0.04, size = 36, normalized size = 0.86 \[ a \arcsin \left (\frac {x}{a}\right ) \mathrm {sgn}\left (a - x\right ) \mathrm {sgn}\relax (a) - \sqrt {a^{2} - x^{2}} \mathrm {sgn}\left (a - x\right ) \]

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)

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maple [A]  time = 0.02, size = 64, normalized size = 1.52 \[ -\frac {\sqrt {-\frac {a +x}{-a +x}}\, \left (-a +x \right ) \left (a \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )-\sqrt {a^{2}-x^{2}}\right )}{\sqrt {-\left (a +x \right ) \left (-a +x \right )}} \]

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

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maxima [A]  time = 1.18, size = 49, normalized size = 1.17 \[ -2 \, a {\left (\frac {\sqrt {\frac {a + x}{a - x}}}{\frac {a + x}{a - x} + 1} - \arctan \left (\sqrt {\frac {a + x}{a - x}}\right )\right )} \]

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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mupad [B]  time = 0.07, size = 49, normalized size = 1.17 \[ 2\,a\,\mathrm {atan}\left (\sqrt {\frac {a+x}{a-x}}\right )-\frac {2\,a\,\sqrt {\frac {a+x}{a-x}}}{\frac {a+x}{a-x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + x)/(a - x))^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {a + x}{a - x}}\, dx \]

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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