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

Optimal. Leaf size=15 \[ 2 \tan ^{-1}\left (\sqrt {-\frac {x}{x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1960, 204} \[ 2 \tan ^{-1}\left (\sqrt {-\frac {x}{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-(x/(1 + x))]/x,x]

[Out]

2*ArcTan[Sqrt[-(x/(1 + x))]]

Rule 204

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

Rule 1960

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

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 32, normalized size = 2.13 \[ \frac {2 \sqrt {-\frac {x}{x+1}} \sqrt {x+1} \sinh ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-(x/(1 + x))]/x,x]

[Out]

(2*Sqrt[-(x/(1 + x))]*Sqrt[1 + x]*ArcSinh[Sqrt[x]])/Sqrt[x]

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fricas [A]  time = 0.44, size = 13, normalized size = 0.87 \[ 2 \, \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(-x/(x + 1)))

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giac [A]  time = 0.30, size = 20, normalized size = 1.33 \[ -\frac {1}{2} \, \pi \mathrm {sgn}\left (x + 1\right ) - \arcsin \left (2 \, x + 1\right ) \mathrm {sgn}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*pi*sgn(x + 1) - arcsin(2*x + 1)*sgn(x + 1)

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maple [B]  time = 0.02, size = 33, normalized size = 2.20 \[ \frac {\sqrt {-\frac {x}{x +1}}\, \left (x +1\right ) \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {\left (x +1\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.95, size = 13, normalized size = 0.87 \[ 2 \, \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(-x/(x + 1)))

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mupad [B]  time = 0.18, size = 13, normalized size = 0.87 \[ 2\,\mathrm {atan}\left (\sqrt {-\frac {x}{x+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x/(x + 1))^(1/2)/x,x)

[Out]

2*atan((-x/(x + 1))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {x}{x + 1}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-x/(x + 1))/x, x)

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