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

Optimal. Leaf size=8 \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*ArcSinh[Sqrt[x]]

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Rubi [A]  time = 0.0113483, antiderivative size = 8, 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 = {1958, 54, 215} \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x/(1 + x)]/x,x]

[Out]

2*ArcSinh[Sqrt[x]]

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0057167, size = 8, normalized size = 1. \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x/(1 + x)]/x,x]

[Out]

2*ArcSinh[Sqrt[x]]

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Maple [B]  time = 0.01, size = 32, normalized size = 4. \begin{align*}{(1+x)\sqrt{{\frac{x}{1+x}}}\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01792, size = 36, normalized size = 4.5 \begin{align*} \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.5946, size = 72, normalized size = 9. \begin{align*} \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{x}{x + 1}}}{x}\, dx \end{align*}

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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Giac [B]  time = 1.13737, size = 30, normalized size = 3.75 \begin{align*} -\log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \mathrm{sgn}\left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-2*x + 2*sqrt(x^2 + x) - 1))*sgn(x + 1)