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(x^(1/2))

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Rubi [A]  time = 0.01, antiderivative size = 8, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ 2 \sinh ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x/(1 + x)]/x,x]

[Out]

2*ArcSinh[Sqrt[x]]

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fricas [B]  time = 0.43, size = 27, normalized size = 3.38 \[ \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]

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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giac [B]  time = 0.40, size = 22, normalized size = 2.75 \[ -\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \mathrm {sgn}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 32, normalized size = 4.00 \[ \frac {\sqrt {\frac {x}{x +1}}\, \left (x +1\right ) \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {\left (x +1\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.88, size = 27, normalized size = 3.38 \[ \log \left (\sqrt {\frac {x}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x}{x + 1}} - 1\right ) \]

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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mupad [B]  time = 0.06, size = 12, normalized size = 1.50 \[ 2\,\mathrm {atanh}\left (\sqrt {\frac {x}{x+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x/(x + 1))^(1/2)/x,x)

[Out]

2*atanh((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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