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

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

[Out]

2*arcsinh(x^(1/2))

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {54, 215} \[ 2 \sinh ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 + 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]

Rubi steps

\begin {align*} \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[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcSinh[Sqrt[x]]

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fricas [B]  time = 0.44, size = 18, normalized size = 2.25 \[ -\log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1)

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giac [B]  time = 0.25, size = 14, normalized size = 1.75 \[ -2 \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(sqrt(x + 1) - sqrt(x))

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maple [B]  time = 0.00, size = 28, normalized size = 3.50 \[ \frac {\sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {x +1}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.87, size = 27, normalized size = 3.38 \[ \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) - \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 14, normalized size = 1.75 \[ 4\,\mathrm {atanh}\left (\frac {\sqrt {x+1}-1}{\sqrt {x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(x + 1)^(1/2)),x)

[Out]

4*atanh(((x + 1)^(1/2) - 1)/x^(1/2))

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sympy [A]  time = 0.95, size = 26, normalized size = 3.25 \[ \begin {cases} 2 \operatorname {acosh}{\left (\sqrt {x + 1} \right )} & \text {for}\: \left |{x + 1}\right | > 1 \\- 2 i \operatorname {asin}{\left (\sqrt {x + 1} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*acosh(sqrt(x + 1)), Abs(x + 1) > 1), (-2*I*asin(sqrt(x + 1)), True))

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