3.8.29 \(\int \frac {1}{\sqrt {x} \sqrt {1+x}} \, dx\) [729]

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 = {56, 221} \begin {gather*} 2 \sinh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcSinh[Sqrt[x]]

Rule 56

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 221

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 \text {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 = 0.02, size = 16, normalized size = 2.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcTanh[Sqrt[x]/Sqrt[1 + x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(27\) vs. \(2(6)=12\).
time = 0.38, size = 28, normalized size = 3.50

method	result	size
meijerg	\(2 \arcsinh \left (\sqrt {x}\right )\)	\(7\)
default	\(\frac {\sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {x}\, \sqrt {1+x}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (6) = 12\).
time = 0.28, size = 27, normalized size = 3.38 \begin {gather*} \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) - \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (6) = 12\).
time = 0.33, size = 18, normalized size = 2.25 \begin {gather*} -\log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 26, normalized size = 3.25 \begin {gather*} \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} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 1.57, size = 14, normalized size = 1.75 \begin {gather*} -2 \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \end {gather*}

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

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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