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

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

[Out]

2*ArcSinh[Sqrt[x]]

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Rubi [A]  time = 0.00914415, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcSinh[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.04692, size = 7, normalized size = 0.88 \[ 2 \operatorname{asinh}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/2)/(1+x)**(1/2),x)

[Out]

2*asinh(sqrt(x))

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Mathematica [A]  time = 0.00561474, size = 8, normalized size = 1. \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcSinh[Sqrt[x]]

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Maple [B]  time = 0.005, size = 28, normalized size = 3.5 \[{1\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.722956, size = 36, normalized size = 4.5 \[ \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) - \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*sqrt(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.300116, size = 24, normalized size = 3. \[ -\log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*sqrt(x)),x, algorithm="fricas")

[Out]

-log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1)

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Sympy [A]  time = 3.57079, 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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GIAC/XCAS [A]  time = 0.308758, size = 20, normalized size = 2.5 \[ -2 \,{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*sqrt(x)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(x + 1) + sqrt(x)))