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

Optimal. Leaf size=10 \[ -2 \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

[Out]

-2*ArcTanh[Sqrt[1 + x]]

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Rubi [A]  time = 0.010595, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -2 \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 + x]),x]

[Out]

-2*ArcTanh[Sqrt[1 + x]]

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Rubi in Sympy [A]  time = 0.975793, size = 10, normalized size = 1. \[ - 2 \operatorname{atanh}{\left (\sqrt{x + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(x + 1))

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Mathematica [B]  time = 0.00410762, size = 25, normalized size = 2.5 \[ \log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[1 + x]),x]

[Out]

Log[1 - Sqrt[1 + x]] - Log[1 + Sqrt[1 + x]]

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Maple [A]  time = 0.006, size = 9, normalized size = 0.9 \[ -2\,{\it Artanh} \left ( \sqrt{1+x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(1+x)^(1/2),x)

[Out]

-2*arctanh((1+x)^(1/2))

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Maxima [A]  time = 1.34975, size = 26, normalized size = 2.6 \[ -\log \left (\sqrt{x + 1} + 1\right ) + \log \left (\sqrt{x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206026, size = 26, normalized size = 2.6 \[ -\log \left (\sqrt{x + 1} + 1\right ) + \log \left (\sqrt{x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.787056, size = 26, normalized size = 2.6 \[ \begin{cases} - 2 \operatorname{acoth}{\left (\sqrt{x + 1} \right )} & \text{for}\: \left |{x + 1}\right | > 1 \\- 2 \operatorname{atanh}{\left (\sqrt{x + 1} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(1+x)**(1/2),x)

[Out]

Piecewise((-2*acoth(sqrt(x + 1)), Abs(x + 1) > 1), (-2*atanh(sqrt(x + 1)), True)
)

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GIAC/XCAS [A]  time = 0.214613, size = 27, normalized size = 2.7 \[ -{\rm ln}\left (\sqrt{x + 1} + 1\right ) +{\rm ln}\left ({\left | \sqrt{x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(sqrt(x + 1) + 1) + ln(abs(sqrt(x + 1) - 1))