3.154 \(\int \frac{1}{\sqrt{x+x^2}} \, dx\)

Optimal. Leaf size=14 \[ 2 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

[Out]

2*ArcTanh[x/Sqrt[x + x^2]]

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Rubi [A]  time = 0.00838707, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ 2 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[x + x^2],x]

[Out]

2*ArcTanh[x/Sqrt[x + x^2]]

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Rubi in Sympy [A]  time = 0.553924, size = 12, normalized size = 0.86 \[ 2 \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} + x}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(x/sqrt(x**2 + x))

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Mathematica [B]  time = 0.010579, size = 29, normalized size = 2.07 \[ \frac{2 \sqrt{x} \sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[x + x^2],x]

[Out]

(2*Sqrt[x]*Sqrt[1 + x]*ArcSinh[Sqrt[x]])/Sqrt[x*(1 + x)]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.9 \[ \ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.34523, size = 20, normalized size = 1.43 \[ \log \left (2 \, x + 2 \, \sqrt{x^{2} + x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + x),x, algorithm="maxima")

[Out]

log(2*x + 2*sqrt(x^2 + x) + 1)

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Fricas [A]  time = 0.197849, size = 23, normalized size = 1.64 \[ -\log \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + x),x, algorithm="fricas")

[Out]

-log(-2*x + 2*sqrt(x^2 + x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+x)**(1/2),x)

[Out]

Integral(1/sqrt(x**2 + x), x)

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GIAC/XCAS [A]  time = 0.215665, size = 24, normalized size = 1.71 \[ -{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + x),x, algorithm="giac")

[Out]

-ln(abs(-2*x + 2*sqrt(x^2 + x) - 1))