3.543 \(\int \frac{1}{x-\sqrt{2+x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]

[Out]

(4*Log[2 - Sqrt[2 + x]])/3 + (2*Log[1 + Sqrt[2 + x]])/3

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Rubi [A]  time = 0.043598, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x - Sqrt[2 + x])^(-1),x]

[Out]

(4*Log[2 - Sqrt[2 + x]])/3 + (2*Log[1 + Sqrt[2 + x]])/3

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Rubi in Sympy [A]  time = 2.3903, size = 26, normalized size = 0.84 \[ \frac{4 \log{\left (- \sqrt{x + 2} + 2 \right )}}{3} + \frac{2 \log{\left (\sqrt{x + 2} + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*log(-sqrt(x + 2) + 2)/3 + 2*log(sqrt(x + 2) + 1)/3

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Mathematica [A]  time = 0.00707898, size = 31, normalized size = 1. \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x - Sqrt[2 + x])^(-1),x]

[Out]

(4*Log[2 - Sqrt[2 + x]])/3 + (2*Log[1 + Sqrt[2 + x]])/3

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Maple [B]  time = 0.02, size = 54, normalized size = 1.7 \[{\frac{\ln \left ( 1+x \right ) }{3}}+{\frac{2\,\ln \left ( x-2 \right ) }{3}}-{\frac{2}{3}\ln \left ( \sqrt{2+x}+2 \right ) }-{\frac{1}{3}\ln \left ( \sqrt{2+x}-1 \right ) }+{\frac{1}{3}\ln \left ( 1+\sqrt{2+x} \right ) }+{\frac{2}{3}\ln \left ( \sqrt{2+x}-2 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(2+x)^(1/2)),x)

[Out]

1/3*ln(1+x)+2/3*ln(x-2)-2/3*ln((2+x)^(1/2)+2)-1/3*ln((2+x)^(1/2)-1)+1/3*ln(1+(2+
x)^(1/2))+2/3*ln((2+x)^(1/2)-2)

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Maxima [A]  time = 0.718476, size = 28, normalized size = 0.9 \[ \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left (\sqrt{x + 2} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*log(sqrt(x + 2) + 1) + 4/3*log(sqrt(x + 2) - 2)

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Fricas [A]  time = 0.262678, size = 28, normalized size = 0.9 \[ \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left (\sqrt{x + 2} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*log(sqrt(x + 2) + 1) + 4/3*log(sqrt(x + 2) - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x-(2+x)**(1/2)),x)

[Out]

Integral(1/(x - sqrt(x + 2)), x)

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GIAC/XCAS [A]  time = 0.278952, size = 30, normalized size = 0.97 \[ \frac{2}{3} \,{\rm ln}\left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \,{\rm ln}\left ({\left | \sqrt{x + 2} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*ln(sqrt(x + 2) + 1) + 4/3*ln(abs(sqrt(x + 2) - 2))