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

Optimal. Leaf size=61 \[ \frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (-2 \sqrt{x+2}-\sqrt{5}+1\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{x+2}+\sqrt{5}+1\right ) \]

[Out]

((5 - Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[2 + x]])/5 + ((5 + Sqrt[5])*Log[1 + Sqrt
[5] - 2*Sqrt[2 + x]])/5

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Rubi [A]  time = 0.0802047, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (-2 \sqrt{x+2}-\sqrt{5}+1\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{x+2}+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

((5 - Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[2 + x]])/5 + ((5 + Sqrt[5])*Log[1 + Sqrt
[5] - 2*Sqrt[2 + x]])/5

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Rubi in Sympy [A]  time = 2.90252, size = 70, normalized size = 1.15 \[ \frac{2 \sqrt{5} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \log{\left (- 2 \sqrt{x + 2} + 1 + \sqrt{5} \right )}}{5} - \frac{2 \sqrt{5} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (- 2 \sqrt{x + 2} - \sqrt{5} + 1 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*(1/2 + sqrt(5)/2)*log(-2*sqrt(x + 2) + 1 + sqrt(5))/5 - 2*sqrt(5)*(-sq
rt(5)/2 + 1/2)*log(-2*sqrt(x + 2) - sqrt(5) + 1)/5

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Mathematica [A]  time = 0.0258799, size = 39, normalized size = 0.64 \[ \log \left (-x+\sqrt{x+2}-1\right )-\frac{2 \tanh ^{-1}\left (\frac{2 \sqrt{x+2}-1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTanh[(-1 + 2*Sqrt[2 + x])/Sqrt[5]])/Sqrt[5] + Log[-1 - x + Sqrt[2 + x]]

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Maple [A]  time = 0.011, size = 91, normalized size = 1.5 \[{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}}-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) }+{\frac{1}{2}\ln \left ( 1+x-\sqrt{2+x} \right ) }-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{2+x}-1 \right ) } \right ) }-{\frac{1}{2}\ln \left ( 1+x+\sqrt{2+x} \right ) }-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{2+x}+1 \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x-(2+x)^(1/2)),x)

[Out]

1/2*ln(x^2+x-1)-1/5*5^(1/2)*arctanh(1/5*(1+2*x)*5^(1/2))+1/2*ln(1+x-(2+x)^(1/2))
-1/5*5^(1/2)*arctanh(1/5*(2*(2+x)^(1/2)-1)*5^(1/2))-1/2*ln(1+x+(2+x)^(1/2))-1/5*
5^(1/2)*arctanh(1/5*(2*(2+x)^(1/2)+1)*5^(1/2))

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Maxima [A]  time = 0.808924, size = 62, normalized size = 1.02 \[ \frac{1}{5} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, \sqrt{x + 2} + 1}{\sqrt{5} + 2 \, \sqrt{x + 2} - 1}\right ) + \log \left (x - \sqrt{x + 2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(x + 2) + 1)/(sqrt(5) + 2*sqrt(x + 2) - 1)) +
log(x - sqrt(x + 2) + 1)

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Fricas [A]  time = 0.270092, size = 78, normalized size = 1.28 \[ \frac{1}{5} \, \sqrt{5}{\left (\sqrt{5} \log \left (x - \sqrt{x + 2} + 1\right ) + \log \left (\frac{\sqrt{5}{\left (2 \, x + 7\right )} - 2 \, \sqrt{x + 2}{\left (\sqrt{5} + 5\right )} + 5}{x - \sqrt{x + 2} + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(5)*(sqrt(5)*log(x - sqrt(x + 2) + 1) + log((sqrt(5)*(2*x + 7) - 2*sqrt(
x + 2)*(sqrt(5) + 5) + 5)/(x - sqrt(x + 2) + 1)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x - sqrt(x + 2) + 1), x)

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GIAC/XCAS [A]  time = 0.301543, size = 68, normalized size = 1.11 \[ \frac{1}{5} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -\sqrt{5} + 2 \, \sqrt{x + 2} - 1 \right |}}{{\left | \sqrt{5} + 2 \, \sqrt{x + 2} - 1 \right |}}\right ) +{\rm ln}\left ({\left | x - \sqrt{x + 2} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sqrt(5)*ln(abs(-sqrt(5) + 2*sqrt(x + 2) - 1)/abs(sqrt(5) + 2*sqrt(x + 2) - 1
)) + ln(abs(x - sqrt(x + 2) + 1))