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

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

[Out]

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

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Rubi [A]  time = 0.0662003, antiderivative size = 61, 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 \[ \frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (-2 \sqrt{x+1}-\sqrt{5}+1\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{x+1}+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x - Sqrt[1 + x])^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.70687, size = 70, normalized size = 1.15 \[ \frac{2 \sqrt{5} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \log{\left (- 2 \sqrt{x + 1} + 1 + \sqrt{5} \right )}}{5} - \frac{2 \sqrt{5} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (- 2 \sqrt{x + 1} - \sqrt{5} + 1 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0190064, size = 38, normalized size = 0.62 \[ \log \left (\sqrt{x+1}-x\right )-\frac{2 \tanh ^{-1}\left (\frac{2 \sqrt{x+1}-1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 91, normalized size = 1.5 \[{\frac{\ln \left ({x}^{2}-x-1 \right ) }{2}}-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{5}}{5}} \right ) }+{\frac{1}{2}\ln \left ( x-\sqrt{1+x} \right ) }-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+x}-1 \right ) } \right ) }-{\frac{1}{2}\ln \left ( x+\sqrt{1+x} \right ) }-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 1+2\,\sqrt{1+x} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(1+x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.794425, size = 61, normalized size = 1. \[ \frac{1}{5} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, \sqrt{x + 1} + 1}{\sqrt{5} + 2 \, \sqrt{x + 1} - 1}\right ) + \log \left (x - \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.267254, size = 76, normalized size = 1.25 \[ \frac{1}{5} \, \sqrt{5}{\left (\sqrt{5} \log \left (x - \sqrt{x + 1}\right ) + \log \left (\frac{\sqrt{5}{\left (2 \, x + 5\right )} - 2 \, \sqrt{x + 1}{\left (\sqrt{5} + 5\right )} + 5}{x - \sqrt{x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x-(1+x)**(1/2)),x)

[Out]

Integral(1/(x - sqrt(x + 1)), x)

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GIAC/XCAS [A]  time = 0.296464, size = 66, normalized size = 1.08 \[ \frac{1}{5} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -\sqrt{5} + 2 \, \sqrt{x + 1} - 1 \right |}}{{\left | \sqrt{5} + 2 \, \sqrt{x + 1} - 1 \right |}}\right ) +{\rm ln}\left ({\left | x - \sqrt{x + 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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