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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]