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

Optimal. Leaf size=73 \[ \frac{2}{5} \left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{\sqrt{x+1}+1}-\sqrt{5}+1\right )+\frac{2}{5} \left (5-\sqrt{5}\right ) \log \left (-2 \sqrt{\sqrt{x+1}+1}+\sqrt{5}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.204207, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2}{5} \left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{\sqrt{x+1}+1}-\sqrt{5}+1\right )+\frac{2}{5} \left (5-\sqrt{5}\right ) \log \left (-2 \sqrt{\sqrt{x+1}+1}+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.00105, size = 80, normalized size = 1.1 \[ - \frac{4 \sqrt{5} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (- 2 \sqrt{\sqrt{x + 1} + 1} + 1 + \sqrt{5} \right )}}{5} + \frac{4 \sqrt{5} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \log{\left (- 2 \sqrt{\sqrt{x + 1} + 1} - \sqrt{5} + 1 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(5)*(-sqrt(5)/2 + 1/2)*log(-2*sqrt(sqrt(x + 1) + 1) + 1 + sqrt(5))/5 + 4*
sqrt(5)*(1/2 + sqrt(5)/2)*log(-2*sqrt(sqrt(x + 1) + 1) - sqrt(5) + 1)/5

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Mathematica [A]  time = 0.165133, size = 136, normalized size = 1.86 \[ \log \left (\sqrt{x+1}-x\right )+2 \sqrt{\frac{2}{5} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3-\sqrt{5}}} \sqrt{\sqrt{x+1}+1}\right )-4 \sqrt{\frac{2}{5 \left (3+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} \sqrt{\sqrt{x+1}+1}\right )+\frac{2 \tanh ^{-1}\left (\frac{2 \sqrt{x+1}-1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Warning: Unable to verify antiderivative.

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

[Out]

2*Sqrt[(2*(3 + Sqrt[5]))/5]*ArcTanh[Sqrt[2/(3 - Sqrt[5])]*Sqrt[1 + Sqrt[1 + x]]]
- 4*Sqrt[2/(5*(3 + Sqrt[5]))]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*Sqrt[1 + Sqrt[1 + x
]]] + (2*ArcTanh[(-1 + 2*Sqrt[1 + x])/Sqrt[5]])/Sqrt[5] + Log[-x + Sqrt[1 + x]]

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Maple [B]  time = 0.19, size = 175, normalized size = 2.4 \[{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{5}}{5}} \right ) }+{\frac{2\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 1+2\,\sqrt{1+\sqrt{1+x}} \right ) } \right ) }+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+x}+1 \right ) } \right ) }+{\frac{\ln \left ({x}^{2}-x-1 \right ) }{2}}+{\frac{2\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+\sqrt{1+x}}-1 \right ) } \right ) }+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 2\,\sqrt{1+x}-1 \right ) } \right ) }+\ln \left ( \sqrt{1+x}-\sqrt{1+\sqrt{1+x}} \right ) -\ln \left ( \sqrt{1+x}+\sqrt{1+\sqrt{1+x}} \right ) -{\frac{1}{2}\ln \left ( x+\sqrt{1+x} \right ) }+{\frac{1}{2}\ln \left ( x-\sqrt{1+x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(1+(1+x)^(1/2))^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.58519, size = 85, normalized size = 1.16 \[ -\frac{2}{5} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, \sqrt{\sqrt{x + 1} + 1} + 1}{\sqrt{5} + 2 \, \sqrt{\sqrt{x + 1} + 1} - 1}\right ) + 2 \, \log \left (\sqrt{x + 1} - \sqrt{\sqrt{x + 1} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.234395, size = 112, normalized size = 1.53 \[ \frac{2}{5} \, \sqrt{5}{\left (\sqrt{5} \log \left (\sqrt{x + 1} - \sqrt{\sqrt{x + 1} + 1}\right ) + \log \left (-\frac{2 \,{\left (\sqrt{5} - 5\right )} \sqrt{\sqrt{x + 1} + 1} - 2 \, \sqrt{5} \sqrt{x + 1} - 5 \, \sqrt{5} + 5}{\sqrt{x + 1} - \sqrt{\sqrt{x + 1} + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x - sqrt(sqrt(x + 1) + 1)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError