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

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 = 0.0311208, 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, Rules used = {632, 31} \[ \frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (-2 \sqrt{x+1}-\sqrt{5}+1\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{x+1}+\sqrt{5}+1\right ) \]

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

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{x-\sqrt{1+x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{-1-x+x^2} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{5} \left (5-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,\sqrt{1+x}\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}-\frac{\sqrt{5}}{2}+x} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{5} \left (5-\sqrt{5}\right ) \log \left (1-\sqrt{5}-2 \sqrt{1+x}\right )+\frac{1}{5} \left (5+\sqrt{5}\right ) \log \left (1+\sqrt{5}-2 \sqrt{1+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0248369, size = 58, normalized size = 0.95 \[ \frac{1}{5} \left (\left (5+\sqrt{5}\right ) \log \left (-2 \sqrt{x+1}+\sqrt{5}+1\right )-\left (\sqrt{5}-5\right ) \log \left (-2 \sqrt{x+1}-\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x - Sqrt[1 + x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.006, size = 91, normalized size = 1.5 \begin{align*}{\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 ( 1+2\,\sqrt{1+x} \right ) } \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 ) } \end{align*}

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*(1+2*(1
+x)^(1/2))*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))

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Maxima [A]  time = 1.46453, size = 61, normalized size = 1. \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-(1+x)^(1/2)),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 = 1.48591, size = 174, normalized size = 2.85 \begin{align*} \frac{1}{5} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5}{\left (x + 2\right )} -{\left (\sqrt{5}{\left (2 \, x - 1\right )} - 5\right )} \sqrt{x + 1} + 3 \, x - 2}{x^{2} - x - 1}\right ) + \log \left (x - \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-(1+x)^(1/2)),x, algorithm="fricas")

[Out]

1/5*sqrt(5)*log((2*x^2 - sqrt(5)*(x + 2) - (sqrt(5)*(2*x - 1) - 5)*sqrt(x + 1) + 3*x - 2)/(x^2 - x - 1)) + log
(x - sqrt(x + 1))

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Sympy [A]  time = 1.09701, size = 92, normalized size = 1.51 \begin{align*} 4 \left (\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left (\frac{2 \sqrt{5} \left (\sqrt{x + 1} - \frac{1}{2}\right )}{5} \right )}}{10} & \text{for}\: \left (\sqrt{x + 1} - \frac{1}{2}\right )^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left (\frac{2 \sqrt{5} \left (\sqrt{x + 1} - \frac{1}{2}\right )}{5} \right )}}{10} & \text{for}\: \left (\sqrt{x + 1} - \frac{1}{2}\right )^{2} < \frac{5}{4} \end{cases}\right ) + \log{\left (x - \sqrt{x + 1} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-(1+x)**(1/2)),x)

[Out]

4*Piecewise((-sqrt(5)*acoth(2*sqrt(5)*(sqrt(x + 1) - 1/2)/5)/10, (sqrt(x + 1) - 1/2)**2 > 5/4), (-sqrt(5)*atan
h(2*sqrt(5)*(sqrt(x + 1) - 1/2)/5)/10, (sqrt(x + 1) - 1/2)**2 < 5/4)) + log(x - sqrt(x + 1))

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Giac [A]  time = 1.15241, size = 66, normalized size = 1.08 \begin{align*} \frac{1}{5} \, \sqrt{5} \log \left (\frac{{\left | -\sqrt{5} + 2 \, \sqrt{x + 1} - 1 \right |}}{{\left | \sqrt{5} + 2 \, \sqrt{x + 1} - 1 \right |}}\right ) + \log \left ({\left | x - \sqrt{x + 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-(1+x)^(1/2)),x, algorithm="giac")

[Out]

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