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

Optimal. Leaf size=31 \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0225466, antiderivative size = 31, 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} \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x - Sqrt[2 + x])^(-1),x]

[Out]

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

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{2+x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{-2-x+x^2} \, dx,x,\sqrt{2+x}\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{2+x}\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-2+x} \, dx,x,\sqrt{2+x}\right )\\ &=\frac{4}{3} \log \left (2-\sqrt{2+x}\right )+\frac{2}{3} \log \left (1+\sqrt{2+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0101808, size = 31, normalized size = 1. \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x - Sqrt[2 + x])^(-1),x]

[Out]

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

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Maple [B]  time = 0.013, size = 54, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{3}}+{\frac{2\,\ln \left ( -2+x \right ) }{3}}-{\frac{2}{3}\ln \left ( \sqrt{2+x}+2 \right ) }+{\frac{1}{3}\ln \left ( 1+\sqrt{2+x} \right ) }-{\frac{1}{3}\ln \left ( -1+\sqrt{2+x} \right ) }+{\frac{2}{3}\ln \left ( \sqrt{2+x}-2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x-(2+x)^(1/2)),x)

[Out]

1/3*ln(1+x)+2/3*ln(-2+x)-2/3*ln((2+x)^(1/2)+2)+1/3*ln(1+(2+x)^(1/2))-1/3*ln(-1+(2+x)^(1/2))+2/3*ln((2+x)^(1/2)
-2)

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Maxima [A]  time = 0.926835, size = 28, normalized size = 0.9 \begin{align*} \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left (\sqrt{x + 2} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*log(sqrt(x + 2) + 1) + 4/3*log(sqrt(x + 2) - 2)

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Fricas [A]  time = 1.89868, size = 72, normalized size = 2.32 \begin{align*} \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left (\sqrt{x + 2} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*log(sqrt(x + 2) + 1) + 4/3*log(sqrt(x + 2) - 2)

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Sympy [A]  time = 1.31173, size = 36, normalized size = 1.16 \begin{align*} \log{\left (x - \sqrt{x + 2} \right )} + \frac{\log{\left (2 \sqrt{x + 2} - 4 \right )}}{3} - \frac{\log{\left (2 \sqrt{x + 2} + 2 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - sqrt(x + 2)) + log(2*sqrt(x + 2) - 4)/3 - log(2*sqrt(x + 2) + 2)/3

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Giac [A]  time = 1.07866, size = 30, normalized size = 0.97 \begin{align*} \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left ({\left | \sqrt{x + 2} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*log(sqrt(x + 2) + 1) + 4/3*log(abs(sqrt(x + 2) - 2))