3.419 \(\int \frac{2+x}{-1+2 x+x^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0119036, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac{1}{4} \left (2+\sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{1}{4} \left (2-\sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(-1 + 2*x + x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0202686, size = 42, normalized size = 0.93 \[ \frac{1}{4} \left (\left (2+\sqrt{2}\right ) \log \left (-x+\sqrt{2}-1\right )-\left (\sqrt{2}-2\right ) \log \left (x+\sqrt{2}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(-1 + 2*x + x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}+2\,x-1 \right ) }{2}}-{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x^2+2*x-1)-1/2*2^(1/2)*arctanh(1/4*(2+2*x)*2^(1/2))

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Maxima [A]  time = 1.48366, size = 47, normalized size = 1.04 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x - \sqrt{2} + 1}{x + \sqrt{2} + 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.982831, size = 128, normalized size = 2.84 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x^{2} - 2 \, \sqrt{2}{\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.100667, size = 39, normalized size = 0.87 \begin{align*} \left (\frac{1}{2} - \frac{\sqrt{2}}{4}\right ) \log{\left (x + 1 + \sqrt{2} \right )} + \left (\frac{\sqrt{2}}{4} + \frac{1}{2}\right ) \log{\left (x - \sqrt{2} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1336, size = 59, normalized size = 1.31 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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