3.228 \(\int \frac{2+x-4 x^2+2 x^3}{1-x+x^2-x^3+x^4} \, dx\)

Optimal. Leaf size=63 \[ -\frac{2 \log \left (2 x^2-\left (1-\sqrt{5}\right ) x+2\right )}{1-\sqrt{5}}-\frac{2 \log \left (2 x^2-\left (1+\sqrt{5}\right ) x+2\right )}{1+\sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.0655793, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2086, 628} \[ -\frac{2 \log \left (2 x^2-\left (1-\sqrt{5}\right ) x+2\right )}{1-\sqrt{5}}-\frac{2 \log \left (2 x^2-\left (1+\sqrt{5}\right ) x+2\right )}{1+\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + x - 4*x^2 + 2*x^3)/(1 - x + x^2 - x^3 + x^4),x]

[Out]

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

Rule 2086

Int[(P3_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4), x_Symbol] :> With[{q = Sqrt[8*a^2
+ b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3]}, Dist[1/q,
Int[(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x)/(2*a + (b + q)*x + 2*a*x^2), x], x] - Dist[1/
q, Int[(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x)/(2*a + (b - q)*x + 2*a*x^2), x], x]] /; Fre
eQ[{a, b, c}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{2+x-4 x^2+2 x^3}{1-x+x^2-x^3+x^4} \, dx &=-\frac{\int \frac{-2 \sqrt{5}+\left (10-2 \sqrt{5}\right ) x}{2+\left (-1-\sqrt{5}\right ) x+2 x^2} \, dx}{\sqrt{5}}+\frac{\int \frac{2 \sqrt{5}+\left (10+2 \sqrt{5}\right ) x}{2+\left (-1+\sqrt{5}\right ) x+2 x^2} \, dx}{\sqrt{5}}\\ &=-\frac{2 \log \left (2-\left (1-\sqrt{5}\right ) x+2 x^2\right )}{1-\sqrt{5}}-\frac{2 \log \left (2-\left (1+\sqrt{5}\right ) x+2 x^2\right )}{1+\sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0273721, size = 55, normalized size = 0.87 \[ \frac{1}{2} \left (\left (1+\sqrt{5}\right ) \log \left (2 x^2+\left (\sqrt{5}-1\right ) x+2\right )-\left (\sqrt{5}-1\right ) \log \left (-2 x^2+\sqrt{5} x+x-2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x - 4*x^2 + 2*x^3)/(1 - x + x^2 - x^3 + x^4),x]

[Out]

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

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Maple [A]  time = 0.024, size = 82, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{2}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{2}}+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{2}}-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3-4*x^2+x+2)/(x^4-x^3+x^2-x+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{3} - 4 \, x^{2} + x + 2}{x^{4} - x^{3} + x^{2} - x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3-4*x^2+x+2)/(x^4-x^3+x^2-x+1),x, algorithm="maxima")

[Out]

integrate((2*x^3 - 4*x^2 + x + 2)/(x^4 - x^3 + x^2 - x + 1), x)

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Fricas [A]  time = 1.4834, size = 193, normalized size = 3.06 \begin{align*} \frac{1}{2} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - 2 \, x^{3} + 7 \, x^{2} + \sqrt{5}{\left (2 \, x^{3} - x^{2} + 2 \, x\right )} - 2 \, x + 2}{x^{4} - x^{3} + x^{2} - x + 1}\right ) + \frac{1}{2} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3-4*x^2+x+2)/(x^4-x^3+x^2-x+1),x, algorithm="fricas")

[Out]

1/2*sqrt(5)*log((2*x^4 - 2*x^3 + 7*x^2 + sqrt(5)*(2*x^3 - x^2 + 2*x) - 2*x + 2)/(x^4 - x^3 + x^2 - x + 1)) + 1
/2*log(x^4 - x^3 + x^2 - x + 1)

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Sympy [A]  time = 0.113659, size = 58, normalized size = 0.92 \begin{align*} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \log{\left (x^{2} + x \left (- \frac{1}{2} + \frac{\sqrt{5}}{2}\right ) + 1 \right )} + \left (\frac{1}{2} - \frac{\sqrt{5}}{2}\right ) \log{\left (x^{2} + x \left (- \frac{\sqrt{5}}{2} - \frac{1}{2}\right ) + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3-4*x**2+x+2)/(x**4-x**3+x**2-x+1),x)

[Out]

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

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Giac [A]  time = 1.20082, size = 78, normalized size = 1.24 \begin{align*} -\frac{1}{2} \, \sqrt{5} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{2} \, \sqrt{5} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) + \frac{1}{2} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3-4*x^2+x+2)/(x^4-x^3+x^2-x+1),x, algorithm="giac")

[Out]

-1/2*sqrt(5)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/2*sqrt(5)*log(x^2 + 1/2*x*(sqrt(5) - 1) + 1) + 1/2*log(x^4
- x^3 + x^2 - x + 1)