3.67 \(\int \frac{2-x-2 x^2+x^3}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=4 \[ \log (x+2) \]

[Out]

Log[2 + x]

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Rubi [A]  time = 0.0105952, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1586, 31} \[ \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(2 - x - 2*x^2 + x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[2 + x]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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

Mathematica [A]  time = 0.0011245, size = 4, normalized size = 1. \[ \log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - x - 2*x^2 + x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[2 + x]

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Maple [A]  time = 0.002, size = 5, normalized size = 1.3 \begin{align*} \ln \left ( 2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-2*x^2-x+2)/(x^4-5*x^2+4),x)

[Out]

ln(2+x)

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Maxima [A]  time = 0.968118, size = 5, normalized size = 1.25 \begin{align*} \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2*x^2-x+2)/(x^4-5*x^2+4),x, algorithm="maxima")

[Out]

log(x + 2)

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Fricas [A]  time = 1.98294, size = 16, normalized size = 4. \begin{align*} \log \left (x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2*x^2-x+2)/(x^4-5*x^2+4),x, algorithm="fricas")

[Out]

log(x + 2)

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Sympy [A]  time = 0.061844, size = 3, normalized size = 0.75 \begin{align*} \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-2*x**2-x+2)/(x**4-5*x**2+4),x)

[Out]

log(x + 2)

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Giac [A]  time = 1.06881, size = 7, normalized size = 1.75 \begin{align*} \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2*x^2-x+2)/(x^4-5*x^2+4),x, algorithm="giac")

[Out]

log(abs(x + 2))