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

Optimal. Leaf size=11 \[ \log (x+1)-\log (x+2) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1586, 616, 31} \begin {gather*} \log (x+1)-\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + x] - Log[2 + x]

Rule 31

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

Rule 616

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \log (x+1)-\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + x] - Log[2 + x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2-3 x+x^2}{4-5 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.22, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (x + 2\right ) + \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 2) + log(x + 1)

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giac [A]  time = 0.28, size = 13, normalized size = 1.18 \begin {gather*} -\log \left ({\left | x + 2 \right |}\right ) + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x + 2)) + log(abs(x + 1))

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maple [A]  time = 0.00, size = 12, normalized size = 1.09 \begin {gather*} -\ln \left (x +2\right )+\ln \left (x +1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x+1)-ln(x+2)

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maxima [A]  time = 0.43, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (x + 2\right ) + \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 2) + log(x + 1)

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mupad [B]  time = 0.08, size = 8, normalized size = 0.73 \begin {gather*} -2\,\mathrm {atanh}\left (2\,x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*atanh(2*x + 3)

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sympy [A]  time = 0.11, size = 8, normalized size = 0.73 \begin {gather*} \log {\left (x + 1 \right )} - \log {\left (x + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 1) - log(x + 2)

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