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

Optimal. Leaf size=29 \[ -\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1) \]

[Out]

-Log[1 - x]/2 + Log[2 - x]/3 + Log[1 + x]/6

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Rubi [A]  time = 0.0207676, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1586, 2058} \[ -\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(4 - 5*x^2 + x^4),x]

[Out]

-Log[1 - x]/2 + Log[2 - x]/3 + Log[1 + x]/6

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 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{2+x}{4-5 x^2+x^4} \, dx &=\int \frac{1}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (\frac{1}{3 (-2+x)}-\frac{1}{2 (-1+x)}+\frac{1}{6 (1+x)}\right ) \, dx\\ &=-\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0067574, size = 29, normalized size = 1. \[ -\frac{1}{2} \log (1-x)+\frac{1}{3} \log (2-x)+\frac{1}{6} \log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x]/2 + Log[2 - x]/3 + Log[1 + x]/6

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Maple [A]  time = 0.006, size = 20, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ( x-2 \right ) }{3}}-{\frac{\ln \left ( x-1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*ln(1+x)+1/3*ln(x-2)-1/2*ln(x-1)

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Maxima [A]  time = 0.975499, size = 26, normalized size = 0.9 \begin{align*} \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{3} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*log(x + 1) - 1/2*log(x - 1) + 1/3*log(x - 2)

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Fricas [A]  time = 1.50185, size = 68, normalized size = 2.34 \begin{align*} \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{3} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*log(x + 1) - 1/2*log(x - 1) + 1/3*log(x - 2)

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Sympy [A]  time = 0.124361, size = 19, normalized size = 0.66 \begin{align*} \frac{\log{\left (x - 2 \right )}}{3} - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 2)/3 - log(x - 1)/2 + log(x + 1)/6

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Giac [A]  time = 1.08238, size = 30, normalized size = 1.03 \begin{align*} \frac{1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*log(abs(x + 1)) - 1/2*log(abs(x - 1)) + 1/3*log(abs(x - 2))