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

Optimal. Leaf size=17 \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

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Rubi [A]  time = 0.0181711, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1593, 801} \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(-2 + 4*x)/(-x + x^3),x]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

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

Mathematica [A]  time = 0.0055575, size = 17, normalized size = 1. \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 4*x)/(-x + x^3),x]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

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Maple [A]  time = 0.004, size = 16, normalized size = 0.9 \begin{align*} \ln \left ( x-1 \right ) +2\,\ln \left ( x \right ) -3\,\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.97238, size = 20, normalized size = 1.18 \begin{align*} -3 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*log(x + 1) + log(x - 1) + 2*log(x)

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Fricas [A]  time = 0.999472, size = 53, normalized size = 3.12 \begin{align*} -3 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*log(x + 1) + log(x - 1) + 2*log(x)

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Sympy [A]  time = 0.110978, size = 15, normalized size = 0.88 \begin{align*} 2 \log{\left (x \right )} + \log{\left (x - 1 \right )} - 3 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x) + log(x - 1) - 3*log(x + 1)

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Giac [A]  time = 1.18553, size = 24, normalized size = 1.41 \begin{align*} -3 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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