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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, size = 17, normalized size = 1.00 \[ \log (1-x)+2 \log (x)-3 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 4*x)/(-x + x^3),x]

[Out]

Log[1 - x] + 2*Log[x] - 3*Log[1 + x]

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 15, normalized size = 0.88 \[ -3 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) + 2 \, \log \relax (x) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 18, normalized size = 1.06 \[ -3 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ 2 \ln \relax (x )+\ln \left (x -1\right )-3 \ln \left (x +1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 15, normalized size = 0.88 \[ -3 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) + 2 \, \log \relax (x) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 15, normalized size = 0.88 \[ \ln \left (x-1\right )-3\,\ln \left (x+1\right )+2\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x - 2)/(x - x^3),x)

[Out]

log(x - 1) - 3*log(x + 1) + 2*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 15, normalized size = 0.88 \[ 2 \log {\relax (x )} + \log {\left (x - 1 \right )} - 3 \log {\left (x + 1 \right )} \]

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)

________________________________________________________________________________________