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

Optimal. Leaf size=18 \[ \log \left (\frac {12 e^{-x^2} x}{2-x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1593, 1620} \begin {gather*} -x^2-\log (2-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2 - Log[2 - x] + Log[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 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.44 \begin {gather*} -2 \left (\frac {x^2}{2}+\frac {1}{2} \log (2-x)-\frac {\log (x)}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*(x^2/2 + Log[2 - x]/2 - Log[x]/2)

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fricas [A]  time = 0.66, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} - \log \left (x - 2\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - log(x - 2) + log(x)

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giac [A]  time = 0.23, size = 16, normalized size = 0.89 \begin {gather*} -x^{2} - \log \left ({\left | x - 2 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - log(abs(x - 2)) + log(abs(x))

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maple [A]  time = 0.40, size = 15, normalized size = 0.83

method	result	size
default	\(-x^{2}+\ln \relax (x )-\ln \left (x -2\right )\)	\(15\)
norman	\(-x^{2}+\ln \relax (x )-\ln \left (x -2\right )\)	\(15\)
risch	\(-x^{2}+\ln \relax (x )-\ln \left (x -2\right )\)	\(15\)
meijerg	\(\ln \relax (x )-\ln \relax (2)+i \pi -\ln \left (1-\frac {x}{2}\right )-\frac {2 x \left (\frac {3 x}{2}+6\right )}{3}+4 x\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2+ln(x)-ln(x-2)

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maxima [A]  time = 0.45, size = 14, normalized size = 0.78 \begin {gather*} -x^{2} - \log \left (x - 2\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - log(x - 2) + log(x)

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mupad [B]  time = 1.37, size = 12, normalized size = 0.67 \begin {gather*} 2\,\mathrm {atanh}\left (x-1\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh(x - 1) - x^2

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sympy [A]  time = 0.09, size = 10, normalized size = 0.56 \begin {gather*} - x^{2} + \log {\relax (x )} - \log {\left (x - 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + log(x) - log(x - 2)

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