3.3.81 \(\int \frac {-3+2 x}{-x^2+x^3} \, dx\) [281]

Optimal. Leaf size=16 \[ -\frac {3}{x}-\log (1-x)+\log (x) \]

[Out]

-3/x-ln(1-x)+ln(x)

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1607, 78} \begin {gather*} -\frac {3}{x}-\log (1-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x)/(-x^2 + x^3),x]

[Out]

-3/x - Log[1 - x] + Log[x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 1607

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3}{x}-\log (1-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*x)/(-x^2 + x^3),x]

[Out]

-3/x - Log[1 - x] + Log[x]

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Maple [A]
time = 0.42, size = 15, normalized size = 0.94

method	result	size
default	\(-\ln \left (x -1\right )+\ln \left (x \right )-\frac {3}{x}\)	\(15\)
norman	\(-\ln \left (x -1\right )+\ln \left (x \right )-\frac {3}{x}\)	\(15\)
risch	\(-\ln \left (x -1\right )+\ln \left (x \right )-\frac {3}{x}\)	\(15\)
meijerg	\(-\ln \left (1-x \right )+\ln \left (x \right )+i \pi -\frac {3}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x-1)+ln(x)-3/x

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} -\frac {3}{x} - \log \left (x - 1\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/x - log(x - 1) + log(x)

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Fricas [A]
time = 2.22, size = 18, normalized size = 1.12 \begin {gather*} -\frac {x \log \left (x - 1\right ) - x \log \left (x\right ) + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*log(x - 1) - x*log(x) + 3)/x

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Sympy [A]
time = 0.03, size = 10, normalized size = 0.62 \begin {gather*} \log {\left (x \right )} - \log {\left (x - 1 \right )} - \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x - 1) - 3/x

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Giac [A]
time = 0.50, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3}{x} - \log \left ({\left | x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/x - log(abs(x - 1)) + log(abs(x))

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Mupad [B]
time = 0.13, size = 14, normalized size = 0.88 \begin {gather*} 2\,\mathrm {atanh}\left (2\,x-1\right )-\frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 3)/(x^2 - x^3),x)

[Out]

2*atanh(2*x - 1) - 3/x

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