3.134 \(\int \frac{1}{-2 x^3+x^4} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8} \]

[Out]

1/(4*x^2) + 1/(4*x) + Log[2 - x]/8 - Log[x]/8

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Rubi [A]  time = 0.0090875, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 44} \[ \frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8} \]

Antiderivative was successfully verified.

[In]

Int[(-2*x^3 + x^4)^(-1),x]

[Out]

1/(4*x^2) + 1/(4*x) + Log[2 - x]/8 - Log[x]/8

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{-2 x^3+x^4} \, dx &=\int \frac{1}{(-2+x) x^3} \, dx\\ &=\int \left (\frac{1}{8 (-2+x)}-\frac{1}{2 x^3}-\frac{1}{4 x^2}-\frac{1}{8 x}\right ) \, dx\\ &=\frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8}\\ \end{align*}

Mathematica [A]  time = 0.0018443, size = 31, normalized size = 1. \[ \frac{1}{4 x^2}+\frac{1}{4 x}+\frac{1}{8} \log (2-x)-\frac{\log (x)}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^3 + x^4)^(-1),x]

[Out]

1/(4*x^2) + 1/(4*x) + Log[2 - x]/8 - Log[x]/8

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Maple [A]  time = 0.006, size = 22, normalized size = 0.7 \begin{align*}{\frac{1}{4\,{x}^{2}}}+{\frac{1}{4\,x}}-{\frac{\ln \left ( x \right ) }{8}}+{\frac{\ln \left ( -2+x \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4-2*x^3),x)

[Out]

1/4/x^2+1/4/x-1/8*ln(x)+1/8*ln(-2+x)

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Maxima [A]  time = 0.937358, size = 26, normalized size = 0.84 \begin{align*} \frac{x + 1}{4 \, x^{2}} + \frac{1}{8} \, \log \left (x - 2\right ) - \frac{1}{8} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-2*x^3),x, algorithm="maxima")

[Out]

1/4*(x + 1)/x^2 + 1/8*log(x - 2) - 1/8*log(x)

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Fricas [A]  time = 1.13007, size = 66, normalized size = 2.13 \begin{align*} \frac{x^{2} \log \left (x - 2\right ) - x^{2} \log \left (x\right ) + 2 \, x + 2}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-2*x^3),x, algorithm="fricas")

[Out]

1/8*(x^2*log(x - 2) - x^2*log(x) + 2*x + 2)/x^2

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Sympy [A]  time = 0.100686, size = 19, normalized size = 0.61 \begin{align*} - \frac{\log{\left (x \right )}}{8} + \frac{\log{\left (x - 2 \right )}}{8} + \frac{x + 1}{4 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4-2*x**3),x)

[Out]

-log(x)/8 + log(x - 2)/8 + (x + 1)/(4*x**2)

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Giac [A]  time = 1.06752, size = 28, normalized size = 0.9 \begin{align*} \frac{x + 1}{4 \, x^{2}} + \frac{1}{8} \, \log \left ({\left | x - 2 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-2*x^3),x, algorithm="giac")

[Out]

1/4*(x + 1)/x^2 + 1/8*log(abs(x - 2)) - 1/8*log(abs(x))