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

Optimal. Leaf size=25 \[ \frac{x^2}{2}-\frac{1}{x}-2 \log (1-x)+2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0356623, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1593, 1620} \[ \frac{x^2}{2}-\frac{1}{x}-2 \log (1-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 - x - x^3 + x^4)/(-x^2 + x^3),x]

[Out]

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

Mathematica [A]  time = 0.0061798, size = 25, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{x}-2 \log (1-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 - x - x^3 + x^4)/(-x^2 + x^3),x]

[Out]

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

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Maple [A]  time = 0.007, size = 22, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}-2\,\ln \left ( x-1 \right ) -{x}^{-1}+2\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02026, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{x} - 2 \, \log \left (x - 1\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.26056, size = 63, normalized size = 2.52 \begin{align*} \frac{x^{3} - 4 \, x \log \left (x - 1\right ) + 4 \, x \log \left (x\right ) - 2}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^3 - 4*x*log(x - 1) + 4*x*log(x) - 2)/x

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Sympy [A]  time = 0.094558, size = 19, normalized size = 0.76 \begin{align*} \frac{x^{2}}{2} + 2 \log{\left (x \right )} - 2 \log{\left (x - 1 \right )} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20626, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{x} - 2 \, \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((x^4-x^3-x-1)/(x^3-x^2),x, algorithm="giac")

[Out]

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