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

Optimal. Leaf size=21 \[ \frac{1}{2 x^2}+\frac{1}{x}+\log (1-x)-\log (x) \]

[Out]

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

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Rubi [A]  time = 0.0111957, antiderivative size = 21, 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} \[ \frac{1}{2 x^2}+\frac{1}{x}+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-x^3 + x^4)^(-1),x]

[Out]

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

Mathematica [A]  time = 0.0019447, size = 21, normalized size = 1. \[ \frac{1}{2 x^2}+\frac{1}{x}+\log (1-x)-\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-x^3 + x^4)^(-1),x]

[Out]

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

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Maple [A]  time = 0.006, size = 18, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{x}^{2}}}+{x}^{-1}-\ln \left ( x \right ) +\ln \left ( -1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.923482, size = 26, normalized size = 1.24 \begin{align*} \frac{2 \, x + 1}{2 \, x^{2}} + \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86409, size = 72, normalized size = 3.43 \begin{align*} \frac{2 \, x^{2} \log \left (x - 1\right ) - 2 \, x^{2} \log \left (x\right ) + 2 \, x + 1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.097411, size = 17, normalized size = 0.81 \begin{align*} - \log{\left (x \right )} + \log{\left (x - 1 \right )} + \frac{2 x + 1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05542, size = 28, normalized size = 1.33 \begin{align*} \frac{2 \, x + 1}{2 \, x^{2}} + \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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