3.29 \(\int \frac{x^2}{x-x^3} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{2} \log \left (1-x^2\right ) \]

[Out]

-Log[1 - x^2]/2

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Rubi [A]  time = 0.0063917, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1584, 260} \[ -\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(x - x^3),x]

[Out]

-Log[1 - x^2]/2

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0018832, size = 12, normalized size = 1. \[ -\frac{1}{2} \log \left (1-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(x - x^3),x]

[Out]

-Log[1 - x^2]/2

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Maple [A]  time = 0.003, size = 14, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( -1+x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0234, size = 18, normalized size = 1.5 \begin{align*} -\frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x + 1) - 1/2*log(x - 1)

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Fricas [A]  time = 1.60973, size = 26, normalized size = 2.17 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x^2 - 1)

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Sympy [A]  time = 0.074027, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (x^{2} - 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x**2 - 1)/2

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Giac [A]  time = 1.27095, size = 20, normalized size = 1.67 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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