3.33 \(\int \frac{1}{x^2 (x-x^3)} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{2 x^2}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]

[Out]

-1/(2*x^2) + Log[x] - Log[1 - x^2]/2

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Rubi [A]  time = 0.0146911, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 266, 44} \[ -\frac{1}{2 x^2}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(x - x^3)),x]

[Out]

-1/(2*x^2) + Log[x] - 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A]  time = 0.0034872, size = 22, normalized size = 1. \[ -\frac{1}{2 x^2}-\frac{1}{2} \log \left (1-x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(x - x^3)),x]

[Out]

-1/(2*x^2) + Log[x] - Log[1 - x^2]/2

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Maple [A]  time = 0.007, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,{x}^{2}}}+\ln \left ( x \right ) -{\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(1/x^2/(-x^3+x),x)

[Out]

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

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Maxima [A]  time = 1.04656, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \, x^{2}} - \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{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^2/(-x^3+x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.385, size = 65, normalized size = 2.95 \begin{align*} -\frac{x^{2} \log \left (x^{2} - 1\right ) - 2 \, x^{2} \log \left (x\right ) + 1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.096854, size = 17, normalized size = 0.77 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} - 1 \right )}}{2} - \frac{1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x**2 - 1)/2 - 1/(2*x**2)

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Giac [A]  time = 1.27046, size = 35, normalized size = 1.59 \begin{align*} -\frac{x^{2} + 1}{2 \, x^{2}} + \frac{1}{2} \, \log \left (x^{2}\right ) - \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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