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

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{2 x^2}-\frac {1}{2} \log \left (1-x^2\right )+\log (x) \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{2 x^2}-\frac {1}{2} \log \left (1-x^2\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(x - x^3)),x]

[Out]

-1/2*1/x^2 + Log[x] - Log[1 - x^2]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (x-x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^2*(x - x^3)),x]

[Out]

IntegrateAlgebraic[1/(x^2*(x - x^3)), x]

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fricas [A]  time = 0.40, size = 24, normalized size = 1.09 \begin {gather*} -\frac {x^{2} \log \left (x^{2} - 1\right ) - 2 \, x^{2} \log \relax (x) + 1}{2 \, x^{2}} \end {gather*}

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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giac [A]  time = 0.15, size = 26, normalized size = 1.18 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} \ln \relax (x )-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}-\frac {1}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.29, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{2 \, x^{2}} - \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) + \log \relax (x) \end {gather*}

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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mupad [B]  time = 0.03, size = 16, normalized size = 0.73 \begin {gather*} \ln \relax (x)-\frac {\ln \left (x^2-1\right )}{2}-\frac {1}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x - x^3)),x)

[Out]

log(x) - log(x^2 - 1)/2 - 1/(2*x^2)

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sympy [A]  time = 0.11, size = 17, normalized size = 0.77 \begin {gather*} \log {\relax (x )} - \frac {\log {\left (x^{2} - 1 \right )}}{2} - \frac {1}{2 x^{2}} \end {gather*}

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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