3.2.26 \(\int \frac {1}{-x+x^3} \, dx\) [126]

Optimal. Leaf size=17 \[ -\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1607, 272, 36, 31, 29} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x + x^3)^(-1),x]

[Out]

-Log[x] + Log[1 - x^2]/2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

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 1607

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^3)^(-1),x]

[Out]

-Log[x] + Log[1 - x^2]/2

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Mathics [A]
time = 1.64, size = 13, normalized size = 0.76 \begin {gather*} -\text {Log}\left [x\right ]+\frac {\text {Log}\left [-1+x^2\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^3 - x),x]')

[Out]

-Log[x] + Log[-1 + x ^ 2] / 2

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Maple [A]
time = 0.07, size = 18, normalized size = 1.06

method	result	size
risch	\(-\ln \left (x \right )+\frac {\ln \left (x^{2}-1\right )}{2}\)	\(14\)
default	\(-\ln \left (x \right )+\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(18\)
norman	\(-\ln \left (x \right )+\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(18\)
meijerg	\(-\ln \left (x \right )-\frac {i \pi }{2}+\frac {\ln \left (-x^{2}+1\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3-x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 - 1) - log(x)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.59 \begin {gather*} - \log {\left (x \right )} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x**2 - 1)/2

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Giac [A]
time = 0.00, size = 19, normalized size = 1.12 \begin {gather*} -\frac {\ln \left (x^{2}\right )}{2}+\frac {\ln \left |x^{2}-1\right |}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-x),x)

[Out]

-1/2*log(x^2) + 1/2*log(abs(x^2 - 1))

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Mupad [B]
time = 0.11, size = 13, normalized size = 0.76 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2}-\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x - x^3),x)

[Out]

log(x^2 - 1)/2 - log(x)

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