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

Optimal. Leaf size=2 \[ \tanh ^{-1}(x) \]

[Out]

ArcTanh[x]

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Rubi [A]  time = 0.0039866, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1584, 206} \[ \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x/(x - x^3),x]

[Out]

ArcTanh[x]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [B]  time = 0.0020419, size = 19, normalized size = 9.5 \[ \frac{1}{2} \log (x+1)-\frac{1}{2} \log (1-x) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(x - x^3),x]

[Out]

-Log[1 - x]/2 + Log[1 + x]/2

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Maple [A]  time = 0.001, size = 3, normalized size = 1.5 \begin{align*}{\it Artanh} \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(x)

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Maxima [B]  time = 1.07402, size = 18, normalized size = 9. \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/(-x^3+x),x, algorithm="maxima")

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

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Fricas [B]  time = 1.60764, size = 45, normalized size = 22.5 \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/(-x^3+x),x, algorithm="fricas")

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

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Sympy [B]  time = 0.082663, size = 12, normalized size = 6. \begin{align*} - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1)/2 + log(x + 1)/2

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Giac [B]  time = 1.23626, size = 20, normalized size = 10. \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/(-x^3+x),x, algorithm="giac")

[Out]

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