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

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

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Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, 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} \begin {gather*} \tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(x - x^3),x]

[Out]

ArcTanh[x]

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])

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

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Mathematica [B]  time = 0.00, size = 19, normalized size = 9.50 \begin {gather*} \frac {1}{2} \log (x+1)-\frac {1}{2} \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(x - x^3),x]

[Out]

-1/2*Log[1 - x] + Log[1 + x]/2

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/(x - x^3),x]

[Out]

IntegrateAlgebraic[x/(x - x^3), x]

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fricas [B]  time = 0.41, size = 13, normalized size = 6.50 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}

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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giac [B]  time = 0.17, size = 15, normalized size = 7.50 \begin {gather*} \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

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))

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maple [A]  time = 0.04, size = 3, normalized size = 1.50 \begin {gather*} \arctanh \relax (x ) \end {gather*}

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 = 1.28, size = 13, normalized size = 6.50 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}

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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mupad [B]  time = 0.03, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {atanh}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x - x^3),x)

[Out]

atanh(x)

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sympy [B]  time = 0.12, size = 12, normalized size = 6.00 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}

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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