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

Optimal. Leaf size=6 \[ \tanh ^{-1}(x)-x \]

[Out]

-x + ArcTanh[x]

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Rubi [A]  time = 0.0080615, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 321, 206} \[ \tanh ^{-1}(x)-x \]

Antiderivative was successfully verified.

[In]

Int[x^3/(x - x^3),x]

[Out]

-x + 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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])

Rubi steps

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

Mathematica [B]  time = 0.0030891, size = 22, normalized size = 3.67 \[ -x-\frac{1}{2} \log (1-x)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(x - x^3),x]

[Out]

-x - Log[1 - x]/2 + Log[1 + x]/2

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Maple [B]  time = 0.001, size = 17, normalized size = 2.8 \begin{align*} -x-{\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(x^3/(-x^3+x),x)

[Out]

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

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Maxima [B]  time = 1.0418, size = 22, normalized size = 3.67 \begin{align*} -x + \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^3/(-x^3+x),x, algorithm="maxima")

[Out]

-x + 1/2*log(x + 1) - 1/2*log(x - 1)

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Fricas [B]  time = 1.68595, size = 51, normalized size = 8.5 \begin{align*} -x + \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^3/(-x^3+x),x, algorithm="fricas")

[Out]

-x + 1/2*log(x + 1) - 1/2*log(x - 1)

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Sympy [B]  time = 0.083079, size = 14, normalized size = 2.33 \begin{align*} - x - \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**3/(-x**3+x),x)

[Out]

-x - log(x - 1)/2 + log(x + 1)/2

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Giac [B]  time = 1.23107, size = 24, normalized size = 4. \begin{align*} -x + \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^3/(-x^3+x),x, algorithm="giac")

[Out]

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