3.256 \(\int \frac{x}{-1+x^4} \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{2} \tanh ^{-1}\left (x^2\right ) \]

[Out]

-ArcTanh[x^2]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0029564, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {275, 207} \[ -\frac{1}{2} \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^4),x]

[Out]

-ArcTanh[x^2]/2

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{x}{-1+x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{2} \tanh ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [B]  time = 0.0027506, size = 23, normalized size = 2.88 \[ \frac{1}{4} \log \left (1-x^2\right )-\frac{1}{4} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^4),x]

[Out]

Log[1 - x^2]/4 - Log[1 + x^2]/4

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 22, normalized size = 2.8 \begin{align*}{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{\ln \left ( 1+x \right ) }{4}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4-1),x)

[Out]

1/4*ln(-1+x)+1/4*ln(1+x)-1/4*ln(x^2+1)

________________________________________________________________________________________

Maxima [B]  time = 0.941232, size = 23, normalized size = 2.88 \begin{align*} -\frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4-1),x, algorithm="maxima")

[Out]

-1/4*log(x^2 + 1) + 1/4*log(x^2 - 1)

________________________________________________________________________________________

Fricas [B]  time = 1.57931, size = 51, normalized size = 6.38 \begin{align*} -\frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4-1),x, algorithm="fricas")

[Out]

-1/4*log(x^2 + 1) + 1/4*log(x^2 - 1)

________________________________________________________________________________________

Sympy [B]  time = 0.089208, size = 15, normalized size = 1.88 \begin{align*} \frac{\log{\left (x^{2} - 1 \right )}}{4} - \frac{\log{\left (x^{2} + 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**4-1),x)

[Out]

log(x**2 - 1)/4 - log(x**2 + 1)/4

________________________________________________________________________________________

Giac [B]  time = 1.08494, size = 24, normalized size = 3. \begin{align*} -\frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4-1),x, algorithm="giac")

[Out]

-1/4*log(x^2 + 1) + 1/4*log(abs(x^2 - 1))