3.3.56 \(\int \frac {x}{-1+x^4} \, dx\) [256]

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

[Out]

-1/2*arctanh(x^2)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {281, 213} \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^4),x]

[Out]

-1/2*ArcTanh[x^2]

Rule 213

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

Rule 281

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]

Rubi steps

\begin {align*} \int \frac {x}{-1+x^4} \, dx &=\frac {1}{2} \text {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] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).
time = 0.00, size = 23, normalized size = 2.88 \begin {gather*} \frac {1}{4} \log \left (1-x^2\right )-\frac {1}{4} \log \left (1+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^4),x]

[Out]

Log[1 - x^2]/4 - Log[1 + x^2]/4

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs. \(2(6)=12\).
time = 0.03, size = 22, normalized size = 2.75

method	result	size
meijerg	\(-\frac {\arctanh \left (x^{2}\right )}{2}\)	\(7\)
risch	\(-\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\ln \left (x^{2}-1\right )}{4}\)	\(18\)
default	\(\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\)	\(22\)
norman	\(\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4-1),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 4.49, size = 17, normalized size = 2.12 \begin {gather*} -\frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.94, size = 17, normalized size = 2.12 \begin {gather*} -\frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
time = 0.03, size = 15, normalized size = 1.88 \begin {gather*} \frac {\log {\left (x^{2} - 1 \right )}}{4} - \frac {\log {\left (x^{2} + 1 \right )}}{4} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (6) = 12\).
time = 0.46, size = 18, normalized size = 2.25 \begin {gather*} -\frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 6, normalized size = 0.75 \begin {gather*} -\frac {\mathrm {atanh}\left (x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 - 1),x)

[Out]

-atanh(x^2)/2

________________________________________________________________________________________