3.4.29 \(\int \frac {x}{-a^4+x^4} \, dx\) [329]

Optimal. Leaf size=15 \[ -\frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]

[Out]

-1/2*arctanh(x^2/a^2)/a^2

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 213} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-a^4 + x^4),x]

[Out]

-1/2*ArcTanh[x^2/a^2]/a^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}{-a^4+x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{-a^4+x^2} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-a^4 + x^4),x]

[Out]

-1/2*ArcTanh[x^2/a^2]/a^2

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Mathics [A]
time = 2.16, size = 26, normalized size = 1.73 \begin {gather*} \frac {\text {Log}\left [-a^2+x^2\right ]-\text {Log}\left [a^2+x^2\right ]}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x/(-a^4 + x^4),x]')

[Out]

(Log[-a ^ 2 + x ^ 2] - Log[a ^ 2 + x ^ 2]) / (4 a ^ 2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(13)=26\).
time = 0.04, size = 30, normalized size = 2.00

method	result	size
default	\(-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{2}}+\frac {\ln \left (a^{2}-x^{2}\right )}{4 a^{2}}\)	\(30\)
risch	\(\frac {\ln \left (-a^{2}+x^{2}\right )}{4 a^{2}}-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{2}}\)	\(30\)
norman	\(\frac {\ln \left (a -x \right )}{4 a^{2}}+\frac {\ln \left (a +x \right )}{4 a^{2}}-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{2}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (13) = 26\).
time = 0.26, size = 29, normalized size = 1.93 \begin {gather*} -\frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} + \frac {\log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 26, normalized size = 1.73 \begin {gather*} -\frac {\log \left (a^{2} + x^{2}\right ) - \log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 22, normalized size = 1.47 \begin {gather*} \frac {\frac {\log {\left (- a^{2} + x^{2} \right )}}{4} - \frac {\log {\left (a^{2} + x^{2} \right )}}{4}}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-a**4+x**4),x)

[Out]

(log(-a**2 + x**2)/4 - log(a**2 + x**2)/4)/a**2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
time = 0.00, size = 39, normalized size = 2.60 \begin {gather*} \frac {\frac {\frac {1}{2} \ln \left |x^{2}-a^{2}\right |}{a^{2}}-\frac {\frac {1}{2} \ln \left (x^{2}+a^{2}\right )}{a^{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-a^4+x^4),x)

[Out]

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

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Mupad [B]
time = 0.06, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x^2}{a^2}\right )}{2\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/(a^4 - x^4),x)

[Out]

-atanh(x^2/a^2)/(2*a^2)

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