3.128 \(\int \frac{x}{a^4-x^4} \, dx\)

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

[Out]

ArcTanh[x^2/a^2]/(2*a^2)

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Rubi [A]  time = 0.0065664, antiderivative size = 15, normalized size of antiderivative = 1., 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 = {275, 206} \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a^4 - x^4),x]

[Out]

ArcTanh[x^2/a^2]/(2*a^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 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}{a^4-x^4} \, dx &=\frac{1}{2} \operatorname{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*}

Mathematica [A]  time = 0.0034625, size = 15, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a^4 - x^4),x]

[Out]

ArcTanh[x^2/a^2]/(2*a^2)

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Maple [B]  time = 0.004, size = 30, normalized size = 2. \begin{align*}{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{4\,{a}^{2}}}-{\frac{\ln \left ( -{a}^{2}+{x}^{2} \right ) }{4\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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]  time = 0.921784, size = 39, normalized size = 2.6 \begin{align*} \frac{\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac{\log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \end{align*}

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 = 1.91185, size = 59, normalized size = 3.93 \begin{align*} \frac{\log \left (a^{2} + x^{2}\right ) - \log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \end{align*}

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 = 0.13667, size = 24, normalized size = 1.6 \begin{align*} - \frac{\frac{\log{\left (- a^{2} + x^{2} \right )}}{4} - \frac{\log{\left (a^{2} + x^{2} \right )}}{4}}{a^{2}} \end{align*}

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]  time = 1.05916, size = 41, normalized size = 2.73 \begin{align*} \frac{\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac{\log \left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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