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

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

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Rubi [A]  time = 0.01, 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 = {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 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])

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]

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*}

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

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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IntegrateAlgebraic [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(a^4 - x^4),x]

[Out]

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

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fricas [A]  time = 0.84, size = 26, normalized size = 1.73 \[ \frac {\log \left (a^{2} + x^{2}\right ) - \log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \]

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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giac [B]  time = 0.81, size = 30, normalized size = 2.00 \[ \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac {\log \left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{2}} \]

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

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maple [B]  time = 0.34, 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]  time = 0.42, size = 29, normalized size = 1.93 \[ \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac {\log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \]

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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mupad [B]  time = 0.05, size = 13, normalized size = 0.87 \[ \frac {\mathrm {atanh}\left (\frac {x^2}{a^2}\right )}{2\,a^2} \]

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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sympy [A]  time = 0.15, size = 24, normalized size = 1.60 \[ - \frac {\frac {\log {\left (- a^{2} + x^{2} \right )}}{4} - \frac {\log {\left (a^{2} + x^{2} \right )}}{4}}{a^{2}} \]

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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