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 = 0.018065, 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 \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a^4 - x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.78894, size = 12, normalized size = 0.8 \[ \frac{\operatorname{atanh}{\left (\frac{x^{2}}{a^{2}} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a**4-x**4),x)

[Out]

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

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Mathematica [A]  time = 0.00490054, 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 = 0.008, size = 30, normalized size = 2. \[{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{4\,{a}^{2}}}-{\frac{\ln \left ( -{a}^{2}+{x}^{2} \right ) }{4\,{a}^{2}}} \]

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 [A]  time = 1.32774, size = 39, normalized size = 2.6 \[ \frac{\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac{\log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \]

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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Fricas [A]  time = 0.19856, size = 35, normalized size = 2.33 \[ \frac{\log \left (a^{2} + x^{2}\right ) - \log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \]

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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Sympy [A]  time = 0.178762, size = 24, normalized size = 1.6 \[ - \frac{\frac{\log{\left (- a^{2} + x^{2} \right )}}{4} - \frac{\log{\left (a^{2} + x^{2} \right )}}{4}}{a^{2}} \]

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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GIAC/XCAS [A]  time = 0.203549, size = 41, normalized size = 2.73 \[ \frac{{\rm ln}\left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac{{\rm ln}\left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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