3.127 \(\int \frac{1}{a^4-x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^3}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^3} \]

[Out]

ArcTan[x/a]/(2*a^3) + ArcTanh[x/a]/(2*a^3)

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Rubi [A]  time = 0.0195542, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^3}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^4 - x^4)^(-1),x]

[Out]

ArcTan[x/a]/(2*a^3) + ArcTanh[x/a]/(2*a^3)

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Rubi in Sympy [A]  time = 1.80589, size = 19, normalized size = 0.7 \[ \frac{\operatorname{atan}{\left (\frac{x}{a} \right )}}{2 a^{3}} + \frac{\operatorname{atanh}{\left (\frac{x}{a} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a**4-x**4),x)

[Out]

atan(x/a)/(2*a**3) + atanh(x/a)/(2*a**3)

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Mathematica [A]  time = 0.00701467, size = 38, normalized size = 1.41 \[ -\frac{\log (a-x)}{4 a^3}+\frac{\log (a+x)}{4 a^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^4 - x^4)^(-1),x]

[Out]

ArcTan[x/a]/(2*a^3) - Log[a - x]/(4*a^3) + Log[a + x]/(4*a^3)

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Maple [A]  time = 0.011, size = 33, normalized size = 1.2 \[{\frac{\ln \left ( a+x \right ) }{4\,{a}^{3}}}+{\frac{1}{2\,{a}^{3}}\arctan \left ({\frac{x}{a}} \right ) }-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a^4-x^4),x)

[Out]

1/4*ln(a+x)/a^3+1/2*arctan(x/a)/a^3-1/4/a^3*ln(-a+x)

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Maxima [A]  time = 1.50392, size = 43, normalized size = 1.59 \[ \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{3}} + \frac{\log \left (a + x\right )}{4 \, a^{3}} - \frac{\log \left (-a + x\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a^4 - x^4),x, algorithm="maxima")

[Out]

1/2*arctan(x/a)/a^3 + 1/4*log(a + x)/a^3 - 1/4*log(-a + x)/a^3

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Fricas [A]  time = 0.203108, size = 35, normalized size = 1.3 \[ \frac{2 \, \arctan \left (\frac{x}{a}\right ) + \log \left (a + x\right ) - \log \left (-a + x\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a^4 - x^4),x, algorithm="fricas")

[Out]

1/4*(2*arctan(x/a) + log(a + x) - log(-a + x))/a^3

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Sympy [A]  time = 0.16196, size = 37, normalized size = 1.37 \[ - \frac{\frac{\log{\left (- a + x \right )}}{4} - \frac{\log{\left (a + x \right )}}{4} + \frac{i \log{\left (- i a + x \right )}}{4} - \frac{i \log{\left (i a + x \right )}}{4}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a**4-x**4),x)

[Out]

-(log(-a + x)/4 - log(a + x)/4 + I*log(-I*a + x)/4 - I*log(I*a + x)/4)/a**3

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GIAC/XCAS [A]  time = 0.202625, size = 46, normalized size = 1.7 \[ \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{3}} + \frac{{\rm ln}\left ({\left | a + x \right |}\right )}{4 \, a^{3}} - \frac{{\rm ln}\left ({\left | -a + x \right |}\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a^4 - x^4),x, algorithm="giac")

[Out]

1/2*arctan(x/a)/a^3 + 1/4*ln(abs(a + x))/a^3 - 1/4*ln(abs(-a + x))/a^3