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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]