Optimal. Leaf size=54 \[ \frac{\log (x)}{a^{12}}+\frac{1}{8 a^4 \left (a^4+x^4\right )^2}-\frac{\log \left (a^4+x^4\right )}{4 a^{12}}+\frac{1}{4 a^8 \left (a^4+x^4\right )} \]
[Out]
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Rubi [A] time = 0.0616274, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\log (x)}{a^{12}}+\frac{1}{8 a^4 \left (a^4+x^4\right )^2}-\frac{\log \left (a^4+x^4\right )}{4 a^{12}}+\frac{1}{4 a^8 \left (a^4+x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a^4 + x^4)^3),x]
[Out]
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Rubi in Sympy [A] time = 4.48663, size = 51, normalized size = 0.94 \[ \frac{1}{8 a^{4} \left (a^{4} + x^{4}\right )^{2}} + \frac{1}{4 a^{8} \left (a^{4} + x^{4}\right )} + \frac{\log{\left (x^{4} \right )}}{4 a^{12}} - \frac{\log{\left (a^{4} + x^{4} \right )}}{4 a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a**4+x**4)**3,x)
[Out]
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Mathematica [A] time = 0.0402737, size = 46, normalized size = 0.85 \[ \frac{-2 \log \left (a^4+x^4\right )+\frac{3 a^8+2 a^4 x^4}{\left (a^4+x^4\right )^2}+8 \log (x)}{8 a^{12}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a^4 + x^4)^3),x]
[Out]
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Maple [A] time = 0.023, size = 49, normalized size = 0.9 \[{\frac{1}{8\,{a}^{4} \left ({a}^{4}+{x}^{4} \right ) ^{2}}}+{\frac{1}{4\,{a}^{8} \left ({a}^{4}+{x}^{4} \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{12}}}-{\frac{\ln \left ({a}^{4}+{x}^{4} \right ) }{4\,{a}^{12}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(a^4+x^4)^3,x)
[Out]
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Maxima [A] time = 1.4597, size = 77, normalized size = 1.43 \[ \frac{3 \, a^{4} + 2 \, x^{4}}{8 \,{\left (a^{16} + 2 \, a^{12} x^{4} + a^{8} x^{8}\right )}} - \frac{\log \left (a^{4} + x^{4}\right )}{4 \, a^{12}} + \frac{\log \left (x^{4}\right )}{4 \, a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a^4 + x^4)^3*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227378, size = 109, normalized size = 2.02 \[ \frac{3 \, a^{8} + 2 \, a^{4} x^{4} - 2 \,{\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )} \log \left (a^{4} + x^{4}\right ) + 8 \,{\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )} \log \left (x\right )}{8 \,{\left (a^{20} + 2 \, a^{16} x^{4} + a^{12} x^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a^4 + x^4)^3*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.93131, size = 51, normalized size = 0.94 \[ \frac{3 a^{4} + 2 x^{4}}{8 a^{16} + 16 a^{12} x^{4} + 8 a^{8} x^{8}} + \frac{\log{\left (x \right )}}{a^{12}} - \frac{\log{\left (a^{4} + x^{4} \right )}}{4 a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(a**4+x**4)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.204906, size = 76, normalized size = 1.41 \[ -\frac{{\rm ln}\left (a^{4} + x^{4}\right )}{4 \, a^{12}} + \frac{{\rm ln}\left (x^{4}\right )}{4 \, a^{12}} + \frac{6 \, a^{8} + 8 \, a^{4} x^{4} + 3 \, x^{8}}{8 \,{\left (a^{4} + x^{4}\right )}^{2} a^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a^4 + x^4)^3*x),x, algorithm="giac")
[Out]