3.173 \(\int \frac{1}{x \left (a^4+x^4\right )^3} \, dx\)

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]

1/(8*a^4*(a^4 + x^4)^2) + 1/(4*a^8*(a^4 + x^4)) + Log[x]/a^12 - Log[a^4 + x^4]/(
4*a^12)

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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]

1/(8*a^4*(a^4 + x^4)^2) + 1/(4*a^8*(a^4 + x^4)) + Log[x]/a^12 - Log[a^4 + x^4]/(
4*a^12)

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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]

1/(8*a**4*(a**4 + x**4)**2) + 1/(4*a**8*(a**4 + x**4)) + log(x**4)/(4*a**12) - l
og(a**4 + x**4)/(4*a**12)

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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]

((3*a^8 + 2*a^4*x^4)/(a^4 + x^4)^2 + 8*Log[x] - 2*Log[a^4 + x^4])/(8*a^12)

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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]

1/8/a^4/(a^4+x^4)^2+1/4/a^8/(a^4+x^4)+ln(x)/a^12-1/4*ln(a^4+x^4)/a^12

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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]

1/8*(3*a^4 + 2*x^4)/(a^16 + 2*a^12*x^4 + a^8*x^8) - 1/4*log(a^4 + x^4)/a^12 + 1/
4*log(x^4)/a^12

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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]

1/8*(3*a^8 + 2*a^4*x^4 - 2*(a^8 + 2*a^4*x^4 + x^8)*log(a^4 + x^4) + 8*(a^8 + 2*a
^4*x^4 + x^8)*log(x))/(a^20 + 2*a^16*x^4 + a^12*x^8)

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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]

(3*a**4 + 2*x**4)/(8*a**16 + 16*a**12*x**4 + 8*a**8*x**8) + log(x)/a**12 - log(a
**4 + x**4)/(4*a**12)

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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]

-1/4*ln(a^4 + x^4)/a^12 + 1/4*ln(x^4)/a^12 + 1/8*(6*a^8 + 8*a^4*x^4 + 3*x^8)/((a
^4 + x^4)^2*a^12)