Optimal. Leaf size=54 \[ \frac {1}{8 a^4 \left (a^4+x^4\right )^2}+\frac {1}{4 a^8 \left (a^4+x^4\right )}+\frac {\log (x)}{a^{12}}-\frac {\log \left (a^4+x^4\right )}{4 a^{12}} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46}
\begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^4+x^4\right )^3} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \left (a^4+x\right )^3} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{a^{12} x}-\frac {1}{a^4 \left (a^4+x\right )^3}-\frac {1}{a^8 \left (a^4+x\right )^2}-\frac {1}{a^{12} \left (a^4+x\right )}\right ) \, dx,x,x^4\right )\\ &=\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 (x)}{a^{12}}-\frac {\log \left (a^4+x^4\right )}{4 a^{12}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.85 \begin {gather*} \frac {\frac {3 a^8+2 a^4 x^4}{\left (a^4+x^4\right )^2}+8 \log (x)-2 \log \left (a^4+x^4\right )}{8 a^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 52, normalized size = 0.96
method | result | size |
norman | \(\frac {\frac {3}{8 a^{4}}+\frac {x^{4}}{4 a^{8}}}{\left (a^{4}+x^{4}\right )^{2}}+\frac {\ln \left (x \right )}{a^{12}}-\frac {\ln \left (a^{4}+x^{4}\right )}{4 a^{12}}\) | \(45\) |
risch | \(\frac {\frac {3}{8 a^{4}}+\frac {x^{4}}{4 a^{8}}}{\left (a^{4}+x^{4}\right )^{2}}+\frac {\ln \left (x \right )}{a^{12}}-\frac {\ln \left (a^{4}+x^{4}\right )}{4 a^{12}}\) | \(45\) |
default | \(-\frac {-\frac {a^{4}}{2 \left (a^{4}+x^{4}\right )}+\frac {\ln \left (a^{4}+x^{4}\right )}{2}-\frac {a^{8}}{4 \left (a^{4}+x^{4}\right )^{2}}}{2 a^{12}}+\frac {\ln \left (x \right )}{a^{12}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.46, size = 57, normalized size = 1.06 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 81, normalized size = 1.50 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 51, normalized size = 0.94 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 56, normalized size = 1.04 \begin {gather*} -\frac {\log \left (a^{4} + x^{4}\right )}{4 \, a^{12}} + \frac {\log \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 52, normalized size = 0.96 \begin {gather*} \frac {\ln \left (x\right )}{a^{12}}+\frac {\frac {3}{8\,a^4}+\frac {x^4}{4\,a^8}}{a^8+2\,a^4\,x^4+x^8}-\frac {\ln \left (a^4+x^4\right )}{4\,a^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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