Optimal. Leaf size=22 \[ \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31}
\begin {gather*} \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^5+x^5\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x \left (a^5+x\right )} \, dx,x,x^5\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^5\right )}{5 a^5}-\frac {\text {Subst}\left (\int \frac {1}{a^5+x} \, dx,x,x^5\right )}{5 a^5}\\ &=\frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs.
\(2(20)=40\).
time = 0.14, size = 49, normalized size = 2.23
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (a^{5}+x^{5}\right )}{5 a^{5}}\) | \(21\) |
default | \(\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (a +x \right )}{5 a^{5}}-\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}\) | \(49\) |
norman | \(\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (a +x \right )}{5 a^{5}}-\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.28, size = 23, normalized size = 1.05 \begin {gather*} -\frac {\log \left (a^{5} + x^{5}\right )}{5 \, a^{5}} + \frac {\log \left (x^{5}\right )}{5 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\log \left (a^{5} + x^{5}\right ) - 5 \, \log \left (x\right )}{5 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} \frac {\log {\left (x \right )}}{a^{5}} - \frac {\log {\left (a^{5} + x^{5} \right )}}{5 a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 22, normalized size = 1.00 \begin {gather*} -\frac {\log \left ({\left | a^{5} + x^{5} \right |}\right )}{5 \, a^{5}} + \frac {\log \left ({\left | x \right |}\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\ln \left (a^5+x^5\right )-5\,\ln \left (x\right )}{5\,a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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