Optimal. Leaf size=201 \[ -\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a}-\frac {\log (a+x)}{5 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a} \]
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Rubi [A]
time = 0.22, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {299, 648, 632,
210, 642, 31} \begin {gather*} \frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a}-\frac {\log (a+x)}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 299
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^3}{a^5+x^5} \, dx &=\frac {2 \int \frac {\frac {1}{4} \left (1+\sqrt {5}\right ) a-\frac {1}{4} \left (-1+\sqrt {5}\right ) x}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{5 a}+\frac {2 \int \frac {\frac {1}{4} \left (1-\sqrt {5}\right ) a-\frac {1}{4} \left (-1-\sqrt {5}\right ) x}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{5 a}-\frac {\int \frac {1}{a+x} \, dx}{5 a}\\ &=-\frac {\log (a+x)}{5 a}+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {1}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {1}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {-\frac {1}{2} \left (1-\sqrt {5}\right ) a+2 x}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{20 a}+\frac {\left (1+\sqrt {5}\right ) \int \frac {-\frac {1}{2} \left (1+\sqrt {5}\right ) a+2 x}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{20 a}\\ &=-\frac {\log (a+x)}{5 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a}+\frac {1}{10} \left (-5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5-\sqrt {5}\right ) a^2-x^2} \, dx,x,-\frac {1}{2} \left (1+\sqrt {5}\right ) a+2 x\right )-\frac {1}{10} \left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5+\sqrt {5}\right ) a^2-x^2} \, dx,x,-\frac {1}{2} \left (1-\sqrt {5}\right ) a+2 x\right )\\ &=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a}-\frac {\log (a+x)}{5 a}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 204, normalized size = 1.01 \begin {gather*} \frac {2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (-1+\sqrt {5}\right ) a+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )+2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {-\left (\left (1+\sqrt {5}\right ) a\right )+4 x}{\sqrt {10-2 \sqrt {5}} a}\right )-4 \log (a+x)+\log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )-\sqrt {5} \log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\right ) a x+x^2\right )+\log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )+\sqrt {5} \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.36, size = 51, normalized size = 0.25 \begin {gather*} \frac {\text {RootSum}\left [1-125 \text {\#1}^3-5 \text {\#1}+25 \text {\#1}^2+625 \text {\#1}^4\&,\text {Log}\left [625 a \text {\#1}^4+x\right ] \text {\#1}\&\right ]-\frac {\text {Log}\left [a+x\right ]}{5}}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.05, size = 97, normalized size = 0.48
method | result | size |
risch | \(-\frac {\ln \left (a +x \right )}{5 a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{4} \textit {\_Z}^{4}-a^{3} \textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3} a^{4}-\textit {\_R}^{2} a^{3}+a^{2} \textit {\_R} -a +x \right )\right )}{5}\) | \(73\) |
default | \(-\frac {\ln \left (a +x \right )}{5 a}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}+3 \textit {\_R}^{2} a -2 a^{2} \textit {\_R} +a^{3}\right ) \ln \left (-\textit {\_R} +x \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 a^{2} \textit {\_R} -a^{3}}}{5 a}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 180, normalized size = 0.90 \begin {gather*} \frac {\sqrt {5} {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {2 \, \sqrt {5} + 10}} + \frac {\sqrt {5} {\left (\sqrt {5} - 1\right )} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a \sqrt {-2 \, \sqrt {5} + 10}} + \frac {{\left (\sqrt {5} + 3\right )} \log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} + 1\right )}} + \frac {{\left (\sqrt {5} - 3\right )} \log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{10 \, a {\left (\sqrt {5} - 1\right )}} - \frac {\log \left (a + x\right )}{5 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 39, normalized size = 0.19 \begin {gather*} \frac {- \frac {\log {\left (a + x \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log {\left (625 t^{4} a + x \right )} \right )\right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 224, normalized size = 1.11 \begin {gather*} -\frac {8 \sqrt {5} \ln \left (x^{2}+\frac {1}{2} \left (a \sqrt {5}-a\right ) x+a a\right )}{160 a}-\frac {8 \sqrt {2 \left (\sqrt {5}+5\right )} \arctan \left (-\frac {x+\frac {1}{4} \left (\sqrt {5}-1\right ) a}{\frac {1}{4} \sqrt {2 \sqrt {5}+10} a}\right )}{80 a}-\frac {\ln \left |x+a\right |}{5 a}+\frac {8 \sqrt {5} \ln \left (x^{2}+\frac {1}{2} \left (-a \sqrt {5}-a\right ) x+a a\right )}{160 a}-\frac {8 \sqrt {2 \left (-\sqrt {5}+5\right )} \arctan \left (-\frac {x-\frac {1}{4} \left (\sqrt {5}+1\right ) a}{\frac {1}{4} \sqrt {-2 \sqrt {5}+10} a}\right )}{80 a}+\frac {\ln \left |x^{4}-x^{3} a+x^{2} a^{2}-x a^{3}+a^{4}\right |}{20 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 202, normalized size = 1.00 \begin {gather*} \frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}-\frac {\ln \left (5\,a^{10}+\frac {5\,x\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )\,a^9}{4}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a}-\frac {\ln \left (a+x\right )}{5\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{4}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a}+\frac {\ln \left (5\,a^{10}-\frac {5\,a^9\,x\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{4}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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