Optimal. Leaf size=201 \[ -\frac {\log (a+x)}{5 a^3}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a^3}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a^3}+\frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a^3}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {293, 634, 618, 204, 628, 31} \[ \frac {\left (1+\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2\right )}{20 a^3}+\frac {\left (1-\sqrt {5}\right ) \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^3}-\frac {\log (a+x)}{5 a^3}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) a-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )}{5 a^3}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (\left (1+\sqrt {5}\right ) a-4 x\right )}{2 a}\right )}{5 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 293
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x}{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^3}+\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^3}-\frac {\int \frac {1}{a+x} \, dx}{5 a^3}\\ &=-\frac {\log (a+x)}{5 a^3}+\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^3}+\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^3}-\frac {\int \frac {1}{a^2-\frac {1}{2} \left (1-\sqrt {5}\right ) a x+x^2} \, dx}{2 \sqrt {5} a^2}+\frac {\int \frac {1}{a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2} \, dx}{2 \sqrt {5} a^2}\\ &=-\frac {\log (a+x)}{5 a^3}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a^3}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a^3}-\frac {\operatorname {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 )}{\sqrt {5} a^2}+\frac {\operatorname {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 )}{\sqrt {5} a^2}\\ &=\frac {\sqrt {\frac {2}{5 \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 )}{a^3}-\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^3}-\frac {\log (a+x)}{5 a^3}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-a x-\sqrt {5} a x+2 x^2\right )}{20 a^3}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2-a x+\sqrt {5} a x+2 x^2\right )}{20 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 204, normalized size = 1.01 \[ \frac {\sqrt {5} \log \left (a^2+\frac {1}{2} \left (\sqrt {5}-1\right ) a x+x^2\right )+\log \left (a^2+\frac {1}{2} \left (\sqrt {5}-1\right ) a x+x^2\right )-\sqrt {5} \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )+\log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )-4 \log (a+x)-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {\left (\sqrt {5}-1\right ) a+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )} a}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {4 x-\left (1+\sqrt {5}\right ) a}{\sqrt {10-2 \sqrt {5}} a}\right )}{20 a^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.15, size = 260, normalized size = 1.29 \[ -\frac {\log (a+x)}{5 a^3}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\frac {2}{\sqrt {5}}} x}{a}-\frac {1}{2} \sqrt {\frac {1}{10} \left (5-\sqrt {5}\right )}+\frac {1}{2} \sqrt {\frac {1}{10} \left (25-5 \sqrt {5}\right )}\right )}{5 a^3}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (-\frac {\sqrt {2+\frac {2}{\sqrt {5}}} x}{a}+\frac {1}{2} \sqrt {\frac {1}{10} \left (25+5 \sqrt {5}\right )}+\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )}\right )}{5 a^3}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 a^2-\sqrt {5} a x-a x+2 x^2\right )}{20 a^3}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 a^2+\sqrt {5} a x-a x+2 x^2\right )}{20 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.89, size = 177, normalized size = 0.88 \[ -\frac {\sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{10 \, a^{3}} + \frac {\sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{10 \, a^{3}} - \frac {\sqrt {5} \log \left (a^{2} - \frac {1}{2} \, {\left (\sqrt {5} a + a\right )} x + x^{2}\right )}{20 \, a^{3}} + \frac {\sqrt {5} \log \left (a^{2} + \frac {1}{2} \, {\left (\sqrt {5} a - a\right )} x + x^{2}\right )}{20 \, a^{3}} + \frac {\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{3}} - \frac {\log \left ({\left | a + x \right |}\right )}{5 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.35, size = 60, normalized size = 0.30
method | result | size |
risch | \(-\frac {\ln \left (a +x \right )}{5 a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{12} \textit {\_Z}^{4}-a^{9} \textit {\_Z}^{3}+a^{6} \textit {\_Z}^{2}-a^{3} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (-a^{10} \textit {\_R}^{3}+x \right )\right )}{5}\) | \(60\) |
default | \(-\frac {\ln \left (a +x \right )}{5 a^{3}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}-2 \textit {\_R}^{2} a +3 \textit {\_R} \,a^{2}+a^{3}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 \textit {\_R} \,a^{2}-a^{3}}}{5 a^{3}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 160, normalized size = 0.80 \[ -\frac {2 \, \sqrt {5} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a^{3} \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a^{3} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {\log \left (a + x\right )}{5 \, a^{3}} - \frac {\log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{5 \, a^{3} {\left (\sqrt {5} + 1\right )}} + \frac {\log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{5 \, a^{3} {\left (\sqrt {5} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.59, size = 182, normalized size = 0.91 \[ \frac {\ln \left (x-\frac {a\,{\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}^3}{64}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^3}-\frac {\ln \left (a+x\right )}{5\,a^3}+\frac {\ln \left (x-\frac {a\,{\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}^3}{64}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a^3}-\frac {\ln \left (x+\frac {a\,{\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}^3}{64}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^3}+\frac {\ln \left (x-\frac {a\,{\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}^3}{64}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 41, normalized size = 0.20 \[ \frac {- \frac {\log {\left (a + x \right )}}{5} + \operatorname {RootSum} {\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left (t \mapsto t \log {\left (- 125 t^{3} a + x \right )} \right )\right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________