Optimal. Leaf size=56 \[ -\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}+\frac {\log (a+x)}{3 a^2}-\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}+\frac {\log (a+x)}{3 a^2}-\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{a^3+x^3} \, dx &=\frac {\int \frac {1}{a+x} \, dx}{3 a^2}+\frac {\int \frac {2 a-x}{a^2-a x+x^2} \, dx}{3 a^2}\\ &=\frac {\log (a+x)}{3 a^2}-\frac {\int \frac {-a+2 x}{a^2-a x+x^2} \, dx}{6 a^2}+\frac {\int \frac {1}{a^2-a x+x^2} \, dx}{2 a}\\ &=\frac {\log (a+x)}{3 a^2}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{a}\right )}{a^2}\\ &=-\frac {\tan ^{-1}\left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2}+\frac {\log (a+x)}{3 a^2}-\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 52, normalized size = 0.93 \[ \frac {-\log \left (a^2-a x+x^2\right )+2 \log (a+x)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-a}{\sqrt {3} a}\right )}{6 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.02, size = 59, normalized size = 1.05 \[ -\frac {\log \left (a^2-a x+x^2\right )}{6 a^2}+\frac {\log (a+x)}{3 a^2}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 45, normalized size = 0.80 \[ \frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left (a^{2} - a x + x^{2}\right ) + 2 \, \log \left (a + x\right )}{6 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.88, size = 50, normalized size = 0.89 \[ \frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{2}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{2}} + \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.30, size = 41, normalized size = 0.73
method | result | size |
risch | \(\frac {\ln \left (a +x \right )}{3 a^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{4} \textit {\_Z}^{2}+a^{2} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} \,a^{3}+x \right )\right )}{3}\) | \(41\) |
default | \(\frac {-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 x -a \right ) \sqrt {3}}{3 a}\right )}{3 a^{2}}+\frac {\ln \left (a +x \right )}{3 a^{2}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 49, normalized size = 0.88 \[ \frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{2}} - \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{2}} + \frac {\log \left (a + x\right )}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.44, size = 64, normalized size = 1.14 \[ \frac {\ln \left (a+x\right )}{3\,a^2}+\frac {\ln \left (x+\frac {a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^2}-\frac {\ln \left (x-\frac {a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.14, size = 73, normalized size = 1.30 \[ \frac {\frac {\log {\left (a + x \right )}}{3} + \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (3 a \left (- \frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) + x \right )} + \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (3 a \left (- \frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) + x \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________