Optimal. Leaf size=28 \[ -\frac {1}{2} \log \left (x^2+3\right )+\log (x)+\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1593, 801, 635, 203, 260} \[ -\frac {1}{2} \log \left (x^2+3\right )+\log (x)+\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 260
Rule 635
Rule 801
Rule 1593
Rubi steps
\begin {align*} \int \frac {3+2 x}{3 x+x^3} \, dx &=\int \frac {3+2 x}{x \left (3+x^2\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {2-x}{3+x^2}\right ) \, dx\\ &=\log (x)+\int \frac {2-x}{3+x^2} \, dx\\ &=\log (x)+2 \int \frac {1}{3+x^2} \, dx-\int \frac {x}{3+x^2} \, dx\\ &=\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}+\log (x)-\frac {1}{2} \log \left (3+x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ -\frac {1}{2} \log \left (x^2+3\right )+\log (x)+\frac {2 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 23, normalized size = 0.82 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{2} \, \log \left (x^{2} + 3\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.96, size = 24, normalized size = 0.86 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{2} \, \log \left (x^{2} + 3\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 24, normalized size = 0.86 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{3}+\ln \relax (x )-\frac {\ln \left (x^{2}+3\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 23, normalized size = 0.82 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) - \frac {1}{2} \, \log \left (x^{2} + 3\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.30, size = 55, normalized size = 1.96 \[ \ln \relax (x)-\frac {\ln \left (x+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (x-\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\sqrt {3}\,\ln \left (x-\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {\sqrt {3}\,\ln \left (x+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 29, normalized size = 1.04 \[ \log {\relax (x )} - \frac {\log {\left (x^{2} + 3 \right )}}{2} + \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________