Optimal. Leaf size=27 \[ -\frac {1}{6} \log \left (x^3+1\right )+\frac {1}{18} \log \left (x^3+3\right )+\frac {\log (x)}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1357, 705, 29, 632, 31} \[ -\frac {1}{6} \log \left (x^3+1\right )+\frac {1}{18} \log \left (x^3+3\right )+\frac {\log (x)}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 632
Rule 705
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x \left (3+4 x^3+x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \left (3+4 x+x^2\right )} \, dx,x,x^3\right )\\ &=\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )+\frac {1}{9} \operatorname {Subst}\left (\int \frac {-4-x}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac {\log (x)}{3}+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{3+x} \, dx,x,x^3\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^3\right )\\ &=\frac {\log (x)}{3}-\frac {1}{6} \log \left (1+x^3\right )+\frac {1}{18} \log \left (3+x^3\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {1}{6} \log \left (x^3+1\right )+\frac {1}{18} \log \left (x^3+3\right )+\frac {\log (x)}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.05, size = 21, normalized size = 0.78 \[ \frac {1}{18} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) + \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 24, normalized size = 0.89 \[ \frac {1}{18} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 31, normalized size = 1.15 \[ \frac {\ln \relax (x )}{3}-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{3}+3\right )}{18}-\frac {\ln \left (x^{2}-x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 23, normalized size = 0.85 \[ \frac {1}{18} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) + \frac {1}{9} \, \log \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.26, size = 21, normalized size = 0.78 \[ \frac {\ln \left (x^3+3\right )}{18}-\frac {\ln \left (x^3+1\right )}{6}+\frac {\ln \relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 20, normalized size = 0.74 \[ \frac {\log {\relax (x )}}{3} - \frac {\log {\left (x^{3} + 1 \right )}}{6} + \frac {\log {\left (x^{3} + 3 \right )}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________