Optimal. Leaf size=32 \[ \frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (3 x^2+4\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1114, 703, 632, 31} \[ \frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (3 x^2+4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 632
Rule 703
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^5}{-4+x^2+3 x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{-4+x+3 x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {4-x}{-4+x+3 x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{6}+\frac {3}{14} \operatorname {Subst}\left (\int \frac {1}{-3+3 x} \, dx,x,x^2\right )-\frac {8}{21} \operatorname {Subst}\left (\int \frac {1}{4+3 x} \, dx,x,x^2\right )\\ &=\frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (4+3 x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ \frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (3 x^2+4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.01, size = 30, normalized size = 0.94 \[ \frac {x^2}{6}+\frac {1}{14} \log \left (x^2-1\right )-\frac {8}{63} \log \left (3 x^2+4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.09, size = 24, normalized size = 0.75 \[ \frac {1}{6} \, x^{2} - \frac {8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac {1}{14} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.91, size = 25, normalized size = 0.78 \[ \frac {1}{6} \, x^{2} - \frac {8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac {1}{14} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 25, normalized size = 0.78
method | result | size |
default | \(\frac {x^{2}}{6}-\frac {8 \ln \left (3 x^{2}+4\right )}{63}+\frac {\ln \left (x^{2}-1\right )}{14}\) | \(25\) |
risch | \(\frac {x^{2}}{6}-\frac {8 \ln \left (3 x^{2}+4\right )}{63}+\frac {\ln \left (x^{2}-1\right )}{14}\) | \(25\) |
norman | \(\frac {x^{2}}{6}+\frac {\ln \left (-1+x \right )}{14}+\frac {\ln \left (1+x \right )}{14}-\frac {8 \ln \left (3 x^{2}+4\right )}{63}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 24, normalized size = 0.75 \[ \frac {1}{6} \, x^{2} - \frac {8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac {1}{14} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 22, normalized size = 0.69 \[ \frac {\ln \left (x^2-1\right )}{14}-\frac {8\,\ln \left (x^2+\frac {4}{3}\right )}{63}+\frac {x^2}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 24, normalized size = 0.75 \[ \frac {x^{2}}{6} + \frac {\log {\left (x^{2} - 1 \right )}}{14} - \frac {8 \log {\left (x^{2} + \frac {4}{3} \right )}}{63} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________