Optimal. Leaf size=24 \[ \frac {2}{4-x^2}+\frac {1}{2} \log \left (4-x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {28, 266, 43} \[ \frac {2}{4-x^2}+\frac {1}{2} \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{16-8 x^2+x^4} \, dx &=\int \frac {x^3}{\left (-4+x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(-4+x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {4}{(-4+x)^2}+\frac {1}{-4+x}\right ) \, dx,x,x^2\right )\\ &=\frac {2}{4-x^2}+\frac {1}{2} \log \left (4-x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 20, normalized size = 0.83 \[ \frac {1}{2} \log \left (x^2-4\right )-\frac {2}{x^2-4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 23, normalized size = 0.96 \[ \frac {{\left (x^{2} - 4\right )} \log \left (x^{2} - 4\right ) - 4}{2 \, {\left (x^{2} - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 19, normalized size = 0.79 \[ -\frac {2}{x^{2} - 4} + \frac {1}{2} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 19, normalized size = 0.79 \[ \frac {\ln \left (x^{2}-4\right )}{2}-\frac {2}{x^{2}-4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 18, normalized size = 0.75 \[ -\frac {2}{x^{2} - 4} + \frac {1}{2} \, \log \left (x^{2} - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.23, size = 18, normalized size = 0.75 \[ \frac {\ln \left (x^2-4\right )}{2}-\frac {2}{x^2-4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 14, normalized size = 0.58 \[ \frac {\log {\left (x^{2} - 4 \right )}}{2} - \frac {2}{x^{2} - 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________