Optimal. Leaf size=50 \[ -\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1504, 1178,
642} \begin {gather*} \frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 642
Rule 1178
Rule 1504
Rubi steps
\begin {align*} \int \frac {x \left (1-x^4\right )}{1-x^4+x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}\\ &=-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 44, normalized size = 0.88 \begin {gather*} \frac {-\log \left (-1+\sqrt {3} x^2-x^4\right )+\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 39, normalized size = 0.78
method | result | size |
default | \(-\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\) | \(39\) |
risch | \(-\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 41, normalized size = 0.82 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (\frac {x^{8} + 5 \, x^{4} + 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.04, size = 42, normalized size = 0.84 \begin {gather*} - \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.75, size = 31, normalized size = 0.62 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \log \left (\frac {x^{2} - \sqrt {3} + \frac {1}{x^{2}}}{x^{2} + \sqrt {3} + \frac {1}{x^{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.89, size = 20, normalized size = 0.40 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x^2}{x^4+1}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________