Optimal. Leaf size=39 \[ -\frac {1}{2} \log (1-2 x)+\frac {1}{2} \log (1-x)-\frac {1}{2} \log (x+1)+\frac {1}{2} \log (2 x+1) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1161, 616, 31} \begin {gather*} -\frac {1}{2} \log (1-2 x)+\frac {1}{2} \log (1-x)-\frac {1}{2} \log (x+1)+\frac {1}{2} \log (2 x+1) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 616
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+2 x^2}{1-5 x^2+4 x^4} \, dx &=\frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {3 x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {3 x}{2}+x^2} \, dx\\ &=\frac {1}{2} \int \frac {1}{-1+x} \, dx-\frac {1}{2} \int \frac {1}{-\frac {1}{2}+x} \, dx+\frac {1}{2} \int \frac {1}{\frac {1}{2}+x} \, dx-\frac {1}{2} \int \frac {1}{1+x} \, dx\\ &=-\frac {1}{2} \log (1-2 x)+\frac {1}{2} \log (1-x)-\frac {1}{2} \log (1+x)+\frac {1}{2} \log (1+2 x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{2} \log \left (-2 x^2+x+1\right )-\frac {1}{2} \log \left (-2 x^2-x+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^2}{1-5 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 25, normalized size = 0.64 \begin {gather*} -\frac {1}{2} \, \log \left (2 \, x^{2} + x - 1\right ) + \frac {1}{2} \, \log \left (2 \, x^{2} - x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 33, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} -\frac {\ln \left (x +1\right )}{2}+\frac {\ln \left (2 x +1\right )}{2}+\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (2 x -1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.04, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x + 1\right ) - \frac {1}{2} \, \log \left (2 \, x - 1\right ) - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.30, size = 14, normalized size = 0.36 \begin {gather*} -\mathrm {atanh}\left (\frac {x}{2\,x^2-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 26, normalized size = 0.67 \begin {gather*} \frac {\log {\left (x^{2} - \frac {x}{2} - \frac {1}{2} \right )}}{2} - \frac {\log {\left (x^{2} + \frac {x}{2} - \frac {1}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________