Optimal. Leaf size=20 \[ e^{\frac {e^2}{x \left (4+x^2\right )^2 \log ^2(81)}} \]
________________________________________________________________________________________
Rubi [A] time = 0.73, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6688, 6706} \begin {gather*} e^{\frac {e^2}{x \left (x^2+4\right )^2 \log ^2(81)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 6688
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{2+\frac {e^2}{\left (16 x+8 x^3+x^5\right ) \log ^2(81)}} \left (-4-5 x^2\right )}{64 x^2+48 x^4+12 x^6+x^8} \, dx}{\log ^2(81)}\\ &=\frac {\int \frac {e^{2+\frac {e^2}{x \left (4+x^2\right )^2 \log ^2(81)}} \left (-4-5 x^2\right )}{x^2 \left (4+x^2\right )^3} \, dx}{\log ^2(81)}\\ &=e^{\frac {e^2}{x \left (4+x^2\right )^2 \log ^2(81)}}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.59, size = 20, normalized size = 1.00 \begin {gather*} e^{\frac {e^2}{x \left (4+x^2\right )^2 \log ^2(81)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 44, normalized size = 2.20 \begin {gather*} e^{\left (\frac {32 \, {\left (x^{5} + 8 \, x^{3} + 16 \, x\right )} \log \relax (3)^{2} + e^{2}}{16 \, {\left (x^{5} + 8 \, x^{3} + 16 \, x\right )} \log \relax (3)^{2}} - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.25, size = 140, normalized size = 7.00 \begin {gather*} e^{\left (\frac {2 \, x^{5} \log \relax (3)^{2}}{x^{5} \log \relax (3)^{2} + 8 \, x^{3} \log \relax (3)^{2} + 16 \, x \log \relax (3)^{2}} + \frac {16 \, x^{3} \log \relax (3)^{2}}{x^{5} \log \relax (3)^{2} + 8 \, x^{3} \log \relax (3)^{2} + 16 \, x \log \relax (3)^{2}} + \frac {32 \, x \log \relax (3)^{2}}{x^{5} \log \relax (3)^{2} + 8 \, x^{3} \log \relax (3)^{2} + 16 \, x \log \relax (3)^{2}} + \frac {e^{2}}{16 \, {\left (x^{5} \log \relax (3)^{2} + 8 \, x^{3} \log \relax (3)^{2} + 16 \, x \log \relax (3)^{2}\right )}} - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.35, size = 20, normalized size = 1.00
method | result | size |
risch | \({\mathrm e}^{\frac {{\mathrm e}^{2}}{16 \left (x^{2}+4\right )^{2} \ln \relax (3)^{2} x}}\) | \(20\) |
gosper | \({\mathrm e}^{\frac {{\mathrm e}^{2}}{16 x \left (x^{4}+8 x^{2}+16\right ) \ln \relax (3)^{2}}}\) | \(25\) |
norman | \(\frac {x^{5} \ln \relax (3) {\mathrm e}^{\frac {{\mathrm e}^{2}}{16 \left (x^{5}+8 x^{3}+16 x \right ) \ln \relax (3)^{2}}}+16 x \ln \relax (3) {\mathrm e}^{\frac {{\mathrm e}^{2}}{16 \left (x^{5}+8 x^{3}+16 x \right ) \ln \relax (3)^{2}}}+8 x^{3} \ln \relax (3) {\mathrm e}^{\frac {{\mathrm e}^{2}}{16 \left (x^{5}+8 x^{3}+16 x \right ) \ln \relax (3)^{2}}}}{x \left (x^{2}+4\right )^{2} \ln \relax (3)}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.62, size = 66, normalized size = 3.30 \begin {gather*} e^{\left (-\frac {x e^{2}}{64 \, {\left (x^{4} \log \relax (3)^{2} + 8 \, x^{2} \log \relax (3)^{2} + 16 \, \log \relax (3)^{2}\right )}} - \frac {x e^{2}}{256 \, {\left (x^{2} \log \relax (3)^{2} + 4 \, \log \relax (3)^{2}\right )}} + \frac {e^{2}}{256 \, x \log \relax (3)^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.51, size = 32, normalized size = 1.60 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^2}{16\,\left ({\ln \relax (3)}^2\,x^5+8\,{\ln \relax (3)}^2\,x^3+16\,{\ln \relax (3)}^2\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.42, size = 22, normalized size = 1.10 \begin {gather*} e^{\frac {e^{2}}{16 \left (x^{5} + 8 x^{3} + 16 x\right ) \log {\relax (3 )}^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________