Optimal. Leaf size=30 \[ \frac {2}{3} \left (-4+\frac {x}{4}\right ) x \left (-\frac {\left (5-\frac {e^4}{2}\right )^2}{x^2}+x\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 14} \begin {gather*} \frac {x^3}{6}-\frac {8 x^2}{3}+\frac {2 \left (10-e^4\right )^2}{3 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{24} e^8 \int \frac {-16+\frac {320}{e^4}+\frac {4 \left (-400-32 x^3+3 x^4\right )}{e^8}}{x^2} \, dx\\ &=\frac {1}{24} e^8 \int \left (-\frac {16 \left (-10+e^4\right )^2}{e^8 x^2}-\frac {128 x}{e^8}+\frac {12 x^2}{e^8}\right ) \, dx\\ &=\frac {2 \left (10-e^4\right )^2}{3 x}-\frac {8 x^2}{3}+\frac {x^3}{6}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{6} \left (\frac {4 \left (100-20 e^4+e^8\right )}{x}-16 x^2+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 39, normalized size = 1.30 \begin {gather*} \frac {{\left ({\left (x^{4} - 16 \, x^{3} + 400\right )} e^{\left (2 \, \log \relax (2) - 8\right )} - 160 \, e^{\left (\log \relax (2) - 4\right )} + 16\right )} e^{\left (-2 \, \log \relax (2) + 8\right )}}{6 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 56, normalized size = 1.87 \begin {gather*} \frac {1}{6} \, {\left (x^{3} e^{\left (2 \, \log \relax (2) - 8\right )} - 16 \, x^{2} e^{\left (2 \, \log \relax (2) - 8\right )} + \frac {16 \, {\left (25 \, e^{\left (2 \, \log \relax (2) - 8\right )} - 10 \, e^{\left (\log \relax (2) - 4\right )} + 1\right )}}{x}\right )} e^{\left (-2 \, \log \relax (2) + 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 37, normalized size = 1.23
method | result | size |
risch | \(\frac {x^{3}}{6}-\frac {8 x^{2}}{3}-\frac {40 \,{\mathrm e}^{8} {\mathrm e}^{-4}}{3 x}+\frac {200 \,{\mathrm e}^{8} {\mathrm e}^{-8}}{3 x}+\frac {2 \,{\mathrm e}^{8}}{3 x}\) | \(37\) |
norman | \(\frac {\left (-\frac {8 x^{3} {\mathrm e}^{4}}{3}+\frac {x^{4} {\mathrm e}^{4}}{6}+\frac {2 \,{\mathrm e}^{4} \left ({\mathrm e}^{8}-20 \,{\mathrm e}^{4}+100\right )}{3}\right ) {\mathrm e}^{-4}}{x}\) | \(38\) |
gosper | \(\frac {\left (4 \,{\mathrm e}^{-8} x^{4}-64 \,{\mathrm e}^{-8} x^{3}+16+1600 \,{\mathrm e}^{-8}-160 \,{\mathrm e}^{\ln \relax (2)-4}\right ) {\mathrm e}^{8}}{24 x}\) | \(54\) |
default | \(\frac {{\mathrm e}^{8} \left (x^{3} {\mathrm e}^{2 \ln \relax (2)-8}-16 \,{\mathrm e}^{2 \ln \relax (2)-8} x^{2}-\frac {160 \,{\mathrm e}^{\ln \relax (2)-4}-400 \,{\mathrm e}^{2 \ln \relax (2)-8}-16}{x}\right )}{24}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 32, normalized size = 1.07 \begin {gather*} \frac {1}{6} \, {\left ({\left (x^{3} - 16 \, x^{2}\right )} e^{\left (-8\right )} + \frac {4 \, {\left (e^{8} - 20 \, e^{4} + 100\right )} e^{\left (-8\right )}}{x}\right )} e^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 23, normalized size = 0.77 \begin {gather*} \frac {x^4-16\,x^3-80\,{\mathrm {e}}^4+4\,{\mathrm {e}}^8+400}{6\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 26, normalized size = 0.87 \begin {gather*} \frac {x^{3}}{6} - \frac {8 x^{2}}{3} + \frac {- 80 e^{4} + 400 + 4 e^{8}}{6 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________