Optimal. Leaf size=34 \[ \frac {e^x}{\log \left (\left (3-e+\frac {4 \left (e^4+x+\frac {e^{-x} x}{3}\right )}{x}\right )^2\right )} \]
________________________________________________________________________________________
Rubi [B] time = 1.05, antiderivative size = 132, normalized size of antiderivative = 3.88, number of steps used = 2, number of rules used = 2, integrand size = 253, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6688, 2288} \begin {gather*} \frac {e^x \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right ) \log \left (\frac {e^{-2 x} \left (21 e^x x-3 e^{x+1} x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )}{\left (3 (7-e) e^x x+4 x+12 e^{x+4}\right ) \log ^2\left (\frac {e^{-2 x} \left (3 (7-e) e^x x+4 x+12 e^{x+4}\right )^2}{9 x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2288
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (24 e^{4+x}+8 x^2+x \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right ) \log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )\right )}{x \left (12 e^{4+x}+4 x+21 \left (1-\frac {e}{7}\right ) e^x x\right ) \log ^2\left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 \left (1-\frac {e}{7}\right ) e^x x\right )^2}{9 x^2}\right )} \, dx\\ &=\frac {e^x \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right ) \log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )}{\left (12 e^{4+x}+4 x+3 (7-e) e^x x\right ) \log ^2\left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+3 (7-e) e^x x\right )^2}{9 x^2}\right )}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 46, normalized size = 1.35 \begin {gather*} \frac {e^x}{\log \left (\frac {e^{-2 x} \left (12 e^{4+x}+4 x+21 e^x x-3 e^{1+x} x\right )^2}{9 x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.63, size = 81, normalized size = 2.38 \begin {gather*} \frac {e^{x}}{\log \left (\frac {{\left (16 \, x^{2} + 9 \, {\left (x^{2} e^{2} - 14 \, x^{2} e + 49 \, x^{2} - 8 \, x e^{5} + 56 \, x e^{4} + 16 \, e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e - 7 \, x^{2} - 4 \, x e^{4}\right )} e^{x}\right )} e^{\left (-2 \, x\right )}}{9 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 26.08, size = 110, normalized size = 3.24 \begin {gather*} -\frac {e^{x}}{2 \, x + 2 \, \log \relax (3) - \log \left (441 \, x^{2} e^{\left (2 \, x\right )} + 9 \, x^{2} e^{\left (2 \, x + 2\right )} - 126 \, x^{2} e^{\left (2 \, x + 1\right )} - 24 \, x^{2} e^{\left (x + 1\right )} + 168 \, x^{2} e^{x} + 16 \, x^{2} - 72 \, x e^{\left (2 \, x + 5\right )} + 504 \, x e^{\left (2 \, x + 4\right )} + 96 \, x e^{\left (x + 4\right )} + 144 \, e^{\left (2 \, x + 8\right )}\right ) + 2 \, \log \left (x \mathrm {sgn}\relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.77, size = 758, normalized size = 22.29
method | result | size |
risch | \(\frac {2 i {\mathrm e}^{x}}{\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )\right )^{2} \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )\right ) \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )-\pi \,\mathrm {csgn}\left (i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+\pi \mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-2 x} \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (-x \,{\mathrm e}^{x +1}+4 \,{\mathrm e}^{4+x}-\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )^{3}+4 i \ln \left (x \,{\mathrm e}^{x +1}-4 \,{\mathrm e}^{4+x}+\left (-7 \,{\mathrm e}^{x}-\frac {4}{3}\right ) x \right )-4 i \ln \left ({\mathrm e}^{x}\right )-4 i \ln \relax (x )}\) | \(758\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.09, size = 34, normalized size = 1.00 \begin {gather*} -\frac {e^{x}}{2 \, {\left (x + \log \relax (3) - \log \left (3 \, {\left (x {\left (e - 7\right )} - 4 \, e^{4}\right )} e^{x} - 4 \, x\right ) + \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.58, size = 326, normalized size = 9.59 \begin {gather*} \frac {{\mathrm {e}}^x+\frac {x\,{\mathrm {e}}^x\,\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )\,\left (4\,x+12\,{\mathrm {e}}^{x+4}-3\,x\,{\mathrm {e}}^{x+1}+21\,x\,{\mathrm {e}}^x\right )}{8\,\left (3\,{\mathrm {e}}^{x+4}+x^2\right )}}{\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (\frac {{\mathrm {e}}^{2\,x}\,\left (144\,{\mathrm {e}}^8+{\mathrm {e}}^4\,\left (504\,x-72\,x\,\mathrm {e}\right )-126\,x^2\,\mathrm {e}+9\,x^2\,{\mathrm {e}}^2+441\,x^2\right )}{9}+\frac {{\mathrm {e}}^x\,\left (96\,x\,{\mathrm {e}}^4-24\,x^2\,\mathrm {e}+168\,x^2\right )}{9}+\frac {16\,x^2}{9}\right )}{x^2}\right )}-\frac {x^2\,{\mathrm {e}}^{-4}}{6}+\frac {x^3\,{\mathrm {e}}^{-4}}{6}-{\mathrm {e}}^x\,\left (\frac {x}{2}-\frac {x^2\,{\mathrm {e}}^{-4}\,\left (\mathrm {e}-7\right )}{8}\right )-\frac {x^4\,{\mathrm {e}}^{-8}\,\left (\mathrm {e}-7\right )}{24}+\frac {{\mathrm {e}}^{-8}\,\left (2\,x^7\,\mathrm {e}+8\,x^5\,{\mathrm {e}}^4-x^8\,\mathrm {e}-12\,x^6\,{\mathrm {e}}^4+4\,x^7\,{\mathrm {e}}^4-14\,x^7+7\,x^8\right )}{72\,\left (2\,x\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^4\right )\,\left ({\mathrm {e}}^x+\frac {x^2\,{\mathrm {e}}^{-4}}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.47, size = 92, normalized size = 2.71 \begin {gather*} \frac {e^{x}}{\log {\left (\frac {\left (\frac {16 x^{2}}{9} + \frac {\left (- 24 e x^{2} + 168 x^{2} + 96 x e^{4}\right ) e^{x}}{9} + \frac {\left (- 126 e x^{2} + 9 x^{2} e^{2} + 441 x^{2} + \left (- 72 e x + 504 x\right ) e^{4} + 144 e^{8}\right ) e^{2 x}}{9}\right ) e^{- 2 x}}{x^{2}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________