∫ elog 2 (−25+e2+2x+e2+x(−40−8x)−10x−x2+e2(525+185x+11x2−x3) e2(25+10x+x2) ) (e2+2x(16+4x)+e2+x(−320−144x−16x2)+e2(−250−150x−30x2−2x3)) log(−25+e2+2x+e2+x(−40−8x)−10x−x2+e2(525+185x+11x2−x3) e2(25+10x+x2) ) _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________−125−75x−15x2 −x3 +e2+2x (5+x)+e2+x (−200−80x−8x2 )+e2 (2625+1450x+240x2 +6x3 −x4 ) dx

Optimal. Leaf size=28 \[ e^{\log ^2\left (5-\frac {1}{e^2}-x+\left (-4+\frac {e^x}{5+x}\right )^2\right )} \]

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Rubi [F] time = 100.78, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )\right ) \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx \end {gather*}

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (2+\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) \left (2 e^{2 x} (4+x)-(5+x)^3-8 e^x \left (20+9 x+x^2\right )\right ) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{(5+x) \left (e^{2+2 x}-8 e^{2+x} (5+x)-(5+x)^2-e^2 (-21+x) (5+x)^2\right )} \, dx\\ &=2 \int \frac {\exp \left (2+\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) \left (2 e^{2 x} (4+x)-(5+x)^3-8 e^x \left (20+9 x+x^2\right )\right ) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{(5+x) \left (e^{2+2 x}-8 e^{2+x} (5+x)-(5+x)^2-e^2 (-21+x) (5+x)^2\right )} \, dx\\ &=2 \int \left (\frac {2 \exp \left (\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) (4+x) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{5+x}+\frac {\exp \left (\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) \left (-32 e^{2+x}-40 \left (1-\frac {173 e^2}{8}\right )-8 e^{2+x} x-18 \left (1-\frac {58 e^2}{3}\right ) x-2 \left (1-\frac {25 e^2}{2}\right ) x^2-2 e^2 x^3\right ) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{40 e^{2+x}-e^{2+2 x}+25 \left (1-21 e^2\right )+8 e^{2+x} x+10 \left (1-\frac {37 e^2}{2}\right ) x+\left (1-11 e^2\right ) x^2+e^2 x^3}\right ) \, dx\\ &=2 \int \frac {\exp \left (\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) \left (-32 e^{2+x}-40 \left (1-\frac {173 e^2}{8}\right )-8 e^{2+x} x-18 \left (1-\frac {58 e^2}{3}\right ) x-2 \left (1-\frac {25 e^2}{2}\right ) x^2-2 e^2 x^3\right ) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{40 e^{2+x}-e^{2+2 x}+25 \left (1-21 e^2\right )+8 e^{2+x} x+10 \left (1-\frac {37 e^2}{2}\right ) x+\left (1-11 e^2\right ) x^2+e^2 x^3} \, dx+4 \int \frac {\exp \left (\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) (4+x) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{5+x} \, dx\\ &=2 \int \frac {\exp \left (\log ^2\left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) \left (8 e^{2+x} (4+x)+2 \left (20+9 x+x^2\right )+e^2 \left (-865-348 x-25 x^2+2 x^3\right )\right ) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{e^{2+2 x}-8 e^{2+x} (5+x)-(5+x)^2-e^2 (-21+x) (5+x)^2} \, dx+4 \int \frac {\exp \left (\log ^2\left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )\right ) (4+x) \log \left (21 \left (1-\frac {1}{21 e^2}\right )-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )}{5+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.14, size = 36, normalized size = 1.29 \begin {gather*} e^{\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )} \end {gather*}

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fricas [B] time = 0.66, size = 63, normalized size = 2.25 \begin {gather*} e^{\left (\log \left (-\frac {{\left ({\left (x^{3} - 11 \, x^{2} - 185 \, x - 525\right )} e^{4} + {\left (x^{2} + 10 \, x + 25\right )} e^{2} + 8 \, {\left (x + 5\right )} e^{\left (x + 4\right )} - e^{\left (2 \, x + 4\right )}\right )} e^{\left (-4\right )}}{x^{2} + 10 \, x + 25}\right )^{2}\right )} \end {gather*}

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giac [B] time = 26.02, size = 238, normalized size = 8.50 \begin {gather*} e^{\left (\log \left (-\frac {x^{3} e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {11 \, x^{2} e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {x^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {185 \, x e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {8 \, x e^{\left (x + 2\right )}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {10 \, x}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {525 \, e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {e^{\left (2 \, x + 2\right )}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {40 \, e^{\left (x + 2\right )}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {25}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}}\right )^{2}\right )} \end {gather*}

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method	result	size

risch	\({\mathrm e}^{\frac {\left (-i \pi \mathrm {csgn}\left (i \left (5+x \right )^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i \left (5+x \right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (5+x \right )\right )-i \pi \,\mathrm {csgn}\left (i \left (5+x \right )^{2}\right ) \mathrm {csgn}\left (i \left (5+x \right )\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{\left (5+x \right )^{2}}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )^{2} \mathrm {csgn}\left (i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\left (5+x \right )^{2}}\right ) \mathrm {csgn}\left (i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )\right )+2 i \mathrm {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )^{2} \pi -2 i \pi +4 \ln \left (5+x \right )-2 \ln \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )+4\right )^{2}}{4}}\)	\(491\)

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maxima [B] time = 1.28, size = 1199, normalized size = 42.82 result too large to display

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mupad [B] time = 6.98, size = 59, normalized size = 2.11 \begin {gather*} {\mathrm {e}}^{{\ln \left ({\mathrm {e}}^{-2}\,\left (21\,{\mathrm {e}}^2-1\right )-\frac {40\,{\mathrm {e}}^2\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2+8\,x\,{\mathrm {e}}^2\,{\mathrm {e}}^x}{{\mathrm {e}}^2\,x^2+10\,{\mathrm {e}}^2\,x+25\,{\mathrm {e}}^2}-x\right )}^2} \end {gather*}

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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