Optimal. Leaf size=26 \[ \left (e^x+\left (x-x^2\right )^2\right ) \log \left (e^2+\frac {1}{2} (-5+x)\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1053\) vs. \(2(26)=52\).
time = 1.12, antiderivative size = 1053, normalized size of antiderivative = 40.50, number
of steps used = 42, number of rules used = 13, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules
used = {6874, 45, 2458, 12, 2388, 2338, 2332, 2465, 2436, 2442, 2437, 2372, 2326}
\begin {gather*} \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^4-\frac {1}{4} \left (-x-2 e^2+5\right )^4-\frac {16}{3} \left (5-2 e^2\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^3+\frac {26}{3} \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^3+\frac {16}{9} \left (5-2 e^2\right ) \left (-x-2 e^2+5\right )^3-\frac {26}{9} \left (-x-2 e^2+5\right )^3+12 \left (5-2 e^2\right )^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^2-39 \left (5-2 e^2\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^2+16 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^2-6 \left (5-2 e^2\right )^2 \left (-x-2 e^2+5\right )^2+\frac {39}{2} \left (5-2 e^2\right ) \left (-x-2 e^2+5\right )^2-8 \left (-x-2 e^2+5\right )^2-8 e^2 \left (18-17 e^2+4 e^4\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )-16 \left (5-2 e^2\right )^3 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )+78 \left (5-2 e^2\right )^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )-64 \left (5-2 e^2\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )+10 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )+\frac {x^4}{4}+\frac {1}{3} \left (5-2 e^2\right ) x^3-\frac {8 e^2 x^3}{9}-\frac {2 x^3}{3}+\frac {1}{2} \left (5-2 e^2\right )^2 x^2-\frac {4}{3} e^2 \left (5-2 e^2\right ) x^2-\left (5-2 e^2\right ) x^2-e^2 \left (7-4 e^2\right ) x^2+\frac {x^2}{2}+4 e^2 \left (90-121 e^2+54 e^4-8 e^6\right ) \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+2 \left (5-2 e^2\right )^4 \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-13 \left (5-2 e^2\right )^3 \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+16 \left (5-2 e^2\right )^2 \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-5 \left (5-2 e^2\right ) \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-8 e^2 \left (18-17 e^2+4 e^4\right ) x-15 \left (5-2 e^2\right )^3 x-\frac {8}{3} e^2 \left (5-2 e^2\right )^2 x+76 \left (5-2 e^2\right )^2 x-2 e^2 \left (7-4 e^2\right ) \left (5-2 e^2\right ) x-63 \left (5-2 e^2\right ) x+10 x+\left (5-2 e^2\right )^4 \log \left (-x-2 e^2+5\right )-\frac {8}{3} e^2 \left (5-2 e^2\right )^3 \log \left (-x-2 e^2+5\right )-2 \left (5-2 e^2\right )^3 \log \left (-x-2 e^2+5\right )-2 e^2 \left (7-4 e^2\right ) \left (5-2 e^2\right )^2 \log \left (-x-2 e^2+5\right )+\left (5-2 e^2\right )^2 \log \left (-x-2 e^2+5\right )+\frac {8}{3} e^2 x^3 \log \left (\frac {x}{2}+\frac {1}{2} \left (-5+2 e^2\right )\right )+2 e^2 \left (7-4 e^2\right ) x^2 \log \left (\frac {x}{2}+\frac {1}{2} \left (-5+2 e^2\right )\right )-\frac {e^x \left (x \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+2 e^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-5 \log \left (x+2 e^2-5\right )+\log (32)\right )}{-x-2 e^2+5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2326
Rule 2332
Rule 2338
Rule 2372
Rule 2388
Rule 2436
Rule 2437
Rule 2442
Rule 2458
Rule 2465
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x^2}{-5+2 e^2+x}-\frac {2 x^3}{-5+2 e^2+x}+\frac {x^4}{-5+2 e^2+x}+\frac {10 x \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{5-2 e^2-x}+\frac {4 e^2 (1-2 x) (-1+x) x \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{5-2 e^2-x}+\frac {26 x^3 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{5-2 e^2-x}+\frac {32 x^2 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{-5+2 e^2+x}+\frac {4 x^4 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{-5+2 e^2+x}+\frac {e^x \left (1+\log (32)+2 e^2 \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )+x \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )-5 \log \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x}\right ) \, dx\\ &=-\left (2 \int \frac {x^3}{-5+2 e^2+x} \, dx\right )+4 \int \frac {x^4 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{-5+2 e^2+x} \, dx+10 \int \frac {x \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{5-2 e^2-x} \, dx+26 \int \frac {x^3 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{5-2 e^2-x} \, dx+32 \int \frac {x^2 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{-5+2 e^2+x} \, dx+\left (4 e^2\right ) \int \frac {(1-2 x) (-1+x) x \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{5-2 e^2-x} \, dx+\int \frac {x^2}{-5+2 e^2+x} \, dx+\int \frac {x^4}{-5+2 e^2+x} \, dx+\int \frac {e^x \left (1+\log (32)+2 e^2 \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )+x \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )-5 \log \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx\\ &=-\frac {e^x \left (\log (32)+2 e^2 \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )+x \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )-5 \log \left (-5+2 e^2+x\right )\right )}{5-2 e^2-x}-2 \int \left (\left (-5+2 e^2\right )^2-\left (-5+2 e^2\right ) x+x^2-\frac {\left (-5+2 e^2\right )^3}{-5+2 e^2+x}\right ) \, dx+8 \text {Subst}\left (\int \frac {\left (5-2 e^2+2 x\right )^4 \log (x)}{2 x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+20 \text {Subst}\left (\int -\frac {\left (5-2 e^2+2 x\right ) \log (x)}{2 x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+52 \text {Subst}\left (\int -\frac {\left (5-2 e^2+2 x\right )^3 \log (x)}{2 x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+64 \text {Subst}\left (\int \frac {\left (5-2 e^2+2 x\right )^2 \log (x)}{2 x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+\left (4 e^2\right ) \int \left (2 \left (18-17 e^2+4 e^4\right ) \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )-\left (-7+4 e^2\right ) x \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+2 x^2 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )-\frac {2 \left (-90+121 e^2-54 e^4+8 e^6\right ) \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{-5+2 e^2+x}\right ) \, dx+\int \left (5 \left (1-\frac {2 e^2}{5}\right )+x+\frac {\left (-5+2 e^2\right )^2}{-5+2 e^2+x}\right ) \, dx+\int \left (-\left (-5+2 e^2\right )^3+\left (-5+2 e^2\right )^2 x-\left (-5+2 e^2\right ) x^2+x^3+\frac {\left (-5+2 e^2\right )^4}{-5+2 e^2+x}\right ) \, dx\\ &=\left (5-2 e^2\right ) x-2 \left (5-2 e^2\right )^2 x+\left (5-2 e^2\right )^3 x+\frac {x^2}{2}-\left (5-2 e^2\right ) x^2+\frac {1}{2} \left (5-2 e^2\right )^2 x^2-\frac {2 x^3}{3}+\frac {1}{3} \left (5-2 e^2\right ) x^3+\frac {x^4}{4}+\left (5-2 e^2\right )^2 \log \left (5-2 e^2-x\right )-2 \left (5-2 e^2\right )^3 \log \left (5-2 e^2-x\right )+\left (5-2 e^2\right )^4 \log \left (5-2 e^2-x\right )-\frac {e^x \left (\log (32)+2 e^2 \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )+x \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )-5 \log \left (-5+2 e^2+x\right )\right )}{5-2 e^2-x}+4 \text {Subst}\left (\int \frac {\left (5-2 e^2+2 x\right )^4 \log (x)}{x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )-10 \text {Subst}\left (\int \frac {\left (5-2 e^2+2 x\right ) \log (x)}{x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )-26 \text {Subst}\left (\int \frac {\left (5-2 e^2+2 x\right )^3 \log (x)}{x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+32 \text {Subst}\left (\int \frac {\left (5-2 e^2+2 x\right )^2 \log (x)}{x} \, dx,x,\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )+\left (8 e^2\right ) \int x^2 \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right ) \, dx+\left (4 e^2 \left (7-4 e^2\right )\right ) \int x \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right ) \, dx+\left (8 e^2 \left (18-17 e^2+4 e^4\right )\right ) \int \log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right ) \, dx+\left (8 e^2 \left (90-121 e^2+54 e^4-8 e^6\right )\right ) \int \frac {\log \left (\frac {1}{2} \left (-5+2 e^2\right )+\frac {x}{2}\right )}{-5+2 e^2+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 29, normalized size = 1.12 \begin {gather*} \left (e^x+x^2-2 x^3+x^4\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs.
\(2(21)=42\).
time = 1.17, size = 346, normalized size = 13.31
method | result | size |
risch | \(\left (x^{4}-2 x^{3}+x^{2}+{\mathrm e}^{x}\right ) \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )\) | \(24\) |
norman | \({\mathrm e}^{x} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+\ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{2}+\ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{4}-2 \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{3}\) | \(50\) |
default | \(\frac {100 \,{\mathrm e}^{8}}{3}-304 \,{\mathrm e}^{6}+1036 \,{\mathrm e}^{4}-\frac {4690 \,{\mathrm e}^{2}}{3}-24 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{2}+24 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x +16 \left ({\mathrm e}^{4}\right )^{2} \ln \left (2 \,{\mathrm e}^{2}+x -5\right )-32 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x -8 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{3}+12 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{2}+400 \ln \left (2 \,{\mathrm e}^{2}+x -5\right )-720 \ln \left (2 \,{\mathrm e}^{2}+x -5\right ) {\mathrm e}^{2}+484 \,{\mathrm e}^{4} \ln \left (2 \,{\mathrm e}^{2}+x -5\right )-80 \,{\mathrm e}^{2} {\mathrm e}^{4} \ln \left (2 \,{\mathrm e}^{2}+x -5\right )-4 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x +\frac {3525}{4}-64 \,{\mathrm e}^{6} \ln \left (2 \,{\mathrm e}^{2}+x -5\right )-32 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+160 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )-488 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+484 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+720 \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+144 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )^{3} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+16 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+720 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+{\mathrm e}^{x} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 23, normalized size = 0.88 \begin {gather*} {\left (x^{4} - 2 \, x^{3} + x^{2} + e^{x}\right )} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 37, normalized size = 1.42 \begin {gather*} \left (x^{4} - 2 x^{3} + x^{2}\right ) \log {\left (\frac {x}{2} - \frac {5}{2} + e^{2} \right )} + e^{x} \log {\left (\frac {x}{2} - \frac {5}{2} + e^{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 263 vs.
\(2 (21) = 42\).
time = 0.45, size = 263, normalized size = 10.12 \begin {gather*} {\left (x + 2 \, e^{2} - 5\right )}^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 8 \, {\left (x + 2 \, e^{2} - 5\right )}^{3} e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 18 \, {\left (x + 2 \, e^{2} - 5\right )}^{3} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 24 \, {\left (x + 2 \, e^{2} - 5\right )}^{2} e^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 108 \, {\left (x + 2 \, e^{2} - 5\right )}^{2} e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 121 \, {\left (x + 2 \, e^{2} - 5\right )}^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 32 \, {\left (x + 2 \, e^{2} - 5\right )} e^{6} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 216 \, {\left (x + 2 \, e^{2} - 5\right )} e^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 484 \, {\left (x + 2 \, e^{2} - 5\right )} e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 360 \, {\left (x + 2 \, e^{2} - 5\right )} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 16 \, e^{8} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 144 \, e^{6} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 484 \, e^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 720 \, e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + e^{x} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 400 \, \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.57, size = 23, normalized size = 0.88 \begin {gather*} \ln \left (\frac {x}{2}+{\mathrm {e}}^2-\frac {5}{2}\right )\,\left ({\mathrm {e}}^x+x^2-2\,x^3+x^4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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