Optimal. Leaf size=27 \[ 5 x \left (x^2+5 (9-x) \log \left (x (1+x) \left (e^2+x\right )\right )\right )^2 \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 3.00, antiderivative size = 1165, normalized size of antiderivative = 43.15, number of
steps used = 127, number of rules used = 17, integrand size = 185, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.092, Rules used = {6820,
12, 6874, 1626, 2608, 2603, 907, 2605, 2604, 2465, 2438, 2437, 2338, 2441, 2440, 2439, 2352}
\begin {gather*} 5 x^5-50 \log \left (x (x+1) \left (x+e^2\right )\right ) x^4+125 \log ^2\left (x (x+1) \left (x+e^2\right )\right ) x^3+450 \log \left (x (x+1) \left (x+e^2\right )\right ) x^3+\frac {50}{3} \left (28+e^2\right ) x^3-\frac {50}{3} \left (1+e^2\right ) x^3-450 x^3-2250 \log ^2\left (x (x+1) \left (x+e^2\right )\right ) x^2-125 \left (55+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right ) x^2+125 \left (1+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right ) x^2+6750 \log \left (x (x+1) \left (x+e^2\right )\right ) x^2-25 \left (10+9 e^2+e^4\right ) x^2+25 \left (1+e^4\right ) x^2+\frac {375}{2} \left (55+e^2\right ) x^2+\frac {75}{2} \left (1+e^2\right ) x^2-10125 x^2+10125 \log ^2\left (x (x+1) \left (x+e^2\right )\right ) x+250 \left (262+18 e^2+e^4\right ) \log \left (x (x+1) \left (x+e^2\right )\right ) x-250 \left (1+e^4\right ) \log \left (x (x+1) \left (x+e^2\right )\right ) x-4500 \left (1+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right ) x-60750 \log \left (x (x+1) \left (x+e^2\right )\right ) x+50 \left (10+9 e^4+e^6\right ) x-50 \left (1+e^6\right ) x-750 \left (262+18 e^2+e^4\right ) x+300 \left (1+e^4\right ) x-125 \left (1+e^2\right ) \left (55+e^2\right ) x+125 \left (1+e^2\right )^2 x+20250 \left (1+e^2\right ) x+182250 x-10250 \log ^2(-x-1)+10250 \log ^2(x+1)+125 e^2 \left (9+e^2\right )^2 \log ^2\left (x+e^2\right )-125 e^6 \log ^2\left (x+e^2\right )-2250 e^4 \log ^2\left (x+e^2\right )-10125 e^2 \log ^2\left (x+e^2\right )-20500 \log (-x-1) \log (-x)+500 e^2 \left (9+e^2\right )^2 \log (x)-500 e^6 \log (x)-9000 e^4 \log (x)-40500 e^2 \log (x)+250 \left (262+18 e^2+e^4\right ) \log (x+1)-250 \left (1+e^4\right ) \log (x+1)+125 \left (55+e^2\right ) \log (x+1)-4625 \left (1+e^2\right ) \log (x+1)-67500 \log (x+1)+250 e^2 \left (9+e^2\right )^2 \log \left (\frac {x+1}{1-e^2}\right ) \log \left (x+e^2\right )-250 e^6 \log \left (\frac {x+1}{1-e^2}\right ) \log \left (x+e^2\right )-4500 e^4 \log \left (\frac {x+1}{1-e^2}\right ) \log \left (x+e^2\right )-20250 e^2 \log \left (\frac {x+1}{1-e^2}\right ) \log \left (x+e^2\right )+250 e^2 \left (262+18 e^2+e^4\right ) \log \left (x+e^2\right )-250 e^2 \left (1+e^4\right ) \log \left (x+e^2\right )+125 e^4 \left (55+e^2\right ) \log \left (x+e^2\right )-50 e^6 \left (9+e^2\right ) \log \left (x+e^2\right )-125 e^4 \left (1+e^2\right ) \log \left (x+e^2\right )-4500 e^2 \left (1+e^2\right ) \log \left (x+e^2\right )+50 e^8 \log \left (x+e^2\right )+450 e^6 \log \left (x+e^2\right )-6750 e^4 \log \left (x+e^2\right )-60750 e^2 \log \left (x+e^2\right )-20500 \log (-x-1) \log \left (-\frac {x+e^2}{1-e^2}\right )+20500 \log (x+1) \log \left (-\frac {x+e^2}{1-e^2}\right )+20500 \log (-x-1) \log \left (x (x+1) \left (x+e^2\right )\right )-20500 \log (x+1) \log \left (x (x+1) \left (x+e^2\right )\right )-250 e^2 \left (9+e^2\right )^2 \log \left (x+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right )+250 e^6 \log \left (x+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right )+4500 e^4 \log \left (x+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right )+20250 e^2 \log \left (x+e^2\right ) \log \left (x (x+1) \left (x+e^2\right )\right )-20500 \text {Li}_2(-x)-250 e^2 \left (9+e^2\right )^2 \text {Li}_2\left (-\frac {x}{e^2}\right )+250 e^6 \text {Li}_2\left (-\frac {x}{e^2}\right )+4500 e^4 \text {Li}_2\left (-\frac {x}{e^2}\right )+20250 e^2 \text {Li}_2\left (-\frac {x}{e^2}\right )-20500 \text {Li}_2(x+1)+250 e^2 \left (9+e^2\right )^2 \text {Li}_2\left (-\frac {x+e^2}{1-e^2}\right )-250 e^6 \text {Li}_2\left (-\frac {x+e^2}{1-e^2}\right )-4500 e^4 \text {Li}_2\left (-\frac {x+e^2}{1-e^2}\right )-20250 e^2 \text {Li}_2\left (-\frac {x+e^2}{1-e^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 907
Rule 1626
Rule 2338
Rule 2352
Rule 2437
Rule 2438
Rule 2439
Rule 2440
Rule 2441
Rule 2465
Rule 2603
Rule 2604
Rule 2605
Rule 2608
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (x^2-5 (-9+x) \log \left (x (1+x) \left (e^2+x\right )\right )\right ) \left (x \left (36+50 x-5 x^2+x^3\right )+e^2 \left (18+34 x-3 x^2+x^3\right )-3 \left (e^2+x\right ) \left (-3-2 x+x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )\right )}{(1+x) \left (e^2+x\right )} \, dx\\ &=25 \int \frac {\left (x^2-5 (-9+x) \log \left (x (1+x) \left (e^2+x\right )\right )\right ) \left (x \left (36+50 x-5 x^2+x^3\right )+e^2 \left (18+34 x-3 x^2+x^3\right )-3 \left (e^2+x\right ) \left (-3-2 x+x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )\right )}{(1+x) \left (e^2+x\right )} \, dx\\ &=25 \int \left (\frac {x^2 \left (18 e^2+2 \left (18+17 e^2\right ) x+\left (50-3 e^2\right ) x^2-\left (5-e^2\right ) x^3+x^4\right )}{(1+x) \left (e^2+x\right )}+\frac {2 \left (405 e^2+810 \left (1+\frac {8 e^2}{9}\right ) x+1035 \left (1-\frac {148 e^2}{1035}\right ) x^2-233 \left (1-\frac {33 e^2}{233}\right ) x^3+38 \left (1-\frac {2 e^2}{19}\right ) x^4-4 x^5\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )}+15 (-9+x) (-3+x) \log ^2\left (x (1+x) \left (e^2+x\right )\right )\right ) \, dx\\ &=25 \int \frac {x^2 \left (18 e^2+2 \left (18+17 e^2\right ) x+\left (50-3 e^2\right ) x^2-\left (5-e^2\right ) x^3+x^4\right )}{(1+x) \left (e^2+x\right )} \, dx+50 \int \frac {\left (405 e^2+810 \left (1+\frac {8 e^2}{9}\right ) x+1035 \left (1-\frac {148 e^2}{1035}\right ) x^2-233 \left (1-\frac {33 e^2}{233}\right ) x^3+38 \left (1-\frac {2 e^2}{19}\right ) x^4-4 x^5\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx+375 \int (-9+x) (-3+x) \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx\\ &=25 \int \left (2 \left (10+9 e^4+e^6\right )-2 \left (10+9 e^2+e^4\right ) x+2 \left (28+e^2\right ) x^2-6 x^3+x^4-\frac {20}{1+x}-\frac {2 e^6 \left (9+e^2\right )}{e^2+x}\right ) \, dx+50 \int \left (5 \left (262+18 e^2+e^4\right ) \log \left (x (1+x) \left (e^2+x\right )\right )-5 \left (55+e^2\right ) x \log \left (x (1+x) \left (e^2+x\right )\right )+42 x^2 \log \left (x (1+x) \left (e^2+x\right )\right )-4 x^3 \log \left (x (1+x) \left (e^2+x\right )\right )-\frac {500 \log \left (x (1+x) \left (e^2+x\right )\right )}{1+x}-\frac {5 e^2 \left (9+e^2\right )^2 \log \left (x (1+x) \left (e^2+x\right )\right )}{e^2+x}\right ) \, dx+375 \int \left (27 \log ^2\left (x (1+x) \left (e^2+x\right )\right )-12 x \log ^2\left (x (1+x) \left (e^2+x\right )\right )+x^2 \log ^2\left (x (1+x) \left (e^2+x\right )\right )\right ) \, dx\\ &=50 \left (10+9 e^4+e^6\right ) x-25 \left (10+9 e^2+e^4\right ) x^2+\frac {50}{3} \left (28+e^2\right ) x^3-\frac {75 x^4}{2}+5 x^5-500 \log (1+x)-50 e^6 \left (9+e^2\right ) \log \left (e^2+x\right )-200 \int x^3 \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx+375 \int x^2 \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx+2100 \int x^2 \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx-4500 \int x \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx+10125 \int \log ^2\left (x (1+x) \left (e^2+x\right )\right ) \, dx-25000 \int \frac {\log \left (x (1+x) \left (e^2+x\right )\right )}{1+x} \, dx-\left (250 e^2 \left (9+e^2\right )^2\right ) \int \frac {\log \left (x (1+x) \left (e^2+x\right )\right )}{e^2+x} \, dx-\left (250 \left (55+e^2\right )\right ) \int x \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx+\left (250 \left (262+18 e^2+e^4\right )\right ) \int \log \left (x (1+x) \left (e^2+x\right )\right ) \, dx\\ &=50 \left (10+9 e^4+e^6\right ) x-25 \left (10+9 e^2+e^4\right ) x^2+\frac {50}{3} \left (28+e^2\right ) x^3-\frac {75 x^4}{2}+5 x^5-500 \log (1+x)-50 e^6 \left (9+e^2\right ) \log \left (e^2+x\right )+250 \left (262+18 e^2+e^4\right ) x \log \left (x (1+x) \left (e^2+x\right )\right )-125 \left (55+e^2\right ) x^2 \log \left (x (1+x) \left (e^2+x\right )\right )+700 x^3 \log \left (x (1+x) \left (e^2+x\right )\right )-50 x^4 \log \left (x (1+x) \left (e^2+x\right )\right )-25000 \log (1+x) \log \left (x (1+x) \left (e^2+x\right )\right )-250 e^2 \left (9+e^2\right )^2 \log \left (e^2+x\right ) \log \left (x (1+x) \left (e^2+x\right )\right )+10125 x \log ^2\left (x (1+x) \left (e^2+x\right )\right )-2250 x^2 \log ^2\left (x (1+x) \left (e^2+x\right )\right )+125 x^3 \log ^2\left (x (1+x) \left (e^2+x\right )\right )+50 \int \frac {x^3 \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right )}{(1+x) \left (e^2+x\right )} \, dx-250 \int \frac {x^2 \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx-700 \int \frac {x^2 \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right )}{(1+x) \left (e^2+x\right )} \, dx+4500 \int \frac {x \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx-20250 \int \frac {\left (e^2+2 \left (1+e^2\right ) x+3 x^2\right ) \log \left (x (1+x) \left (e^2+x\right )\right )}{(1+x) \left (e^2+x\right )} \, dx+25000 \int \frac {\left (x (1+x)+x \left (e^2+x\right )+(1+x) \left (e^2+x\right )\right ) \log (1+x)}{x (1+x) \left (e^2+x\right )} \, dx+\left (250 e^2 \left (9+e^2\right )^2\right ) \int \frac {\left (x (1+x)+x \left (e^2+x\right )+(1+x) \left (e^2+x\right )\right ) \log \left (e^2+x\right )}{x (1+x) \left (e^2+x\right )} \, dx+\left (125 \left (55+e^2\right )\right ) \int \frac {x \left (e^2+2 \left (1+e^2\right ) x+3 x^2\right )}{(1+x) \left (e^2+x\right )} \, dx-\left (250 \left (262+18 e^2+e^4\right )\right ) \int \frac {e^2+2 \left (1+e^2\right ) x+3 x^2}{(1+x) \left (e^2+x\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.93 \begin {gather*} 5 x \left (x^2-5 (-9+x) \log \left (x (1+x) \left (e^2+x\right )\right )\right )^2 \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. \(67\) vs.
\(2(25)=50\).
time = 0.85, size = 68, normalized size = 2.52
method | result | size |
risch | \(\left (125 x^{3}-2250 x^{2}+10125 x \right ) \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}+\left (-50 x^{4}+450 x^{3}\right ) \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )+5 x^{5}\) | \(68\) |
norman | \(5 x^{5}+10125 x \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}-2250 x^{2} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}+450 x^{3} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )+125 x^{3} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )^{2}-50 x^{4} \ln \left (\left (x^{2}+x \right ) {\mathrm e}^{2}+x^{3}+x^{2}\right )\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 587 vs.
\(2 (24) = 48\).
time = 0.33, size = 587, normalized size = 21.74 \begin {gather*} 5 \, x^{5} - \frac {25}{4} \, x^{4} {\left (e^{2} + 1\right )} + \frac {25}{4} \, x^{4} + \frac {25}{3} \, x^{3} {\left (e^{4} + e^{2} + 1\right )} - \frac {50}{3} \, x^{3} {\left (e^{2} + 28\right )} + \frac {125}{3} \, x^{3} {\left (e^{2} + 1\right )} + \frac {1250}{3} \, x^{3} - \frac {25}{2} \, x^{2} {\left (e^{6} + e^{4} + e^{2} + 1\right )} + 25 \, x^{2} {\left (e^{4} + 9 \, e^{2} + 10\right )} - \frac {125}{2} \, x^{2} {\left (e^{4} + e^{2} + 1\right )} - 625 \, x^{2} {\left (e^{2} + 1\right )} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x + e^{2}\right )^{2} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x + 1\right )^{2} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x\right )^{2} + 450 \, x^{2} + 25 \, x {\left (e^{8} + e^{6} + e^{4} + e^{2} + 1\right )} - 50 \, x {\left (e^{6} + 9 \, e^{4} + 10\right )} + 125 \, x {\left (e^{6} + e^{4} + e^{2} + 1\right )} + 1250 \, x {\left (e^{4} + e^{2} + 1\right )} - 900 \, x {\left (e^{2} + 1\right )} + \frac {25}{12} \, {\left (3 \, x^{4} - 4 \, x^{3} {\left (e^{2} + 1\right )} + 6 \, x^{2} {\left (e^{4} + e^{2} + 1\right )} - 12 \, x {\left (e^{6} + e^{4} + e^{2} + 1\right )} + \frac {12 \, e^{10} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {12 \, \log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} - \frac {25}{2} \, {\left (2 \, x^{3} - 3 \, x^{2} {\left (e^{2} + 1\right )} + 6 \, x {\left (e^{4} + e^{2} + 1\right )} - \frac {6 \, e^{8} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {6 \, \log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} + 425 \, {\left (x^{2} - 2 \, x {\left (e^{2} + 1\right )} + \frac {2 \, e^{6} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {2 \, \log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} + 450 \, {\left (x - \frac {e^{4} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {\log \left (x + 1\right )}{e^{2} - 1}\right )} e^{2} - 50 \, {\left (x^{4} - 9 \, x^{3} - 5 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x + 1\right ) - 5 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x\right ) - e^{8} - 9 \, e^{6}\right )} \log \left (x + e^{2}\right ) - 50 \, {\left (x^{4} - 9 \, x^{3} - 5 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x\right ) - 10\right )} \log \left (x + 1\right ) - 50 \, {\left (x^{4} - 9 \, x^{3}\right )} \log \left (x\right ) - \frac {25 \, e^{12} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {125 \, e^{10} \log \left (x + e^{2}\right )}{e^{2} - 1} - \frac {1250 \, e^{8} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {900 \, e^{6} \log \left (x + e^{2}\right )}{e^{2} - 1} + \frac {500 \, \log \left (x + 1\right )}{e^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (24) = 48\).
time = 0.36, size = 65, normalized size = 2.41 \begin {gather*} 5 \, x^{5} + 125 \, {\left (x^{3} - 18 \, x^{2} + 81 \, x\right )} \log \left (x^{3} + x^{2} + {\left (x^{2} + x\right )} e^{2}\right )^{2} - 50 \, {\left (x^{4} - 9 \, x^{3}\right )} \log \left (x^{3} + x^{2} + {\left (x^{2} + x\right )} e^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (24) = 48\).
time = 0.16, size = 63, normalized size = 2.33 \begin {gather*} 5 x^{5} + \left (- 50 x^{4} + 450 x^{3}\right ) \log {\left (x^{3} + x^{2} + \left (x^{2} + x\right ) e^{2} \right )} + \left (125 x^{3} - 2250 x^{2} + 10125 x\right ) \log {\left (x^{3} + x^{2} + \left (x^{2} + x\right ) e^{2} \right )}^{2} \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. 125 vs.
\(2 (24) = 48\).
time = 0.54, size = 125, normalized size = 4.63 \begin {gather*} 5 \, x^{5} - 50 \, x^{4} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right ) + 125 \, x^{3} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right )^{2} + 450 \, x^{3} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right ) - 2250 \, x^{2} \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right )^{2} + 10125 \, x \log \left (x^{3} + x^{2} e^{2} + x^{2} + x e^{2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 46, normalized size = 1.70 \begin {gather*} 5\,x\,{\left (45\,\ln \left ({\mathrm {e}}^2\,\left (x^2+x\right )+x^2+x^3\right )-5\,x\,\ln \left ({\mathrm {e}}^2\,\left (x^2+x\right )+x^2+x^3\right )+x^2\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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