Optimal. Leaf size=25 \[ x \left (-\frac {x}{-1+x}+2 e^3 x^3-\log (x)\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(25)=50\).
time = 0.38, antiderivative size = 115, normalized size of antiderivative = 4.60, number of steps
used = 17, number of rules used = 10, integrand size = 141, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6874, 37,
45, 907, 2404, 2332, 2351, 31, 2341, 2333} \begin {gather*} 4 e^6 x^7-4 e^3 x^4-4 e^3 x^4 \log (x)-4 e^3 x^3-\frac {x^2}{(1-x)^2}-4 e^3 x^2-4 e^3 x+x+\frac {4 e^3}{1-x}-\frac {5}{1-x}+\frac {2}{(1-x)^2}+x \log ^2(x)-\frac {2 x \log (x)}{1-x}+2 x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 37
Rule 45
Rule 907
Rule 2332
Rule 2333
Rule 2341
Rule 2351
Rule 2404
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 x}{(-1+x)^3}-\frac {7 x^2}{(-1+x)^3}+\frac {3 x^3}{(-1+x)^3}+28 e^6 x^6-\frac {4 e^3 x^3 \left (1-7 x+5 x^2\right )}{(-1+x)^2}-\frac {2 \left (-1+4 x-2 x^2+8 e^3 x^3-16 e^3 x^4+8 e^3 x^5\right ) \log (x)}{(-1+x)^2}+\log ^2(x)\right ) \, dx\\ &=4 e^6 x^7+2 \int \frac {x}{(-1+x)^3} \, dx-2 \int \frac {\left (-1+4 x-2 x^2+8 e^3 x^3-16 e^3 x^4+8 e^3 x^5\right ) \log (x)}{(-1+x)^2} \, dx+3 \int \frac {x^3}{(-1+x)^3} \, dx-7 \int \frac {x^2}{(-1+x)^3} \, dx-\left (4 e^3\right ) \int \frac {x^3 \left (1-7 x+5 x^2\right )}{(-1+x)^2} \, dx+\int \log ^2(x) \, dx\\ &=-\frac {x^2}{(1-x)^2}+4 e^6 x^7+x \log ^2(x)-2 \int \log (x) \, dx-2 \int \left (-2 \log (x)+\frac {\log (x)}{(-1+x)^2}+8 e^3 x^3 \log (x)\right ) \, dx+3 \int \left (1+\frac {1}{(-1+x)^3}+\frac {3}{(-1+x)^2}+\frac {3}{-1+x}\right ) \, dx-7 \int \left (\frac {1}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx-\left (4 e^3\right ) \int \left (1-\frac {1}{(-1+x)^2}+2 x+3 x^2+5 x^3\right ) \, dx\\ &=\frac {2}{(1-x)^2}-\frac {5}{1-x}+\frac {4 e^3}{1-x}+5 x-4 e^3 x-4 e^3 x^2-\frac {x^2}{(1-x)^2}-4 e^3 x^3-5 e^3 x^4+4 e^6 x^7+2 \log (1-x)-2 x \log (x)+x \log ^2(x)-2 \int \frac {\log (x)}{(-1+x)^2} \, dx+4 \int \log (x) \, dx-\left (16 e^3\right ) \int x^3 \log (x) \, dx\\ &=\frac {2}{(1-x)^2}-\frac {5}{1-x}+\frac {4 e^3}{1-x}+x-4 e^3 x-4 e^3 x^2-\frac {x^2}{(1-x)^2}-4 e^3 x^3-4 e^3 x^4+4 e^6 x^7+2 \log (1-x)+2 x \log (x)-\frac {2 x \log (x)}{1-x}-4 e^3 x^4 \log (x)+x \log ^2(x)-2 \int \frac {1}{-1+x} \, dx\\ &=\frac {2}{(1-x)^2}-\frac {5}{1-x}+\frac {4 e^3}{1-x}+x-4 e^3 x-4 e^3 x^2-\frac {x^2}{(1-x)^2}-4 e^3 x^3-4 e^3 x^4+4 e^6 x^7+2 x \log (x)-\frac {2 x \log (x)}{1-x}-4 e^3 x^4 \log (x)+x \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(25)=50\).
time = 0.07, size = 121, normalized size = 4.84 \begin {gather*} -3+17 e^3-4 e^6+\frac {1}{(-1+x)^2}+\frac {3}{-1+x}-\frac {4 e^3}{-1+x}+x-4 e^3 x-4 e^3 x^2-4 e^3 x^3-4 e^3 x^4+4 e^6 x^7-2 \log (1-x)+2 \log (-1+x)+2 \log (x)+\frac {2 \log (x)}{-1+x}+2 x \log (x)-4 e^3 x^4 \log (x)+x \log ^2(x) \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. \(86\) vs.
\(2(24)=48\).
time = 0.07, size = 87, normalized size = 3.48
method | result | size |
default | \(x \ln \left (x \right )^{2}+2 x \ln \left (x \right )+x +4 x^{7} {\mathrm e}^{6}-4 x^{4} {\mathrm e}^{3}-4 x^{3} {\mathrm e}^{3}-4 x^{2} {\mathrm e}^{3}-4 x \,{\mathrm e}^{3}+\frac {1}{\left (x -1\right )^{2}}-\frac {4 \,{\mathrm e}^{3}}{x -1}+\frac {3}{x -1}-4 \ln \left (x \right ) {\mathrm e}^{3} x^{4}+\frac {2 \ln \left (x \right ) x}{x -1}\) | \(87\) |
risch | \(x \ln \left (x \right )^{2}-\frac {2 \left (2 x^{5} {\mathrm e}^{3}-2 x^{4} {\mathrm e}^{3}-x^{2}+x -1\right ) \ln \left (x \right )}{x -1}+\frac {4 \,{\mathrm e}^{6} x^{9}-8 \,{\mathrm e}^{6} x^{8}+4 x^{7} {\mathrm e}^{6}-4 x^{6} {\mathrm e}^{3}+4 x^{5} {\mathrm e}^{3}+2 x^{2} \ln \left (x \right )+4 x^{2} {\mathrm e}^{3}+x^{3}-4 x \ln \left (x \right )-8 x \,{\mathrm e}^{3}-2 x^{2}+2 \ln \left (x \right )+4 \,{\mathrm e}^{3}+4 x -2}{\left (x -1\right )^{2}}\) | \(125\) |
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. 557 vs.
\(2 (24) = 48\).
time = 0.32, size = 557, normalized size = 22.28 \begin {gather*} \frac {2}{5} \, {\left (10 \, x^{7} + 35 \, x^{6} + 84 \, x^{5} + 175 \, x^{4} + 350 \, x^{3} + 735 \, x^{2} + 1960 \, x - \frac {35 \, {\left (18 \, x - 17\right )}}{x^{2} - 2 \, x + 1} + 2520 \, \log \left (x - 1\right )\right )} e^{6} - \frac {14}{5} \, {\left (5 \, x^{6} + 18 \, x^{5} + 45 \, x^{4} + 100 \, x^{3} + 225 \, x^{2} + 630 \, x - \frac {15 \, {\left (16 \, x - 15\right )}}{x^{2} - 2 \, x + 1} + 840 \, \log \left (x - 1\right )\right )} e^{6} + \frac {21}{5} \, {\left (4 \, x^{5} + 15 \, x^{4} + 40 \, x^{3} + 100 \, x^{2} + 300 \, x - \frac {10 \, {\left (14 \, x - 13\right )}}{x^{2} - 2 \, x + 1} + 420 \, \log \left (x - 1\right )\right )} e^{6} - 7 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 40 \, x - \frac {2 \, {\left (12 \, x - 11\right )}}{x^{2} - 2 \, x + 1} + 60 \, \log \left (x - 1\right )\right )} e^{6} - 5 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 40 \, x - \frac {2 \, {\left (12 \, x - 11\right )}}{x^{2} - 2 \, x + 1} + 60 \, \log \left (x - 1\right )\right )} e^{3} + 8 \, {\left (2 \, x^{3} + 9 \, x^{2} + 36 \, x - \frac {3 \, {\left (10 \, x - 9\right )}}{x^{2} - 2 \, x + 1} + 60 \, \log \left (x - 1\right )\right )} e^{3} - 16 \, {\left (x^{2} + 6 \, x - \frac {8 \, x - 7}{x^{2} - 2 \, x + 1} + 12 \, \log \left (x - 1\right )\right )} e^{3} + 2 \, {\left (2 \, x - \frac {6 \, x - 5}{x^{2} - 2 \, x + 1} + 6 \, \log \left (x - 1\right )\right )} e^{3} + 3 \, x - \frac {5 \, {\left (2 \, x - 1\right )} \log \left (x\right )}{x^{2} - 2 \, x + 1} + \frac {x^{6} e^{3} - 2 \, x^{5} e^{3} + x^{4} e^{3} - 2 \, x^{3} + {\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 2 \, {\left (2 \, x^{6} e^{3} - 4 \, x^{5} e^{3} + 2 \, x^{4} e^{3} - x^{3} - 2 \, x^{2} + x\right )} \log \left (x\right ) + 2 \, x - 4}{x^{2} - 2 \, x + 1} - \frac {3 \, {\left (6 \, x - 5\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7 \, {\left (4 \, x - 3\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, x - 1}{x^{2} - 2 \, x + 1} + \frac {\log \left (x\right )}{x^{2} - 2 \, x + 1} - \frac {4}{x - 1} - 6 \, \log \left (x\right ) \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. 106 vs.
\(2 (24) = 48\).
time = 0.38, size = 106, normalized size = 4.24 \begin {gather*} \frac {x^{3} + {\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (x\right )^{2} - 2 \, x^{2} + 4 \, {\left (x^{9} - 2 \, x^{8} + x^{7}\right )} e^{6} - 4 \, {\left (x^{6} - x^{5} - x^{2} + 2 \, x - 1\right )} e^{3} + 2 \, {\left (x^{3} - x^{2} - 2 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} e^{3}\right )} \log \left (x\right ) + 4 \, x - 2}{x^{2} - 2 \, x + 1} \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. 110 vs.
\(2 (19) = 38\).
time = 0.44, size = 110, normalized size = 4.40 \begin {gather*} 4 x^{7} e^{6} - 4 x^{4} e^{3} - 4 x^{3} e^{3} - 4 x^{2} e^{3} + x \log {\left (x \right )}^{2} + x \left (1 - 4 e^{3}\right ) + 2 \log {\left (x \right )} + \frac {x \left (3 - 4 e^{3}\right ) - 2 + 4 e^{3}}{x^{2} - 2 x + 1} + \frac {\left (- 4 x^{5} e^{3} + 4 x^{4} e^{3} + 2 x^{2} - 2 x + 2\right ) \log {\left (x \right )}}{x - 1} \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. 139 vs.
\(2 (24) = 48\).
time = 0.40, size = 139, normalized size = 5.56 \begin {gather*} \frac {4 \, x^{9} e^{6} - 8 \, x^{8} e^{6} + 4 \, x^{7} e^{6} - 4 \, x^{6} e^{3} \log \left (x\right ) - 4 \, x^{6} e^{3} + 8 \, x^{5} e^{3} \log \left (x\right ) + 4 \, x^{5} e^{3} - 4 \, x^{4} e^{3} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + 2 \, x^{3} \log \left (x\right ) - 2 \, x^{2} \log \left (x\right )^{2} + x^{3} + 4 \, x^{2} e^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} - 2 \, x^{2} - 8 \, x e^{3} + 4 \, x + 4 \, e^{3} - 2}{x^{2} - 2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.85, size = 33, normalized size = 1.32 \begin {gather*} \frac {x\,{\left (x-\ln \left (x\right )+2\,x^3\,{\mathrm {e}}^3-2\,x^4\,{\mathrm {e}}^3+x\,\ln \left (x\right )\right )}^2}{{\left (x-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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