Optimal. Leaf size=32 \[ \left (-e^3+\frac {(5-\log (-2+x))^4}{e^5}+\log \left (-\frac {1}{x}+x^2\right )\right )^2 \]
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Rubi [A]
time = 0.38, antiderivative size = 58, normalized size of antiderivative = 1.81, number of steps
used = 4, number of rules used = 3, integrand size = 407, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {12, 6820,
6818} \begin {gather*} \frac {\left (e^5 \log \left (-\frac {1-x^3}{x}\right )+\log ^4(x-2)-20 \log ^3(x-2)+150 \log ^2(x-2)-500 \log (x-2)-e^8+625\right )^2}{e^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {625000 x-625000 x^4+e^{13} \left (4-2 x+8 x^3-4 x^4\right )+e^5 \left (-2500+1250 x-5000 x^3+2500 x^4+e^3 \left (-1000 x+1000 x^4\right )\right )+\left (-875000 x+875000 x^4+e^5 \left (2000-1000 x+4000 x^3-2000 x^4+e^3 \left (600 x-600 x^4\right )\right )\right ) \log (-2+x)+\left (525000 x-525000 x^4+e^5 \left (-600+300 x-1200 x^3+600 x^4+e^3 \left (-120 x+120 x^4\right )\right )\right ) \log ^2(-2+x)+\left (-175000 x+175000 x^4+e^5 \left (80-40 x+160 x^3-80 x^4+e^3 \left (8 x-8 x^4\right )\right )\right ) \log ^3(-2+x)+\left (35000 x-35000 x^4+e^5 \left (-4+2 x-8 x^3+4 x^4\right )\right ) \log ^4(-2+x)+\left (-4200 x+4200 x^4\right ) \log ^5(-2+x)+\left (280 x-280 x^4\right ) \log ^6(-2+x)+\left (-8 x+8 x^4\right ) \log ^7(-2+x)+\left (e^5 \left (1000 x-1000 x^4\right )+e^{10} \left (-4+2 x-8 x^3+4 x^4\right )+e^5 \left (-600 x+600 x^4\right ) \log (-2+x)+e^5 \left (120 x-120 x^4\right ) \log ^2(-2+x)+e^5 \left (-8 x+8 x^4\right ) \log ^3(-2+x)\right ) \log \left (\frac {-1+x^3}{x}\right )}{2 x-x^2-2 x^4+x^5} \, dx}{e^{10}}\\ &=\frac {\int \frac {2 \left (500 x \left (-1+x^3\right )-e^5 \left (-2+x-4 x^3+2 x^4\right )-300 x \left (-1+x^3\right ) \log (-2+x)+60 x \left (-1+x^3\right ) \log ^2(-2+x)-4 x \left (-1+x^3\right ) \log ^3(-2+x)\right ) \left (-625 \left (1-\frac {e^8}{625}\right )+500 \log (-2+x)-150 \log ^2(-2+x)+20 \log ^3(-2+x)-\log ^4(-2+x)-e^5 \log \left (\frac {-1+x^3}{x}\right )\right )}{x \left (2-x-2 x^3+x^4\right )} \, dx}{e^{10}}\\ &=\frac {2 \int \frac {\left (500 x \left (-1+x^3\right )-e^5 \left (-2+x-4 x^3+2 x^4\right )-300 x \left (-1+x^3\right ) \log (-2+x)+60 x \left (-1+x^3\right ) \log ^2(-2+x)-4 x \left (-1+x^3\right ) \log ^3(-2+x)\right ) \left (-625 \left (1-\frac {e^8}{625}\right )+500 \log (-2+x)-150 \log ^2(-2+x)+20 \log ^3(-2+x)-\log ^4(-2+x)-e^5 \log \left (\frac {-1+x^3}{x}\right )\right )}{x \left (2-x-2 x^3+x^4\right )} \, dx}{e^{10}}\\ &=\frac {\left (625-e^8-500 \log (-2+x)+150 \log ^2(-2+x)-20 \log ^3(-2+x)+\log ^4(-2+x)+e^5 \log \left (-\frac {1-x^3}{x}\right )\right )^2}{e^{10}}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(32)=64\).
time = 0.15, size = 182, normalized size = 5.69 \begin {gather*} -\frac {2 \left (-500 \left (-625+e^8\right ) \log (2-x)+50 \left (-4375+3 e^8\right ) \log ^2(-2+x)-20 \left (-4375+e^8\right ) \log ^3(-2+x)+\left (-21875+e^8\right ) \log ^4(-2+x)+3500 \log ^5(-2+x)-350 \log ^6(-2+x)+20 \log ^7(-2+x)-\frac {1}{2} \log ^8(-2+x)-e^5 \left (-625+e^8\right ) \log (x)+e^5 \left (-625+e^8\right ) \log \left (1-x^3\right )-e^5 \log (-2+x) \left (-500+150 \log (-2+x)-20 \log ^2(-2+x)+\log ^3(-2+x)\right ) \log \left (\frac {-1+x^3}{x}\right )-\frac {1}{2} e^{10} \log ^2\left (\frac {-1+x^3}{x}\right )\right )}{e^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 18.28, size = 218135, normalized size = 6816.72
method | result | size |
risch | \(\text {Expression too large to display}\) | \(218135\) |
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. 392 vs.
\(2 (28) = 56\).
time = 0.82, size = 392, normalized size = 12.25 \begin {gather*} \frac {1}{21} \, {\left (21 \, \log \left (x - 2\right )^{8} - 840 \, \log \left (x - 2\right )^{7} + 14700 \, \log \left (x - 2\right )^{6} - 42 \, {\left (e^{5} \log \left (x\right ) + e^{8} - 21875\right )} \log \left (x - 2\right )^{4} - 147000 \, \log \left (x - 2\right )^{5} + 840 \, {\left (e^{5} \log \left (x\right ) + e^{8} - 4375\right )} \log \left (x - 2\right )^{3} + 21 \, e^{10} \log \left (x^{2} + x + 1\right )^{2} + 21 \, e^{10} \log \left (x - 1\right )^{2} - 2100 \, {\left (3 \, e^{5} \log \left (x\right ) + 3 \, e^{8} - 4375\right )} \log \left (x - 2\right )^{2} + 21 \, e^{10} \log \left (x\right )^{2} - 4 \, \sqrt {3} {\left (e^{13} - 625 \, e^{5}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + 2 \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 5 \, \log \left (x^{2} + x + 1\right ) - 14 \, \log \left (x - 1\right ) + 3 \, \log \left (x - 2\right ) + 21 \, \log \left (x\right )\right )} e^{13} - 1250 \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 5 \, \log \left (x^{2} + x + 1\right ) - 14 \, \log \left (x - 1\right ) + 3 \, \log \left (x - 2\right ) + 21 \, \log \left (x\right )\right )} e^{5} + 2 \, {\left (21 \, e^{5} \log \left (x - 2\right )^{4} - 420 \, e^{5} \log \left (x - 2\right )^{3} + 3150 \, e^{5} \log \left (x - 2\right )^{2} + 21 \, e^{10} \log \left (x - 1\right ) - 10500 \, e^{5} \log \left (x - 2\right ) - 21 \, e^{10} \log \left (x\right ) - 16 \, e^{13} + 10000 \, e^{5}\right )} \log \left (x^{2} + x + 1\right ) + 14 \, {\left (3 \, e^{5} \log \left (x - 2\right )^{4} - 60 \, e^{5} \log \left (x - 2\right )^{3} + 450 \, e^{5} \log \left (x - 2\right )^{2} - 1500 \, e^{5} \log \left (x - 2\right ) - 3 \, e^{10} \log \left (x\right ) - e^{13} + 625 \, e^{5}\right )} \log \left (x - 1\right ) + 6 \, {\left (3500 \, e^{5} \log \left (x\right ) - e^{13} + 3500 \, e^{8} + 625 \, e^{5} - 2187500\right )} \log \left (x - 2\right )\right )} e^{\left (-10\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. 155 vs.
\(2 (28) = 56\).
time = 0.37, size = 155, normalized size = 4.84 \begin {gather*} {\left (\log \left (x - 2\right )^{8} - 40 \, \log \left (x - 2\right )^{7} + 700 \, \log \left (x - 2\right )^{6} - 2 \, {\left (e^{8} - 21875\right )} \log \left (x - 2\right )^{4} - 7000 \, \log \left (x - 2\right )^{5} + 40 \, {\left (e^{8} - 4375\right )} \log \left (x - 2\right )^{3} - 100 \, {\left (3 \, e^{8} - 4375\right )} \log \left (x - 2\right )^{2} + e^{10} \log \left (\frac {x^{3} - 1}{x}\right )^{2} + 1000 \, {\left (e^{8} - 625\right )} \log \left (x - 2\right ) + 2 \, {\left (e^{5} \log \left (x - 2\right )^{4} - 20 \, e^{5} \log \left (x - 2\right )^{3} + 150 \, e^{5} \log \left (x - 2\right )^{2} - 500 \, e^{5} \log \left (x - 2\right ) - e^{13} + 625 \, e^{5}\right )} \log \left (\frac {x^{3} - 1}{x}\right )\right )} e^{\left (-10\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. 272 vs.
\(2 (24) = 48\).
time = 13.63, size = 272, normalized size = 8.50 \begin {gather*} \frac {\left (2 \log {\left (x - 2 \right )}^{4} - 40 \log {\left (x - 2 \right )}^{3} + 300 \log {\left (x - 2 \right )}^{2} - 1000 \log {\left (x - 2 \right )}\right ) \log {\left (\frac {x^{3} - 1}{x} \right )}}{e^{5}} + \log {\left (\frac {x^{3} - 1}{x} \right )}^{2} + \frac {\log {\left (x - 2 \right )}^{8}}{e^{10}} - \frac {40 \log {\left (x - 2 \right )}^{7}}{e^{10}} + \frac {700 \log {\left (x - 2 \right )}^{6}}{e^{10}} - \frac {7000 \log {\left (x - 2 \right )}^{5}}{e^{10}} + \frac {\left (43750 - 2 e^{8}\right ) \log {\left (x - 2 \right )}^{4}}{e^{10}} + \frac {\left (-175000 + 40 e^{8}\right ) \log {\left (x - 2 \right )}^{3}}{e^{10}} + \frac {\left (437500 - 300 e^{8}\right ) \log {\left (x - 2 \right )}^{2}}{e^{10}} + \frac {\left (-1250 + 2 e^{8}\right ) \log {\left (x + \frac {- 2 e^{13} + 1250 e^{5} + \left (-1250 + 2 e^{8}\right ) e^{5}}{- 500 e^{8} - 625 e^{5} + 312500 + e^{13}} \right )}}{e^{5}} + \frac {\left (-625000 + 1000 e^{8}\right ) \log {\left (x + \frac {- 2 e^{13} - 625000 + 1250 e^{5} + 1000 e^{8}}{- 500 e^{8} - 625 e^{5} + 312500 + e^{13}} \right )}}{e^{10}} - \frac {2 \left (-5 + e^{2}\right ) \left (5 + e^{2}\right ) \left (25 + e^{4}\right ) \log {\left (x^{3} - 1 \right )}}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.36, size = 238, normalized size = 7.44 \begin {gather*} 1000\,\ln \left (x-2\right )\,{\mathrm {e}}^{-2}-625000\,\ln \left (x-2\right )\,{\mathrm {e}}^{-10}+{\ln \left (\frac {x^3-1}{x}\right )}^2-300\,{\ln \left (x-2\right )}^2\,{\mathrm {e}}^{-2}-2\,{\mathrm {e}}^3\,\ln \left (x^3-1\right )+40\,{\ln \left (x-2\right )}^3\,{\mathrm {e}}^{-2}-2\,{\ln \left (x-2\right )}^4\,{\mathrm {e}}^{-2}+1250\,{\mathrm {e}}^{-5}\,\ln \left (x^3-1\right )+437500\,{\ln \left (x-2\right )}^2\,{\mathrm {e}}^{-10}-175000\,{\ln \left (x-2\right )}^3\,{\mathrm {e}}^{-10}+43750\,{\ln \left (x-2\right )}^4\,{\mathrm {e}}^{-10}-7000\,{\ln \left (x-2\right )}^5\,{\mathrm {e}}^{-10}+700\,{\ln \left (x-2\right )}^6\,{\mathrm {e}}^{-10}-40\,{\ln \left (x-2\right )}^7\,{\mathrm {e}}^{-10}+{\ln \left (x-2\right )}^8\,{\mathrm {e}}^{-10}+2\,{\mathrm {e}}^3\,\ln \left (x\right )-1250\,{\mathrm {e}}^{-5}\,\ln \left (x\right )+300\,{\ln \left (x-2\right )}^2\,{\mathrm {e}}^{-5}\,\ln \left (\frac {x^3-1}{x}\right )-40\,{\ln \left (x-2\right )}^3\,{\mathrm {e}}^{-5}\,\ln \left (\frac {x^3-1}{x}\right )+2\,{\ln \left (x-2\right )}^4\,{\mathrm {e}}^{-5}\,\ln \left (\frac {x^3-1}{x}\right )-1000\,\ln \left (x-2\right )\,{\mathrm {e}}^{-5}\,\ln \left (\frac {x^3-1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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