Optimal. Leaf size=27 \[ \left (-\log \left (-\frac {15 x}{16}\right )+\frac {e^3}{x+\log (x)-\log (2 x)}\right )^2 \]
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Rubi [F]
time = 2.55, 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 {2 e^6 x+2 e^3 x^2+\left (-2 e^3 x^2-2 x^3\right ) \log \left (-\frac {15 x}{16}\right )+\left (4 e^3 x+\left (-2 e^3 x-6 x^2\right ) \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )\right ) \log ^2(x)-2 \log \left (-\frac {15 x}{16}\right ) \log ^3(x)+\left (-4 e^3 x+\left (2 e^3 x+6 x^2\right ) \log \left (-\frac {15 x}{16}\right )+\left (-4 e^3+12 x \log \left (-\frac {15 x}{16}\right )\right ) \log (x)+6 \log \left (-\frac {15 x}{16}\right ) \log ^2(x)\right ) \log (2 x)+\left (2 e^3-6 x \log \left (-\frac {15 x}{16}\right )-6 \log \left (-\frac {15 x}{16}\right ) \log (x)\right ) \log ^2(2 x)+2 \log \left (-\frac {15 x}{16}\right ) \log ^3(2 x)}{-x^4-3 x^3 \log (x)-3 x^2 \log ^2(x)-x \log ^3(x)+\left (3 x^3+6 x^2 \log (x)+3 x \log ^2(x)\right ) \log (2 x)+\left (-3 x^2-3 x \log (x)\right ) \log ^2(2 x)+x \log ^3(2 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x^2+\log ^2(2)+x \left (e^3-\log (4)\right )\right ) \left (-e^3-\log \left (\frac {16}{15}\right ) (x-\log (2))-(-x+\log (2)) \log (-x)\right )}{x (x+\log (x)-\log (2 x))^3} \, dx\\ &=2 \int \frac {\left (x^2+\log ^2(2)+x \left (e^3-\log (4)\right )\right ) \left (-e^3-\log \left (\frac {16}{15}\right ) (x-\log (2))-(-x+\log (2)) \log (-x)\right )}{x (x+\log (x)-\log (2 x))^3} \, dx\\ &=2 \int \left (\frac {x \left (-x \log \left (\frac {16}{15}\right )-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )+x \log (-x)-\log (2) \log (-x)\right )}{(x+\log (x)-\log (2 x))^3}+\frac {\log ^2(2) \left (-x \log \left (\frac {16}{15}\right )-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )+x \log (-x)-\log (2) \log (-x)\right )}{x (x+\log (x)-\log (2 x))^3}+\frac {e^3 \left (1-\frac {\log (4)}{e^3}\right ) \left (-x \log \left (\frac {16}{15}\right )-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )+x \log (-x)-\log (2) \log (-x)\right )}{(x+\log (x)-\log (2 x))^3}\right ) \, dx\\ &=2 \int \frac {x \left (-x \log \left (\frac {16}{15}\right )-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )+x \log (-x)-\log (2) \log (-x)\right )}{(x+\log (x)-\log (2 x))^3} \, dx+\left (2 \log ^2(2)\right ) \int \frac {-x \log \left (\frac {16}{15}\right )-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )+x \log (-x)-\log (2) \log (-x)}{x (x+\log (x)-\log (2 x))^3} \, dx+\left (2 \left (e^3-\log (4)\right )\right ) \int \frac {-x \log \left (\frac {16}{15}\right )-e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )+x \log (-x)-\log (2) \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx\\ &=2 \int \left (-\frac {x^2 \log \left (\frac {16}{15}\right )}{(x+\log (x)-\log (2 x))^3}-\frac {x \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right )}{(x+\log (x)-\log (2 x))^3}+\frac {x^2 \log (-x)}{(x+\log (x)-\log (2 x))^3}-\frac {x \log (2) \log (-x)}{(x+\log (x)-\log (2 x))^3}\right ) \, dx+\left (2 \log ^2(2)\right ) \int \frac {-e^3+\log \left (\frac {16}{15}\right ) (-x+\log (2))+(x-\log (2)) \log (-x)}{x (x+\log (x)-\log (2 x))^3} \, dx+\left (2 \left (e^3-\log (4)\right )\right ) \int \frac {-e^3+\log \left (\frac {16}{15}\right ) (-x+\log (2))+(x-\log (2)) \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx\\ &=2 \int \frac {x^2 \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log \left (\frac {16}{15}\right )\right ) \int \frac {x^2}{(x+\log (x)-\log (2 x))^3} \, dx-(2 \log (2)) \int \frac {x \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx+\left (2 \log ^2(2)\right ) \int \left (-\frac {\log \left (\frac {16}{15}\right )}{(x+\log (x)-\log (2 x))^3}-\frac {e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )}{x (x+\log (x)-\log (2 x))^3}+\frac {\log (-x)}{(x+\log (x)-\log (2 x))^3}-\frac {\log (2) \log (-x)}{x (x+\log (x)-\log (2 x))^3}\right ) \, dx-\left (2 \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right )\right ) \int \frac {x}{(x+\log (x)-\log (2 x))^3} \, dx+\left (2 \left (e^3-\log (4)\right )\right ) \int \left (-\frac {x \log \left (\frac {16}{15}\right )}{(x+\log (x)-\log (2 x))^3}-\frac {e^3 \left (1-\frac {\log \left (\frac {16}{15}\right ) \log (2)}{e^3}\right )}{(x+\log (x)-\log (2 x))^3}+\frac {x \log (-x)}{(x+\log (x)-\log (2 x))^3}-\frac {\log (2) \log (-x)}{(x+\log (x)-\log (2 x))^3}\right ) \, dx\\ &=2 \int \frac {x^2 \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log \left (\frac {16}{15}\right )\right ) \int \frac {x^2}{(x+\log (x)-\log (2 x))^3} \, dx-(2 \log (2)) \int \frac {x \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx+\left (2 \log ^2(2)\right ) \int \frac {\log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log \left (\frac {16}{15}\right ) \log ^2(2)\right ) \int \frac {1}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log ^3(2)\right ) \int \frac {\log (-x)}{x (x+\log (x)-\log (2 x))^3} \, dx-\left (2 \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right )\right ) \int \frac {x}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log ^2(2) \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right )\right ) \int \frac {1}{x (x+\log (x)-\log (2 x))^3} \, dx+\left (2 \left (e^3-\log (4)\right )\right ) \int \frac {x \log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log \left (\frac {16}{15}\right ) \left (e^3-\log (4)\right )\right ) \int \frac {x}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \log (2) \left (e^3-\log (4)\right )\right ) \int \frac {\log (-x)}{(x+\log (x)-\log (2 x))^3} \, dx-\left (2 \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right ) \left (e^3-\log (4)\right )\right ) \int \frac {1}{(x+\log (x)-\log (2 x))^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.87, size = 1463, normalized size = 54.19 \begin {gather*} -\frac {e^6 \log (2) (\log (x)-\log (2 x))^2+2 \log ^2(2) \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right ) \log (x) (x+\log (x)-\log (2 x))^2+2 (\log (x)-\log (2 x)) (x+\log (x)-\log (2 x)) \left (\log ^2(2) \left (e^3-\log \left (\frac {16}{15}\right ) \log (2)\right )+2 \log \left (\frac {16}{15}\right ) \log ^3(x)+\left (-e^3 \left (1+\log \left (\frac {16}{15}\right )\right )+\log \left (\frac {16}{15}\right ) \log (8)\right ) \log ^2(2 x)-2 \log \left (\frac {16}{15}\right ) \log ^3(2 x)+2 \log (x) \log (2 x) \left (e^3 \left (1+\log \left (\frac {16}{15}\right )\right )-\log \left (\frac {16}{15}\right ) \log (8)+3 \log \left (\frac {16}{15}\right ) \log (2 x)\right )-\log ^2(x) \left (e^3 \left (1+\log \left (\frac {16}{15}\right )\right )-\log \left (\frac {16}{15}\right ) \log (8)+6 \log \left (\frac {16}{15}\right ) \log (2 x)\right )\right )-2 \log (-x) (\log (x)-\log (2 x))^3 (x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )-4 (\log (-x)-\log (x)) (\log (x)-\log (2 x))^2 (x+\log (x)-\log (2 x)) \left (x \log (-x)-(x+\log (x)-\log (2 x)) \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )+4 (\log (-x)-\log (2 x)) (\log (x)-\log (2 x))^2 (x+\log (x)-\log (2 x)) \left (x \log (-x)-(x+\log (x)-\log (2 x)) \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )-e^3 \log (2) (\log (x)-\log (2 x)) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )+\log ^2(2) (\log (x)-\log (2 x)) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )+\log (2) \log (4) (\log (x)-\log (2 x)) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )+(\log (-x)-\log (x))^2 (\log (x)-\log (2 x)) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )-2 (\log (-x)-\log (x)) (\log (-x)-\log (2 x)) (\log (x)-\log (2 x)) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )+(\log (-x)-\log (2 x))^2 (\log (x)-\log (2 x)) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )+2 \left (-e^3 \log ^2(2)+\left (\log \left (\frac {4096}{3375}\right ) \log (2)-\log \left (\frac {16}{15}\right ) \log (8)\right ) \log ^2(x)\right ) (x+\log (x)-\log (2 x))^2 \log (x+\log (x)-\log (2 x))+e^3 (\log (x)-\log (2 x))^2 \left (2 \log (-x) (\log (x)-\log (2 x)) (x+\log (x)-\log (2 x))-\log (x) (x+\log (x)-\log (2 x))^2+(\log (x)-\log (2 x)) (x+\log (2) \log (-x)+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log (x+\log (x)-\log (2 x))\right )-\log (2) (\log (x)-\log (2 x))^2 \left (2 \log (-x) (\log (x)-\log (2 x)) (x+\log (x)-\log (2 x))-\log (x) (x+\log (x)-\log (2 x))^2+(\log (x)-\log (2 x)) (x+\log (2) \log (-x)+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log (x+\log (x)-\log (2 x))\right )-\log (4) (\log (x)-\log (2 x))^2 \left (2 \log (-x) (\log (x)-\log (2 x)) (x+\log (x)-\log (2 x))-\log (x) (x+\log (x)-\log (2 x))^2+(\log (x)-\log (2 x)) (x+\log (2) \log (-x)+\log (x)-\log (2 x))+(x+\log (x)-\log (2 x))^2 \log (x+\log (x)-\log (2 x))\right )-2 (\log (x)-\log (2 x))^3 (x+\log (x)-\log (2 x))^2 \text {PolyLog}\left (2,-\frac {x}{\log (x)-\log (2 x)}\right )+\log ^3(2) \left (-x (x-\log (4)) \log (-x)+x (x+\log (x)-\log (2 x))+\log ^2(-x) (x+\log (x)-\log (2 x))^2+(x+\log (x)-\log (2 x))^2 \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )-2 (x+\log (x)-\log (2 x)) \left (x \log (-x)-(x+\log (x)-\log (2 x)) \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )\right )-2 (x+\log (x)-\log (2 x))^2 \left (\log (-x) \log \left (1+\frac {x}{\log (x)-\log (2 x)}\right )+\text {PolyLog}\left (2,-\frac {x}{\log (x)-\log (2 x)}\right )\right )\right )}{(\log (x)-\log (2 x))^3 (x+\log (x)-\log (2 x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs.
\(2(24)=48\).
time = 1.02, size = 120, normalized size = 4.44
method | result | size |
default | \(-8 \ln \left (2\right ) \ln \left (x \right )+\frac {2 \ln \left (x \right ) {\mathrm e}^{3}}{\ln \left (2\right )}+\frac {{\mathrm e}^{6}}{\left (x -\ln \left (2\right )\right )^{2}}-\frac {2 \,{\mathrm e}^{3} \ln \left (x -\ln \left (2\right )\right )}{\ln \left (2\right )}+\frac {8 \,{\mathrm e}^{3} \ln \left (2\right )}{x -\ln \left (2\right )}+2 \ln \left (x \right ) \ln \left (15\right )-\frac {2 \ln \left (15\right ) {\mathrm e}^{3}}{x -\ln \left (2\right )}+\ln \left (-x \right )^{2}+\frac {2 \,{\mathrm e}^{3} \ln \left (\ln \left (2\right )-x \right )}{\ln \left (2\right )}+\frac {2 \,{\mathrm e}^{3} \ln \left (-x \right ) x}{\ln \left (2\right ) \left (\ln \left (2\right )-x \right )}\) | \(120\) |
risch | \(\text {Expression too large to display}\) | \(1096\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.57, size = 315, normalized size = 11.67 \begin {gather*} {\left (\frac {\log \left (x - \log \left (2\right )\right )}{\log \left (2\right )} - \frac {\log \left (x\right )}{\log \left (2\right )} - \frac {1}{x - \log \left (2\right )}\right )} e^{3} - \frac {{\left (2 \, x - \log \left (2\right )\right )} e^{3} \log \left (-\frac {15}{16} \, x\right )}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} + \frac {{\left (2 \, x - \log \left (2\right )\right )} e^{3}}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} - \frac {e^{3} \log \left (x - \log \left (2\right )\right )}{\log \left (2\right )} - \frac {x e^{3} \log \left (2\right ) - {\left (x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )^{2} + \log \left (2\right )^{3}\right )} \log \left (x\right )^{2} - {\left ({\left (i \, \pi + \log \left (5\right )\right )} \log \left (2\right )^{2} + \log \left (3\right ) \log \left (2\right )^{2} - 4 \, \log \left (2\right )^{3}\right )} e^{3} + {\left (2 \, {\left (-i \, \pi - \log \left (5\right )\right )} \log \left (2\right )^{3} - 2 \, \log \left (3\right ) \log \left (2\right )^{3} + 8 \, \log \left (2\right )^{4} + {\left (2 \, {\left (-i \, \pi - \log \left (5\right )\right )} \log \left (2\right ) - 2 \, \log \left (3\right ) \log \left (2\right ) + 8 \, \log \left (2\right )^{2} - e^{3}\right )} x^{2} - 2 \, e^{3} \log \left (2\right )^{2} + 2 \, {\left (2 \, {\left (i \, \pi + \log \left (5\right )\right )} \log \left (2\right )^{2} + 2 \, \log \left (3\right ) \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + e^{3} \log \left (2\right )\right )} x\right )} \log \left (x\right )}{x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )^{2} + \log \left (2\right )^{3}} + \frac {e^{6}}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} \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. 85 vs.
\(2 (24) = 48\).
time = 0.38, size = 85, normalized size = 3.15 \begin {gather*} \frac {{\left (x^{2} - 2 \, {\left (x + \log \left (\frac {16}{15}\right )\right )} \log \left (\frac {32}{15}\right ) + \log \left (\frac {32}{15}\right )^{2} + 2 \, x \log \left (\frac {16}{15}\right ) + \log \left (\frac {16}{15}\right )^{2}\right )} \log \left (-\frac {15}{16} \, x\right )^{2} - 2 \, {\left (x e^{3} - e^{3} \log \left (\frac {32}{15}\right ) + e^{3} \log \left (\frac {16}{15}\right )\right )} \log \left (-\frac {15}{16} \, x\right ) + e^{6}}{x^{2} - 2 \, {\left (x + \log \left (\frac {16}{15}\right )\right )} \log \left (\frac {32}{15}\right ) + \log \left (\frac {32}{15}\right )^{2} + 2 \, x \log \left (\frac {16}{15}\right ) + \log \left (\frac {16}{15}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.41, size = 114, normalized size = 4.22 \begin {gather*} \log {\left (x \right )}^{2} + 2 \left (- 4 \log {\left (2 \right )} + \log {\left (15 \right )} + i \pi \right ) \log {\left (x \right )} + \frac {x \left (- 2 e^{3} \log {\left (15 \right )} + 8 e^{3} \log {\left (2 \right )} - 2 i \pi e^{3}\right ) - 8 e^{3} \log {\left (2 \right )}^{2} + 2 e^{3} \log {\left (2 \right )} \log {\left (15 \right )} + e^{6} + 2 i \pi e^{3} \log {\left (2 \right )}}{x^{2} - 2 x \log {\left (2 \right )} + \log {\left (2 \right )}^{2}} - \frac {2 e^{3} \log {\left (x \right )}}{x - \log {\left (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. 70 vs.
\(2 (24) = 48\).
time = 0.43, size = 70, normalized size = 2.59 \begin {gather*} \frac {x^{2} \log \left (-\frac {15}{16} \, x\right )^{2} - 2 \, x \log \left (2\right ) \log \left (-\frac {15}{16} \, x\right )^{2} + \log \left (2\right )^{2} \log \left (-\frac {15}{16} \, x\right )^{2} - 2 \, x e^{3} \log \left (-\frac {15}{16} \, x\right ) + 2 \, e^{3} \log \left (2\right ) \log \left (-\frac {15}{16} \, x\right ) + e^{6}}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.56, size = 105, normalized size = 3.89 \begin {gather*} 2\,\ln \left (x\right )\,\left (\ln \left (-\frac {15\,x}{16}\right )-\ln \left (x\right )\right )+{\ln \left (x\right )}^2+\frac {{\mathrm {e}}^6-2\,x\,{\mathrm {e}}^3\,\left (\ln \left (-\frac {15\,x}{16}\right )-\ln \left (x\right )\right )+2\,{\mathrm {e}}^3\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )\,\left (\ln \left (-\frac {15\,x}{16}\right )-\ln \left (x\right )\right )}{{\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )}^2-2\,x\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )+x^2}-\frac {2\,{\mathrm {e}}^3\,\ln \left (x\right )}{x-\ln \left (2\,x\right )+\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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