Optimal. Leaf size=30 \[ \log \left (x \left (-4+x+x^2 \left (e^x+x\right )^2-\log (4)\right )\right ) (5-x+\log (\log (x))) \]
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Rubi [F]
time = 9.20, 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 {\left (-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} \left (15 x^2+7 x^3-2 x^4\right )+e^x \left (40 x^3+2 x^4-2 x^5\right )+(-5+x) \log (4)\right ) \log (x)+\left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+\left (4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)\right ) \log (x)\right ) \log \left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right )+\left (-4+2 x+5 x^4+e^{2 x} \left (3 x^2+2 x^3\right )+e^x \left (8 x^3+2 x^4\right )-\log (4)\right ) \log (x) \log (\log (x))}{\left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right ) \log (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 {\left (-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} \left (15 x^2+7 x^3-2 x^4\right )+e^x \left (40 x^3+2 x^4-2 x^5\right )+(-5+x) \log (4)\right ) \log (x)+\left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+\left (4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)\right ) \log (x)\right ) \log \left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right )+\left (-4+2 x+5 x^4+e^{2 x} \left (3 x^2+2 x^3\right )+e^x \left (8 x^3+2 x^4\right )-\log (4)\right ) \log (x) \log (\log (x))}{\left (x^2+e^{2 x} x^3+2 e^x x^4+x^5+x (-4-\log (4))\right ) \log (x)} \, dx\\ &=\int \left (\frac {\left (2 x^2-2 e^x x^3-2 x^4+2 e^x x^4+2 x^5-8 \left (1+\frac {\log (2)}{2}\right )-7 x \left (1+\frac {4 \log (2)}{7}\right )\right ) (-5+x-\log (\log (x)))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )}+\frac {15 \log (x)+7 x \log (x)-2 x^2 \log (x)+\log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )-x \log (x) \log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )+3 \log (x) \log (\log (x))+2 x \log (x) \log (\log (x))}{x \log (x)}\right ) \, dx\\ &=\int \frac {\left (2 x^2-2 e^x x^3-2 x^4+2 e^x x^4+2 x^5-8 \left (1+\frac {\log (2)}{2}\right )-7 x \left (1+\frac {4 \log (2)}{7}\right )\right ) (-5+x-\log (\log (x)))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )} \, dx+\int \frac {15 \log (x)+7 x \log (x)-2 x^2 \log (x)+\log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )-x \log (x) \log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )+3 \log (x) \log (\log (x))+2 x \log (x) \log (\log (x))}{x \log (x)} \, dx\\ &=\int \left (\frac {15 \log (x)+7 x \log (x)-2 x^2 \log (x)+\log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )-x \log (x) \log \left (x \left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)\right )\right )}{x \log (x)}+\frac {(3+2 x) \log (\log (x))}{x}\right ) \, dx+\int \left (\frac {2 x^2}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {10 e^x x^2}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {10 x^3}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 e^x x^4}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x^5}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {12 e^x x^3}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {12 x^4}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {20 (2+\log (2))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )}+\frac {10 x \left (1+\frac {1}{10} (7+\log (16))\right )}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {4 (-2-\log (2)) \left (1-\frac {5 (7+\log (16))}{8+\log (16)}\right )}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 e^x x^2 \log (\log (x))}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x^3 \log (\log (x))}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x \log (\log (x))}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 e^x x^3 \log (\log (x))}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {2 x^4 \log (\log (x))}{-x-e^{2 x} x^2-2 e^x x^3-x^4+4 \left (1+\frac {\log (2)}{2}\right )}+\frac {4 (2+\log (2)) \log (\log (x))}{x \left (x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )\right )}+\frac {(7+\log (16)) \log (\log (x))}{x+e^{2 x} x^2+2 e^x x^3+x^4-4 \left (1+\frac {\log (2)}{2}\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F]
time = 2.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-20+14 x-2 x^2+25 x^4-5 x^5+e^{2 x} \left (15 x^2+7 x^3-2 x^4\right )+e^x \left (40 x^3+2 x^4-2 x^5\right )+(-5+x) \log (4)\right ) \log (x)+\left (-4+x+e^{2 x} x^2+2 e^x x^3+x^4-\log (4)+\left (4 x-x^2-e^{2 x} x^3-2 e^x x^4-x^5+x \log (4)\right ) \log (x)\right ) \log \left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right )+\left (-4+2 x+5 x^4+e^{2 x} \left (3 x^2+2 x^3\right )+e^x \left (8 x^3+2 x^4\right )-\log (4)\right ) \log (x) \log (\log (x))}{\left (-4 x+x^2+e^{2 x} x^3+2 e^x x^4+x^5-x \log (4)\right ) \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.56, size = 666, normalized size = 22.20
method | result | size |
risch | \(\left (\ln \left (\ln \left (x \right )\right )-x \right ) \ln \left (-\frac {x^{4}}{2}-{\mathrm e}^{x} x^{3}-\frac {{\mathrm e}^{2 x} x^{2}}{2}+\ln \left (2\right )-\frac {x}{2}+2\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x \ln \left (x \right )+\frac {i \pi x \,\mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )}{2}+\frac {i \pi \ln \left (\ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{2}}{2}+i \pi x \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{2}-i \pi \ln \left (\ln \left (x \right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{2}-i \pi x -\frac {i \pi \ln \left (\ln \left (x \right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{2}}{2}-\frac {i \pi \ln \left (\ln \left (x \right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{3}}{2}-\frac {i \pi \ln \left (\ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right )\right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )}{2}+15 \ln \left (x \right )+5 \ln \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x -\frac {-x^{4}+2 \ln \left (2\right )-x +4}{x^{2}}\right )-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left (\frac {x^{4}}{2}+{\mathrm e}^{x} x^{3}+\frac {{\mathrm e}^{2 x} x^{2}}{2}-\ln \left (2\right )+\frac {x}{2}-2\right ) x \right )^{3}}{2}+i \pi \ln \left (\ln \left (x \right )\right )\) | \(666\) |
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. 81 vs.
\(2 (30) = 60\).
time = 0.57, size = 81, normalized size = 2.70 \begin {gather*} -{\left (x - \log \left (\log \left (x\right )\right )\right )} \log \left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} + x - 2 \, \log \left (2\right ) - 4\right ) - {\left (x - 15\right )} \log \left (x\right ) + \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, \log \left (\frac {x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )} + x - 2 \, \log \left (2\right ) - 4}{x^{2}}\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. 72 vs.
\(2 (30) = 60\).
time = 0.32, size = 72, normalized size = 2.40 \begin {gather*} -{\left (x - 5\right )} \log \left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \left (2\right ) - 4 \, x\right ) + \log \left (x^{5} + 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} + x^{2} - 2 \, x \log \left (2\right ) - 4 \, x\right ) \log \left (\log \left (x\right )\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. 76 vs.
\(2 (29) = 58\).
time = 47.41, size = 76, normalized size = 2.53 \begin {gather*} \left (- x + \log {\left (\log {\left (x \right )} \right )}\right ) \log {\left (x^{5} + 2 x^{4} e^{x} + x^{3} e^{2 x} + x^{2} - 4 x - 2 x \log {\left (2 \right )} \right )} + 15 \log {\left (x \right )} + 5 \log {\left (2 x e^{x} + e^{2 x} + \frac {x^{4} + x - 4 - 2 \log {\left (2 \right )}}{x^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (2\,x^4\,{\mathrm {e}}^x-4\,x-2\,x\,\ln \left (2\right )+x^3\,{\mathrm {e}}^{2\,x}+x^2+x^5\right )\,\left (x-2\,\ln \left (2\right )+2\,x^3\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+x^4-\ln \left (x\right )\,\left (2\,x^4\,{\mathrm {e}}^x-4\,x-2\,x\,\ln \left (2\right )+x^3\,{\mathrm {e}}^{2\,x}+x^2+x^5\right )-4\right )+\ln \left (x\right )\,\left (14\,x+{\mathrm {e}}^x\,\left (-2\,x^5+2\,x^4+40\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (-2\,x^4+7\,x^3+15\,x^2\right )+2\,\ln \left (2\right )\,\left (x-5\right )-2\,x^2+25\,x^4-5\,x^5-20\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x-2\,\ln \left (2\right )+{\mathrm {e}}^x\,\left (2\,x^4+8\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (2\,x^3+3\,x^2\right )+5\,x^4-4\right )}{\ln \left (x\right )\,\left (2\,x^4\,{\mathrm {e}}^x-4\,x-2\,x\,\ln \left (2\right )+x^3\,{\mathrm {e}}^{2\,x}+x^2+x^5\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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