Optimal. Leaf size=18 \[ -\frac {2 \log \left (-3+e+\frac {4}{x}\right ) (5+\log (x))}{x} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.94, antiderivative size = 185, normalized size of antiderivative = 10.28, number of steps
used = 29, number of rules used = 19, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {6, 1607,
6820, 6874, 14, 46, 2504, 2436, 2332, 2380, 2341, 2379, 2438, 2423, 2525, 2458, 45, 2393, 2354}
\begin {gather*} \frac {1}{2} (3-e) \text {PolyLog}\left (2,1-\frac {4}{(3-e) x}\right )+\frac {1}{2} (3-e) \text {PolyLog}\left (2,\frac {4}{(3-e) x}\right )+\frac {5}{2} \left (-\frac {4}{x}-e+3\right ) \log \left (\frac {4}{x}+e-3\right )+\frac {1}{2} (3-e) \log \left (\frac {4}{(3-e) x}\right ) \log \left (\frac {4}{x}+e-3\right )+\frac {1}{2} \left (-\frac {4}{x}-e+3\right ) \log (x) \log \left (\frac {4}{x}+e-3\right )-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) \log (x)+\frac {5}{2} (3-e) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 14
Rule 45
Rule 46
Rule 1607
Rule 2332
Rule 2341
Rule 2354
Rule 2379
Rule 2380
Rule 2393
Rule 2423
Rule 2436
Rule 2438
Rule 2458
Rule 2504
Rule 2525
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2+(-3+e) x^3} \, dx\\ &=\int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{x^2 (4+(-3+e) x)} \, dx\\ &=\int \frac {40+8 \log (x)+2 (4+(-3+e) x) \log \left (-3+e+\frac {4}{x}\right ) (4+\log (x))}{x^2 (4+(-3+e) x)} \, dx\\ &=\int \left (\frac {8 \left (5+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )\right )}{x^2 (4-(3-e) x)}+\frac {2 \left (4+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )\right ) \log (x)}{x^2 (4-(3-e) x)}\right ) \, dx\\ &=2 \int \frac {\left (4+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )\right ) \log (x)}{x^2 (4-(3-e) x)} \, dx+8 \int \frac {5+4 \log \left (-3+e+\frac {4}{x}\right )-3 \left (1-\frac {e}{3}\right ) x \log \left (-3+e+\frac {4}{x}\right )}{x^2 (4-(3-e) x)} \, dx\\ &=2 \int \frac {\left (\frac {4}{4+(-3+e) x}+\log \left (-3+e+\frac {4}{x}\right )\right ) \log (x)}{x^2} \, dx+8 \int \frac {\frac {5}{4+(-3+e) x}+\log \left (-3+e+\frac {4}{x}\right )}{x^2} \, dx\\ &=2 \int \left (\frac {4 \log (x)}{x^2 (4-(3-e) x)}+\frac {\log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x^2}\right ) \, dx+8 \int \left (\frac {5}{x^2 (4-(3-e) x)}+\frac {\log \left (-3+e+\frac {4}{x}\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {\log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x^2} \, dx+8 \int \frac {\log \left (-3+e+\frac {4}{x}\right )}{x^2} \, dx+8 \int \frac {\log (x)}{x^2 (4+(-3+e) x)} \, dx+40 \int \frac {1}{x^2 (4-(3-e) x)} \, dx\\ &=\frac {2 \log (x)}{x}+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-2 \int \left (\frac {1}{x^2}+\frac {\left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )}{4 x}\right ) \, dx+8 \int \left (\frac {\log (x)}{4 x^2}+\frac {(3-e) \log (x)}{16 x}+\frac {(3-e)^2 \log (x)}{16 (4-(3-e) x)}\right ) \, dx-8 \text {Subst}\left (\int \log (-3+e+4 x) \, dx,x,\frac {1}{x}\right )+40 \int \left (\frac {1}{4 x^2}+\frac {3-e}{16 x}+\frac {(3-e)^2}{16 (4-(3-e) x)}\right ) \, dx\\ &=-\frac {8}{x}+\frac {5}{2} (3-e) \log (x)+\frac {2 \log (x)}{x}+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} \int \frac {\left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )}{x} \, dx+2 \int \frac {\log (x)}{x^2} \, dx-2 \text {Subst}\left (\int \log (x) \, dx,x,-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \int \frac {\log (x)}{x} \, dx+\frac {1}{2} (3-e)^2 \int \frac {\log (x)}{4+(-3+e) x} \, dx\\ &=-\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)+\frac {1}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} \text {Subst}\left (\int \frac {(3-e-4 x) \log (-3+e+4 x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} (3-e)^2 \int \frac {\log \left (\frac {1}{4} (3-e) x\right )}{4+(-3+e) x} \, dx\\ &=-\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)+\frac {1}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{8} \text {Subst}\left (\int \frac {x \log (x)}{\frac {3-e}{4}+\frac {x}{4}} \, dx,x,-3+e+\frac {4}{x}\right )\\ &=-\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)+\frac {1}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{8} \text {Subst}\left (\int \left (4 \log (x)-\frac {4 (-3+e) \log (x)}{-3+e-x}\right ) \, dx,x,-3+e+\frac {4}{x}\right )\\ &=-\frac {2}{x}+2 \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)+\frac {1}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{2} \text {Subst}\left (\int \log (x) \, dx,x,-3+e+\frac {4}{x}\right )-\frac {1}{2} (3-e) \text {Subst}\left (\int \frac {\log (x)}{-3+e-x} \, dx,x,-3+e+\frac {4}{x}\right )\\ &=\frac {5}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \log \left (-3+e+\frac {4}{x}\right ) \log \left (\frac {4}{(3-e) x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)+\frac {1}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )-\frac {1}{2} (3-e) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{-3+e}\right )}{x} \, dx,x,-3+e+\frac {4}{x}\right )\\ &=\frac {5}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right )+\frac {1}{2} (3-e) \log \left (-3+e+\frac {4}{x}\right ) \log \left (\frac {4}{(3-e) x}\right )+\frac {5}{2} (3-e) \log (x)+\frac {1}{2} \left (3-e-\frac {4}{x}\right ) \log \left (-3+e+\frac {4}{x}\right ) \log (x)+\frac {1}{4} (3-e) \log ^2(x)-\frac {5}{2} (3-e) \log (4-(3-e) x)-\frac {1}{2} (3-e) \log \left (\frac {4}{3-e}\right ) \log (4-(3-e) x)+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {4}{(3-e) x}\right )+\frac {1}{2} (3-e) \text {Li}_2\left (1-\frac {1}{4} (3-e) x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 31, normalized size = 1.72 \begin {gather*} -\frac {10 \log \left (-3+e+\frac {4}{x}\right )}{x}-\frac {2 \log \left (-3+e+\frac {4}{x}\right ) \log (x)}{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. \(174\) vs.
\(2(19)=38\).
time = 2.35, size = 175, normalized size = 9.72
method | result | size |
default | \(\frac {2+2 \left (-\frac {{\mathrm e}}{4}+\frac {3}{4}\right ) x \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )-2 \ln \left (x \right ) \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{x}-\frac {2}{x}-\frac {5 \,{\mathrm e} \ln \left (x \right )}{2}+\frac {15 \ln \left (x \right )}{2}-\frac {5 \ln \left (-x \,{\mathrm e}-4+3 x \right ) {\mathrm e}^{2}}{2 \left (3-{\mathrm e}\right )}+\frac {15 \ln \left (-x \,{\mathrm e}-4+3 x \right ) {\mathrm e}}{3-{\mathrm e}}-\frac {45 \ln \left (-x \,{\mathrm e}-4+3 x \right )}{2 \left (3-{\mathrm e}\right )}-2 \ln \left (\frac {4}{x}+{\mathrm e}-3\right ) \left (\frac {4}{x}+{\mathrm e}-3\right )+2 \,{\mathrm e}-6\) | \(175\) |
risch | \(-\frac {2 \left (5+\ln \left (x \right )\right ) \ln \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {-i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \ln \left (x \right )+i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \ln \left (x \right )+i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3} \ln \left (x \right )-i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \ln \left (x \right )-5 i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2}+5 i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+5 i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3}-5 i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+2 \ln \left (x \right )^{2}+10 \ln \left (x \right )}{x}\) | \(296\) |
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. 304 vs.
\(2 (19) = 38\).
time = 0.32, size = 304, normalized size = 16.89 \begin {gather*} -2 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \left (x\right )\right )} e \log \left (\frac {4}{x} + e - 3\right ) + {\left (\log \left (x {\left (e - 3\right )} + 4\right )^{2} - 2 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e + \frac {5}{2} \, {\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - 3 \, \log \left (x {\left (e - 3\right )} + 4\right )^{2} - \frac {5}{2} \, {\left (e - 3\right )} \log \left (x\right ) + 6 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \left (x\right ) - 3 \, \log \left (x\right )^{2} + 2 \, {\left ({\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (e - 3\right )} \log \left (x\right ) - \frac {4}{x}\right )} \log \left (\frac {4}{x} + e - 3\right ) + 6 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \left (x\right )\right )} \log \left (\frac {4}{x} + e - 3\right ) - \frac {x {\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right )^{2} + x {\left (e - 3\right )} \log \left (x\right )^{2} - 2 \, x {\left (e - 3\right )} \log \left (x\right ) - 2 \, {\left (x {\left (e - 3\right )} \log \left (x\right ) - x {\left (e - 3\right )}\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - 8}{x} - \frac {{\left (x {\left (e - 3\right )} + 4 \, \log \left (x\right ) + 4\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (x {\left (e - 3\right )} + 4\right )} \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - 4}{2 \, x} - \frac {10}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (\log \left (x\right ) + 5\right )} \log \left (\frac {x e - 3 \, x + 4}{x}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 22, normalized size = 1.22 \begin {gather*} \frac {\left (- 2 \log {\left (x \right )} - 10\right ) \log {\left (\frac {- 3 x + e x + 4}{x} \right )}}{x} \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. 41 vs.
\(2 (19) = 38\).
time = 0.40, size = 41, normalized size = 2.28 \begin {gather*} -\frac {2 \, {\left (\log \left (x e - 3 \, x + 4\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 5 \, \log \left (x e - 3 \, x + 4\right ) - 5 \, \log \left (x\right )\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.75, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2\,\ln \left (\frac {x\,\mathrm {e}-3\,x+4}{x}\right )\,\left (\ln \left (x\right )+5\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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