Optimal. Leaf size=40 \[ \frac {-x+5 \log \left (\frac {x}{3}\right )}{e^x-x-\frac {(3-x) \log ^2\left (3 x^2\right )}{x}} \]
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Rubi [F]
time = 26.15, 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 {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (-5 x+e^x \left (5-x+x^2\right )\right )+4 (-3+x) x \log \left (3 x^2\right )-\left (15-11 x+x^2\right ) \log ^2\left (3 x^2\right )-5 \log \left (\frac {x}{3}\right ) \left (\left (-1+e^x\right ) x^2+4 (-3+x) \log \left (3 x^2\right )+3 \log ^2\left (3 x^2\right )\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {5-x+x^2-5 x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}+\frac {\left (x-5 \log \left (\frac {x}{3}\right )\right ) \left (-x^2+x^3-12 \log \left (3 x^2\right )+4 x \log \left (3 x^2\right )+3 \log ^2\left (3 x^2\right )+3 x \log ^2\left (3 x^2\right )-x^2 \log ^2\left (3 x^2\right )\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}\right ) \, dx\\ &=-\int \frac {5-x+x^2-5 x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx+\int \frac {\left (x-5 \log \left (\frac {x}{3}\right )\right ) \left (-x^2+x^3-12 \log \left (3 x^2\right )+4 x \log \left (3 x^2\right )+3 \log ^2\left (3 x^2\right )+3 x \log ^2\left (3 x^2\right )-x^2 \log ^2\left (3 x^2\right )\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx\\ &=\int \frac {\left (x-5 \log \left (\frac {x}{3}\right )\right ) \left ((-1+x) x^2+4 (-3+x) \log \left (3 x^2\right )+\left (3+3 x-x^2\right ) \log ^2\left (3 x^2\right )\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-\int \left (-\frac {x}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}+\frac {x^2}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}-\frac {5 x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}-\frac {5}{e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.20, size = 37, normalized size = 0.92 \begin {gather*} \frac {x \left (x-5 \log \left (\frac {x}{3}\right )\right )}{x \left (-e^x+x\right )-(-3+x) \log ^2\left (3 x^2\right )} \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 = 36.66, size = 492, normalized size = 12.30
method | result | size |
risch | \(\frac {-20 x \ln \left (x \right )+20 x \ln \left (3\right )+4 x^{2}}{-16 x \ln \left (3\right ) \ln \left (x \right )-4 x \ln \left (3\right )^{2}+48 \ln \left (3\right ) \ln \left (x \right )-16 x \ln \left (x \right )^{2}-4 \,{\mathrm e}^{x} x -24 i \ln \left (x \right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+12 \ln \left (3\right )^{2}+48 \ln \left (x \right )^{2}+4 x^{2}-3 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}-16 i x \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-24 i \ln \left (x \right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-3 \pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}+12 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}-18 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}+12 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}+\pi ^{2} x \mathrm {csgn}\left (i x^{2}\right )^{6}+48 i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+4 i x \ln \left (3\right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-8 i x \ln \left (3\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+8 i x \ln \left (x \right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+6 \pi ^{2} x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}-4 \pi ^{2} x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}+\pi ^{2} x \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} x \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}-12 i \ln \left (3\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-12 i \ln \left (3\right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+24 i \ln \left (3\right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+4 i x \ln \left (3\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+8 i x \ln \left (x \right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}\) | \(492\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.68, size = 62, normalized size = 1.55 \begin {gather*} -\frac {x^{2} + 5 \, x \log \left (3\right ) - 5 \, x \log \left (x\right )}{x \log \left (3\right )^{2} + 4 \, {\left (x - 3\right )} \log \left (x\right )^{2} - x^{2} + x e^{x} - 3 \, \log \left (3\right )^{2} + 4 \, {\left (x \log \left (3\right ) - 3 \, \log \left (3\right )\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 56, normalized size = 1.40 \begin {gather*} -\frac {x^{2} - 5 \, x \log \left (\frac {1}{3} \, x\right )}{9 \, {\left (x - 3\right )} \log \left (3\right )^{2} + 12 \, {\left (x - 3\right )} \log \left (3\right ) \log \left (\frac {1}{3} \, x\right ) + 4 \, {\left (x - 3\right )} \log \left (\frac {1}{3} \, x\right )^{2} - x^{2} + x e^{x}} \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. 75 vs.
\(2 (26) = 52\).
time = 0.31, size = 75, normalized size = 1.88 \begin {gather*} \frac {- x^{2} + 5 x \log {\left (\frac {x}{3} \right )}}{- x^{2} + x e^{x} + 4 x \log {\left (\frac {x}{3} \right )}^{2} + 12 x \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )} + 9 x \log {\left (3 \right )}^{2} - 12 \log {\left (\frac {x}{3} \right )}^{2} - 36 \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )} - 27 \log {\left (3 \right )}^{2}} \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. 1267 vs.
\(2 (33) = 66\).
time = 143.40, size = 1267, normalized size = 31.68 \begin {gather*} \text {Too large to display} \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 = -0.02 \begin {gather*} \int -\frac {{\ln \left (3\,x^2\right )}^2\,\left (15\,\ln \left (\frac {x}{3}\right )-11\,x+x^2+15\right )+\ln \left (3\,x^2\right )\,\left (12\,x-4\,x^2+\ln \left (\frac {x}{3}\right )\,\left (20\,x-60\right )\right )+\ln \left (\frac {x}{3}\right )\,\left (5\,x^2\,{\mathrm {e}}^x-5\,x^2\right )-{\mathrm {e}}^x\,\left (x^3-x^2+5\,x\right )+5\,x^2}{{\ln \left (3\,x^2\right )}^4\,\left (x^2-6\,x+9\right )-2\,x^3\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+x^4-{\ln \left (3\,x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (6\,x-2\,x^2\right )-6\,x^2+2\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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