Optimal. Leaf size=33 \[ \frac {x}{\left (4-3 x+\frac {x^2}{\log \left (x^2\right )}\right ) \left (3-\log \left (x+e^x x\right )\right )} \]
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Rubi [F] time = 21.55, 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 {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \left (1+e^x\right ) x^2+\left (-2+e^x (-2+x)\right ) x^2 \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (1+e^x\right ) \log \left (\left (1+e^x\right ) x\right ) \left (-2 x^2+x^2 \log \left (x^2\right )-4 \log ^2\left (x^2\right )\right )}{\left (1+e^x\right ) \left (3-\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+(4-3 x) \log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )}+\frac {6 x^2-2 x^2 \log \left (\left (1+e^x\right ) x\right )-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )+x^2 \log \left (\left (1+e^x\right ) x\right ) \log \left (x^2\right )+16 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )-3 x^2 \log ^2\left (x^2\right )-4 \log \left (\left (1+e^x\right ) x\right ) \log ^2\left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx+\int \frac {6 x^2-2 x^2 \log \left (\left (1+e^x\right ) x\right )-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )+x^2 \log \left (\left (1+e^x\right ) x\right ) \log \left (x^2\right )+16 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )-3 x^2 \log ^2\left (x^2\right )-4 \log \left (\left (1+e^x\right ) x\right ) \log ^2\left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2} \, dx\\ &=-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx+\int \frac {6 x^2+(-2+x) x^2 \log \left (x^2\right )+\left (16+x-3 x^2\right ) \log ^2\left (x^2\right )+\log \left (\left (1+e^x\right ) x\right ) \left (-2 x^2+x^2 \log \left (x^2\right )-4 \log ^2\left (x^2\right )\right )}{\left (3-\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+(4-3 x) \log \left (x^2\right )\right )^2} \, dx\\ &=-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx+\int \left (\frac {16+x-3 x^2-4 \log \left (\left (1+e^x\right ) x\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2}+\frac {x^2 \left (-32+48 x-26 x^2+3 x^3\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2}+\frac {x^2 \left (-40+8 x+3 x^2+12 \log \left (\left (1+e^x\right ) x\right )-3 x \log \left (\left (1+e^x\right ) x\right )\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )}\right ) \, dx\\ &=\int \frac {16+x-3 x^2-4 \log \left (\left (1+e^x\right ) x\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2} \, dx+\int \frac {x^2 \left (-32+48 x-26 x^2+3 x^3\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2} \, dx+\int \frac {x^2 \left (-40+8 x+3 x^2+12 \log \left (\left (1+e^x\right ) x\right )-3 x \log \left (\left (1+e^x\right ) x\right )\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (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 = 0.26, size = 35, normalized size = 1.06 \begin {gather*} -\frac {x \log \left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+(4-3 x) \log \left (x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 49, normalized size = 1.48 \begin {gather*} \frac {x \log \left (x^{2}\right )}{3 \, x^{2} - 3 \, {\left (3 \, x - 4\right )} \log \left (x^{2}\right ) - {\left (x^{2} - {\left (3 \, x - 4\right )} \log \left (x^{2}\right )\right )} \log \left (x e^{x} + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.23, size = 69, normalized size = 2.09 \begin {gather*} -\frac {2 \, x \log \relax (x)}{x^{2} \log \relax (x) - 6 \, x \log \relax (x)^{2} + x^{2} \log \left (e^{x} + 1\right ) - 6 \, x \log \relax (x) \log \left (e^{x} + 1\right ) - 3 \, x^{2} + 18 \, x \log \relax (x) + 8 \, \log \relax (x)^{2} + 8 \, \log \relax (x) \log \left (e^{x} + 1\right ) - 24 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 30.77, size = 266, normalized size = 8.06
method | result | size |
risch | \(\frac {2 i x \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )\right )}{\left (3 \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-6 \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+3 \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}-4 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+8 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i x^{2}+12 i x \ln \relax (x )-16 i \ln \relax (x )\right ) \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )^{3}+2 i \ln \relax (x )+2 i \ln \left ({\mathrm e}^{x}+1\right )-6 i\right )}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 56, normalized size = 1.70 \begin {gather*} \frac {2 \, x \log \relax (x)}{2 \, {\left (3 \, x - 4\right )} \log \relax (x)^{2} + 3 \, x^{2} - {\left (x^{2} + 18 \, x - 24\right )} \log \relax (x) - {\left (x^{2} - 2 \, {\left (3 \, x - 4\right )} \log \relax (x)\right )} \log \left (e^{x} + 1\right )} \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 {6\,x^2\,{\mathrm {e}}^x-\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (2\,x^2-x^3\right )+2\,x^2\right )+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (-3\,x^2+x+16\right )-3\,x+16\right )-\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (2\,x^2\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left (4\,{\mathrm {e}}^x+4\right )+2\,x^2-\ln \left (x^2\right )\,\left (x^2\,{\mathrm {e}}^x+x^2\right )\right )+6\,x^2}{9\,x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (81\,x^2-216\,x+144\right )-216\,x+81\,x^2+144\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (72\,x^2-54\,x^3\right )+72\,x^2-54\,x^3\right )-\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (6\,x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (54\,x^2-144\,x+96\right )-144\,x+54\,x^2+96\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (48\,x^2-36\,x^3\right )+48\,x^2-36\,x^3\right )+6\,x^4\right )+{\ln \left (x+x\,{\mathrm {e}}^x\right )}^2\,\left (x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (9\,x^2-24\,x+16\right )-24\,x+9\,x^2+16\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (8\,x^2-6\,x^3\right )+8\,x^2-6\,x^3\right )+x^4\right )+9\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.63, size = 54, normalized size = 1.64 \begin {gather*} - \frac {x \log {\left (x^{2} \right )}}{- 3 x^{2} + 9 x \log {\left (x^{2} \right )} + \left (x^{2} - 3 x \log {\left (x^{2} \right )} + 4 \log {\left (x^{2} \right )}\right ) \log {\left (x e^{x} + x \right )} - 12 \log {\left (x^{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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