Optimal. Leaf size=23 \[ e^{\frac {1}{16} \left (-1+2 x+\log \left (\frac {x^2}{1+x}\right )\right )^4} \]
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Rubi [F] time = 62.22, 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 {\exp \left (\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{x (4+4 x)} \, dx\\ &=\int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (-2-3 x-2 x^2\right ) \left (1-2 x-\log \left (\frac {x^2}{1+x}\right )\right )^3}{4 x^3} \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (-2-3 x-2 x^2\right ) \left (1-2 x-\log \left (\frac {x^2}{1+x}\right )\right )^3}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^3 \left (2+3 x+2 x^2\right )}{x^3}+\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^2 \left (2+3 x+2 x^2\right ) \log \left (\frac {x^2}{1+x}\right )}{x^3}+\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x) \left (2+3 x+2 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )}{x^3}+\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )}{x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^3 \left (2+3 x+2 x^2\right )}{x^3} \, dx+\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} \left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )}{x^3} \, dx+\frac {3}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x)^2 \left (2+3 x+2 x^2\right ) \log \left (\frac {x^2}{1+x}\right )}{x^3} \, dx+\frac {3}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )\right ) \left (\frac {x^2}{1+x}\right )^{\frac {3}{4}+\frac {3 x}{2}-3 x^2+2 x^3} (-1+2 x) \left (2+3 x+2 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )}{x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.39, size = 87, normalized size = 3.78 \begin {gather*} e^{\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (\frac {x^2}{1+x}\right )^{\frac {1}{4} (-1+2 x)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 103, normalized size = 4.48 \begin {gather*} e^{\left (x^{4} + \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} + \frac {3}{8} \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {3}{2} \, x^{2} + \frac {1}{4} \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {1}{2} \, x + \frac {1}{16}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 18.45, size = 163, normalized size = 7.09 \begin {gather*} e^{\left (x^{4} + 2 \, x^{3} \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {3}{2} \, x^{2} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {1}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {3}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right )^{2} - \frac {1}{4} \, \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {3}{8} \, \log \left (\frac {x^{2}}{x + 1}\right )^{2} - \frac {1}{2} \, x - \frac {1}{4} \, \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {1}{16}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 129, normalized size = 5.61
method | result | size |
risch | \(\left (\frac {x^{2}}{x +1}\right )^{\frac {\left (2 x -1\right )^{3}}{4}} {\mathrm e}^{\frac {\ln \left (\frac {x^{2}}{x +1}\right )^{4}}{16}+\frac {\ln \left (\frac {x^{2}}{x +1}\right )^{3} x}{2}-\frac {\ln \left (\frac {x^{2}}{x +1}\right )^{3}}{4}+\frac {3 \ln \left (\frac {x^{2}}{x +1}\right )^{2} x^{2}}{2}-\frac {3 \ln \left (\frac {x^{2}}{x +1}\right )^{2} x}{2}+\frac {3 \ln \left (\frac {x^{2}}{x +1}\right )^{2}}{8}+\frac {1}{16}+x^{4}-2 x^{3}+\frac {3 x^{2}}{2}-\frac {x}{2}}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{4} \, \int \frac {{\left (16 \, x^{5} + {\left (2 \, x^{2} + 3 \, x + 2\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} - 8 \, x^{3} + 3 \, {\left (4 \, x^{3} + 4 \, x^{2} + x - 2\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} - 8 \, x^{2} + 3 \, {\left (8 \, x^{4} + 4 \, x^{3} - 2 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {x^{2}}{x + 1}\right ) + 9 \, x - 2\right )} e^{\left (x^{4} + \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} + \frac {3}{8} \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {3}{2} \, x^{2} + \frac {1}{4} \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {1}{2} \, x + \frac {1}{16}\right )}}{x^{2} + x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.81, size = 338, normalized size = 14.70 \begin {gather*} {\mathrm {e}}^{\frac {\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^3}{4}}\,{\mathrm {e}}^{\frac {{\ln \left (x^2\right )}^3\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {3\,\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^2}{4}}\,{\mathrm {e}}^{-\frac {3\,{\ln \left (x^2\right )}^2\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{\frac {x\,{\ln \left (\frac {1}{x+1}\right )}^3}{2}}\,{\mathrm {e}}^{-\frac {3\,x\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,x^2\,{\ln \left (x^2\right )}^2}{2}}\,{\mathrm {e}}^{1/16}\,{\mathrm {e}}^{-\frac {{\ln \left (x^2\right )}^3}{4}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (x^2\right )}^2}{8}}\,{\mathrm {e}}^{\frac {{\ln \left (x^2\right )}^4}{16}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (x^2\right )}^2\,{\ln \left (\frac {1}{x+1}\right )}^2}{8}}\,{\mathrm {e}}^{\frac {3\,x\,\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,x\,{\ln \left (x^2\right )}^2\,\ln \left (\frac {1}{x+1}\right )}{2}}\,{\mathrm {e}}^{-2\,x^3}\,{\mathrm {e}}^{\frac {3\,x^2}{2}}\,{\mathrm {e}}^{\frac {3\,x^2\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,\ln \left (x^2\right )\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x+1}\right )}^3}{4}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (\frac {1}{x+1}\right )}^2}{8}}\,{\mathrm {e}}^{\frac {{\ln \left (\frac {1}{x+1}\right )}^4}{16}}\,{\mathrm {e}}^{\frac {3\,x\,\ln \left (\frac {1}{x+1}\right )}{2}}\,{\mathrm {e}}^{\frac {x\,{\ln \left (x^2\right )}^3}{2}}\,{\mathrm {e}}^{-\frac {3\,x\,{\ln \left (x^2\right )}^2}{2}}\,{\left (\frac {1}{x+1}\right )}^{3\,x^2\,\ln \left (x^2\right )-3\,x\,\ln \left (x^2\right )-3\,x^2+2\,x^3-\frac {1}{4}}\,{\left (x^2\right )}^{2\,x^3-3\,x^2+\frac {3\,x}{2}-\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.08, size = 104, normalized size = 4.52 \begin {gather*} e^{x^{4} - 2 x^{3} + \frac {3 x^{2}}{2} - \frac {x}{2} + \left (\frac {x}{2} - \frac {1}{4}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )}^{3} + \left (\frac {3 x^{2}}{2} - \frac {3 x}{2} + \frac {3}{8}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )}^{2} + \left (2 x^{3} - 3 x^{2} + \frac {3 x}{2} - \frac {1}{4}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )} + \frac {\log {\left (\frac {x^{2}}{x + 1} \right )}^{4}}{16} + \frac {1}{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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