Optimal. Leaf size=28 \[ -x+\frac {2}{\log \left (-6+e^{-4 x} \left (\frac {3}{x}+x\right )^2+\log (4)\right )} \]
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Rubi [F] time = 5.91, 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 {36+72 x+48 x^3-4 x^4+8 x^5+\left (-9 x-6 x^3-x^5+e^{4 x} \left (6 x^3-x^3 \log (4)\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+6 x^2+x^4+e^{4 x} \left (-6 x^2+x^2 \log (4)\right )\right )}{x^2}\right )}{\left (9 x+6 x^3+x^5+e^{4 x} \left (-6 x^3+x^3 \log (4)\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+6 x^2+x^4+e^{4 x} \left (-6 x^2+x^2 \log (4)\right )\right )}{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 \left (-1+\frac {4 \left (3+x^2\right ) \left (3+6 x-x^2+2 x^3\right )}{x \left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )}\right ) \, dx\\ &=-x+4 \int \frac {\left (3+x^2\right ) \left (3+6 x-x^2+2 x^3\right )}{x \left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \, dx\\ &=-x+4 \int \left (\frac {18}{\left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )}+\frac {9}{x \left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )}+\frac {12 x^2}{\left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )}+\frac {2 x^4}{\left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )}+\frac {x^3}{\left (-9-6 x^2-x^4+6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )}\right ) \, dx\\ &=-x+4 \int \frac {x^3}{\left (-9-6 x^2-x^4+6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \, dx+8 \int \frac {x^4}{\left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \, dx+36 \int \frac {1}{x \left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \, dx+48 \int \frac {x^2}{\left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \, dx+72 \int \frac {1}{\left (9+6 x^2+x^4-6 e^{4 x} x^2 \left (1-\frac {\log (2)}{3}\right )\right ) \log ^2\left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 39, normalized size = 1.39 \begin {gather*} -x+\frac {2}{\log \left (\frac {e^{-4 x} \left (9+x^4+x^2 \left (6+e^{4 x} (-6+\log (4))\right )\right )}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 82, normalized size = 2.93 \begin {gather*} -\frac {x \log \left (\frac {{\left (x^{4} + 6 \, x^{2} + 2 \, {\left (x^{2} \log \relax (2) - 3 \, x^{2}\right )} e^{\left (4 \, x\right )} + 9\right )} e^{\left (-4 \, x\right )}}{x^{2}}\right ) - 2}{\log \left (\frac {{\left (x^{4} + 6 \, x^{2} + 2 \, {\left (x^{2} \log \relax (2) - 3 \, x^{2}\right )} e^{\left (4 \, x\right )} + 9\right )} e^{\left (-4 \, x\right )}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.07, size = 94, normalized size = 3.36 \begin {gather*} -\frac {x \log \left (x^{2}\right ) - x \log \left ({\left (x^{4} + 2 \, x^{2} e^{\left (4 \, x\right )} \log \relax (2) - 6 \, x^{2} e^{\left (4 \, x\right )} + 6 \, x^{2} + 9\right )} e^{\left (-4 \, x\right )}\right ) + 2}{\log \left (x^{2}\right ) - \log \left ({\left (x^{4} + 2 \, x^{2} e^{\left (4 \, x\right )} \log \relax (2) - 6 \, x^{2} e^{\left (4 \, x\right )} + 6 \, x^{2} + 9\right )} e^{\left (-4 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.81, size = 83, normalized size = 2.96
method | result | size |
norman | \(\frac {2-x \ln \left (\frac {\left (\left (2 x^{2} \ln \relax (2)-6 x^{2}\right ) {\mathrm e}^{4 x}+x^{4}+6 x^{2}+9\right ) {\mathrm e}^{-4 x}}{x^{2}}\right )}{\ln \left (\frac {\left (\left (2 x^{2} \ln \relax (2)-6 x^{2}\right ) {\mathrm e}^{4 x}+x^{4}+6 x^{2}+9\right ) {\mathrm e}^{-4 x}}{x^{2}}\right )}\) | \(83\) |
risch | \(-x +\frac {4 i}{\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-4 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-4 x} \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right )+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-4 x} \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right ) {\mathrm e}^{-4 x}}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )+2 i \ln \relax (2)-4 i \ln \relax (x )-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )^{3}-\pi \mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )^{3}-8 i \ln \left ({\mathrm e}^{x}\right )+\pi \mathrm {csgn}\left (i {\mathrm e}^{-4 x} \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right )^{3}-\pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right ) {\mathrm e}^{-4 x}}{x^{2}}\right )^{3}+2 i \ln \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )-\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 \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-4 x} \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-4 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-4 x} \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right )^{2}+2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{3 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )^{2}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )^{2}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-4 x} \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right ) {\mathrm e}^{-4 x}}{x^{2}}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{4 x} \ln \relax (2) x^{2}+\frac {9}{2}+\frac {x^{4}}{2}+\left (-3 \,{\mathrm e}^{4 x}+3\right ) x^{2}\right ) {\mathrm e}^{-4 x}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}\) | \(819\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.73, size = 77, normalized size = 2.75 \begin {gather*} -\frac {4 \, x^{2} - x \log \left (x^{4} + 2 \, x^{2} {\left (\log \relax (2) - 3\right )} e^{\left (4 \, x\right )} + 6 \, x^{2} + 9\right ) + 2 \, x \log \relax (x) + 2}{4 \, x - \log \left (x^{4} + 2 \, x^{2} {\left (\log \relax (2) - 3\right )} e^{\left (4 \, x\right )} + 6 \, x^{2} + 9\right ) + 2 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 45, normalized size = 1.61 \begin {gather*} \frac {2}{\ln \left (\frac {{\mathrm {e}}^{-4\,x}\,\left ({\mathrm {e}}^{4\,x}\,\left (2\,x^2\,\ln \relax (2)-6\,x^2\right )+6\,x^2+x^4+9\right )}{x^2}\right )}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 41, normalized size = 1.46 \begin {gather*} - x + \frac {2}{\log {\left (\frac {\left (x^{4} + 6 x^{2} + \left (- 6 x^{2} + 2 x^{2} \log {\relax (2 )}\right ) e^{4 x} + 9\right ) e^{- 4 x}}{x^{2}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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