Optimal. Leaf size=24 \[ \left (6+e^{-\frac {3}{x}+x}-2 x\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right ) \]
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Rubi [B] time = 1.16, antiderivative size = 97, normalized size of antiderivative = 4.04, number of steps used = 14, number of rules used = 9, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6742, 6688, 2353, 2300, 2178, 2302, 29, 2520, 2288} \begin {gather*} -2 x \log \left (\log \left (\frac {x^4}{16}\right )\right )+6 \log \left (\log \left (\frac {x^4}{16}\right )\right )+\frac {e^{x-\frac {3}{x}} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (\frac {3}{x^2}+1\right ) x^2 \log \left (\frac {x^4}{16}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2178
Rule 2288
Rule 2300
Rule 2302
Rule 2353
Rule 2520
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (-12+4 x+x \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{x \log \left (\frac {x^4}{16}\right )}+\frac {e^{-\frac {3}{x}+x} \left (4 x+3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{x^2 \log \left (\frac {x^4}{16}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-12+4 x+x \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )}{x \log \left (\frac {x^4}{16}\right )} \, dx\right )+\int \frac {e^{-\frac {3}{x}+x} \left (4 x+3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{x^2 \log \left (\frac {x^4}{16}\right )} \, dx\\ &=\frac {e^{-\frac {3}{x}+x} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (1+\frac {3}{x^2}\right ) x^2 \log \left (\frac {x^4}{16}\right )}-2 \int \left (\frac {4 (-3+x)}{x \log \left (\frac {x^4}{16}\right )}+\log \left (\log \left (\frac {x^4}{16}\right )\right )\right ) \, dx\\ &=\frac {e^{-\frac {3}{x}+x} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (1+\frac {3}{x^2}\right ) x^2 \log \left (\frac {x^4}{16}\right )}-2 \int \log \left (\log \left (\frac {x^4}{16}\right )\right ) \, dx-8 \int \frac {-3+x}{x \log \left (\frac {x^4}{16}\right )} \, dx\\ &=-2 x \log \left (\log \left (\frac {x^4}{16}\right )\right )+\frac {e^{-\frac {3}{x}+x} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (1+\frac {3}{x^2}\right ) x^2 \log \left (\frac {x^4}{16}\right )}-8 \int \left (\frac {1}{\log \left (\frac {x^4}{16}\right )}-\frac {3}{x \log \left (\frac {x^4}{16}\right )}\right ) \, dx+8 \int \frac {1}{\log \left (\frac {x^4}{16}\right )} \, dx\\ &=-2 x \log \left (\log \left (\frac {x^4}{16}\right )\right )+\frac {e^{-\frac {3}{x}+x} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (1+\frac {3}{x^2}\right ) x^2 \log \left (\frac {x^4}{16}\right )}-8 \int \frac {1}{\log \left (\frac {x^4}{16}\right )} \, dx+24 \int \frac {1}{x \log \left (\frac {x^4}{16}\right )} \, dx+\frac {(4 x) \operatorname {Subst}\left (\int \frac {e^{x/4}}{x} \, dx,x,\log \left (\frac {x^4}{16}\right )\right )}{\sqrt [4]{x^4}}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{4} \log \left (\frac {x^4}{16}\right )\right )}{\sqrt [4]{x^4}}-2 x \log \left (\log \left (\frac {x^4}{16}\right )\right )+\frac {e^{-\frac {3}{x}+x} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (1+\frac {3}{x^2}\right ) x^2 \log \left (\frac {x^4}{16}\right )}+6 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {x^4}{16}\right )\right )-\frac {(4 x) \operatorname {Subst}\left (\int \frac {e^{x/4}}{x} \, dx,x,\log \left (\frac {x^4}{16}\right )\right )}{\sqrt [4]{x^4}}\\ &=6 \log \left (\log \left (\frac {x^4}{16}\right )\right )-2 x \log \left (\log \left (\frac {x^4}{16}\right )\right )+\frac {e^{-\frac {3}{x}+x} \left (3 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )+x^2 \log \left (\frac {x^4}{16}\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right )\right )}{\left (1+\frac {3}{x^2}\right ) x^2 \log \left (\frac {x^4}{16}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.44, size = 24, normalized size = 1.00 \begin {gather*} \left (6+e^{-\frac {3}{x}+x}-2 x\right ) \log \left (\log \left (\frac {x^4}{16}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 26, normalized size = 1.08 \begin {gather*} -{\left (2 \, x - e^{\left (\frac {x^{2} - 3}{x}\right )} - 6\right )} \log \left (\log \left (\frac {1}{16} \, x^{4}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x^{2} - {\left (x^{2} + 3\right )} e^{\left (\frac {x^{2} - 3}{x}\right )}\right )} \log \left (\frac {1}{16} \, x^{4}\right ) \log \left (\log \left (\frac {1}{16} \, x^{4}\right )\right ) + 8 \, x^{2} - 4 \, x e^{\left (\frac {x^{2} - 3}{x}\right )} - 24 \, x}{x^{2} \log \left (\frac {1}{16} \, x^{4}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.12, size = 336, normalized size = 14.00
method | result | size |
risch | \(\left (-2 x +{\mathrm e}^{\frac {x^{2}-3}{x}}\right ) \ln \left (-4 \ln \relax (2)+4 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{4}\right ) \left (-\mathrm {csgn}\left (i x^{4}\right )+\mathrm {csgn}\left (i x^{3}\right )\right ) \left (-\mathrm {csgn}\left (i x^{4}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}\right )+6 \ln \left (\ln \relax (x )-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\pi \mathrm {csgn}\left (i x^{4}\right )^{3}+\pi \mathrm {csgn}\left (i x^{3}\right )^{3}-\pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}-8 i \ln \relax (2)\right )}{8}\right )\) | \(336\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 43, normalized size = 1.79 \begin {gather*} -4 \, x \log \relax (2) + 2 \, e^{\left (x - \frac {3}{x}\right )} \log \relax (2) - {\left (2 \, x - e^{\left (x - \frac {3}{x}\right )} - 6\right )} \log \left (-\log \relax (2) + \log \relax (x)\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 = -0.04 \begin {gather*} \int \frac {24\,x+4\,x\,{\mathrm {e}}^{\frac {x^2-3}{x}}-8\,x^2+\ln \left (\ln \left (\frac {x^4}{16}\right )\right )\,\ln \left (\frac {x^4}{16}\right )\,\left ({\mathrm {e}}^{\frac {x^2-3}{x}}\,\left (x^2+3\right )-2\,x^2\right )}{x^2\,\ln \left (\frac {x^4}{16}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.61, size = 37, normalized size = 1.54 \begin {gather*} - 2 x \log {\left (\log {\left (\frac {x^{4}}{16} \right )} \right )} + e^{\frac {x^{2} - 3}{x}} \log {\left (\log {\left (\frac {x^{4}}{16} \right )} \right )} + 6 \log {\left (\log {\left (\frac {x^{4}}{16} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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