Optimal. Leaf size=26 \[ 4+\frac {3 e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \]
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Rubi [F] time = 7.52, 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 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^x \left (2+(-2+x) x \left (4+\log \left (x^2\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )+\left (4+\log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right ) \left (-1+(-2+x) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\\ &=3 \int \frac {e^x \left (2+(-2+x) x \left (4+\log \left (x^2\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )+\left (4+\log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right ) \left (-1+(-2+x) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\\ &=3 \int \left (\frac {e^x \left (2-4 \log \left (4+\log \left (x^2\right )\right )-\log \left (x^2\right ) \log \left (4+\log \left (x^2\right )\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}+\frac {e^x (-2+x)}{x^3 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}\right ) \, dx\\ &=3 \int \frac {e^x \left (2-4 \log \left (4+\log \left (x^2\right )\right )-\log \left (x^2\right ) \log \left (4+\log \left (x^2\right )\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx+3 \int \frac {e^x (-2+x)}{x^3 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\\ &=3 \int \left (-\frac {2 e^x}{x^3 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}+\frac {e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}\right ) \, dx+3 \int \frac {e^x \left (2-\left (4+\log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\\ &=3 \int \left (\frac {2 e^x}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}-\frac {4 e^x \log \left (4+\log \left (x^2\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}-\frac {e^x \log \left (x^2\right ) \log \left (4+\log \left (x^2\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}\right ) \, dx+3 \int \frac {e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx-6 \int \frac {e^x}{x^3 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\\ &=-\left (3 \int \frac {e^x \log \left (x^2\right ) \log \left (4+\log \left (x^2\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\right )+3 \int \frac {e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx+6 \int \frac {e^x}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx-6 \int \frac {e^x}{x^3 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx-12 \int \frac {e^x \log \left (4+\log \left (x^2\right )\right )}{x^3 \left (4+\log \left (x^2\right )\right ) \left (x+\log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 24, normalized size = 0.92 \begin {gather*} \frac {3 e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 23, normalized size = 0.88 \begin {gather*} \frac {3 \, e^{x}}{x^{2} \log \left (\frac {x}{x + \log \left (\log \left (x^{2}\right ) + 4\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.36, size = 288, normalized size = 11.08 \begin {gather*} -\frac {3 \, {\left (e^{x} \log \left (x^{2}\right ) \log \relax (x) \log \left (\log \left (x^{2}\right ) + 4\right ) + 2 \, e^{x} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) + 4 \, e^{x} \log \relax (x) \log \left (\log \left (x^{2}\right ) + 4\right ) - e^{x} \log \left (x^{2}\right ) + 8 \, e^{x} \log \left (\log \left (x^{2}\right ) + 4\right ) - 4 \, e^{x}\right )}}{x^{2} \log \left (x^{2}\right ) \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \relax (x) \log \left (\log \left (x^{2}\right ) + 4\right ) - x^{2} \log \left (x^{2}\right ) \log \relax (x)^{2} \log \left (\log \left (x^{2}\right ) + 4\right ) + 2 \, x^{2} \log \left (x^{2}\right ) \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - 2 \, x^{2} \log \left (x^{2}\right ) \log \relax (x) \log \left (\log \left (x^{2}\right ) + 4\right ) + 4 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \relax (x) \log \left (\log \left (x^{2}\right ) + 4\right ) - 4 \, x^{2} \log \relax (x)^{2} \log \left (\log \left (x^{2}\right ) + 4\right ) - 2 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \relax (x) + 2 \, x^{2} \log \relax (x)^{2} + 8 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - 8 \, x^{2} \log \relax (x) \log \left (\log \left (x^{2}\right ) + 4\right ) - 4 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) + 4 \, x^{2} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.37, size = 346, normalized size = 13.31
method | result | size |
risch | \(\frac {6 i {\mathrm e}^{x}}{x^{2} \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left (\ln \left (4+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+x \right )\right )}\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 28, normalized size = 1.08 \begin {gather*} -\frac {3 \, e^{x}}{x^{2} \log \left (x + \log \relax (2) + \log \left (\log \relax (x) + 2\right )\right ) - x^{2} \log \relax (x)} \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 {\ln \left (\ln \left (x^2\right )+4\right )\,\left (12\,{\mathrm {e}}^x+3\,\ln \left (x^2\right )\,{\mathrm {e}}^x\right )-6\,{\mathrm {e}}^x+\ln \left (\frac {x}{x+\ln \left (\ln \left (x^2\right )+4\right )}\right )\,\left ({\mathrm {e}}^x\,\left (24\,x-12\,x^2\right )-\ln \left (\ln \left (x^2\right )+4\right )\,\left ({\mathrm {e}}^x\,\left (12\,x-24\right )+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (3\,x-6\right )\right )+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (6\,x-3\,x^2\right )\right )}{{\ln \left (\frac {x}{x+\ln \left (\ln \left (x^2\right )+4\right )}\right )}^2\,\left (x^4\,\ln \left (x^2\right )+\ln \left (\ln \left (x^2\right )+4\right )\,\left (x^3\,\ln \left (x^2\right )+4\,x^3\right )+4\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.67, size = 20, normalized size = 0.77 \begin {gather*} \frac {3 e^{x}}{x^{2} \log {\left (\frac {x}{x + \log {\left (\log {\left (x^{2} \right )} + 4 \right )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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