Optimal. Leaf size=31 \[ -5+\frac {x}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \]
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Rubi [F] time = 7.00, 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 {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\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 {-16-27 x-2 x^2+x (3+x) \log \left (e^{4 x} x^4\right )-\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5-2 x-(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx\\ &=\int \left (\frac {16+27 x+2 x^2-3 x \log \left (e^{4 x} x^4\right )-x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}+\frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}\right ) \, dx\\ &=\int \frac {16+27 x+2 x^2-3 x \log \left (e^{4 x} x^4\right )-x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx\\ &=\int \frac {-16-27 x-2 x^2+x (3+x) \log \left (e^{4 x} x^4\right )}{\left (5-2 x-(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx\\ &=\int \left (\frac {16}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}+\frac {27 x}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}+\frac {2 x^2}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}-\frac {3 x \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}-\frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx\\ &=2 \int \frac {x^2}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-3 \int \frac {x \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+16 \int \frac {1}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+27 \int \frac {x}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-\int \frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx\\ &=2 \int \frac {x^2}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-3 \int \frac {x \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+16 \int \frac {1}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+27 \int \frac {x}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-\int \frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 29, normalized size = 0.94 \begin {gather*} \frac {x}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 28, normalized size = 0.90 \begin {gather*} \frac {x}{\log \left (x - \log \left ({\left (x + 4\right )} \log \left (x^{4} e^{\left (4 \, x\right )}\right ) + 2 \, x - 5\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 6.72, size = 32, normalized size = 1.03 \begin {gather*} \frac {x}{\log \left (x - \log \left (4 \, x^{2} + x \log \left (x^{4}\right ) + 18 \, x + 4 \, \log \left (x^{4}\right ) - 5\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.58, size = 329, normalized size = 10.61
method | result | size |
risch | \(\frac {x}{\ln \left (-\ln \left (\left (4+x \right ) \left (4 \ln \relax (x )+4 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{3 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{4 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{3 x}\right )\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{4}\right ) \left (-\mathrm {csgn}\left (i x^{4}\right )+\mathrm {csgn}\left (i x^{3}\right )\right ) \left (-\mathrm {csgn}\left (i x^{4}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right ) \left (-\mathrm {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right )+\mathrm {csgn}\left (i x^{4}\right )\right ) \left (-\mathrm {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{4 x}\right )\right )}{2}\right )+2 x -5\right )+x \right )}\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 27, normalized size = 0.87 \begin {gather*} \frac {x}{\log \left (x - \log \left (4 \, x^{2} + 4 \, {\left (x + 4\right )} \log \relax (x) + 18 \, x - 5\right )\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 {27\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+3\,x\right )+2\,x^2-\ln \left (x-\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\right )\,\left (5\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+4\,x\right )+\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\,\left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )-2\,x^2\right )+16}{{\ln \left (x-\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\right )}^2\,\left (5\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+4\,x\right )+\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\,\left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )-2\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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