Optimal. Leaf size=24 \[ e^{x \log (8) (3 (1+x)+\log (x)) \log \left (e^{1+x} (4+x)\right )} \]
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Rubi [F] time = 3.38, 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 {\left (e^{1+x} (4+x)\right )^{\left (3 x+3 x^2\right ) \log (8)+x \log (8) \log (x)} \left (\left (15 x+18 x^2+3 x^3\right ) \log (8)+\left (5 x+x^2\right ) \log (8) \log (x)+\left (\left (16+28 x+6 x^2\right ) \log (8)+(4+x) \log (8) \log (x)\right ) \log \left (e^{1+x} (4+x)\right )\right )}{4+x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \left (\left (15 x+18 x^2+3 x^3\right ) \log (8)+\left (5 x+x^2\right ) \log (8) \log (x)+\left (\left (16+28 x+6 x^2\right ) \log (8)+(4+x) \log (8) \log (x)\right ) \log \left (e^{1+x} (4+x)\right )\right )}{4+x} \, dx\\ &=\int \left (\frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) \log (8) (3+3 x+\log (x))}{4+x}+\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (8) (4+6 x+\log (x)) \log \left (e^{1+x} (4+x)\right )\right ) \, dx\\ &=\log (8) \int \frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) (3+3 x+\log (x))}{4+x} \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (4+6 x+\log (x)) \log \left (e^{1+x} (4+x)\right ) \, dx\\ &=\log (8) \int \left (\frac {3 x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \left (5+6 x+x^2\right )}{4+x}+\frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) \log (x)}{4+x}\right ) \, dx+\log (8) \int \left (4 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right )+6 x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right )+\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right )\right ) \, dx\\ &=\log (8) \int \frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} (5+x) \log (x)}{4+x} \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right ) \, dx+(3 \log (8)) \int \frac {x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \left (5+6 x+x^2\right )}{4+x} \, dx+(4 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx\\ &=\log (8) \int \left (\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)+x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)-\frac {4 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)}{4+x}\right ) \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right ) \, dx+(3 \log (8)) \int \left (-3 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}+2 x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}+x^2 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}+\frac {12 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}}{4+x}\right ) \, dx+(4 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx\\ &=\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \, dx+\log (8) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \, dx+\log (8) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x) \log \left (e^{1+x} (4+x)\right ) \, dx+(3 \log (8)) \int x^2 \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \, dx-(4 \log (8)) \int \frac {\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log (x)}{4+x} \, dx+(4 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \, dx+(6 \log (8)) \int x \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \log \left (e^{1+x} (4+x)\right ) \, dx-(9 \log (8)) \int \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \, dx+(36 \log (8)) \int \frac {\left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))}}{4+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.22, size = 21, normalized size = 0.88 \begin {gather*} \left (e^{1+x} (4+x)\right )^{x \log (8) (3+3 x+\log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 26, normalized size = 1.08 \begin {gather*} \left ({\left (x + 4\right )} e^{\left (x + 1\right )}\right )^{3 \, x \log \relax (2) \log \relax (x) + 9 \, {\left (x^{2} + x\right )} \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 34, normalized size = 1.42 \begin {gather*} {\left (x e^{\left (x + 1\right )} + 4 \, e^{\left (x + 1\right )}\right )}^{9 \, x^{2} \log \relax (2) + 3 \, x \log \relax (2) \log \relax (x) + 9 \, x \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 95, normalized size = 3.96
method | result | size |
risch | \(2^{-\frac {3 x \left (3 x +3+\ln \relax (x )\right ) \left (i \mathrm {csgn}\left (i {\mathrm e}^{x +1} \left (4+x \right )\right ) \pi -i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x +1}\right )-i \pi \,\mathrm {csgn}\left (i \left (4+x \right )\right )+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x +1} \left (4+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x +1}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right )-2 \ln \left ({\mathrm e}^{x +1}\right )-2 \ln \left (4+x \right )\right )}{2}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 68, normalized size = 2.83 \begin {gather*} e^{\left (9 \, x^{3} \log \relax (2) + 9 \, x^{2} \log \relax (2) \log \left (x + 4\right ) + 3 \, x^{2} \log \relax (2) \log \relax (x) + 3 \, x \log \relax (2) \log \left (x + 4\right ) \log \relax (x) + 18 \, x^{2} \log \relax (2) + 9 \, x \log \relax (2) \log \left (x + 4\right ) + 3 \, x \log \relax (2) \log \relax (x) + 9 \, x \log \relax (2)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.97, size = 49, normalized size = 2.04 \begin {gather*} x^{3\,x\,\ln \relax (2)\,\ln \left (4\,\mathrm {e}\,{\mathrm {e}}^x+x\,\mathrm {e}\,{\mathrm {e}}^x\right )}\,{\left (4\,{\mathrm {e}}^{x+1}+x\,{\mathrm {e}}^{x+1}\right )}^{9\,\ln \relax (2)\,x^2+9\,\ln \relax (2)\,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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