Optimal. Leaf size=24 \[ x^4 \left (1+\log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right ) \]
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Rubi [A] time = 0.72, antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 23, number of rules used = 10, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {6742, 2184, 2190, 2531, 6609, 2282, 6589, 14, 43, 2551} \begin {gather*} x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (e^{3 x}+7\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 43
Rule 2184
Rule 2190
Rule 2282
Rule 2531
Rule 2551
Rule 6589
Rule 6609
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {21 x^4}{7+e^{3 x}}-x^3 \left (-5+2 x-4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right )\right ) \, dx\\ &=21 \int \frac {x^4}{7+e^{3 x}} \, dx-\int x^3 \left (-5+2 x-4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right ) \, dx\\ &=\frac {3 x^5}{5}-3 \int \frac {e^{3 x} x^4}{7+e^{3 x}} \, dx-\int \left (x^3 (-5+2 x)-4 x^3 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\right ) \, dx\\ &=\frac {3 x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+4 \int x^3 \log \left (1+\frac {e^{3 x}}{7}\right ) \, dx+4 \int x^3 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right ) \, dx-\int x^3 (-5+2 x) \, dx\\ &=\frac {3 x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+4 \int x^2 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right ) \, dx-\int \left (-5 x^3+2 x^4\right ) \, dx-\int \frac {x^3 \left (7 (1+x)-e^{3 x} (-1+2 x)\right )}{7+e^{3 x}} \, dx\\ &=\frac {5 x^4}{4}+\frac {x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{3} \int x \text {Li}_3\left (-\frac {e^{3 x}}{7}\right ) \, dx-\int \left (x^3-2 x^4+\frac {21 x^4}{7+e^{3 x}}\right ) \, dx\\ &=x^4+\frac {3 x^5}{5}-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{9} \int \text {Li}_4\left (-\frac {e^{3 x}}{7}\right ) \, dx-21 \int \frac {x^4}{7+e^{3 x}} \, dx\\ &=x^4-x^4 \log \left (1+\frac {e^{3 x}}{7}\right )+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {x}{7}\right )}{x} \, dx,x,e^{3 x}\right )+3 \int \frac {e^{3 x} x^4}{7+e^{3 x}} \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {4}{3} x^3 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-4 \int x^3 \log \left (1+\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )+\frac {4}{3} x^2 \text {Li}_3\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-4 \int x^2 \text {Li}_2\left (-\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )-\frac {8}{9} x \text {Li}_4\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )+\frac {8}{3} \int x \text {Li}_3\left (-\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{9} \int \text {Li}_4\left (-\frac {e^{3 x}}{7}\right ) \, dx\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )+\frac {8}{27} \text {Li}_5\left (-\frac {e^{3 x}}{7}\right )-\frac {8}{27} \operatorname {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {x}{7}\right )}{x} \, dx,x,e^{3 x}\right )\\ &=x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 26, normalized size = 1.08 \begin {gather*} x^4+x^4 \log \left (\frac {3 e^x x}{4 \left (7+e^{3 x}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 22, normalized size = 0.92 \begin {gather*} x^{4} \log \left (\frac {3 \, x e^{x}}{4 \, {\left (e^{\left (3 \, x\right )} + 7\right )}}\right ) + x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 23, normalized size = 0.96 \begin {gather*} x^{5} + x^{4} \log \left (\frac {3 \, x}{4 \, {\left (e^{\left (3 \, x\right )} + 7\right )}}\right ) + x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 301, normalized size = 12.54
method | result | size |
risch | \(x^{4} \ln \left ({\mathrm e}^{x}\right )-x^{4} \ln \left ({\mathrm e}^{3 x}+7\right )+x^{4} \ln \relax (x )-\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{3}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{2}}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )^{3}}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{x}}{{\mathrm e}^{3 x}+7}\right )}{2}+x^{4} \ln \relax (3)-2 x^{4} \ln \relax (2)+x^{4}\) | \(301\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 34, normalized size = 1.42 \begin {gather*} x^{5} + x^{4} {\left (\log \relax (3) - 2 \, \log \relax (2) + 1\right )} + x^{4} \log \relax (x) - x^{4} \log \left (e^{\left (3 \, x\right )} + 7\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.20, size = 33, normalized size = 1.38 \begin {gather*} x^4\,\ln \relax (x)-x^4\,\ln \left (4\,{\mathrm {e}}^{3\,x}+28\right )+x^4\,\ln \relax (3)+x^4+x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 22, normalized size = 0.92 \begin {gather*} x^{4} \log {\left (\frac {3 x e^{x}}{4 e^{3 x} + 28} \right )} + x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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