Optimal. Leaf size=21 \[ \frac {2 x \left (2+x^2\right )^2 \log (27)}{e^{9 x}+x} \]
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Rubi [F]
time = 0.85, 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 {e^{9 x} \left (8-72 x+24 x^2-72 x^3+10 x^4-18 x^5\right ) \log (27)+\left (16 x^3+8 x^5\right ) \log (27)}{e^{18 x}+2 e^{9 x} x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (2+x^2\right ) \left (4 x^3-e^{9 x} \left (-2+18 x-5 x^2+9 x^3\right )\right ) \log (27)}{\left (e^{9 x}+x\right )^2} \, dx\\ &=(2 \log (27)) \int \frac {\left (2+x^2\right ) \left (4 x^3-e^{9 x} \left (-2+18 x-5 x^2+9 x^3\right )\right )}{\left (e^{9 x}+x\right )^2} \, dx\\ &=(2 \log (27)) \int \left (\frac {x (-1+9 x) \left (2+x^2\right )^2}{\left (e^{9 x}+x\right )^2}-\frac {-4+36 x-12 x^2+36 x^3-5 x^4+9 x^5}{e^{9 x}+x}\right ) \, dx\\ &=(2 \log (27)) \int \frac {x (-1+9 x) \left (2+x^2\right )^2}{\left (e^{9 x}+x\right )^2} \, dx-(2 \log (27)) \int \frac {-4+36 x-12 x^2+36 x^3-5 x^4+9 x^5}{e^{9 x}+x} \, dx\\ &=(2 \log (27)) \int \left (-\frac {4 x}{\left (e^{9 x}+x\right )^2}+\frac {36 x^2}{\left (e^{9 x}+x\right )^2}-\frac {4 x^3}{\left (e^{9 x}+x\right )^2}+\frac {36 x^4}{\left (e^{9 x}+x\right )^2}-\frac {x^5}{\left (e^{9 x}+x\right )^2}+\frac {9 x^6}{\left (e^{9 x}+x\right )^2}\right ) \, dx-(2 \log (27)) \int \left (-\frac {4}{e^{9 x}+x}+\frac {36 x}{e^{9 x}+x}-\frac {12 x^2}{e^{9 x}+x}+\frac {36 x^3}{e^{9 x}+x}-\frac {5 x^4}{e^{9 x}+x}+\frac {9 x^5}{e^{9 x}+x}\right ) \, dx\\ &=-\left ((2 \log (27)) \int \frac {x^5}{\left (e^{9 x}+x\right )^2} \, dx\right )-(8 \log (27)) \int \frac {x}{\left (e^{9 x}+x\right )^2} \, dx-(8 \log (27)) \int \frac {x^3}{\left (e^{9 x}+x\right )^2} \, dx+(8 \log (27)) \int \frac {1}{e^{9 x}+x} \, dx+(10 \log (27)) \int \frac {x^4}{e^{9 x}+x} \, dx+(18 \log (27)) \int \frac {x^6}{\left (e^{9 x}+x\right )^2} \, dx-(18 \log (27)) \int \frac {x^5}{e^{9 x}+x} \, dx+(24 \log (27)) \int \frac {x^2}{e^{9 x}+x} \, dx+(72 \log (27)) \int \frac {x^2}{\left (e^{9 x}+x\right )^2} \, dx+(72 \log (27)) \int \frac {x^4}{\left (e^{9 x}+x\right )^2} \, dx-(72 \log (27)) \int \frac {x}{e^{9 x}+x} \, dx-(72 \log (27)) \int \frac {x^3}{e^{9 x}+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.05, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 x \left (2+x^2\right )^2 \log (27)}{e^{9 x}+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.60, size = 24, normalized size = 1.14
method | result | size |
risch | \(\frac {6 \left (x^{4}+4 x^{2}+4\right ) \ln \left (3\right ) x}{x +{\mathrm e}^{9 x}}\) | \(24\) |
norman | \(\frac {-24 \ln \left (3\right ) {\mathrm e}^{9 x}+24 x^{3} \ln \left (3\right )+6 x^{5} \ln \left (3\right )}{x +{\mathrm e}^{9 x}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 29, normalized size = 1.38 \begin {gather*} \frac {6 \, {\left (x^{5} \log \left (3\right ) + 4 \, x^{3} \log \left (3\right ) + 4 \, x \log \left (3\right )\right )}}{x + e^{\left (9 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 24, normalized size = 1.14 \begin {gather*} \frac {6 \, {\left (x^{5} + 4 \, x^{3} + 4 \, x\right )} \log \left (3\right )}{x + e^{\left (9 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 29, normalized size = 1.38 \begin {gather*} \frac {6 x^{5} \log {\left (3 \right )} + 24 x^{3} \log {\left (3 \right )} + 24 x \log {\left (3 \right )}}{x + e^{9 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 29, normalized size = 1.38 \begin {gather*} \frac {6 \, {\left (x^{5} \log \left (3\right ) + 4 \, x^{3} \log \left (3\right ) + 4 \, x \log \left (3\right )\right )}}{x + e^{\left (9 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.30, size = 20, normalized size = 0.95 \begin {gather*} \frac {6\,x\,\ln \left (3\right )\,{\left (x^2+2\right )}^2}{x+{\mathrm {e}}^{9\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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