Optimal. Leaf size=23 \[ \frac {x}{-\frac {9}{4}-e^{x+x^2 \log (3 x)}+x} \]
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Rubi [F] time = 1.69, 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 {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 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 {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=\int \left (-\frac {36}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}-\frac {16 e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {16 e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {16\ 3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {32\ 3^{x^2} e^x x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}\right ) \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+32 \int \frac {3^{x^2} e^x x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+32 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-32 \int \frac {\int \frac {3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx}{x} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+(32 \log (3 x)) \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-32 \int \frac {\int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx}{x} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+(32 \log (3 x)) \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.55, size = 25, normalized size = 1.09 \begin {gather*} -\frac {4 x}{9-4 x+4\ 3^{x^2} e^x x^{x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 23, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3 \, x\right ) + x\right )} - 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 23, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3 \, x\right ) + x\right )} - 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 0.96
method | result | size |
risch | \(\frac {4 x}{4 x -4 \left (3 x \right )^{x^{2}} {\mathrm e}^{x}-9}\) | \(22\) |
norman | \(\frac {4 x}{4 x -4 \,{\mathrm e}^{x^{2} \ln \left (3 x \right )+x}-9}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 27, normalized size = 1.17 \begin {gather*} \frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \relax (3) + x^{2} \log \relax (x) + x\right )} - 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.32, size = 80, normalized size = 3.48 \begin {gather*} -\frac {52\,x+72\,x^2\,\ln \left (3\,x\right )-32\,x^3\,\ln \left (3\,x\right )+20\,x^2-16\,x^3}{\left (4\,{\mathrm {e}}^x\,{\left (3\,x\right )}^{x^2}-4\,x+9\right )\,\left (5\,x+18\,x\,\ln \left (3\,x\right )-8\,x^2\,\ln \left (3\,x\right )-4\,x^2+13\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 22, normalized size = 0.96 \begin {gather*} - \frac {4 x}{- 4 x + 4 e^{x^{2} \log {\left (3 x \right )} + x} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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