Optimal. Leaf size=24 \[ \frac {1}{8 (3+x) \left (\frac {e^{-x} x}{4}+\log (3)\right )} \]
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Rubi [F] time = 1.56, 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^x \left (-3+x+x^2\right )-4 e^{2 x} \log (3)}{18 x^2+12 x^3+2 x^4+e^x \left (144 x+96 x^2+16 x^3\right ) \log (3)+e^{2 x} \left (288+192 x+32 x^2\right ) \log ^2(3)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-3+x+x^2-4 e^x \log (3)\right )}{2 (3+x)^2 \left (x+e^x \log (81)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^x \left (-3+x+x^2-4 e^x \log (3)\right )}{(3+x)^2 \left (x+e^x \log (81)\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {e^x (-1+x)}{(3+x) \left (x+e^x \log (81)\right )^2}-\frac {4 e^x \log (3)}{(3+x)^2 \log (81) \left (x+e^x \log (81)\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^x (-1+x)}{(3+x) \left (x+e^x \log (81)\right )^2} \, dx-\frac {(2 \log (3)) \int \frac {e^x}{(3+x)^2 \left (x+e^x \log (81)\right )} \, dx}{\log (81)}\\ &=\frac {1}{2} \int \left (\frac {e^x}{\left (x+e^x \log (81)\right )^2}-\frac {4 e^x}{(3+x) \left (x+e^x \log (81)\right )^2}\right ) \, dx-\frac {(2 \log (3)) \int \frac {e^x}{(3+x)^2 \left (x+e^x \log (81)\right )} \, dx}{\log (81)}\\ &=\frac {1}{2} \int \frac {e^x}{\left (x+e^x \log (81)\right )^2} \, dx-2 \int \frac {e^x}{(3+x) \left (x+e^x \log (81)\right )^2} \, dx-\frac {(2 \log (3)) \int \frac {e^x}{(3+x)^2 \left (x+e^x \log (81)\right )} \, dx}{\log (81)}\\ &=-\frac {1}{2 \log (81) \left (x+e^x \log (81)\right )}-2 \int \frac {e^x}{(3+x) \left (x+e^x \log (81)\right )^2} \, dx-\frac {\int \frac {1}{\left (x+e^x \log (81)\right )^2} \, dx}{2 \log (81)}-\frac {(2 \log (3)) \int \frac {e^x}{(3+x)^2 \left (x+e^x \log (81)\right )} \, dx}{\log (81)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.51, size = 46, normalized size = 1.92 \begin {gather*} \frac {e^x \left (x^2 \log (81)+x \log (6561)-\log (531441)\right )}{2 (-1+x) (3+x)^2 \log (81) \left (x+e^x \log (81)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 22, normalized size = 0.92 \begin {gather*} \frac {e^{x}}{2 \, {\left (4 \, {\left (x + 3\right )} e^{x} \log \relax (3) + x^{2} + 3 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 26, normalized size = 1.08 \begin {gather*} \frac {e^{x}}{2 \, {\left (4 \, x e^{x} \log \relax (3) + x^{2} + 12 \, e^{x} \log \relax (3) + 3 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 20, normalized size = 0.83
method | result | size |
norman | \(\frac {{\mathrm e}^{x}}{2 \left (3+x \right ) \left (4 \ln \relax (3) {\mathrm e}^{x}+x \right )}\) | \(20\) |
risch | \(\frac {1}{8 \ln \relax (3) \left (3+x \right )}-\frac {x}{8 \ln \relax (3) \left (3+x \right ) \left (4 \ln \relax (3) {\mathrm e}^{x}+x \right )}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 26, normalized size = 1.08 \begin {gather*} \frac {e^{x}}{2 \, {\left (x^{2} + 4 \, {\left (x \log \relax (3) + 3 \, \log \relax (3)\right )} e^{x} + 3 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 19, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {e}}^x}{2\,\left (x+4\,{\mathrm {e}}^x\,\ln \relax (3)\right )\,\left (x+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 48, normalized size = 2.00 \begin {gather*} - \frac {x}{8 x^{2} \log {\relax (3 )} + 24 x \log {\relax (3 )} + \left (32 x \log {\relax (3 )}^{2} + 96 \log {\relax (3 )}^{2}\right ) e^{x}} + \frac {1}{8 x \log {\relax (3 )} + 24 \log {\relax (3 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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