Optimal. Leaf size=21 \[ \log \left (x \log ^2\left (e^{2 x}+\frac {1}{5} \left (8+x^2\right )\right )\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps
used = 3, number of rules used = 2, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6820, 6816}
\begin {gather*} 2 \log \left (\log \left (\frac {1}{5} \left (x^2+5 e^{2 x}+8\right )\right )\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}+\frac {4 \left (5 e^{2 x}+x\right )}{\left (8+5 e^{2 x}+x^2\right ) \log \left (\frac {1}{5} \left (8+5 e^{2 x}+x^2\right )\right )}\right ) \, dx\\ &=\log (x)+4 \int \frac {5 e^{2 x}+x}{\left (8+5 e^{2 x}+x^2\right ) \log \left (\frac {1}{5} \left (8+5 e^{2 x}+x^2\right )\right )} \, dx\\ &=\log (x)+2 \log \left (\log \left (\frac {1}{5} \left (8+5 e^{2 x}+x^2\right )\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 23, normalized size = 1.10 \begin {gather*} \log (x)+2 \log \left (\log \left (\frac {1}{5} \left (8+5 e^{2 x}+x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 19, normalized size = 0.90
method | result | size |
risch | \(\ln \left (x \right )+2 \ln \left (\ln \left ({\mathrm e}^{2 x}+\frac {x^{2}}{5}+\frac {8}{5}\right )\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 23, normalized size = 1.10 \begin {gather*} \log \left (x\right ) + 2 \, \log \left (-\log \left (5\right ) + \log \left (x^{2} + 5 \, e^{\left (2 \, x\right )} + 8\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 18, normalized size = 0.86 \begin {gather*} \log \left (x\right ) + 2 \, \log \left (\log \left (\frac {1}{5} \, x^{2} + e^{\left (2 \, x\right )} + \frac {8}{5}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 20, normalized size = 0.95 \begin {gather*} \log {\left (x \right )} + 2 \log {\left (\log {\left (\frac {x^{2}}{5} + e^{2 x} + \frac {8}{5} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 18, normalized size = 0.86 \begin {gather*} \log \left (x\right ) + 2 \, \log \left (\log \left (\frac {1}{5} \, x^{2} + e^{\left (2 \, x\right )} + \frac {8}{5}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 18, normalized size = 0.86 \begin {gather*} 2\,\ln \left (\ln \left ({\mathrm {e}}^{2\,x}+\frac {x^2}{5}+\frac {8}{5}\right )\right )+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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