Optimal. Leaf size=24 \[ \frac {e}{3+e^{2 x}+x}+\log \left (\frac {3 x}{3+82 x}\right ) \]
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Rubi [F] time = 1.21, 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 {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{4 x}+6 e^{2 x} (3+x)+3 (3+x)^2-e x (3+82 x)-2 e^{1+2 x} x (3+82 x)}{x \left (3+e^{2 x}+x\right )^2 (3+82 x)} \, dx\\ &=\int \left (-\frac {2 e}{3+e^{2 x}+x}+\frac {e (5+2 x)}{\left (3+e^{2 x}+x\right )^2}+\frac {3}{x (3+82 x)}\right ) \, dx\\ &=3 \int \frac {1}{x (3+82 x)} \, dx+e \int \frac {5+2 x}{\left (3+e^{2 x}+x\right )^2} \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx\\ &=-\left (82 \int \frac {1}{3+82 x} \, dx\right )+e \int \left (\frac {5}{\left (3+e^{2 x}+x\right )^2}+\frac {2 x}{\left (3+e^{2 x}+x\right )^2}\right ) \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx+\int \frac {1}{x} \, dx\\ &=\log (x)-\log (3+82 x)+(2 e) \int \frac {x}{\left (3+e^{2 x}+x\right )^2} \, dx-(2 e) \int \frac {1}{3+e^{2 x}+x} \, dx+(5 e) \int \frac {1}{\left (3+e^{2 x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.21, size = 23, normalized size = 0.96 \begin {gather*} \frac {e}{3+e^{2 x}+x}+\log (x)-\log (3+82 x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 41, normalized size = 1.71 \begin {gather*} -\frac {{\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \left (82 \, x + 3\right ) - {\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \relax (x) - e}{x + e^{\left (2 \, x\right )} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 60, normalized size = 2.50 \begin {gather*} -\frac {x \log \left (82 \, x + 3\right ) + e^{\left (2 \, x\right )} \log \left (82 \, x + 3\right ) - x \log \relax (x) - e^{\left (2 \, x\right )} \log \relax (x) - e + 3 \, \log \left (82 \, x + 3\right ) - 3 \, \log \relax (x)}{x + e^{\left (2 \, x\right )} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 24, normalized size = 1.00
method | result | size |
norman | \(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \relax (x )\) | \(24\) |
risch | \(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \relax (x )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 23, normalized size = 0.96 \begin {gather*} \frac {e}{x + e^{\left (2 \, x\right )} + 3} - \log \left (82 \, x + 3\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.43, size = 34, normalized size = 1.42 \begin {gather*} \ln \relax (x)-\ln \left (x+\frac {3}{82}\right )-\frac {\frac {{\mathrm {e}}^{2\,x+1}}{3}+\frac {x\,\mathrm {e}}{3}}{x+{\mathrm {e}}^{2\,x}+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 20, normalized size = 0.83 \begin {gather*} \log {\relax (x )} - \log {\left (x + \frac {3}{82} \right )} + \frac {e}{x + e^{2 x} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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