Optimal. Leaf size=26 \[ \frac {x}{x+\log \left (2-\frac {e \left (e^x+5 x^2\right )}{3 x}\right )} \]
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Rubi [A] time = 0.77, antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, integrand size = 172, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6688, 6711, 32} \begin {gather*} -\frac {1}{\frac {x}{\log \left (-\frac {5 e x}{3}-\frac {e^{x+1}}{3 x}+2\right )}+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6688
Rule 6711
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e \left (e^x (-1+x)+5 x^2\right )+\left (e^{1+x}-6 x+5 e x^2\right ) \log \left (2-\frac {e^{1+x}}{3 x}-\frac {5 e x}{3}\right )}{\left (e^{1+x}-6 x+5 e x^2\right ) \left (x+\log \left (2-\frac {e^{1+x}}{3 x}-\frac {5 e x}{3}\right )\right )^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {x}{\log \left (2-\frac {e^{1+x}}{3 x}-\frac {5 e x}{3}\right )}\right )\\ &=-\frac {1}{1+\frac {x}{\log \left (2-\frac {e^{1+x}}{3 x}-\frac {5 e x}{3}\right )}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 27, normalized size = 1.04 \begin {gather*} \frac {x}{x+\log \left (2-\frac {e^{1+x}}{3 x}-\frac {5 e x}{3}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 27, normalized size = 1.04 \begin {gather*} \frac {x}{x + \log \left (-\frac {5 \, x^{2} e - 6 \, x + e^{\left (x + 1\right )}}{3 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 27, normalized size = 1.04 \begin {gather*} \frac {x}{x + \log \left (-\frac {5 \, x^{2} e - 6 \, x + e^{\left (x + 1\right )}}{3 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 234, normalized size = 9.00
method | result | size |
risch | \(\frac {2 x}{-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )}{x}\right )^{2}-2 i \pi \mathrm {csgn}\left (\frac {i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )}{x}\right )^{3}+2 i \pi +2 \ln \relax (5)-2 \ln \relax (3)+2 x -2 \ln \relax (x )+2 \ln \left (\left (x^{2}+\frac {{\mathrm e}^{x}}{5}\right ) {\mathrm e}-\frac {6 x}{5}\right )}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 32, normalized size = 1.23 \begin {gather*} \frac {x}{x - \log \relax (3) + \log \left (-5 \, x^{2} e + 6 \, x - e^{\left (x + 1\right )}\right ) - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {5\,x^2\,\mathrm {e}-\ln \left (-\frac {\frac {5\,x^2\,\mathrm {e}}{3}-2\,x+\frac {\mathrm {e}\,{\mathrm {e}}^x}{3}}{x}\right )\,\left (5\,x^2\,\mathrm {e}-6\,x+\mathrm {e}\,{\mathrm {e}}^x\right )+\mathrm {e}\,{\mathrm {e}}^x\,\left (x-1\right )}{{\ln \left (-\frac {\frac {5\,x^2\,\mathrm {e}}{3}-2\,x+\frac {\mathrm {e}\,{\mathrm {e}}^x}{3}}{x}\right )}^2\,\left (5\,x^2\,\mathrm {e}-6\,x+\mathrm {e}\,{\mathrm {e}}^x\right )+5\,x^4\,\mathrm {e}+\ln \left (-\frac {\frac {5\,x^2\,\mathrm {e}}{3}-2\,x+\frac {\mathrm {e}\,{\mathrm {e}}^x}{3}}{x}\right )\,\left (10\,x^3\,\mathrm {e}-12\,x^2+2\,x\,\mathrm {e}\,{\mathrm {e}}^x\right )-6\,x^3+x^2\,\mathrm {e}\,{\mathrm {e}}^x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 27, normalized size = 1.04 \begin {gather*} \frac {x}{x + \log {\left (\frac {- \frac {5 e x^{2}}{3} + 2 x - \frac {e e^{x}}{3}}{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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