Optimal. Leaf size=17 \[ 1+\frac {2}{x \left (2+\frac {1}{-3+\log (x)}\right )} \]
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Rubi [A] time = 0.28, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6688, 12, 6742, 2306, 2309, 2178} \begin {gather*} \frac {1}{x}+\frac {1}{x (5-2 \log (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2306
Rule 2309
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-14+11 \log (x)-2 \log ^2(x)\right )}{x^2 (5-2 \log (x))^2} \, dx\\ &=2 \int \frac {-14+11 \log (x)-2 \log ^2(x)}{x^2 (5-2 \log (x))^2} \, dx\\ &=2 \int \left (-\frac {1}{2 x^2}+\frac {1}{x^2 (-5+2 \log (x))^2}+\frac {1}{2 x^2 (-5+2 \log (x))}\right ) \, dx\\ &=\frac {1}{x}+2 \int \frac {1}{x^2 (-5+2 \log (x))^2} \, dx+\int \frac {1}{x^2 (-5+2 \log (x))} \, dx\\ &=\frac {1}{x}+\frac {1}{x (5-2 \log (x))}-\int \frac {1}{x^2 (-5+2 \log (x))} \, dx+\operatorname {Subst}\left (\int \frac {e^{-x}}{-5+2 x} \, dx,x,\log (x)\right )\\ &=\frac {1}{x}+\frac {\text {Ei}\left (\frac {1}{2} (5-2 \log (x))\right )}{2 e^{5/2}}+\frac {1}{x (5-2 \log (x))}-\operatorname {Subst}\left (\int \frac {e^{-x}}{-5+2 x} \, dx,x,\log (x)\right )\\ &=\frac {1}{x}+\frac {1}{x (5-2 \log (x))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{x}-\frac {1}{x (-5+2 \log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 17, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (\log \relax (x) - 3\right )}}{2 \, x \log \relax (x) - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{2 \, x \log \relax (x) - 5 \, x} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 18, normalized size = 1.06
method | result | size |
risch | \(\frac {1}{x}-\frac {1}{x \left (2 \ln \relax (x )-5\right )}\) | \(18\) |
norman | \(\frac {2 \ln \relax (x )-6}{x \left (2 \ln \relax (x )-5\right )}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 17, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (\log \relax (x) - 3\right )}}{2 \, x \log \relax (x) - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.61, size = 17, normalized size = 1.00 \begin {gather*} \frac {2\,\left (\ln \relax (x)-3\right )}{x\,\left (2\,\ln \relax (x)-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 14, normalized size = 0.82 \begin {gather*} - \frac {1}{2 x \log {\relax (x )} - 5 x} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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