Optimal. Leaf size=26 \[ -\frac {1+\left (2-e^x\right ) x}{x \log (2) (-3+\log (x))} \]
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Rubi [B] time = 0.82, antiderivative size = 60, normalized size of antiderivative = 2.31, number of steps used = 16, number of rules used = 10, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6741, 12, 6742, 2353, 2306, 2309, 2178, 2302, 30, 2288} \begin {gather*} -\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}+\frac {1}{x \log (2) (3-\log (x))}+\frac {2}{\log (2) (3-\log (x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2178
Rule 2288
Rule 2302
Rule 2306
Rule 2309
Rule 2353
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+2 x+e^x \left (-x-3 x^2\right )+\left (1+e^x x^2\right ) \log (x)}{x^2 \log (2) (3-\log (x))^2} \, dx\\ &=\frac {\int \frac {-2+2 x+e^x \left (-x-3 x^2\right )+\left (1+e^x x^2\right ) \log (x)}{x^2 (3-\log (x))^2} \, dx}{\log (2)}\\ &=\frac {\int \left (\frac {-2+2 x+\log (x)}{x^2 (-3+\log (x))^2}+\frac {e^x (-1-3 x+x \log (x))}{x (-3+\log (x))^2}\right ) \, dx}{\log (2)}\\ &=\frac {\int \frac {-2+2 x+\log (x)}{x^2 (-3+\log (x))^2} \, dx}{\log (2)}+\frac {\int \frac {e^x (-1-3 x+x \log (x))}{x (-3+\log (x))^2} \, dx}{\log (2)}\\ &=-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}+\frac {\int \left (\frac {1+2 x}{x^2 (-3+\log (x))^2}+\frac {1}{x^2 (-3+\log (x))}\right ) \, dx}{\log (2)}\\ &=-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}+\frac {\int \frac {1+2 x}{x^2 (-3+\log (x))^2} \, dx}{\log (2)}+\frac {\int \frac {1}{x^2 (-3+\log (x))} \, dx}{\log (2)}\\ &=-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}+\frac {\int \left (\frac {1}{x^2 (-3+\log (x))^2}+\frac {2}{x (-3+\log (x))^2}\right ) \, dx}{\log (2)}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{-3+x} \, dx,x,\log (x)\right )}{\log (2)}\\ &=\frac {\text {Ei}(3-\log (x))}{e^3 \log (2)}-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}+\frac {\int \frac {1}{x^2 (-3+\log (x))^2} \, dx}{\log (2)}+\frac {2 \int \frac {1}{x (-3+\log (x))^2} \, dx}{\log (2)}\\ &=\frac {\text {Ei}(3-\log (x))}{e^3 \log (2)}+\frac {1}{x \log (2) (3-\log (x))}-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}-\frac {\int \frac {1}{x^2 (-3+\log (x))} \, dx}{\log (2)}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,-3+\log (x)\right )}{\log (2)}\\ &=\frac {\text {Ei}(3-\log (x))}{e^3 \log (2)}+\frac {2}{\log (2) (3-\log (x))}+\frac {1}{x \log (2) (3-\log (x))}-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{-3+x} \, dx,x,\log (x)\right )}{\log (2)}\\ &=\frac {2}{\log (2) (3-\log (x))}+\frac {1}{x \log (2) (3-\log (x))}-\frac {e^x (3 x-x \log (x))}{x \log (2) (3-\log (x))^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 24, normalized size = 0.92 \begin {gather*} \frac {-1-2 x+e^x x}{x \log (2) (-3+\log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 24, normalized size = 0.92 \begin {gather*} \frac {x e^{x} - 2 \, x - 1}{x \log \relax (2) \log \relax (x) - 3 \, x \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 24, normalized size = 0.92 \begin {gather*} \frac {x e^{x} - 2 \, x - 1}{x \log \relax (2) \log \relax (x) - 3 \, x \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 24, normalized size = 0.92
method | result | size |
risch | \(\frac {{\mathrm e}^{x} x -2 x -1}{x \ln \relax (2) \left (\ln \relax (x )-3\right )}\) | \(24\) |
norman | \(\frac {-\frac {2 x}{\ln \relax (2)}+\frac {x \,{\mathrm e}^{x}}{\ln \relax (2)}-\frac {1}{\ln \relax (2)}}{x \left (\ln \relax (x )-3\right )}\) | \(33\) |
default | \(\frac {-\frac {2 x}{\ln \relax (2)}-\frac {1}{\ln \relax (2)}}{x \left (\ln \relax (x )-3\right )}+\frac {{\mathrm e}^{x}}{\ln \relax (2) \left (\ln \relax (x )-3\right )}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 24, normalized size = 0.92 \begin {gather*} \frac {x e^{x} - 2 \, x - 1}{x \log \relax (2) \log \relax (x) - 3 \, x \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 25, normalized size = 0.96 \begin {gather*} -\frac {2\,x-x\,{\mathrm {e}}^x+1}{x\,\ln \relax (2)\,\left (\ln \relax (x)-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 36, normalized size = 1.38 \begin {gather*} \frac {- 2 x - 1}{x \log {\relax (2 )} \log {\relax (x )} - 3 x \log {\relax (2 )}} + \frac {e^{x}}{\log {\relax (2 )} \log {\relax (x )} - 3 \log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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