Optimal. Leaf size=10 \[ \frac {-7+x}{1+\log (x)} \]
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Rubi [A] time = 0.21, antiderivative size = 17, normalized size of antiderivative = 1.70, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6688, 6742, 2353, 2297, 2299, 2178, 2302, 30} \begin {gather*} \frac {x}{\log (x)+1}-\frac {7}{\log (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2178
Rule 2297
Rule 2299
Rule 2302
Rule 2353
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7+x \log (x)}{x (1+\log (x))^2} \, dx\\ &=\int \left (\frac {7-x}{x (1+\log (x))^2}+\frac {1}{1+\log (x)}\right ) \, dx\\ &=\int \frac {7-x}{x (1+\log (x))^2} \, dx+\int \frac {1}{1+\log (x)} \, dx\\ &=\int \left (-\frac {1}{(1+\log (x))^2}+\frac {7}{x (1+\log (x))^2}\right ) \, dx+\operatorname {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,\log (x)\right )\\ &=\frac {\text {Ei}(1+\log (x))}{e}+7 \int \frac {1}{x (1+\log (x))^2} \, dx-\int \frac {1}{(1+\log (x))^2} \, dx\\ &=\frac {\text {Ei}(1+\log (x))}{e}+\frac {x}{1+\log (x)}+7 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,1+\log (x)\right )-\int \frac {1}{1+\log (x)} \, dx\\ &=\frac {\text {Ei}(1+\log (x))}{e}-\frac {7}{1+\log (x)}+\frac {x}{1+\log (x)}-\operatorname {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,\log (x)\right )\\ &=-\frac {7}{1+\log (x)}+\frac {x}{1+\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} \frac {-7+x}{1+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 10, normalized size = 1.00 \begin {gather*} \frac {x - 7}{\log \relax (x) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 10, normalized size = 1.00 \begin {gather*} \frac {x - 7}{\log \relax (x) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 11, normalized size = 1.10
method | result | size |
norman | \(\frac {x -7}{\ln \relax (x )+1}\) | \(11\) |
risch | \(\frac {x -7}{\ln \relax (x )+1}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 17, normalized size = 1.70 \begin {gather*} \frac {x}{\log \relax (x) + 1} - \frac {7}{\log \relax (x) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.84, size = 10, normalized size = 1.00 \begin {gather*} \frac {x-7}{\ln \relax (x)+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 7, normalized size = 0.70 \begin {gather*} \frac {x - 7}{\log {\relax (x )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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