Optimal. Leaf size=9 \[ \frac {4 x}{\log (-5+x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(9)=18\).
time = 0.11, antiderivative size = 22, normalized size of antiderivative = 2.44, number of steps
used = 11, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6874, 2458,
2395, 2334, 2335, 2339, 30, 2436} \begin {gather*} \frac {20}{\log (x-5)}-\frac {4 (5-x)}{\log (x-5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2334
Rule 2335
Rule 2339
Rule 2395
Rule 2436
Rule 2458
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 x}{(-5+x) \log ^2(-5+x)}+\frac {4}{\log (-5+x)}\right ) \, dx\\ &=-\left (4 \int \frac {x}{(-5+x) \log ^2(-5+x)} \, dx\right )+4 \int \frac {1}{\log (-5+x)} \, dx\\ &=-\left (4 \text {Subst}\left (\int \frac {5+x}{x \log ^2(x)} \, dx,x,-5+x\right )\right )+4 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )\\ &=4 \text {li}(-5+x)-4 \text {Subst}\left (\int \left (\frac {1}{\log ^2(x)}+\frac {5}{x \log ^2(x)}\right ) \, dx,x,-5+x\right )\\ &=4 \text {li}(-5+x)-4 \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,-5+x\right )-20 \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=-\frac {4 (5-x)}{\log (-5+x)}+4 \text {li}(-5+x)-4 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )-20 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (-5+x)\right )\\ &=\frac {20}{\log (-5+x)}-\frac {4 (5-x)}{\log (-5+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 9, normalized size = 1.00 \begin {gather*} \frac {4 x}{\log (-5+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(20\) vs.
\(2(9)=18\).
time = 1.00, size = 21, normalized size = 2.33
method | result | size |
norman | \(\frac {4 x}{\ln \left (x -5\right )}\) | \(10\) |
risch | \(\frac {4 x}{\ln \left (x -5\right )}\) | \(10\) |
derivativedivides | \(\frac {4 x -20}{\ln \left (x -5\right )}+\frac {20}{\ln \left (x -5\right )}\) | \(21\) |
default | \(\frac {4 x -20}{\ln \left (x -5\right )}+\frac {20}{\ln \left (x -5\right )}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 11, normalized size = 1.22 \begin {gather*} \frac {4 \, x}{\log \left (x - 5\right )} + 20 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 9, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{\log \left (x - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 7, normalized size = 0.78 \begin {gather*} \frac {4 x}{\log {\left (x - 5 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 9, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{\log \left (x - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 9, normalized size = 1.00 \begin {gather*} \frac {4\,x}{\ln \left (x-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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