Optimal. Leaf size=23 \[ \frac {\log (a+b \log (c (e+f x)))}{b d f} \]
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Rubi [A]
time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2437, 12, 2339,
29} \begin {gather*} \frac {\log (a+b \log (c (e+f x)))}{b d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2339
Rule 2437
Rubi steps
\begin {align*} \int \frac {1}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f}\\ &=\frac {\log (a+b \log (c (e+f x)))}{b d f}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log (a+b \log (c (e+f x)))}{b d f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 25, normalized size = 1.09
method | result | size |
norman | \(\frac {\ln \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}{b d f}\) | \(24\) |
derivativedivides | \(\frac {\ln \left (a +b \ln \left (c f x +c e \right )\right )}{f d b}\) | \(25\) |
default | \(\frac {\ln \left (a +b \ln \left (c f x +c e \right )\right )}{f d b}\) | \(25\) |
risch | \(\frac {\ln \left (\ln \left (c \left (f x +e \right )\right )+\frac {a}{b}\right )}{b d f}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 1.30 \begin {gather*} \frac {\log \left (\frac {b \log \left (f x + e\right ) + b \log \left (c\right ) + a}{b}\right )}{b d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log \left (b \log \left (c f x + c e\right ) + a\right )}{b d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 17, normalized size = 0.74 \begin {gather*} \frac {\log {\left (\frac {a}{b} + \log {\left (c \left (e + f x\right ) \right )} \right )}}{b d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.64, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log \left (b \log \left (c f x + c e\right ) + a\right )}{b d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 23, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}{b\,d\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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