Integrand size = 25, antiderivative size = 42 \[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\frac {(e+f x)^{1-p} (g (e+f x))^{-1+p} \operatorname {LogIntegral}\left (d (e+f x)^p\right )}{d f p} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2437, 2345, 2344, 2335} \[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\frac {(e+f x)^{1-p} (g (e+f x))^{p-1} \operatorname {LogIntegral}\left (d (e+f x)^p\right )}{d f p} \]
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Rule 2335
Rule 2344
Rule 2345
Rule 2437
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(g x)^{-1+p}}{\log \left (d x^p\right )} \, dx,x,e+f x\right )}{f} \\ & = \frac {\left ((e+f x)^{1-p} (g (e+f x))^{-1+p}\right ) \text {Subst}\left (\int \frac {x^{-1+p}}{\log \left (d x^p\right )} \, dx,x,e+f x\right )}{f} \\ & = \frac {\left ((e+f x)^{1-p} (g (e+f x))^{-1+p}\right ) \text {Subst}\left (\int \frac {1}{\log (d x)} \, dx,x,(e+f x)^p\right )}{f p} \\ & = \frac {(e+f x)^{1-p} (g (e+f x))^{-1+p} \text {li}\left (d (e+f x)^p\right )}{d f p} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\frac {(e+f x)^{1-p} (g (e+f x))^{-1+p} \operatorname {LogIntegral}\left (d (e+f x)^p\right )}{d f p} \]
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\[\int \frac {\left (f g x +e g \right )^{-1+p}}{\ln \left (d \left (f x +e \right )^{p}\right )}d x\]
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none
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64 \[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\frac {g^{p - 1} {\rm Ei}\left (p \log \left (f x + e\right ) + \log \left (d\right )\right )}{d f p} \]
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\[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\int \frac {\left (g \left (e + f x\right )\right )^{p - 1}}{\log {\left (d \left (e + f x\right )^{p} \right )}}\, dx \]
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\[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\int { \frac {{\left (f g x + e g\right )}^{p - 1}}{\log \left ({\left (f x + e\right )}^{p} d\right )} \,d x } \]
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\[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\int { \frac {{\left (f g x + e g\right )}^{p - 1}}{\log \left ({\left (f x + e\right )}^{p} d\right )} \,d x } \]
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Timed out. \[ \int \frac {(e g+f g x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx=\int \frac {{\left (e\,g+f\,g\,x\right )}^{p-1}}{\ln \left (d\,{\left (e+f\,x\right )}^p\right )} \,d x \]
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