Integrand size = 16, antiderivative size = 40 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=-\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )-p \operatorname {PolyLog}\left (2,1+\frac {b}{a x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2441, 2352} \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=\log \left (-\frac {b}{a x}\right ) \left (-\log \left (c \left (a+\frac {b}{x}\right )^p\right )\right )-p \operatorname {PolyLog}\left (2,\frac {b}{a x}+1\right ) \]
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Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )+(b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x}\right ) \\ & = -\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )-p \text {Li}_2\left (1+\frac {b}{a x}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=-\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )-p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x}}{a}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.68
method | result | size |
parts | \(\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (x \right )+p b \left (\frac {\ln \left (x \right )^{2}}{2 b}-\frac {a \left (\frac {\operatorname {dilog}\left (\frac {a x +b}{b}\right )}{a}+\frac {\ln \left (x \right ) \ln \left (\frac {a x +b}{b}\right )}{a}\right )}{b}\right )\) | \(67\) |
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=\int \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (39) = 78\).
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.08 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=\frac {1}{2} \, b p {\left (\frac {2 \, \log \left (a + \frac {b}{x}\right ) \log \left (x\right )}{b} + \frac {\log \left (x\right )^{2}}{b} - \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )}}{b}\right )} - p \log \left (a + \frac {b}{x}\right ) \log \left (x\right ) + \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (39) = 78\).
Time = 0.41 (sec) , antiderivative size = 152, normalized size of antiderivative = 3.80 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=-\frac {\frac {b^{3} p \log \left (\frac {a x + b}{x}\right )}{a^{2} - \frac {2 \, {\left (a x + b\right )} a}{x} + \frac {{\left (a x + b\right )}^{2}}{x^{2}}} + \frac {b^{3} p \log \left (-a + \frac {a x + b}{x}\right )}{a^{2}} - \frac {b^{3} p \log \left (\frac {a x + b}{x}\right )}{a^{2}} - \frac {a b^{3} p - a b^{3} \log \left (c\right ) - \frac {{\left (a x + b\right )} b^{3} p}{x}}{a^{3} - \frac {2 \, {\left (a x + b\right )} a^{2}}{x} + \frac {{\left (a x + b\right )}^{2} a}{x^{2}}}}{2 \, b^{2}} \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x} \,d x \]
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