Integrand size = 12, antiderivative size = 27 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (b+a x)}{a} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2498, 269, 31} \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (a x+b)}{a} \]
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Rule 31
Rule 269
Rule 2498
Rubi steps \begin{align*} \text {integral}& = x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x} \, dx \\ & = x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+(b p) \int \frac {1}{b+a x} \, dx \\ & = x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (b+a x)}{a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {b p \log \left (a+\frac {b}{x}\right )}{a}+x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (x)}{a} \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
parts | \(x \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )+\frac {b p \ln \left (a x +b \right )}{a}\) | \(28\) |
default | \(x \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right )+\frac {b p \ln \left (a x +b \right )}{a}\) | \(30\) |
parallelrisch | \(-\frac {-\ln \left (x \right ) b^{2} p^{2}-x \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) a b p -\ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) b^{2} p}{a b p}\) | \(63\) |
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Time = 0.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {a p x \log \left (\frac {a x + b}{x}\right ) + b p \log \left (a x + b\right ) + a x \log \left (c\right )}{a} \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\begin {cases} x \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {b p \log {\left (a x + b \right )}}{a} & \text {for}\: a \neq 0 \\p x + x \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=x \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) + \frac {b p \log \left (a x + b\right )}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (27) = 54\).
Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.56 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=-\frac {\frac {b^{2} p \log \left (-a + \frac {a x + b}{x}\right )}{a} + \frac {b^{2} p \log \left (\frac {a x + b}{x}\right )}{a - \frac {a x + b}{x}} - \frac {b^{2} p \log \left (\frac {a x + b}{x}\right )}{a} + \frac {b^{2} \log \left (c\right )}{a - \frac {a x + b}{x}}}{b} \]
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Time = 1.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=x\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )+\frac {b\,p\,\ln \left (b+a\,x\right )}{a} \]
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