Integrand size = 14, antiderivative size = 29 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=-p q x+\frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2436, 2332, 2495} \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=\frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
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Rule 2332
Rule 2436
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \log \left (c d^q (e+f x)^{p q}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -p q x+\frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=-p q x+\frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \]
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Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41
method | result | size |
default | \(\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) x -q p f \left (\frac {x}{f}-\frac {e \ln \left (f x +e \right )}{f^{2}}\right )\) | \(41\) |
parts | \(\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) x -q p f \left (\frac {x}{f}-\frac {e \ln \left (f x +e \right )}{f^{2}}\right )\) | \(41\) |
parallelrisch | \(\frac {2 \ln \left (f x +e \right ) e^{2} p q -x e f p q +x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) e f -\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) e^{2}}{e f}\) | \(66\) |
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Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=-\frac {f p q x - f q x \log \left (d\right ) - f x \log \left (c\right ) - {\left (f p q x + e p q\right )} \log \left (f x + e\right )}{f} \]
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Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=\begin {cases} \frac {e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - p q x + x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{p}\right )^{q} \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=-f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) \]
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Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=\frac {{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} p q}{f} + \frac {{\left (f x + e\right )} q \log \left (d\right )}{f} + \frac {{\left (f x + e\right )} \log \left (c\right )}{f} \]
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Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx=x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )+\frac {p\,q\,\left (e\,\ln \left (e+f\,x\right )-f\,x\right )}{f} \]
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