Integrand size = 16, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^{1+p}}{b n (1+p)} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2339, 30} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]
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Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^p \, dx,x,a+b \log \left (c x^n\right )\right )}{b n} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^{1+p}}{b n (1+p)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^{1+p}}{b n (1+p)} \]
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Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p +1}}{b n \left (p +1\right )}\) | \(27\) |
default | \(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p +1}}{b n \left (p +1\right )}\) | \(27\) |
parallelrisch | \(-\frac {-\ln \left (c \,x^{n}\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p} b^{2}-{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p} a b}{b^{2} n \left (p +1\right )}\) | \(54\) |
risch | \(\frac {{\left (\ln \left (x^{n}\right ) b +a +b \left (\ln \left (c \right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )\right )}^{p +1}}{n b \left (p +1\right )}\) | \(76\) |
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} {\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{p}}{b n p + b n} \]
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Time = 0.67 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=- \begin {cases} - a^{p} \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (a + b \log {\left (c \right )}\right )^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{b n} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
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Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
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Time = 0.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]
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