Integrand size = 16, antiderivative size = 22 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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Time = 0.02 (sec) , antiderivative size = 22, 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} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{3 b n}\) | \(21\) |
default | \(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{3 b n}\) | \(21\) |
parts | \(\ln \left (x \right ) a^{2}+\frac {b^{2} \ln \left (c \,x^{n}\right )^{3}}{3 n}+\frac {a b \ln \left (c \,x^{n}\right )^{2}}{n}\) | \(38\) |
parallelrisch | \(\frac {b^{2} \ln \left (c \,x^{n}\right )^{3}+3 \ln \left (x \right ) a^{2} n +3 a b \ln \left (c \,x^{n}\right )^{2}}{3 n}\) | \(39\) |
risch | \(\text {Expression too large to display}\) | \(774\) |
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {1}{3} \, b^{2} n^{2} \log \left (x\right )^{3} + {\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )^{2} + {\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (15) = 30\).
Time = 5.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{3 \, b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {1}{3} \, b^{2} n^{2} \log \left (x\right )^{3} + b^{2} n \log \left (c\right ) \log \left (x\right )^{2} + b^{2} \log \left (c\right )^{2} \log \left (x\right ) + a b n \log \left (x\right )^{2} + 2 \, a b \log \left (c\right ) \log \left (x\right ) + a^{2} \log \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=a^2\,\ln \left (x\right )+\frac {b^2\,{\ln \left (c\,x^n\right )}^3}{3\,n}+\frac {a\,b\,{\ln \left (c\,x^n\right )}^2}{n} \]
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