Integrand size = 14, antiderivative size = 22 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2338} \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
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Rule 2338
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=a \log (x)+\frac {b \log ^2\left (c x^n\right )}{2 n} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
method | result | size |
parts | \(\ln \left (x \right ) a +\frac {b \ln \left (c \,x^{n}\right )^{2}}{2 n}\) | \(20\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) a n +b \ln \left (c \,x^{n}\right )^{2}}{2 n}\) | \(23\) |
derivativedivides | \(\frac {\frac {b \ln \left (c \,x^{n}\right )^{2}}{2}+\ln \left (c \,x^{n}\right ) a}{n}\) | \(25\) |
default | \(\frac {\frac {b \ln \left (c \,x^{n}\right )^{2}}{2}+\ln \left (c \,x^{n}\right ) a}{n}\) | \(25\) |
norman | \(\frac {a \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{n}+\frac {b \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}}{2 n}\) | \(31\) |
risch | \(b \ln \left (x \right ) \ln \left (x^{n}\right )-\frac {b n \ln \left (x \right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \ln \left (x \right ) \pi b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \ln \left (x \right ) \pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \left (x \right ) \ln \left (c \right ) b +\ln \left (x \right ) a\) | \(118\) |
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Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=\frac {1}{2} \, b n \log \left (x\right )^{2} + {\left (b \log \left (c\right ) + a\right )} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).
Time = 1.54 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{2 \, b n} \]
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Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=\frac {1}{2} \, b n \log \left (x\right )^{2} + b \log \left (c\right ) \log \left (x\right ) + a \log \left (x\right ) \]
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c x^n\right )}{x} \, dx=a\,\ln \left (x\right )+\frac {b\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]
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