Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {31} \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rule 31
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {\ln \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
parallelrisch | \(\frac {\ln \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\) | \(19\) |
norman | \(\frac {\ln \left (b \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+a \right )}{b n}\) | \(21\) |
risch | \(\frac {\ln \left (\ln \left (x^{n}\right )-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 b \ln \left (c \right )-2 a}{2 b}\right )}{b n}\) | \(110\) |
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none
Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\log \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).
Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\frac {\log {\left (x \right )}}{a + b \log {\left (c \right )}} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\log \left (\frac {b \log \left (c\right ) + b \log \left (x^{n}\right ) + a}{b}\right )}{b n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (18) = 36\).
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.50 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + a\right )}^{2}\right )}{2 \, b n} \]
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Time = 1.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a x+b x \log \left (c x^n\right )} \, dx=\frac {\ln \left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \]
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