Integrand size = 18, antiderivative size = 100 \[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {a (1+m)}{b n}} (1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2343, 2347, 2209} \[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {(m+1) (d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac {(d x)^{m+1}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Rule 2209
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {(1+m) \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx}{b n} \\ & = -\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left ((1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1+m) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b d n^2} \\ & = \frac {e^{-\frac {a (1+m)}{b n}} (1+m) (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \text {Ei}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 d n^2}-\frac {(d x)^{1+m}}{b d n \left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89 \[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {(d x)^m \left (e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} (1+m) x^{-m} \operatorname {ExpIntegralEi}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n x}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]
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\[\int \frac {\left (d x \right )^{m}}{{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.31 \[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {b n x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} - {\left ({\left (b m + b\right )} n \log \left (x\right ) + a m + {\left (b m + b\right )} \log \left (c\right ) + a\right )} {\rm Ei}\left (\frac {{\left (b m + b\right )} n \log \left (x\right ) + a m + {\left (b m + b\right )} \log \left (c\right ) + a}{b n}\right ) e^{\left (\frac {b m n \log \left (d\right ) - a m - {\left (b m + b\right )} \log \left (c\right ) - a}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} \]
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\[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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