Integrand size = 19, antiderivative size = 48 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\frac {b p \left (d x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {\log \left (d x^n\right )}{n}\right )}{x}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2602, 2347, 2209} \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\frac {b p \left (d x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {\log \left (d x^n\right )}{n}\right )}{x}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]
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Rule 2209
Rule 2347
Rule 2602
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}+(b n p) \int \frac {1}{x^2 \log \left (d x^n\right )} \, dx \\ & = -\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x}+\frac {\left (b p \left (d x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {e^{-\frac {x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right )}{x} \\ & = \frac {b p \left (d x^n\right )^{\frac {1}{n}} \text {Ei}\left (-\frac {\log \left (d x^n\right )}{n}\right )}{x}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=-\frac {a-b p \left (d x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {\log \left (d x^n\right )}{n}\right )+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]
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\[\int \frac {a +b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )}{x^{2}}d x\]
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none
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\frac {b d^{\left (\frac {1}{n}\right )} p x \operatorname {log\_integral}\left (\frac {1}{d^{\left (\frac {1}{n}\right )} x}\right ) - b p \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b \log \left (c\right ) - a}{x} \]
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\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int \frac {a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int { \frac {b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int { \frac {b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )}{x^2} \,d x \]
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