Integrand size = 15, antiderivative size = 45 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=a x-b p x \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right )+b x \log \left (c \log ^p\left (d x^n\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2600, 2337, 2209} \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-b p x \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right ) \]
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Rule 2209
Rule 2337
Rule 2600
Rubi steps \begin{align*} \text {integral}& = a x+b \int \log \left (c \log ^p\left (d x^n\right )\right ) \, dx \\ & = a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-(b n p) \int \frac {1}{\log \left (d x^n\right )} \, dx \\ & = a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-\left (b p x \left (d x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right ) \\ & = a x-b p x \left (d x^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (d x^n\right )}{n}\right )+b x \log \left (c \log ^p\left (d x^n\right )\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=x \left (a-b p \left (d x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {\log \left (d x^n\right )}{n}\right )+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \]
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\[\int \left (a +b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )\right )d x\]
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none
Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=\frac {b d^{\left (\frac {1}{n}\right )} p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p \operatorname {log\_integral}\left (d^{\left (\frac {1}{n}\right )} x\right ) + {\left (b x \log \left (c\right ) + a x\right )} d^{\left (\frac {1}{n}\right )}}{d^{\left (\frac {1}{n}\right )}} \]
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\[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=\int \left (a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}\right )\, dx \]
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\[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=\int { b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a \,d x } \]
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none
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx={\left (p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) + x \log \left (c\right ) - \frac {p {\rm Ei}\left (\frac {\log \left (d\right )}{n} + \log \left (x\right )\right )}{d^{\left (\frac {1}{n}\right )}}\right )} b + a x \]
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Timed out. \[ \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx=\int a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right ) \,d x \]
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