Optimal. Leaf size=45 \[ a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-b p x \left (d x^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (d x^n\right )}{n}\right ) \]
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Rubi [A] time = 0.0300297, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2520, 2300, 2178} \[ a x+b x \log \left (c \log ^p\left (d x^n\right )\right )-b p x \left (d x^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (d x^n\right )}{n}\right ) \]
Antiderivative was successfully verified.
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Rule 2520
Rule 2300
Rule 2178
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx &=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 ) \operatorname{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*}
Mathematica [A] time = 0.0478073, size = 43, normalized size = 0.96 \[ x \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (d x^n\right )}{n}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (n p \int \frac{1}{\log \left (d\right ) + \log \left (x^{n}\right )}\,{d x} - x \log \left (c\right ) - x \log \left ({\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}^{p}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97502, size = 144, normalized size = 3.2 \begin{align*} \frac{b d^{\left (\frac{1}{n}\right )} p x \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p \logintegral \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \log{\left (d x^{n} \right )}^{p} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32369, size = 57, normalized size = 1.27 \begin{align*}{\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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