Optimal. Leaf size=48 \[ \frac{b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x}-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]
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Rubi [A] time = 0.046958, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2522, 2310, 2178} \[ \frac{b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x}-\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2522
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^2} \, dx &=-\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 ) \operatorname{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*}
Mathematica [A] time = 0.0405497, size = 45, normalized size = 0.94 \[ -\frac{a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (d x^n\right )}{n}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) }{{x}^{2}}}\, 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}{x^{2} \log \left (d\right ) + x^{2} \log \left (x^{n}\right )}\,{d x} - \frac{\log \left (c\right ) + \log \left ({\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}^{p}\right )}{x}\right )} b - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89519, size = 123, normalized size = 2.56 \begin{align*} \frac{b d^{\left (\frac{1}{n}\right )} p x \logintegral \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} \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 \frac{a + b \log{\left (c \log{\left (d x^{n} \right )}^{p} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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