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 = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 2178
Rule 2310
Rule 2522
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.44, size = 46, normalized size = 0.96 \[ \frac {b d^{\left (\frac {1}{n}\right )} p x \operatorname {log\_integral}\left (\frac {1}{d^{\left (\frac {1}{n}\right )} x}\right ) - b p \log \left (n \log \relax (x) + \log \relax (d)\right ) - b \log \relax (c) - a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c \log \left (d x^{n}\right )^{p}\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )+a}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (n p \int \frac {1}{x^{2} \log \relax (d) + x^{2} \log \left (x^{n}\right )}\,{d x} - \frac {\log \relax (c) + \log \left ({\left (\log \relax (d) + \log \left (x^{n}\right )\right )}^{p}\right )}{x}\right )} b - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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