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 = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2178
Rule 2300
Rule 2520
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.45, size = 53, normalized size = 1.18 \[ \frac {b d^{\left (\frac {1}{n}\right )} p x \log \left (n \log \relax (x) + \log \relax (d)\right ) - b p \operatorname {log\_integral}\left (d^{\left (\frac {1}{n}\right )} x\right ) + {\left (b x \log \relax (c) + a x\right )} d^{\left (\frac {1}{n}\right )}}{d^{\left (\frac {1}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 42, normalized size = 0.93 \[ {\left (p x \log \left (n \log \relax (x) + \log \relax (d)\right ) + x \log \relax (c) - \frac {p {\rm Ei}\left (\frac {\log \relax (d)}{n} + \log \relax (x)\right )}{d^{\left (\frac {1}{n}\right )}}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )+a\, dx \]
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 \[ -{\left (n p \int \frac {1}{\log \relax (d) + \log \left (x^{n}\right )}\,{d x} - x \log \relax (c) - x \log \left ({\left (\log \relax (d) + \log \left (x^{n}\right )\right )}^{p}\right )\right )} b + a x \]
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 \[ \int a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right ) \,d x \]
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 \[ \int \left (a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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