Optimal. Leaf size=20 \[ \frac {\text {li}\left (d (e+f x)^p\right )}{d f p} \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2390, 2307, 2298} \[ \frac {\text {li}\left (d (e+f x)^p\right )}{d f p} \]
Antiderivative was successfully verified.
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Rule 2298
Rule 2307
Rule 2390
Rubi steps
\begin {align*} \int \frac {(e+f x)^{-1+p}}{\log \left (d (e+f x)^p\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^{-1+p}}{\log \left (d x^p\right )} \, dx,x,e+f x\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\log (d x)} \, dx,x,(e+f x)^p\right )}{f p}\\ &=\frac {\text {li}\left (d (e+f x)^p\right )}{d f p}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 1.00 \[ \frac {\text {li}\left (d (e+f x)^p\right )}{d f p} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 22, normalized size = 1.10 \[ \frac {{\rm Ei}\left (p \log \left (f x + e\right ) + \log \relax (d)\right )}{d f p} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 23, normalized size = 1.15 \[ \frac {{\rm Ei}\left (p \log \left (f x + e\right ) + \log \relax (d)\right )}{d f p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 26, normalized size = 1.30 \[ -\frac {\Ei \left (1, -\ln \left (d \left (f x +e \right )^{p}\right )\right )}{d f p} \]
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 \[ \int \frac {{\left (f x + e\right )}^{p - 1}}{\log \left ({\left (f x + e\right )}^{p} d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 20, normalized size = 1.00 \[ \frac {\mathrm {logint}\left (d\,{\left (e+f\,x\right )}^p\right )}{d\,f\,p} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.18, size = 42, normalized size = 2.10 \[ \begin {cases} - \frac {\begin {cases} - \frac {\log {\left (e + f x \right )}}{\log {\relax (d )}} & \text {for}\: p = 0 \\- \frac {\operatorname {li}{\left (d \left (e + f x\right )^{p} \right )}}{d p} & \text {otherwise} \end {cases}}{f} & \text {for}\: f \neq 0 \\\frac {e^{p - 1} x}{\log {\left (d e^{p} \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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