Optimal. Leaf size=25 \[ -\frac {\text {Li}_2\left (1-c \left (e x^n+d\right )\right )}{c e n} \]
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Rubi [A] time = 0.16, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2475, 2412, 2393, 2391} \[ -\frac {\text {PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2412
Rule 2475
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x \left (c e-(1-c d) x^{-n}\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log (c (d+e x))}{\left (c e+\frac {-1+c d}{x}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,-1+c d+c e x^n\right )}{c e n}\\ &=-\frac {\text {Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 26, normalized size = 1.04 \[ -\frac {\text {Li}_2\left (-c e x^n-c d+1\right )}{c e n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 25, normalized size = 1.00 \[ -\frac {{\rm Li}_2\left (-c e x^{n} - c d + 1\right )}{c e n} \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )} c\right )}{{\left (c e + \frac {c d - 1}{x^{n}}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 23, normalized size = 0.92 \[ -\frac {\dilog \left (c e \,x^{n}+c d \right )}{c e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.88, size = 106, normalized size = 4.24 \[ {\left (\frac {\log \left (c e + \frac {c d - 1}{x^{n}}\right )}{c e n} - \frac {\log \left (\frac {1}{x^{n}}\right )}{c e n}\right )} \log \left ({\left (e x^{n} + d\right )} c\right ) - \frac {\log \left (c e x^{n} + c d\right ) \log \left (c e x^{n} + c d - 1\right ) + {\rm Li}_2\left (-c e x^{n} - c d + 1\right )}{c e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (c\,\left (d+e\,x^n\right )\right )}{x\,\left (c\,e+\frac {c\,d-1}{x^n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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