Optimal. Leaf size=31 \[ -\frac {\text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2279, 2391} \[ -\frac {\text {PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log (1+e x)}{x} \, dx,x,\left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}\\ &=-\frac {\text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.00 \[ -\frac {\text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 31, normalized size = 1.00 \[ -\frac {{\rm Li}_2\left (-e f^{b c n x + a c n}\right )}{b c n \log \relax (f)} \]
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 \[ \int \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 32, normalized size = 1.03 \[ -\frac {\dilog \left (e \left (f^{\left (b x +a \right ) c}\right )^{n}+1\right )}{b c n \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 76, normalized size = 2.45 \[ x \log \left (e f^{{\left (b x + a\right )} c n} + 1\right ) - \frac {b c n x \log \left (e f^{b c n x} f^{a c n} + 1\right ) \log \relax (f) + {\rm Li}_2\left (-e f^{b c n x} f^{a c n}\right )}{b c n \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \ln \left (e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n+1\right ) \,d x \]
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 \[ - b c e n e^{a c n \log {\relax (f )}} \log {\relax (f )} \int \frac {x e^{b c n x \log {\relax (f )}}}{e e^{a c n \log {\relax (f )}} e^{b c n x \log {\relax (f )}} + 1}\, dx + x \log {\left (e \left (f^{c \left (a + b x\right )}\right )^{n} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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