Optimal. Leaf size=118 \[ \frac {\text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{2} x^2 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2532, 2531, 2282, 6589} \[ \frac {\text {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x \text {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{2} x^2 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2532
Rule 6589
Rubi steps
\begin {align*} \int x \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\int \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^2 c^2 n \log ^2(f)}\\ &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 118, normalized size = 1.00 \[ \frac {\text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{2} x^2 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 1.22, size = 169, normalized size = 1.43 \[ -\frac {2 \, b c n x {\rm Li}_2\left (-\frac {e f^{b c n x + a c n} + d}{d} + 1\right ) \log \relax (f) - {\left (b^{2} c^{2} n^{2} x^{2} - a^{2} c^{2} n^{2}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \relax (f)^{2} + {\left (b^{2} c^{2} n^{2} x^{2} - a^{2} c^{2} n^{2}\right )} \log \relax (f)^{2} \log \left (\frac {e f^{b c n x + a c n} + d}{d}\right ) - 2 \, {\rm polylog}\left (3, -\frac {e f^{b c n x + a c n}}{d}\right )}{2 \, b^{2} c^{2} n^{2} \log \relax (f)^{2}} \]
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 \[ \int x \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 558, normalized size = 4.73 \[ \frac {x^{2} \ln \left (e \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{2}-\frac {x^{2} \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{2}-\frac {x \ln \left (f^{\left (b x +a \right ) c}\right ) \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b c \ln \relax (f )}+\frac {x \ln \left (f^{\left (b x +a \right ) c}\right ) \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{b c \ln \relax (f )}-\frac {x \dilog \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b c n \ln \relax (f )}+\frac {\ln \left (f^{\left (b x +a \right ) c}\right )^{2} \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right )}{b^{2} c^{2} \ln \relax (f )^{2}}-\frac {\ln \left (f^{\left (b x +a \right ) c}\right )^{2} \ln \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}+1\right )}{2 b^{2} c^{2} \ln \relax (f )^{2}}-\frac {\ln \left (f^{\left (b x +a \right ) c}\right )^{2} \ln \left (e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d \right )}{2 b^{2} c^{2} \ln \relax (f )^{2}}+\frac {\dilog \left (\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}+d}{d}\right ) \ln \left (f^{\left (b x +a \right ) c}\right )}{b^{2} c^{2} n \ln \relax (f )^{2}}-\frac {\polylog \left (2, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}\right ) \ln \left (f^{\left (b x +a \right ) c}\right )}{b^{2} c^{2} n \ln \relax (f )^{2}}+\frac {\polylog \left (3, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{\left (b x +a \right ) c}\right )^{n}}{d}\right )}{b^{2} c^{2} n^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 126, normalized size = 1.07 \[ \frac {1}{2} \, x^{2} \log \left (e f^{{\left (b x + a\right )} c n} + d\right ) - \frac {b^{2} c^{2} n^{2} x^{2} \log \left (\frac {e f^{b c n x} f^{a c n}}{d} + 1\right ) \log \relax (f)^{2} + 2 \, b c n x {\rm Li}_2\left (-\frac {e f^{b c n x} f^{a c n}}{d}\right ) \log \relax (f) - 2 \, {\rm Li}_{3}(-\frac {e f^{b c n x} f^{a c n}}{d})}{2 \, b^{2} c^{2} n^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\ln \left (d+e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n\right ) \,d x \]
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 \[ - \frac {b c e n e^{a c n \log {\relax (f )}} \log {\relax (f )} \int \frac {x^{2} e^{b c n x \log {\relax (f )}}}{d + e e^{a c n \log {\relax (f )}} e^{b c n x \log {\relax (f )}}}\, dx}{2} + \frac {x^{2} \log {\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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