Integrand size = 20, antiderivative size = 193 \[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \operatorname {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \operatorname {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \operatorname {PolyLog}\left (4,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \operatorname {PolyLog}\left (5,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)} \]
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Time = 0.09 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2612, 2611, 6744, 2320, 6724} \[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\frac {6 \operatorname {PolyLog}\left (5,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac {6 x \operatorname {PolyLog}\left (4,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {3 x^2 \operatorname {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x^3 \operatorname {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{4} x^4 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{4} x^4 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]
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Rule 2320
Rule 2611
Rule 2612
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x^3 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx \\ & = \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 \int x^2 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)} \\ & = \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \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 {6 \int x \text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^2 c^2 n^2 \log ^2(f)} \\ & = \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \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 {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \int \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b^3 c^3 n^3 \log ^3(f)} \\ & = \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \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 {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \text {Subst}\left (\int \frac {\text {Li}_4\left (-\frac {e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^4 c^4 n^3 \log ^4(f)} \\ & = \frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \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 {6 x \text {Li}_4\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \text {Li}_5\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00 \[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\frac {1}{4} x^4 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{4} x^4 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x^3 \operatorname {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {3 x^2 \operatorname {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {6 x \operatorname {PolyLog}\left (4,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac {6 \operatorname {PolyLog}\left (5,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 n^4 \log ^4(f)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1275\) vs. \(2(189)=378\).
Time = 2.12 (sec) , antiderivative size = 1276, normalized size of antiderivative = 6.61
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Time = 0.35 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.27 \[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=-\frac {4 \, b^{3} c^{3} n^{3} x^{3} {\rm Li}_2\left (-\frac {e f^{b c n x + a c n} + d}{d} + 1\right ) \log \left (f\right )^{3} - 12 \, b^{2} c^{2} n^{2} x^{2} \log \left (f\right )^{2} {\rm polylog}\left (3, -\frac {e f^{b c n x + a c n}}{d}\right ) - {\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \left (f\right )^{4} + {\left (b^{4} c^{4} n^{4} x^{4} - a^{4} c^{4} n^{4}\right )} \log \left (f\right )^{4} \log \left (\frac {e f^{b c n x + a c n} + d}{d}\right ) + 24 \, b c n x \log \left (f\right ) {\rm polylog}\left (4, -\frac {e f^{b c n x + a c n}}{d}\right ) - 24 \, {\rm polylog}\left (5, -\frac {e f^{b c n x + a c n}}{d}\right )}{4 \, b^{4} c^{4} n^{4} \log \left (f\right )^{4}} \]
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\[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\int x^{3} \log {\left (d + e \left (f^{a c + b c x}\right )^{n} \right )}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.06 \[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\frac {1}{4} \, x^{4} \log \left (e f^{{\left (b x + a\right )} c n} + d\right ) - \frac {b^{4} c^{4} n^{4} x^{4} \log \left (\frac {e f^{b c n x} f^{a c n}}{d} + 1\right ) \log \left (f\right )^{4} + 4 \, b^{3} c^{3} n^{3} x^{3} {\rm Li}_2\left (-\frac {e f^{b c n x} f^{a c n}}{d}\right ) \log \left (f\right )^{3} - 12 \, b^{2} c^{2} n^{2} x^{2} \log \left (f\right )^{2} {\rm Li}_{3}(-\frac {e f^{b c n x} f^{a c n}}{d}) + 24 \, b c n x \log \left (f\right ) {\rm Li}_{4}(-\frac {e f^{b c n x} f^{a c n}}{d}) - 24 \, {\rm Li}_{5}(-\frac {e f^{b c n x} f^{a c n}}{d})}{4 \, b^{4} c^{4} n^{4} \log \left (f\right )^{4}} \]
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\[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\int { x^{3} \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + d\right ) \,d x } \]
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Timed out. \[ \int x^3 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\int x^3\,\ln \left (d+e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n\right ) \,d x \]
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