Optimal. Leaf size=135 \[ -\frac {6 \text {Li}_{n+4}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^4 c^4 p^4 \log ^4(F)}+\frac {6 x \text {Li}_{n+3}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}-\frac {3 x^2 \text {Li}_{n+2}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {x^3 \text {Li}_{n+1}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
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Rubi [A] time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6609, 2282, 6589} \[ -\frac {3 x^2 \text {PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {6 x \text {PolyLog}\left (n+3,d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}-\frac {6 \text {PolyLog}\left (n+4,d \left (F^{c (a+b x)}\right )^p\right )}{b^4 c^4 p^4 \log ^4(F)}+\frac {x^3 \text {PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \text {Li}_n\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx &=\frac {x^3 \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {3 \int x^2 \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b c p \log (F)}\\ &=\frac {x^3 \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {3 x^2 \text {Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {6 \int x \text {Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b^2 c^2 p^2 \log ^2(F)}\\ &=\frac {x^3 \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {3 x^2 \text {Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {6 x \text {Li}_{3+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}-\frac {6 \int \text {Li}_{3+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b^3 c^3 p^3 \log ^3(F)}\\ &=\frac {x^3 \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {3 x^2 \text {Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {6 x \text {Li}_{3+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}-\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_{3+n}\left (d x^p\right )}{x} \, dx,x,F^{c (a+b x)}\right )}{b^4 c^4 p^3 \log ^4(F)}\\ &=\frac {x^3 \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {3 x^2 \text {Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {6 x \text {Li}_{3+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}-\frac {6 \text {Li}_{4+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^4 c^4 p^4 \log ^4(F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 135, normalized size = 1.00 \[ -\frac {6 \text {Li}_{n+4}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^4 c^4 p^4 \log ^4(F)}+\frac {6 x \text {Li}_{n+3}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}-\frac {3 x^2 \text {Li}_{n+2}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac {x^3 \text {Li}_{n+1}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} {\rm polylog}\left (n, {\left (F^{b c x + a c}\right )}^{p} d\right ), x\right ) \]
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 \[ \int x^{3} {\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int x^{3} \polylog \left (n , d \left (F^{c \left (b x +a \right )}\right )^{p}\right )\, dx \]
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 \[ \int x^{3} {\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d)\,{d x} \]
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 \[ \int x^3\,\mathrm {polylog}\left (n,d\,{\left (F^{c\,\left (a+b\,x\right )}\right )}^p\right ) \,d x \]
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 \[ \int x^{3} \operatorname {Li}_{n}\left (d \left (F^{a c} F^{b c x}\right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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