Optimal. Leaf size=65 \[ \frac {x \text {PolyLog}\left (1+n,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\text {PolyLog}\left (2+n,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)} \]
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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6744, 2320,
6724} \begin {gather*} \frac {x \text {Li}_{n+1}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\text {Li}_{n+2}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x \text {Li}_n\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx &=\frac {x \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\int \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b c p \log (F)}\\ &=\frac {x \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\text {Subst}\left (\int \frac {\text {Li}_{1+n}\left (d x^p\right )}{x} \, dx,x,F^{c (a+b x)}\right )}{b^2 c^2 p \log ^2(F)}\\ &=\frac {x \text {Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\text {Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} \frac {x \text {PolyLog}\left (1+n,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac {\text {PolyLog}\left (2+n,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x \polylog \left (n , d \left (F^{c \left (b x +a \right )}\right )^{p}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x \operatorname {Li}_{n}\left (d \left (F^{a c} F^{b c x}\right )^{p}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\mathrm {polylog}\left (n,d\,{\left (F^{c\,\left (a+b\,x\right )}\right )}^p\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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