Optimal. Leaf size=121 \[ \frac{a (d x)^{m+2} \text{Hypergeometric2F1}(1,m+2,m+3,a x)}{d^2 (m+1)^4 (m+2)}+\frac{(d x)^{m+1} \text{PolyLog}(2,a x)}{d (m+1)^3}-\frac{(d x)^{m+1} \text{PolyLog}(3,a x)}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}(4,a x)}{d (m+1)}+\frac{\log (1-a x) (d x)^{m+1}}{d (m+1)^4} \]
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Rubi [A] time = 0.0857405, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2395, 64} \[ \frac{(d x)^{m+1} \text{PolyLog}(2,a x)}{d (m+1)^3}-\frac{(d x)^{m+1} \text{PolyLog}(3,a x)}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}(4,a x)}{d (m+1)}+\frac{a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^4 (m+2)}+\frac{\log (1-a x) (d x)^{m+1}}{d (m+1)^4} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 64
Rubi steps
\begin{align*} \int (d x)^m \text{Li}_4(a x) \, dx &=\frac{(d x)^{1+m} \text{Li}_4(a x)}{d (1+m)}-\frac{\int (d x)^m \text{Li}_3(a x) \, dx}{1+m}\\ &=-\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4(a x)}{d (1+m)}+\frac{\int (d x)^m \text{Li}_2(a x) \, dx}{(1+m)^2}\\ &=\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)^3}-\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4(a x)}{d (1+m)}+\frac{\int (d x)^m \log (1-a x) \, dx}{(1+m)^3}\\ &=\frac{(d x)^{1+m} \log (1-a x)}{d (1+m)^4}+\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)^3}-\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4(a x)}{d (1+m)}+\frac{a \int \frac{(d x)^{1+m}}{1-a x} \, dx}{d (1+m)^4}\\ &=\frac{a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^4 (2+m)}+\frac{(d x)^{1+m} \log (1-a x)}{d (1+m)^4}+\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)^3}-\frac{(d x)^{1+m} \text{Li}_3(a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4(a x)}{d (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0726869, size = 119, normalized size = 0.98 \[ \frac{x \text{Gamma}(m+2) (d x)^m \left (a (m+1) x \text{Gamma}(m+1) \, _2\tilde{F}_1(1,m+2;m+3;a x)+m^3 \text{PolyLog}(4,a x)-m^2 \text{PolyLog}(3,a x)+3 m^2 \text{PolyLog}(4,a x)-2 m \text{PolyLog}(3,a x)+3 m \text{PolyLog}(4,a x)+(m+1) \text{PolyLog}(2,a x)-\text{PolyLog}(3,a x)+\text{PolyLog}(4,a x)+\log (1-a x)\right )}{(m+1)^5 \text{Gamma}(m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.794, size = 198, normalized size = 1.6 \begin{align*}{\frac{ \left ( dx \right ) ^{m}{x}^{-m} \left ( -a \right ) ^{-m}}{a} \left ({\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( -a{m}^{2}x-2\,amx-{m}^{2}-3\,m-2 \right ) }{ \left ( 1+m \right ) ^{5} \left ( 2+m \right ) m}}-{\frac{{x}^{1+m}a \left ( -a \right ) ^{m} \left ( -m-2 \right ) \ln \left ( -ax+1 \right ) }{ \left ( 1+m \right ) ^{4} \left ( 2+m \right ) }}+{\frac{{x}^{1+m}a \left ( -a \right ) ^{m}{\it polylog} \left ( 2,ax \right ) }{ \left ( 1+m \right ) ^{3}}}+{\frac{{x}^{1+m}a \left ( -a \right ) ^{m} \left ( -m-2 \right ){\it polylog} \left ( 3,ax \right ) }{ \left ( 1+m \right ) ^{2} \left ( 2+m \right ) }}+{\frac{{x}^{1+m}a \left ( -a \right ) ^{m}{\it polylog} \left ( 4,ax \right ) }{1+m}}+{\frac{{x}^{m} \left ( -a \right ) ^{m}{\it LerchPhi} \left ( ax,1,m \right ) }{ \left ( 1+m \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a d^{m} \int -\frac{x x^{m}}{m^{4} + 4 \, m^{3} + 6 \, m^{2} -{\left (a m^{4} + 4 \, a m^{3} + 6 \, a m^{2} + 4 \, a m + a\right )} x + 4 \, m + 1}\,{d x} + \frac{{\left (d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_2\left (a x\right ) + d^{m} x x^{m} \log \left (-a x + 1\right ) +{\left (d^{m} m^{3} + 3 \, d^{m} m^{2} + 3 \, d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_{4}(a x) -{\left (d^{m} m^{2} + 2 \, d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_{3}(a x)}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m}{\rm polylog}\left (4, a x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \operatorname{Li}_{4}\left (a x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m}{\rm Li}_{4}(a x)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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