3.102 \(\int (d x)^m \text {Li}_2(a x) \, dx\)

Optimal. Leaf size=78 \[ \frac {a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^2 (m+2)}+\frac {\text {Li}_2(a x) (d x)^{m+1}}{d (m+1)}+\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)^2} \]

[Out]

a*(d*x)^(2+m)*hypergeom([1, 2+m],[3+m],a*x)/d^2/(1+m)^2/(2+m)+(d*x)^(1+m)*ln(-a*x+1)/d/(1+m)^2+(d*x)^(1+m)*pol
ylog(2,a*x)/d/(1+m)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6591, 2395, 64} \[ \frac {(d x)^{m+1} \text {PolyLog}(2,a x)}{d (m+1)}+\frac {a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^2 (m+2)}+\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*PolyLog[2, a*x],x]

[Out]

(a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)^2*(2 + m)) + ((d*x)^(1 + m)*Log[1 - a*x
])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[2, a*x])/(d*(1 + m))

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rubi steps

\begin {align*} \int (d x)^m \text {Li}_2(a x) \, dx &=\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)}+\frac {\int (d x)^m \log (1-a x) \, dx}{1+m}\\ &=\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)}+\frac {a \int \frac {(d x)^{1+m}}{1-a x} \, dx}{d (1+m)^2}\\ &=\frac {a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^2 (2+m)}+\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 53, normalized size = 0.68 \[ \frac {x (d x)^m (a x \, _2F_1(1,m+2;m+3;a x)+(m+2) ((m+1) \text {Li}_2(a x)+\log (1-a x)))}{(m+1)^2 (m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*PolyLog[2, a*x],x]

[Out]

(x*(d*x)^m*(a*x*Hypergeometric2F1[1, 2 + m, 3 + m, a*x] + (2 + m)*(Log[1 - a*x] + (1 + m)*PolyLog[2, a*x])))/(
(1 + m)^2*(2 + m))

________________________________________________________________________________________

fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} {\rm Li}_2\left (a x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*polylog(2,a*x),x, algorithm="fricas")

[Out]

integral((d*x)^m*dilog(a*x), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} {\rm Li}_2\left (a x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*polylog(2,a*x),x, algorithm="giac")

[Out]

integrate((d*x)^m*dilog(a*x), x)

________________________________________________________________________________________

maple [C]  time = 0.14, size = 144, normalized size = 1.85 \[ \frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-m} \left (\frac {x^{m} \left (-a \right )^{m} \left (-a \,m^{2} x -2 a m x -m^{2}-3 m -2\right )}{\left (m +2\right ) \left (1+m \right )^{3} m}-\frac {x^{1+m} \left (-a \right )^{m} a \left (-2-m \right ) \ln \left (-a x +1\right )}{\left (m +2\right ) \left (1+m \right )^{2}}+\frac {x^{1+m} \left (-a \right )^{m} a \polylog \left (2, a x \right )}{1+m}+\frac {x^{m} \left (-a \right )^{m} \Phi \left (a x , 1, m\right )}{\left (1+m \right )^{2}}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*polylog(2,a*x),x)

[Out]

(d*x)^m*x^(-m)*(-a)^(-m)/a*(1/(m+2)*x^m*(-a)^m*(-a*m^2*x-2*a*m*x-m^2-3*m-2)/(1+m)^3/m-1/(m+2)*x^(1+m)*(-a)^m*a
*(-2-m)/(1+m)^2*ln(-a*x+1)+x^(1+m)*(-a)^m*a/(1+m)*polylog(2,a*x)+x^m*(-a)^m/(1+m)^2*LerchPhi(a*x,1,m))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -a d^{m} \int -\frac {x x^{m}}{m^{2} - {\left (a m^{2} + 2 \, a m + a\right )} x + 2 \, m + 1}\,{d x} + \frac {{\left (d^{m} m + d^{m}\right )} x x^{m} {\rm Li}_2\left (a x\right ) + d^{m} x x^{m} \log \left (-a x + 1\right )}{m^{2} + 2 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*polylog(2,a*x),x, algorithm="maxima")

[Out]

-a*d^m*integrate(-x*x^m/(m^2 - (a*m^2 + 2*a*m + a)*x + 2*m + 1), x) + ((d^m*m + d^m)*x*x^m*dilog(a*x) + d^m*x*
x^m*log(-a*x + 1))/(m^2 + 2*m + 1)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (2,a\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*polylog(2, a*x),x)

[Out]

int((d*x)^m*polylog(2, a*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \operatorname {Li}_{2}\left (a x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*polylog(2,a*x),x)

[Out]

Integral((d*x)**m*polylog(2, a*x), x)

________________________________________________________________________________________