Integrand size = 11, antiderivative size = 121 \[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\frac {a (d x)^{2+m} \operatorname {Hypergeometric2F1}(1,2+m,3+m,a x)}{d^2 (1+m)^4 (2+m)}+\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^4}+\frac {(d x)^{1+m} \operatorname {PolyLog}(2,a x)}{d (1+m)^3}-\frac {(d x)^{1+m} \operatorname {PolyLog}(3,a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \operatorname {PolyLog}(4,a x)}{d (1+m)} \]
a*(d*x)^(2+m)*hypergeom([1, 2+m],[3+m],a*x)/d^2/(1+m)^4/(2+m)+(d*x)^(1+m)* ln(-a*x+1)/d/(1+m)^4+(d*x)^(1+m)*polylog(2,a*x)/d/(1+m)^3-(d*x)^(1+m)*poly log(3,a*x)/d/(1+m)^2+(d*x)^(1+m)*polylog(4,a*x)/d/(1+m)
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\frac {x (d x)^m \operatorname {Gamma}(2+m) \left (a (1+m) x \operatorname {Gamma}(1+m) \, _2\tilde {F}_1(1,2+m;3+m;a x)+\log (1-a x)+(1+m) \operatorname {PolyLog}(2,a x)-\operatorname {PolyLog}(3,a x)-2 m \operatorname {PolyLog}(3,a x)-m^2 \operatorname {PolyLog}(3,a x)+\operatorname {PolyLog}(4,a x)+3 m \operatorname {PolyLog}(4,a x)+3 m^2 \operatorname {PolyLog}(4,a x)+m^3 \operatorname {PolyLog}(4,a x)\right )}{(1+m)^5 \operatorname {Gamma}(1+m)} \]
(x*(d*x)^m*Gamma[2 + m]*(a*(1 + m)*x*Gamma[1 + m]*HypergeometricPFQRegular ized[{1, 2 + m}, {3 + m}, a*x] + Log[1 - a*x] + (1 + m)*PolyLog[2, a*x] - PolyLog[3, a*x] - 2*m*PolyLog[3, a*x] - m^2*PolyLog[3, a*x] + PolyLog[4, a *x] + 3*m*PolyLog[4, a*x] + 3*m^2*PolyLog[4, a*x] + m^3*PolyLog[4, a*x]))/ ((1 + m)^5*Gamma[1 + m])
Time = 0.45 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {7145, 7145, 7145, 25, 2842, 74}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \operatorname {PolyLog}(4,a x) (d x)^m \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {\operatorname {PolyLog}(4,a x) (d x)^{m+1}}{d (m+1)}-\frac {\int (d x)^m \operatorname {PolyLog}(3,a x)dx}{m+1}\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {\operatorname {PolyLog}(4,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\operatorname {PolyLog}(3,a x) (d x)^{m+1}}{d (m+1)}-\frac {\int (d x)^m \operatorname {PolyLog}(2,a x)dx}{m+1}}{m+1}\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {\operatorname {PolyLog}(4,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\operatorname {PolyLog}(3,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\operatorname {PolyLog}(2,a x) (d x)^{m+1}}{d (m+1)}-\frac {\int -(d x)^m \log (1-a x)dx}{m+1}}{m+1}}{m+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\operatorname {PolyLog}(4,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\operatorname {PolyLog}(3,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\int (d x)^m \log (1-a x)dx}{m+1}+\frac {\operatorname {PolyLog}(2,a x) (d x)^{m+1}}{d (m+1)}}{m+1}}{m+1}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {\operatorname {PolyLog}(4,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\operatorname {PolyLog}(3,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\frac {a \int \frac {(d x)^{m+1}}{1-a x}dx}{d (m+1)}+\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)}}{m+1}+\frac {\operatorname {PolyLog}(2,a x) (d x)^{m+1}}{d (m+1)}}{m+1}}{m+1}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {\operatorname {PolyLog}(4,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\operatorname {PolyLog}(3,a x) (d x)^{m+1}}{d (m+1)}-\frac {\frac {\frac {a (d x)^{m+2} \operatorname {Hypergeometric2F1}(1,m+2,m+3,a x)}{d^2 (m+1) (m+2)}+\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)}}{m+1}+\frac {\operatorname {PolyLog}(2,a x) (d x)^{m+1}}{d (m+1)}}{m+1}}{m+1}\) |
-((-((((a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)*(2 + m)) + ((d*x)^(1 + m)*Log[1 - a*x])/(d*(1 + m)))/(1 + m) + ((d*x)^ (1 + m)*PolyLog[2, a*x])/(d*(1 + m)))/(1 + m)) + ((d*x)^(1 + m)*PolyLog[3, a*x])/(d*(1 + m)))/(1 + m)) + ((d*x)^(1 + m)*PolyLog[4, a*x])/(d*(1 + m))
3.2.4.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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])))
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] - Simp[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]
Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.63 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.64
method | result | size |
meijerg | \(\frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-m} \left (\frac {x^{m} \left (-a \right )^{m} \left (-a \,m^{2} x -2 a m x -m^{2}-3 m -2\right )}{\left (2+m \right ) \left (1+m \right )^{5} m}-\frac {x^{1+m} a \left (-a \right )^{m} \left (-m -2\right ) \ln \left (-a x +1\right )}{\left (2+m \right ) \left (1+m \right )^{4}}+\frac {x^{1+m} a \left (-a \right )^{m} \operatorname {polylog}\left (2, a x \right )}{\left (1+m \right )^{3}}+\frac {x^{1+m} a \left (-a \right )^{m} \left (-m -2\right ) \operatorname {polylog}\left (3, a x \right )}{\left (2+m \right ) \left (1+m \right )^{2}}+\frac {x^{1+m} a \left (-a \right )^{m} \operatorname {polylog}\left (4, a x \right )}{1+m}+\frac {x^{m} \left (-a \right )^{m} \operatorname {LerchPhi}\left (a x , 1, m\right )}{\left (1+m \right )^{4}}\right )}{a}\) | \(198\) |
(d*x)^m*x^(-m)*(-a)^(-m)/a*(1/(2+m)*x^m*(-a)^m*(-a*m^2*x-2*a*m*x-m^2-3*m-2 )/(1+m)^5/m-1/(2+m)*x^(1+m)*a*(-a)^m*(-m-2)/(1+m)^4*ln(-a*x+1)+x^(1+m)*a*( -a)^m/(1+m)^3*polylog(2,a*x)+1/(2+m)*x^(1+m)*a*(-a)^m*(-m-2)/(1+m)^2*polyl og(3,a*x)+x^(1+m)*a*(-a)^m/(1+m)*polylog(4,a*x)+x^m*(-a)^m/(1+m)^4*LerchPh i(a*x,1,m))
\[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\int { \left (d x\right )^{m} {\rm Li}_{4}(a x) \,d x } \]
\[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\int \left (d x\right )^{m} \operatorname {Li}_{4}\left (a x\right )\, dx \]
\[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\int { \left (d x\right )^{m} {\rm Li}_{4}(a x) \,d x } \]
-a*d^m*integrate(-x*x^m/(m^4 + 4*m^3 + 6*m^2 - (a*m^4 + 4*a*m^3 + 6*a*m^2 + 4*a*m + a)*x + 4*m + 1), x) + ((d^m*m + d^m)*x*x^m*dilog(a*x) + d^m*x*x^ m*log(-a*x + 1) + (d^m*m^3 + 3*d^m*m^2 + 3*d^m*m + d^m)*x*x^m*polylog(4, a *x) - (d^m*m^2 + 2*d^m*m + d^m)*x*x^m*polylog(3, a*x))/(m^4 + 4*m^3 + 6*m^ 2 + 4*m + 1)
\[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\int { \left (d x\right )^{m} {\rm Li}_{4}(a x) \,d x } \]
Timed out. \[ \int (d x)^m \operatorname {PolyLog}(4,a x) \, dx=\int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (4,a\,x\right ) \,d x \]