Integrand size = 23, antiderivative size = 23 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\frac {2 b e n q^3 x^{1+q} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+q}{q},\frac {1+m+2 q}{q},e x^q\right )}{(1+m)^4 (1+m+q)}+\frac {2 b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac {2 b n q (d x)^{1+m} \operatorname {PolyLog}\left (2,e x^q\right )}{d (1+m)^3}-\frac {q (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right )}{d (1+m)^2}-\frac {b n (d x)^{1+m} \operatorname {PolyLog}\left (3,e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right )}{d (1+m)}-\frac {q^2 \text {Int}\left ((d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ),x\right )}{(1+m)^2} \]
2*b*e*n*q^3*x^(1+q)*(d*x)^m*hypergeom([1, (1+m+q)/q],[(1+m+2*q)/q],e*x^q)/ (1+m)^4/(1+m+q)+2*b*n*q^2*(d*x)^(1+m)*ln(1-e*x^q)/d/(1+m)^4+2*b*n*q*(d*x)^ (1+m)*polylog(2,e*x^q)/d/(1+m)^3-q*(d*x)^(1+m)*(a+b*ln(c*x^n))*polylog(2,e *x^q)/d/(1+m)^2-b*n*(d*x)^(1+m)*polylog(3,e*x^q)/d/(1+m)^2+(d*x)^(1+m)*(a+ b*ln(c*x^n))*polylog(3,e*x^q)/d/(1+m)-q^2*Unintegrable((d*x)^m*(a+b*ln(c*x ^n))*ln(1-e*x^q),x)/(1+m)^2
Not integrable
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx \]
Not integrable
Time = 0.97 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2832, 2832, 25, 2826, 2905, 30, 888, 7145, 25, 2905, 30, 888}
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.
| \(\displaystyle \int (d x)^m \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2832 |
| \(\displaystyle -\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right )dx}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 2832 |
| \(\displaystyle -\frac {q \left (-\frac {q \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {b n q \int -(d x)^m \log \left (1-e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \int (d x)^m \log \left (1-e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 2826 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \int (d x)^m \log \left (1-e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \left (\frac {e q \int \frac {x^{q-1} (d x)^{m+1}}{1-e x^q}dx}{d (m+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \left (\frac {e q x^{-m} (d x)^m \int \frac {x^{m+q}}{1-e x^q}dx}{m+1}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \left (\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)}-\frac {q \int -(d x)^m \log \left (1-e x^q\right )dx}{m+1}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \left (\frac {q \int (d x)^m \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \left (\frac {q \left (\frac {e q \int \frac {x^{q-1} (d x)^{m+1}}{1-e x^q}dx}{d (m+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {b n q \left (\frac {q \left (\frac {e q x^{-m} (d x)^m \int \frac {x^{m+q}}{1-e x^q}dx}{m+1}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle -\frac {q \left (\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\right )}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}+\frac {b n q \left (\frac {q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (3,e x^q\right )}{d (m+1)^2}\) |
3.3.22.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> Unintegrable[(g*x)^q*(a + b*Log[c*x^n])^p*Log[d*(e + f*x^m)^r], x] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, p, q}, x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.)*PolyLog[k_, (e _.)*(x_)^(q_.)], x_Symbol] :> Simp[(-b)*n*(d*x)^(m + 1)*(PolyLog[k, e*x^q]/ (d*(m + 1)^2)), x] + (Simp[(d*x)^(m + 1)*PolyLog[k, e*x^q]*((a + b*Log[c*x^ n])/(d*(m + 1))), x] - Simp[q/(m + 1) Int[(d*x)^m*PolyLog[k - 1, e*x^q]*( a + b*Log[c*x^n]), x], x] + Simp[b*n*(q/(m + 1)^2) Int[(d*x)^m*PolyLog[k - 1, e*x^q], x], x]) /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -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]
Time = 2.00 (sec) , antiderivative size = 1065, normalized size of antiderivative = 46.30
-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*a/q*(q^3*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^3*ln (1-e*x^q)+q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(- e)^(m/q+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^3 *LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*b*ln(c)/q*(q^ 3*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(m/q+1/q)/(1 +m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/(1+m)*polylog(3,e*x^q)+q^3 *x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))+((-e)^(-m /q-1/q)*ln(-e)/q^2*(d*x)^m*x^(-m)*b*n*(q^3*x^m*(-e)^(m/q+1/q)/(1+m)^3*ln(1 -e*x^q)+q^2*x^m*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^m*(-e)^(m/q+1/ q)/(1+m)*polylog(3,e*x^q)+q^3*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^3*LerchPhi(e* x^q,1,(1+m+q)/q))-(-e)^(-m/q-1/q)*(d*x)^m*x^(-m)*b*n/q*(q^3*x^m*(-e)^(m/q+ 1/q)*ln(x)/(1+m)^3*ln(1-e*x^q)+q^2*x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)^3*ln(1- e*x^q)-3*q^3*x^m*(-e)^(m/q+1/q)/(1+m)^4*ln(1-e*x^q)+q^2*x^m*(-e)^(m/q+1/q) *ln(x)/(1+m)^2*polylog(2,e*x^q)+q*x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*polylo g(2,e*x^q)-2*q^2*x^m*(-e)^(m/q+1/q)/(1+m)^3*polylog(2,e*x^q)-q*x^m*(-e)^(m /q+1/q)*ln(x)/(1+m)*polylog(3,e*x^q)-x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)*polyl og(3,e*x^q)+q*x^m*(-e)^(m/q+1/q)/(1+m)^2*polylog(3,e*x^q)+q^3*x^(q+m)*e*(- e)^(m/q+1/q)*ln(x)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q^2*x^(q+m)*e*(-e)^ (m/q+1/q)*ln(-e)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)-3*q^3*x^(q+m)*e*(-e)^ (m/q+1/q)/(1+m)^4*LerchPhi(e*x^q,1,(1+m+q)/q)-q^2*x^(q+m)*e*(-e)^(m/q+1...
Not integrable
Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_{3}(e x^{q}) \,d x } \]
Not integrable
Time = 11.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\int \left (d x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x^{q}\right )\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 427, normalized size of antiderivative = 18.57 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_{3}(e x^{q}) \,d x } \]
-(((m^2*q + 2*m*q + q)*b*d^m*x*x^m*log(x^n) + ((m^2*q + 2*m*q + q)*a*d^m + ((m^2*q + 2*m*q + q)*d^m*log(c) - 2*(m*n*q + n*q)*d^m)*b)*x*x^m)*dilog(e* x^q) + ((m*q^2 + q^2)*b*d^m*x*x^m*log(x^n) + ((m*q^2 + q^2)*a*d^m - (3*d^m *n*q^2 - (m*q^2 + q^2)*d^m*log(c))*b)*x*x^m)*log(-e*x^q + 1) - ((m^3 + 3*m ^2 + 3*m + 1)*b*d^m*x*x^m*log(x^n) + ((m^3 + 3*m^2 + 3*m + 1)*a*d^m + ((m^ 3 + 3*m^2 + 3*m + 1)*d^m*log(c) - (m^2*n + 2*m*n + n)*d^m)*b)*x*x^m)*polyl og(3, e*x^q))/(m^4 + 4*m^3 + 6*m^2 + 4*m + 1) + integrate(-((m*q^3 + q^3)* b*d^m*e*e^(m*log(x) + q*log(x))*log(x^n) + ((m*q^3 + q^3)*a*d^m*e - (3*d^m *e*n*q^3 - (m*q^3 + q^3)*d^m*e*log(c))*b)*e^(m*log(x) + q*log(x)))/(m^4 + 4*m^3 - (m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*e*x^q + 6*m^2 + 4*m + 1), x)
Not integrable
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_{3}(e x^{q}) \,d x } \]
Not integrable
Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,e x^q\right ) \, dx=\int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (3,e\,x^q\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]