Integrand size = 19, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(4,e x)-b n \operatorname {PolyLog}(5,e x) \] Output:
(a+b*ln(c*x^n))*polylog(4,e*x)-b*n*polylog(5,e*x)
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=a \operatorname {PolyLog}(4,e x)+b \log \left (c x^n\right ) \operatorname {PolyLog}(4,e x)-b n \operatorname {PolyLog}(5,e x) \] Input:
Integrate[((a + b*Log[c*x^n])*PolyLog[3, e*x])/x,x]
Output:
a*PolyLog[4, e*x] + b*Log[c*x^n]*PolyLog[4, e*x] - b*n*PolyLog[5, e*x]
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2830, 7143}
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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 2830 |
\(\displaystyle \operatorname {PolyLog}(4,e x) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}(4,e x)}{x}dx\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \operatorname {PolyLog}(4,e x) \left (a+b \log \left (c x^n\right )\right )-b n \operatorname {PolyLog}(5,e x)\) |
Input:
Int[((a + b*Log[c*x^n])*PolyLog[3, e*x])/x,x]
Output:
(a + b*Log[c*x^n])*PolyLog[4, e*x] - b*n*PolyLog[5, e*x]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[b*n*(p/q) Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \operatorname {polylog}\left (3, e x \right )}{x}d x\]
Input:
int((a+b*ln(c*x^n))*polylog(3,e*x)/x,x)
Output:
int((a+b*ln(c*x^n))*polylog(3,e*x)/x,x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x} \,d x } \] Input:
integrate((a+b*log(c*x^n))*polylog(3,e*x)/x,x, algorithm="fricas")
Output:
integral((b*log(c*x^n)*polylog(3, e*x) + a*polylog(3, e*x))/x, x)
Time = 3.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=a \operatorname {Li}_{4}\left (e x\right ) + b \left (- n \operatorname {Li}_{5}\left (e x\right ) + \log {\left (c x^{n} \right )} \operatorname {Li}_{4}\left (e x\right )\right ) \] Input:
integrate((a+b*ln(c*x**n))*polylog(3,e*x)/x,x)
Output:
a*polylog(4, e*x) + b*(-n*polylog(5, e*x) + log(c*x**n)*polylog(4, e*x))
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x} \,d x } \] Input:
integrate((a+b*log(c*x^n))*polylog(3,e*x)/x,x, algorithm="maxima")
Output:
1/6*(2*b*n*log(x)^3 - 3*b*log(x)^2*log(x^n) - 3*(b*log(c) + a)*log(x)^2)*d ilog(e*x) - 1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log (x))*polylog(3, e*x) - 1/6*integrate((3*b*log(-e*x + 1)*log(x)^2*log(x^n) - (2*b*n*log(x)^3 - 3*(b*log(c) + a)*log(x)^2)*log(-e*x + 1))/x, x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x)}{x} \,d x } \] Input:
integrate((a+b*log(c*x^n))*polylog(3,e*x)/x,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*polylog(3, e*x)/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=\text {Hanged} \] Input:
int((polylog(3, e*x)*(a + b*log(c*x^n)))/x,x)
Output:
\text{Hanged}
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x)}{x} \, dx=\left (\int \frac {\mathit {polylog}\left (3, e x \right )}{x}d x \right ) a +\left (\int \frac {\mathrm {log}\left (x^{n} c \right ) \mathit {polylog}\left (3, e x \right )}{x}d x \right ) b \] Input:
int((a+b*log(c*x^n))*polylog(3,e*x)/x,x)
Output:
int(polylog(3,e*x)/x,x)*a + int((log(x**n*c)*polylog(3,e*x))/x,x)*b