\(\int (a+b \log (c x^n)) \operatorname {PolyLog}(3,e x) \, dx\) [222]

Optimal result

Integrand size = 16, antiderivative size = 131 \[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=-4 b n x+x \left (a+b \log \left (c x^n\right )\right )-\frac {3 b n (1-e x) \log (1-e x)}{e}+\frac {(1-e x) \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{e}+\frac {b n \operatorname {PolyLog}(2,e x)}{e}+2 b n x \operatorname {PolyLog}(2,e x)-x \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+x \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \] Output:

-4*b*n*x+x*(a+b*ln(c*x^n))-3*b*n*(-e*x+1)*ln(-e*x+1)/e+(-e*x+1)*(a+b*ln(c* 
x^n))*ln(-e*x+1)/e+b*n*polylog(2,e*x)/e+2*b*n*x*polylog(2,e*x)-x*(a+b*ln(c 
*x^n))*polylog(2,e*x)-b*n*x*polylog(3,e*x)+x*(a+b*ln(c*x^n))*polylog(3,e*x 
)
 

Mathematica [F]

\[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx \] Input:

Integrate[(a + b*Log[c*x^n])*PolyLog[3, e*x],x]
 

Output:

Integrate[(a + b*Log[c*x^n])*PolyLog[3, e*x], x]
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.44, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {2828, 2828, 25, 2817, 2009, 2836, 2732, 7140, 25, 2836, 2732}

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.

\(\displaystyle \int \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2828

\(\displaystyle -\int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)dx+b n \int \operatorname {PolyLog}(2,e x)dx+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-b n x \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2828

\(\displaystyle \int -\left (\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)\right )dx+b n \int \operatorname {PolyLog}(2,e x)dx-b n \int -\log (1-e x)dx-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)dx+b n \int \operatorname {PolyLog}(2,e x)dx+b n \int \log (1-e x)dx-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2817

\(\displaystyle b n \int \operatorname {PolyLog}(2,e x)dx+b n \int \log (1-e x)dx+b n \int \left (-\frac {(1-e x) \log (1-e x)}{e x}-1\right )dx-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2009

\(\displaystyle b n \int \operatorname {PolyLog}(2,e x)dx+b n \int \log (1-e x)dx-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )\)

\(\Big \downarrow \) 2836

\(\displaystyle b n \int \operatorname {PolyLog}(2,e x)dx-\frac {b n \int \log (1-e x)d(1-e x)}{e}-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )\)

\(\Big \downarrow \) 2732

\(\displaystyle b n \int \operatorname {PolyLog}(2,e x)dx-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )-\frac {b n (e x+(1-e x) \log (1-e x)-1)}{e}\)

\(\Big \downarrow \) 7140

\(\displaystyle b n (x \operatorname {PolyLog}(2,e x)-\int -\log (1-e x)dx)-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )-\frac {b n (e x+(1-e x) \log (1-e x)-1)}{e}\)

\(\Big \downarrow \) 25

\(\displaystyle b n (\int \log (1-e x)dx+x \operatorname {PolyLog}(2,e x))-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )-\frac {b n (e x+(1-e x) \log (1-e x)-1)}{e}\)

\(\Big \downarrow \) 2836

\(\displaystyle b n \left (x \operatorname {PolyLog}(2,e x)-\frac {\int \log (1-e x)d(1-e x)}{e}\right )-x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )-\frac {b n (e x+(1-e x) \log (1-e x)-1)}{e}\)

\(\Big \downarrow \) 2732

\(\displaystyle -x \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+x \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {(1-e x) \log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{e}+x \left (a+b \log \left (c x^n\right )\right )+b n x \operatorname {PolyLog}(2,e x)-b n x \operatorname {PolyLog}(3,e x)+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{e}-\frac {(1-e x) \log (1-e x)}{e}-2 x\right )+b n \left (x \operatorname {PolyLog}(2,e x)-\frac {e x+(1-e x) \log (1-e x)-1}{e}\right )-\frac {b n (e x+(1-e x) \log (1-e x)-1)}{e}\)

Input:

Int[(a + b*Log[c*x^n])*PolyLog[3, e*x],x]
 

Output:

x*(a + b*Log[c*x^n]) + ((1 - e*x)*(a + b*Log[c*x^n])*Log[1 - e*x])/e - (b* 
n*(-1 + e*x + (1 - e*x)*Log[1 - e*x]))/e + b*n*x*PolyLog[2, e*x] - x*(a + 
b*Log[c*x^n])*PolyLog[2, e*x] + b*n*(-2*x - ((1 - e*x)*Log[1 - e*x])/e + P 
olyLog[2, e*x]/e) + b*n*(-((-1 + e*x + (1 - e*x)*Log[1 - e*x])/e) + x*Poly 
Log[2, e*x]) - b*n*x*PolyLog[3, e*x] + x*(a + b*Log[c*x^n])*PolyLog[3, e*x 
]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2732

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x 
] /; FreeQ[{c, n}, x]
 

rule 2817

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, 
Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p 
- 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] 
&& RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 
 1] && EqQ[m, 1] && EqQ[d*e, 1]))
 

rule 2828

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*PolyLog[k_, (e_.)*(x_)^(q_.)], x_ 
Symbol] :> Simp[(-b)*n*x*PolyLog[k, e*x^q], x] + (Simp[x*PolyLog[k, e*x^q]* 
(a + b*Log[c*x^n]), x] - Simp[q   Int[PolyLog[k - 1, e*x^q]*(a + b*Log[c*x^ 
n]), x], x] + Simp[b*n*q   Int[PolyLog[k - 1, e*x^q], x], x]) /; FreeQ[{a, 
b, c, e, n, q}, x] && IGtQ[k, 0]
 

rule 2836

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : 
> Simp[1/e   Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ 
a, b, c, d, e, n, p}, x]
 

rule 7140

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

Maple [F]

\[\int \left (a +b \ln \left (c \,x^{n}\right )\right ) \operatorname {polylog}\left (3, e x \right )d x\]

Input:

int((a+b*ln(c*x^n))*polylog(3,e*x),x)
 

Output:

int((a+b*ln(c*x^n))*polylog(3,e*x),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.31 \[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=-\frac {{\left (4 \, b e n - a e\right )} x - {\left (b n + {\left (2 \, b e n - a e\right )} x\right )} {\rm Li}_2\left (e x\right ) + {\left (3 \, b n - {\left (3 \, b e n - a e\right )} x - a\right )} \log \left (-e x + 1\right ) + {\left (b e x {\rm Li}_2\left (e x\right ) - b e x + {\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + {\left (b e n x {\rm Li}_2\left (e x\right ) - b e n x + {\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right ) - {\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) - {\left (b e n - a e\right )} x\right )} {\rm polylog}\left (3, e x\right )}{e} \] Input:

integrate((a+b*log(c*x^n))*polylog(3,e*x),x, algorithm="fricas")
 

Output:

-((4*b*e*n - a*e)*x - (b*n + (2*b*e*n - a*e)*x)*dilog(e*x) + (3*b*n - (3*b 
*e*n - a*e)*x - a)*log(-e*x + 1) + (b*e*x*dilog(e*x) - b*e*x + (b*e*x - b) 
*log(-e*x + 1))*log(c) + (b*e*n*x*dilog(e*x) - b*e*n*x + (b*e*n*x - b*n)*l 
og(-e*x + 1))*log(x) - (b*e*n*x*log(x) + b*e*x*log(c) - (b*e*n - a*e)*x)*p 
olylog(3, e*x))/e
 

Sympy [F]

\[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )\, dx \] Input:

integrate((a+b*ln(c*x**n))*polylog(3,e*x),x)
 

Output:

Integral((a + b*log(c*x**n))*polylog(3, e*x), x)
 

Maxima [F]

\[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x) \,d x } \] Input:

integrate((a+b*log(c*x^n))*polylog(3,e*x),x, algorithm="maxima")
 

Output:

-b*(((e*x*log(x^n) - (2*e*n - e*log(c))*x)*dilog(e*x) - ((3*e*n - e*log(c) 
)*x - n*log(x))*log(-e*x + 1) - (e*x - (e*x - 1)*log(-e*x + 1))*log(x^n) - 
 (e*x*log(x^n) - (e*n - e*log(c))*x)*polylog(3, e*x))/e - integrate(-((4*e 
*n - e*log(c))*x - n*log(x) - n)/(e*x - 1), x)) - (e*x*dilog(e*x) - e*x*po 
lylog(3, e*x) - e*x + (e*x - 1)*log(-e*x + 1))*a/e
 

Giac [F]

\[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} {\rm Li}_{3}(e x) \,d x } \] Input:

integrate((a+b*log(c*x^n))*polylog(3,e*x),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)*polylog(3, e*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\text {Hanged} \] Input:

int(polylog(3, e*x)*(a + b*log(c*x^n)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\left (\int \mathit {polylog}\left (3, e x \right )d x \right ) a +\left (\int \mathrm {log}\left (x^{n} c \right ) \mathit {polylog}\left (3, e x \right )d x \right ) b \] Input:

int((a+b*log(c*x^n))*polylog(3,e*x),x)
 

Output:

int(polylog(3,e*x),x)*a + int(log(x**n*c)*polylog(3,e*x),x)*b