3.1.6 \(\int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 9, antiderivative size = 5 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx=\operatorname {PolyLog}(3,a x) \]

output

polylog(3,a*x)
 

3.1.6.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx=\operatorname {PolyLog}(3,a x) \]

input

Integrate[PolyLog[2, a*x]/x,x]
 

output

PolyLog[3, a*x]
 

3.1.6.3 Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.

\(\displaystyle \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx\)

\(\Big \downarrow \) 7143

\(\displaystyle \operatorname {PolyLog}(3,a x)\)

input

Int[PolyLog[2, a*x]/x,x]
 

output

PolyLog[3, a*x]
 

3.1.6.3.1 Defintions of rubi rules used

rule 7143

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]
 

3.1.6.4 Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\operatorname {polylog}\left (3, a x \right )\)	\(6\)
default	\(\operatorname {polylog}\left (3, a x \right )\)	\(6\)
meijerg	\(\operatorname {polylog}\left (3, a x \right )\)	\(6\)
parts	\(\operatorname {polylog}\left (3, a x \right )\)	\(6\)

input

int(polylog(2,a*x)/x,x,method=_RETURNVERBOSE)
 

output

polylog(3,a*x)
 

3.1.6.5 Fricas [F]

\[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx=\int { \frac {{\rm Li}_2\left (a x\right )}{x} \,d x } \]

input

integrate(polylog(2,a*x)/x,x, algorithm="fricas")
 

output

integral(dilog(a*x)/x, x)
 

3.1.6.6 Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx=\operatorname {Li}_{3}\left (a x\right ) \]

input

integrate(polylog(2,a*x)/x,x)
 

output

polylog(3, a*x)
 

3.1.6.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx={\rm Li}_{3}(a x) \]

input

integrate(polylog(2,a*x)/x,x, algorithm="maxima")
 

output

polylog(3, a*x)
 

3.1.6.8 Giac [F]

\[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx=\int { \frac {{\rm Li}_2\left (a x\right )}{x} \,d x } \]

input

integrate(polylog(2,a*x)/x,x, algorithm="giac")
 

output

integrate(dilog(a*x)/x, x)
 

3.1.6.9 Mupad [B] (verification not implemented)

Time = 5.06 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{x} \, dx=\mathrm {polylog}\left (3,a\,x\right ) \]

input

int(polylog(2, a*x)/x,x)
 

output

polylog(3, a*x)