3.1.24 \(\int x^4 \coth ^{-1}(a x)^3 \, dx\) [24]

3.1.24.1 Optimal result

Integrand size = 10, antiderivative size = 196 \[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\frac {x^2}{20 a^3}+\frac {9 x \coth ^{-1}(a x)}{10 a^4}+\frac {x^3 \coth ^{-1}(a x)}{10 a^2}-\frac {9 \coth ^{-1}(a x)^2}{20 a^5}+\frac {3 x^2 \coth ^{-1}(a x)^2}{10 a^3}+\frac {3 x^4 \coth ^{-1}(a x)^2}{20 a}+\frac {\coth ^{-1}(a x)^3}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{5 a^5}+\frac {\log \left (1-a^2 x^2\right )}{2 a^5}-\frac {3 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{10 a^5} \]

output

1/20*x^2/a^3+9/10*x*arccoth(a*x)/a^4+1/10*x^3*arccoth(a*x)/a^2-9/20*arccot 
h(a*x)^2/a^5+3/10*x^2*arccoth(a*x)^2/a^3+3/20*x^4*arccoth(a*x)^2/a+1/5*arc 
coth(a*x)^3/a^5+1/5*x^5*arccoth(a*x)^3-3/5*arccoth(a*x)^2*ln(2/(-a*x+1))/a 
^5+1/2*ln(-a^2*x^2+1)/a^5-3/5*arccoth(a*x)*polylog(2,1-2/(-a*x+1))/a^5+3/1 
0*polylog(3,1-2/(-a*x+1))/a^5
 

3.1.24.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.91 \[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\frac {-2-i \pi ^3+2 a^2 x^2+36 a x \coth ^{-1}(a x)+4 a^3 x^3 \coth ^{-1}(a x)-18 \coth ^{-1}(a x)^2+12 a^2 x^2 \coth ^{-1}(a x)^2+6 a^4 x^4 \coth ^{-1}(a x)^2+8 \coth ^{-1}(a x)^3+8 a^5 x^5 \coth ^{-1}(a x)^3-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-40 \log \left (\frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )-40 \log \left (\frac {1}{a x}\right )-24 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )}{40 a^5} \]

input

Integrate[x^4*ArcCoth[a*x]^3,x]
 

output

(-2 - I*Pi^3 + 2*a^2*x^2 + 36*a*x*ArcCoth[a*x] + 4*a^3*x^3*ArcCoth[a*x] - 
18*ArcCoth[a*x]^2 + 12*a^2*x^2*ArcCoth[a*x]^2 + 6*a^4*x^4*ArcCoth[a*x]^2 + 
 8*ArcCoth[a*x]^3 + 8*a^5*x^5*ArcCoth[a*x]^3 - 24*ArcCoth[a*x]^2*Log[1 - E 
^(2*ArcCoth[a*x])] - 40*Log[1/Sqrt[1 - 1/(a^2*x^2)]] - 40*Log[1/(a*x)] - 2 
4*ArcCoth[a*x]*PolyLog[2, E^(2*ArcCoth[a*x])] + 12*PolyLog[3, E^(2*ArcCoth 
[a*x])])/(40*a^5)
 

3.1.24.3 Rubi [A] (verified)

Time = 2.57 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.55, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {6453, 6543, 6453, 6543, 6453, 243, 49, 2009, 6543, 6437, 240, 6511, 6547, 6471, 6621, 7164}

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.

\(\displaystyle \int x^4 \coth ^{-1}(a x)^3 \, dx\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \int \frac {x^5 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx\)

\(\Big \downarrow \) 6543

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x^3 \coth ^{-1}(a x)^2dx}{a^2}\right )\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}\right )\)

\(\Big \downarrow \) 6543

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)^2dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x^2 \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6453

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{1-a^2 x^2}dx}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \int \frac {x^2}{1-a^2 x^2}dx^2}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (a^2 x^2-1\right )}\right )dx^2}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6543

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6437

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2}dx}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6511

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6547

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6471

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a}-2 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 6621

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{2 a}\right )}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^3-\frac {3}{5} a \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{2 a}\right )}{a}-\frac {\coth ^{-1}(a x)^3}{3 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)^2-\frac {1}{2} a \left (\frac {\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \left (-\frac {x^2}{a^2}-\frac {\log \left (1-a^2 x^2\right )}{a^4}\right )}{a^2}\right )}{a^2}\right )\)

input

Int[x^4*ArcCoth[a*x]^3,x]
 

output

(x^5*ArcCoth[a*x]^3)/5 - (3*a*(-(((x^4*ArcCoth[a*x]^2)/4 - (a*(-(((x^3*Arc 
Coth[a*x])/3 - (a*(-(x^2/a^2) - Log[1 - a^2*x^2]/a^4))/6)/a^2) + (ArcCoth[ 
a*x]^2/(2*a^3) - (x*ArcCoth[a*x] + Log[1 - a^2*x^2]/(2*a))/a^2)/a^2))/2)/a 
^2) + (-(((x^2*ArcCoth[a*x]^2)/2 - a*(ArcCoth[a*x]^2/(2*a^3) - (x*ArcCoth[ 
a*x] + Log[1 - a^2*x^2]/(2*a))/a^2))/a^2) + (-1/3*ArcCoth[a*x]^3/a^2 + ((A 
rcCoth[a*x]^2*Log[2/(1 - a*x)])/a - 2*(-1/2*(ArcCoth[a*x]*PolyLog[2, 1 - 2 
/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/(4*a)))/a)/a^2)/a^2))/5
 

3.1.24.3.1 Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

rule 6437

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcCoth[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 1])
 

rule 6453

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcCoth[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -1]
 

rule 6471

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6511

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b 
, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
 

rule 6543

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcCoth[c* 
x])^p, x], x] - Simp[d*(f^2/e)   Int[(f*x)^(m - 2)*((a + b*ArcCoth[c*x])^p/ 
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 
 1]
 

rule 6547

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 6621

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcCoth[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( 
d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d 
 + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

3.1.24.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.11 (sec) , antiderivative size = 737, normalized size of antiderivative = 3.76

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(737\)
default	\(\text {Expression too large to display}\)	\(737\)
parts	\(\text {Expression too large to display}\)	\(737\)

input

int(x^4*arccoth(a*x)^3,x,method=_RETURNVERBOSE)
 

output

1/a^5*(6/5*polylog(3,1/((a*x-1)/(a*x+1))^(1/2))+6/5*polylog(3,-1/((a*x-1)/ 
(a*x+1))^(1/2))+9/10*a*x*arccoth(a*x)+3/10*a^2*x^2*arccoth(a*x)^2+1/10*a^3 
*x^3*arccoth(a*x)+3/20*a^4*x^4*arccoth(a*x)^2+3/20*I*csgn(I/(a*x-1)*(a*x+1 
)/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x+1)/(a*x-1)-1)) 
*arccoth(a*x)^2*Pi-1/20+1/20*a^2*x^2-ln(1+1/((a*x-1)/(a*x+1))^(1/2))+1/5*a 
rccoth(a*x)^3+arccoth(a*x)-ln(1/((a*x-1)/(a*x+1))^(1/2)-1)-9/20*arccoth(a* 
x)^2+1/5*x^5*arccoth(a*x)^3*a^5+3/5*arccoth(a*x)^2*ln((a*x+1)/(a*x-1)-1)-3 
/5*arccoth(a*x)^2*ln(1-1/((a*x-1)/(a*x+1))^(1/2))-6/5*arccoth(a*x)*polylog 
(2,1/((a*x-1)/(a*x+1))^(1/2))-3/5*arccoth(a*x)^2*ln(1+1/((a*x-1)/(a*x+1))^ 
(1/2))-6/5*arccoth(a*x)*polylog(2,-1/((a*x-1)/(a*x+1))^(1/2))+3/20*I*csgn( 
I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)^2*Pi-3/1 
0*I*csgn(I/((a*x-1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x) 
^2*Pi-3/20*I*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*csgn(I*(a*x+1)/ 
(a*x-1))*arccoth(a*x)^2*Pi-3/20*I*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)- 
1))^2*csgn(I/((a*x+1)/(a*x-1)-1))*arccoth(a*x)^2*Pi+3/10*arccoth(a*x)^2*ln 
((a*x-1)/(a*x+1))+3/10*arccoth(a*x)^2*ln(a*x-1)+3/10*arccoth(a*x)^2*ln(a*x 
+1)+3/20*I*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^3*arccoth(a*x)^2*Pi 
+3/20*I*csgn(I*(a*x+1)/(a*x-1))^3*arccoth(a*x)^2*Pi-3/5*arccoth(a*x)^2*ln( 
2))
 

3.1.24.5 Fricas [F]

\[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]

input

integrate(x^4*arccoth(a*x)^3,x, algorithm="fricas")
 

output

integral(x^4*arccoth(a*x)^3, x)
 

3.1.24.6 Sympy [F]

\[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\int x^{4} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]

input

integrate(x**4*acoth(a*x)**3,x)
 

output

Integral(x**4*acoth(a*x)**3, x)
 

3.1.24.7 Maxima [F]

\[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]

input

integrate(x^4*arccoth(a*x)^3,x, algorithm="maxima")
 

output

1/80*(2*(a^5*x^5 + 1)*log(a*x + 1)^3 + 3*(a^4*x^4 + 2*a^2*x^2 - 2*(a^5*x^5 
 - 1)*log(a*x - 1))*log(a*x + 1)^2)/a^5 + 1/8*integrate(-1/5*(5*(a^5*x^5 + 
 a^4*x^4)*log(a*x - 1)^3 + 3*(a^4*x^4 + 2*a^2*x^2 - 5*(a^5*x^5 + a^4*x^4)* 
log(a*x - 1)^2 - 2*(a^5*x^5 - 1)*log(a*x - 1))*log(a*x + 1))/(a^5*x + a^4) 
, x)
 

3.1.24.8 Giac [F]

\[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]

input

integrate(x^4*arccoth(a*x)^3,x, algorithm="giac")
 

output

integrate(x^4*arccoth(a*x)^3, x)
 

3.1.24.9 Mupad [F(-1)]

Timed out. \[ \int x^4 \coth ^{-1}(a x)^3 \, dx=\int x^4\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]

input

int(x^4*acoth(a*x)^3,x)
 

output

int(x^4*acoth(a*x)^3, x)