\(\int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx\) [127]

Optimal result

Integrand size = 18, antiderivative size = 309 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {3 \text {arctanh}(a x)^2}{2 a^3 c}+\frac {3 x \text {arctanh}(a x)^2}{2 a^2 c}-\frac {3 \text {arctanh}(a x)^3}{2 a^3 c}-\frac {x \text {arctanh}(a x)^3}{a^2 c}+\frac {x^2 \text {arctanh}(a x)^3}{2 a c}-\frac {3 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}-\frac {3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )}{2 a^3 c}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+a x}\right )}{4 a^3 c} \] Output:

3/2*arctanh(a*x)^2/a^3/c+3/2*x*arctanh(a*x)^2/a^2/c-3/2*arctanh(a*x)^3/a^3 
/c-x*arctanh(a*x)^3/a^2/c+1/2*x^2*arctanh(a*x)^3/a/c-3*arctanh(a*x)*ln(2/( 
-a*x+1))/a^3/c+3*arctanh(a*x)^2*ln(2/(-a*x+1))/a^3/c-arctanh(a*x)^3*ln(2/( 
a*x+1))/a^3/c-3/2*polylog(2,1-2/(-a*x+1))/a^3/c+3*arctanh(a*x)*polylog(2,1 
-2/(-a*x+1))/a^3/c+3/2*arctanh(a*x)^2*polylog(2,1-2/(a*x+1))/a^3/c-3/2*pol 
ylog(3,1-2/(-a*x+1))/a^3/c+3/2*arctanh(a*x)*polylog(3,1-2/(a*x+1))/a^3/c+3 
/4*polylog(4,1-2/(a*x+1))/a^3/c
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.56 \[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {-6 \text {arctanh}(a x)^2+6 a x \text {arctanh}(a x)^2+2 \text {arctanh}(a x)^3-4 a x \text {arctanh}(a x)^3+2 a^2 x^2 \text {arctanh}(a x)^3-12 \text {arctanh}(a x) \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+12 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )-4 \text {arctanh}(a x)^3 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+6 (-1+\text {arctanh}(a x))^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+6 (-1+\text {arctanh}(a x)) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )+3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}(a x)}\right )}{4 a^3 c} \] Input:

Integrate[(x^2*ArcTanh[a*x]^3)/(c + a*c*x),x]
 

Output:

(-6*ArcTanh[a*x]^2 + 6*a*x*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]^3 - 4*a*x*ArcTa 
nh[a*x]^3 + 2*a^2*x^2*ArcTanh[a*x]^3 - 12*ArcTanh[a*x]*Log[1 + E^(-2*ArcTa 
nh[a*x])] + 12*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] - 4*ArcTanh[a*x 
]^3*Log[1 + E^(-2*ArcTanh[a*x])] + 6*(-1 + ArcTanh[a*x])^2*PolyLog[2, -E^( 
-2*ArcTanh[a*x])] + 6*(-1 + ArcTanh[a*x])*PolyLog[3, -E^(-2*ArcTanh[a*x])] 
 + 3*PolyLog[4, -E^(-2*ArcTanh[a*x])])/(4*a^3*c)
 

Rubi [A] (verified)

Time = 4.02 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.944, Rules used = {6492, 27, 6452, 6492, 6436, 6470, 6542, 6436, 6510, 6546, 6470, 2849, 2752, 6618, 6620, 6622, 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.

\(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^3}{a c x+c} \, dx\)

\(\Big \downarrow \) 6492

\(\displaystyle \frac {\int x \text {arctanh}(a x)^3dx}{a c}-\frac {\int \frac {x \text {arctanh}(a x)^3}{c (a x+1)}dx}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x \text {arctanh}(a x)^3dx}{a c}-\frac {\int \frac {x \text {arctanh}(a x)^3}{a x+1}dx}{a c}\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a c}-\frac {\int \frac {x \text {arctanh}(a x)^3}{a x+1}dx}{a c}\)

\(\Big \downarrow \) 6492

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a c}-\frac {\frac {\int \text {arctanh}(a x)^3dx}{a}-\frac {\int \frac {\text {arctanh}(a x)^3}{a x+1}dx}{a}}{a c}\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a}-\frac {\int \frac {\text {arctanh}(a x)^3}{a x+1}dx}{a}}{a c}\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int \text {arctanh}(a x)^2dx}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \text {arctanh}(a x)^2-2 a \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 6510

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^3}{3 a^3}-\frac {x \text {arctanh}(a x)^2-2 a \int \frac {x \text {arctanh}(a x)}{1-a^2 x^2}dx}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^3}{3 a^3}-\frac {x \text {arctanh}(a x)^2-2 a \left (\frac {\int \frac {\text {arctanh}(a x)}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^3}{3 a^3}-\frac {x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^3}{3 a^3}-\frac {x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a}-\frac {3 \int \frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{a x+1}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 6622

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

\(\Big \downarrow \) 7164

\(\displaystyle \frac {\frac {1}{2} x^2 \text {arctanh}(a x)^3-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^3}{3 a^3}-\frac {x \text {arctanh}(a x)^2-2 a \left (\frac {\frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}}{a}-\frac {\text {arctanh}(a x)^2}{2 a^2}\right )}{a^2}\right )}{a c}-\frac {\frac {x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )}{a}-\frac {3 \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{a x+1}\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \log \left (\frac {2}{a x+1}\right )}{a}}{a}}{a c}\)

Input:

Int[(x^2*ArcTanh[a*x]^3)/(c + a*c*x),x]
 

Output:

((x^2*ArcTanh[a*x]^3)/2 - (3*a*(ArcTanh[a*x]^3/(3*a^3) - (x*ArcTanh[a*x]^2 
 - 2*a*(-1/2*ArcTanh[a*x]^2/a^2 + ((ArcTanh[a*x]*Log[2/(1 - a*x)])/a + Pol 
yLog[2, 1 - 2/(1 - a*x)]/(2*a))/a))/a^2))/2)/(a*c) - ((x*ArcTanh[a*x]^3 - 
3*a*(-1/3*ArcTanh[a*x]^3/a^2 + ((ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - 2*(- 
1/2*(ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a 
*x)]/(4*a)))/a))/a - (-((ArcTanh[a*x]^3*Log[2/(1 + a*x)])/a) + 3*((ArcTanh 
[a*x]^2*PolyLog[2, 1 - 2/(1 + a*x)])/(2*a) + (ArcTanh[a*x]*PolyLog[3, 1 - 
2/(1 + a*x)])/(2*a) + PolyLog[4, 1 - 2/(1 + a*x)]/(4*a)))/a)/(a*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[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 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[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 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[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 6492

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

rule 6510

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcTanh[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 6542

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

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[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 6618

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*(p/2)   Int[(a + b*ArcTanh[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 6620

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcTanh[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 6622

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

Maple [A] (verified)

Time = 6.73 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )-\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{2 c}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{2 c}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}-\frac {3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{c}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}-\frac {3 \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}-\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}}{a^{3}}\)	\(352\)
default	\(\frac {\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )-\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{2 c}+\frac {\operatorname {arctanh}\left (a x \right )^{4}}{2 c}-\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}-\frac {3 \operatorname {polylog}\left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{c}-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}-\frac {3 \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}-\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}{c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{c}-\frac {3 \operatorname {polylog}\left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 c}}{a^{3}}\)	\(352\)

Input:

int(x^2*arctanh(a*x)^3/(a*c*x+c),x,method=_RETURNVERBOSE)
 

Output:

1/a^3*(1/2/c*arctanh(a*x)^2*(a*x*arctanh(a*x)-arctanh(a*x)+3)*(a*x-1)+1/2/ 
c*arctanh(a*x)^4-1/c*arctanh(a*x)^3*ln((a*x+1)^2/(-a^2*x^2+1)+1)-3/2/c*arc 
tanh(a*x)^2*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+3/2/c*arctanh(a*x)*polylog( 
3,-(a*x+1)^2/(-a^2*x^2+1))-3/4/c*polylog(4,-(a*x+1)^2/(-a^2*x^2+1))+3/c*ar 
ctanh(a*x)^2-3/c*arctanh(a*x)*ln((a*x+1)^2/(-a^2*x^2+1)+1)-3/2/c*polylog(2 
,-(a*x+1)^2/(-a^2*x^2+1))-2/c*arctanh(a*x)^3+3/c*arctanh(a*x)^2*ln((a*x+1) 
^2/(-a^2*x^2+1)+1)+3/c*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-3/2 
/c*polylog(3,-(a*x+1)^2/(-a^2*x^2+1)))
 

Fricas [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \] Input:

integrate(x^2*arctanh(a*x)^3/(a*c*x+c),x, algorithm="fricas")
 

Output:

integral(x^2*arctanh(a*x)^3/(a*c*x + c), x)
 

Sympy [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {\int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a x + 1}\, dx}{c} \] Input:

integrate(x**2*atanh(a*x)**3/(a*c*x+c),x)
 

Output:

Integral(x**2*atanh(a*x)**3/(a*x + 1), x)/c
 

Maxima [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \] Input:

integrate(x^2*arctanh(a*x)^3/(a*c*x+c),x, algorithm="maxima")
 

Output:

-1/16*(a^2*x^2 - 2*a*x + 2*log(a*x + 1))*log(-a*x + 1)^3/(a^3*c) + 1/8*int 
egrate(1/2*(2*(a^3*x^3 - a^2*x^2)*log(a*x + 1)^3 - 6*(a^3*x^3 - a^2*x^2)*l 
og(a*x + 1)^2*log(-a*x + 1) + 3*(a^3*x^3 - a^2*x^2 - 2*a*x + 2*(a^3*x^3 - 
a^2*x^2 + a*x + 1)*log(a*x + 1))*log(-a*x + 1)^2)/(a^4*c*x^2 - a^2*c), x)
 

Giac [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\int { \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a c x + c} \,d x } \] Input:

integrate(x^2*arctanh(a*x)^3/(a*c*x+c),x, algorithm="giac")
 

Output:

integrate(x^2*arctanh(a*x)^3/(a*c*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{c+a\,c\,x} \,d x \] Input:

int((x^2*atanh(a*x)^3)/(c + a*c*x),x)
 

Output:

int((x^2*atanh(a*x)^3)/(c + a*c*x), x)
 

Reduce [F]

\[ \int \frac {x^2 \text {arctanh}(a x)^3}{c+a c x} \, dx=\frac {\int \frac {\mathit {atanh} \left (a x \right )^{3} x^{2}}{a x +1}d x}{c} \] Input:

int(x^2*atanh(a*x)^3/(a*c*x+c),x)
 

Output:

int((atanh(a*x)**3*x**2)/(a*x + 1),x)/c