3.1.25 \(\int x^4 (a+b \text {arctanh}(c x))^3 \, dx\) [25]

3.1.25.1 Optimal result

Integrand size = 14, antiderivative size = 262 \[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\frac {9 a b^2 x}{10 c^4}+\frac {b^3 x^2}{20 c^3}+\frac {9 b^3 x \text {arctanh}(c x)}{10 c^4}+\frac {b^2 x^3 (a+b \text {arctanh}(c x))}{10 c^2}-\frac {9 b (a+b \text {arctanh}(c x))^2}{20 c^5}+\frac {3 b x^2 (a+b \text {arctanh}(c x))^2}{10 c^3}+\frac {3 b x^4 (a+b \text {arctanh}(c x))^2}{20 c}+\frac {(a+b \text {arctanh}(c x))^3}{5 c^5}+\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3 b (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{5 c^5}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c^5}-\frac {3 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^5}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{10 c^5} \]

output

9/10*a*b^2*x/c^4+1/20*b^3*x^2/c^3+9/10*b^3*x*arctanh(c*x)/c^4+1/10*b^2*x^3 
*(a+b*arctanh(c*x))/c^2-9/20*b*(a+b*arctanh(c*x))^2/c^5+3/10*b*x^2*(a+b*ar 
ctanh(c*x))^2/c^3+3/20*b*x^4*(a+b*arctanh(c*x))^2/c+1/5*(a+b*arctanh(c*x)) 
^3/c^5+1/5*x^5*(a+b*arctanh(c*x))^3-3/5*b*(a+b*arctanh(c*x))^2*ln(2/(-c*x+ 
1))/c^5+1/2*b^3*ln(-c^2*x^2+1)/c^5-3/5*b^2*(a+b*arctanh(c*x))*polylog(2,1- 
2/(-c*x+1))/c^5+3/10*b^3*polylog(3,1-2/(-c*x+1))/c^5
 

3.1.25.2 Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.46 \[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-b^3+18 a b^2 c x+6 a^2 b c^2 x^2+b^3 c^2 x^2+2 a b^2 c^3 x^3+3 a^2 b c^4 x^4+4 a^3 c^5 x^5-18 a b^2 \text {arctanh}(c x)+18 b^3 c x \text {arctanh}(c x)+12 a b^2 c^2 x^2 \text {arctanh}(c x)+2 b^3 c^3 x^3 \text {arctanh}(c x)+6 a b^2 c^4 x^4 \text {arctanh}(c x)+12 a^2 b c^5 x^5 \text {arctanh}(c x)-12 a b^2 \text {arctanh}(c x)^2-9 b^3 \text {arctanh}(c x)^2+6 b^3 c^2 x^2 \text {arctanh}(c x)^2+3 b^3 c^4 x^4 \text {arctanh}(c x)^2+12 a b^2 c^5 x^5 \text {arctanh}(c x)^2-4 b^3 \text {arctanh}(c x)^3+4 b^3 c^5 x^5 \text {arctanh}(c x)^3-24 a b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-12 b^3 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+6 a^2 b \log \left (1-c^2 x^2\right )+10 b^3 \log \left (1-c^2 x^2\right )+12 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+6 b^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{20 c^5} \]

input

Integrate[x^4*(a + b*ArcTanh[c*x])^3,x]
 

output

(-b^3 + 18*a*b^2*c*x + 6*a^2*b*c^2*x^2 + b^3*c^2*x^2 + 2*a*b^2*c^3*x^3 + 3 
*a^2*b*c^4*x^4 + 4*a^3*c^5*x^5 - 18*a*b^2*ArcTanh[c*x] + 18*b^3*c*x*ArcTan 
h[c*x] + 12*a*b^2*c^2*x^2*ArcTanh[c*x] + 2*b^3*c^3*x^3*ArcTanh[c*x] + 6*a* 
b^2*c^4*x^4*ArcTanh[c*x] + 12*a^2*b*c^5*x^5*ArcTanh[c*x] - 12*a*b^2*ArcTan 
h[c*x]^2 - 9*b^3*ArcTanh[c*x]^2 + 6*b^3*c^2*x^2*ArcTanh[c*x]^2 + 3*b^3*c^4 
*x^4*ArcTanh[c*x]^2 + 12*a*b^2*c^5*x^5*ArcTanh[c*x]^2 - 4*b^3*ArcTanh[c*x] 
^3 + 4*b^3*c^5*x^5*ArcTanh[c*x]^3 - 24*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*Ar 
cTanh[c*x])] - 12*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 6*a^2* 
b*Log[1 - c^2*x^2] + 10*b^3*Log[1 - c^2*x^2] + 12*b^2*(a + b*ArcTanh[c*x]) 
*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 6*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])] 
)/(20*c^5)
 

3.1.25.3 Rubi [A] (verified)

Time = 2.82 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.39, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6452, 6542, 6452, 6542, 6452, 243, 49, 2009, 6542, 2009, 6510, 6546, 6470, 6620, 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.

\(\displaystyle \int x^4 (a+b \text {arctanh}(c x))^3 \, dx\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \int \frac {x^5 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x^3 (a+b \text {arctanh}(c x))^2dx}{c^2}\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}\right )\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))^2dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x^2 (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{3} b c \int \frac {x^3}{1-c^2 x^2}dx}{c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \frac {x^2}{1-c^2 x^2}dx^2}{c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2-1\right )}\right )dx^2}{c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6620

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

\(\Big \downarrow \) 7164

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

input

Int[x^4*(a + b*ArcTanh[c*x])^3,x]
 

output

(x^5*(a + b*ArcTanh[c*x])^3)/5 - (3*b*c*(-(((x^4*(a + b*ArcTanh[c*x])^2)/4 
 - (b*c*(-(((x^3*(a + b*ArcTanh[c*x]))/3 - (b*c*(-(x^2/c^2) - Log[1 - c^2* 
x^2]/c^4))/6)/c^2) + ((a + b*ArcTanh[c*x])^2/(2*b*c^3) - (a*x + b*x*ArcTan 
h[c*x] + (b*Log[1 - c^2*x^2])/(2*c))/c^2)/c^2))/2)/c^2) + (-(((x^2*(a + b* 
ArcTanh[c*x])^2)/2 - b*c*((a + b*ArcTanh[c*x])^2/(2*b*c^3) - (a*x + b*x*Ar 
cTanh[c*x] + (b*Log[1 - c^2*x^2])/(2*c))/c^2))/c^2) + (-1/3*(a + b*ArcTanh 
[c*x])^3/(b*c^2) + (((a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c - 2*b*(-1/ 
2*((a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (b*PolyLog[3, 1 - 
 2/(1 - c*x)])/(4*c)))/c)/c^2)/c^2))/5
 

3.1.25.3.1 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 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 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 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 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 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]
 

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

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

Time = 13.38 (sec) , antiderivative size = 1038, normalized size of antiderivative = 3.96

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1038\)
default	\(\text {Expression too large to display}\)	\(1038\)
parts	\(\text {Expression too large to display}\)	\(1040\)

input

int(x^4*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
 

output

1/c^5*(1/5*c^5*x^5*a^3+b^3*(9/10*c*x*arctanh(c*x)-3/10*I*Pi*arctanh(c*x)^2 
+1/10*c^3*x^3*arctanh(c*x)+1/20*c^2*x^2+3/20*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2 
*x^2-1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c 
*x+1)^2/(c^2*x^2-1)))*arctanh(c*x)^2-1/20+3/10*c^2*x^2*arctanh(c*x)^2+3/10 
*arctanh(c*x)^2*ln(c*x-1)+3/10*arctanh(c*x)^2*ln(c*x+1)-3/5*arctanh(c*x)^2 
*ln(2)-3/5*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-9/20*arctanh(c*x) 
^2+3/20*c^4*x^4*arctanh(c*x)^2-ln(1+(c*x+1)^2/(-c^2*x^2+1))+arctanh(c*x)+1 
/5*arctanh(c*x)^3+3/10*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+1/5*c^5*x^5*arct 
anh(c*x)^3-3/5*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-3/20*I*Pi*c 
sgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2 
/(c^2*x^2-1)))^2*arctanh(c*x)^2+3/20*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*cs 
gn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2-3/2 
0*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))* 
arctanh(c*x)^2-3/10*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1) 
^2/(c^2*x^2-1))^2*arctanh(c*x)^2+3/10*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1) 
))^2*arctanh(c*x)^2-3/10*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh( 
c*x)^2-3/20*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2-3/20*I*Pi* 
csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*x)^2)+ 
3*a*b^2*(1/5*c^5*x^5*arctanh(c*x)^2+1/10*c^4*x^4*arctanh(c*x)+1/5*c^2*x^2* 
arctanh(c*x)+1/5*arctanh(c*x)*ln(c*x-1)+1/5*arctanh(c*x)*ln(c*x+1)+1/30...
 

3.1.25.5 Fricas [F]

\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{4} \,d x } \]

input

integrate(x^4*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
 

output

integral(b^3*x^4*arctanh(c*x)^3 + 3*a*b^2*x^4*arctanh(c*x)^2 + 3*a^2*b*x^4 
*arctanh(c*x) + a^3*x^4, x)
 

3.1.25.6 Sympy [F]

\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int x^{4} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \]

input

integrate(x**4*(a+b*atanh(c*x))**3,x)
 

output

Integral(x**4*(a + b*atanh(c*x))**3, x)
 

3.1.25.7 Maxima [F]

\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{4} \,d x } \]

input

integrate(x^4*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
 

output

1/5*a^3*x^5 + 3/20*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log( 
c^2*x^2 - 1)/c^6))*a^2*b - 1/80*(2*(b^3*c^5*x^5 - b^3)*log(-c*x + 1)^3 - 3 
*(4*a*b^2*c^5*x^5 + b^3*c^4*x^4 + 2*b^3*c^2*x^2 + 2*(b^3*c^5*x^5 + b^3)*lo 
g(c*x + 1))*log(-c*x + 1)^2)/c^5 - integrate(-1/40*(5*(b^3*c^5*x^5 - b^3*c 
^4*x^4)*log(c*x + 1)^3 + 30*(a*b^2*c^5*x^5 - a*b^2*c^4*x^4)*log(c*x + 1)^2 
 - 3*(4*a*b^2*c^5*x^5 + b^3*c^4*x^4 + 2*b^3*c^2*x^2 + 5*(b^3*c^5*x^5 - b^3 
*c^4*x^4)*log(c*x + 1)^2 - 2*(10*a*b^2*c^4*x^4 - (10*a*b^2*c^5 + b^3*c^5)* 
x^5 - b^3)*log(c*x + 1))*log(-c*x + 1))/(c^5*x - c^4), x)
 

3.1.25.8 Giac [F]

\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{4} \,d x } \]

input

integrate(x^4*(a+b*arctanh(c*x))^3,x, algorithm="giac")
 

output

integrate((b*arctanh(c*x) + a)^3*x^4, x)
 

3.1.25.9 Mupad [F(-1)]

Timed out. \[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \]

input

int(x^4*(a + b*atanh(c*x))^3,x)
 

output

int(x^4*(a + b*atanh(c*x))^3, x)