\(\int \frac {x^3 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx\) [95]

Optimal result

Integrand size = 22, antiderivative size = 329 \[ \int \frac {x^3 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=-\frac {a b x}{c^3 d}+\frac {b^2 x}{3 c^3 d}-\frac {b^2 \text {arctanh}(c x)}{3 c^4 d}-\frac {b^2 x \text {arctanh}(c x)}{c^3 d}+\frac {b x^2 (a+b \text {arctanh}(c x))}{3 c^2 d}+\frac {11 (a+b \text {arctanh}(c x))^2}{6 c^4 d}+\frac {x (a+b \text {arctanh}(c x))^2}{c^3 d}-\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 c^2 d}+\frac {x^3 (a+b \text {arctanh}(c x))^2}{3 c d}-\frac {8 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^4 d}+\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^4 d}-\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^4 d}-\frac {4 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^4 d}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^4 d}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^4 d} \] Output:

-a*b*x/c^3/d+1/3*b^2*x/c^3/d-1/3*b^2*arctanh(c*x)/c^4/d-b^2*x*arctanh(c*x) 
/c^3/d+1/3*b*x^2*(a+b*arctanh(c*x))/c^2/d+11/6*(a+b*arctanh(c*x))^2/c^4/d+ 
x*(a+b*arctanh(c*x))^2/c^3/d-1/2*x^2*(a+b*arctanh(c*x))^2/c^2/d+1/3*x^3*(a 
+b*arctanh(c*x))^2/c/d-8/3*b*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c^4/d+(a+b* 
arctanh(c*x))^2*ln(2/(c*x+1))/c^4/d-1/2*b^2*ln(-c^2*x^2+1)/c^4/d-4/3*b^2*p 
olylog(2,1-2/(-c*x+1))/c^4/d-b*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/c 
^4/d-1/2*b^2*polylog(3,1-2/(c*x+1))/c^4/d
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.05 \[ \int \frac {x^3 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {a^2 x}{c^3 d}-\frac {a^2 x^2}{2 c^2 d}+\frac {a^2 x^3}{3 c d}-\frac {a^2 \log (1+c x)}{c^4 d}+\frac {a b \left (-3 c x+8 c x \text {arctanh}(c x)+\left (1-c^2 x^2\right ) (-1+3 \text {arctanh}(c x)-2 c x \text {arctanh}(c x))+6 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-8 \log \left (\frac {1}{\sqrt {1-c^2 x^2}}\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{3 c^4 d}+\frac {b^2 \left (2 c x-6 c x \text {arctanh}(c x)-2 \left (1-c^2 x^2\right ) \text {arctanh}(c x)-8 \text {arctanh}(c x)^2+8 c x \text {arctanh}(c x)^2+3 \left (1-c^2 x^2\right ) \text {arctanh}(c x)^2-2 c x \left (1-c^2 x^2\right ) \text {arctanh}(c x)^2-16 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+6 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+6 \log \left (\frac {1}{\sqrt {1-c^2 x^2}}\right )+(8-6 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{6 c^4 d} \] Input:

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

Output:

(a^2*x)/(c^3*d) - (a^2*x^2)/(2*c^2*d) + (a^2*x^3)/(3*c*d) - (a^2*Log[1 + c 
*x])/(c^4*d) + (a*b*(-3*c*x + 8*c*x*ArcTanh[c*x] + (1 - c^2*x^2)*(-1 + 3*A 
rcTanh[c*x] - 2*c*x*ArcTanh[c*x]) + 6*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c 
*x])] - 8*Log[1/Sqrt[1 - c^2*x^2]] - 3*PolyLog[2, -E^(-2*ArcTanh[c*x])]))/ 
(3*c^4*d) + (b^2*(2*c*x - 6*c*x*ArcTanh[c*x] - 2*(1 - c^2*x^2)*ArcTanh[c*x 
] - 8*ArcTanh[c*x]^2 + 8*c*x*ArcTanh[c*x]^2 + 3*(1 - c^2*x^2)*ArcTanh[c*x] 
^2 - 2*c*x*(1 - c^2*x^2)*ArcTanh[c*x]^2 - 16*ArcTanh[c*x]*Log[1 + E^(-2*Ar 
cTanh[c*x])] + 6*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 6*Log[1/Sqr 
t[1 - c^2*x^2]] + (8 - 6*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 
3*PolyLog[3, -E^(-2*ArcTanh[c*x])]))/(6*c^4*d)
 

Rubi [A] (verified)

Time = 5.10 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.29, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {6492, 27, 6452, 6492, 6452, 6492, 6436, 6470, 6542, 2009, 6452, 262, 219, 6510, 6546, 6470, 2849, 2752, 6618, 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.

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

\(\Big \downarrow \) 6492

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6492

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6492

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

((x^3*(a + b*ArcTanh[c*x])^2)/3 - (2*b*c*(-(((x^2*(a + b*ArcTanh[c*x]))/2 
- (b*c*(-(x/c^2) + ArcTanh[c*x]/c^3))/2)/c^2) + (-1/2*(a + b*ArcTanh[c*x]) 
^2/(b*c^2) + (((a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c + (b*PolyLog[2, 1 
- 2/(1 - c*x)])/(2*c))/c)/c^2))/3)/(c*d) - (((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*ArcTanh[c*x] + (b*L 
og[1 - c^2*x^2])/(2*c))/c^2))/c - ((x*(a + b*ArcTanh[c*x])^2 - 2*b*c*(-1/2 
*(a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c*x])*Log[2/(1 - c*x)]) 
/c + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c))/c))/c - (-(((a + b*ArcTanh[c*x 
])^2*Log[2/(1 + c*x)])/c) + 2*b*(((a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 
 + c*x)])/(2*c) + (b*PolyLog[3, 1 - 2/(1 + c*x)])/(4*c)))/c)/c)/(c*d)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 262

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

rule 2009

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

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 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 [C] (warning: unable to verify)

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

Time = 5.90 (sec) , antiderivative size = 967, normalized size of antiderivative = 2.94

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(967\)
default	\(\text {Expression too large to display}\)	\(967\)
parts	\(\text {Expression too large to display}\)	\(973\)

Input:

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

Output:

1/c^4*(a^2/d*(1/3*x^3*c^3-1/2*c^2*x^2+c*x-ln(c*x+1))+b^2/d*(-1/2*I*Pi*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)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*arctanh(c*x)^2-1/3-arctanh(c*x)^ 
2*ln(c*x+1)-8/3*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-8/3*arctan 
h(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-8/3*dilog(1+I*(c*x+1)/(-c^2*x^2+ 
1)^(1/2))-8/3*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+1/3*c*x+1/3*arctanh(c* 
x)^2*c^3*x^3-2/3*arctanh(c*x)^3-1/2*arctanh(c*x)^2*c^2*x^2+arctanh(c*x)^2* 
c*x+11/6*arctanh(c*x)^2-1/2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+1/2*I*Pi*cs 
gn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*csgn(I/(1-(c*x+1)^ 
2/(c^2*x^2-1)))*arctanh(c*x)^2-1/2*I*Pi*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)))^2*arctanh(c*x)^2-1/3*( 
c*x+1)*arctanh(c*x)+ln(1+(c*x+1)^2/(-c^2*x^2+1))+arctanh(c*x)*polylog(2,-( 
c*x+1)^2/(-c^2*x^2+1))+1/3*(c*x-3)*(c*x+1)*arctanh(c*x)+1/2*I*Pi*csgn(I*(c 
*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2+1/2*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+1/2*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+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+2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))+ln(2)*arctanh(c*x)^2)+2* 
b*a/d*(1/3*arctanh(c*x)*c^3*x^3-1/2*arctanh(c*x)*c^2*x^2+arctanh(c*x)*c*x- 
arctanh(c*x)*ln(c*x+1)+1/4*ln(c*x+1)^2-1/2*(ln(c*x+1)-ln(1/2*c*x+1/2))*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/6*a^2*((2*c^2*x^3 - 3*c*x^2 + 6*x)/(c^3*d) - 6*log(c*x + 1)/(c^4*d)) + 1 
/24*(2*b^2*c^3*x^3 - 3*b^2*c^2*x^2 + 6*b^2*c*x - 6*b^2*log(c*x + 1))*log(- 
c*x + 1)^2/(c^4*d) - integrate(-1/12*(3*(b^2*c^4*x^4 - b^2*c^3*x^3)*log(c* 
x + 1)^2 + 12*(a*b*c^4*x^4 - a*b*c^3*x^3)*log(c*x + 1) - (3*b^2*c^2*x^2 + 
2*(6*a*b*c^4 + b^2*c^4)*x^4 + 6*b^2*c*x - (12*a*b*c^3 + b^2*c^3)*x^3 + 6*( 
b^2*c^4*x^4 - b^2*c^3*x^3 - b^2*c*x - b^2)*log(c*x + 1))*log(-c*x + 1))/(c 
^5*d*x^2 - c^3*d), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {12 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{3}}{c x +1}d x \right ) a b \,c^{4}+6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{3}}{c x +1}d x \right ) b^{2} c^{4}-6 \,\mathrm {log}\left (c x +1\right ) a^{2}+2 a^{2} c^{3} x^{3}-3 a^{2} c^{2} x^{2}+6 a^{2} c x}{6 c^{4} d} \] Input:

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

Output:

(12*int((atanh(c*x)*x**3)/(c*x + 1),x)*a*b*c**4 + 6*int((atanh(c*x)**2*x** 
3)/(c*x + 1),x)*b**2*c**4 - 6*log(c*x + 1)*a**2 + 2*a**2*c**3*x**3 - 3*a** 
2*c**2*x**2 + 6*a**2*c*x)/(6*c**4*d)