\(\int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)} \, dx\) [101]

Optimal result

Integrand size = 22, antiderivative size = 250 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)} \, dx=-\frac {b c (a+b \text {arctanh}(c x))}{d x}-\frac {c^2 (a+b \text {arctanh}(c x))^2}{2 d}-\frac {(a+b \text {arctanh}(c x))^2}{2 d x^2}+\frac {c (a+b \text {arctanh}(c x))^2}{d x}+\frac {b^2 c^2 \log (x)}{d}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}-\frac {2 b c^2 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {c^2 (a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b^2 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d}-\frac {b c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x}\right )}{2 d} \] Output:

-b*c*(a+b*arctanh(c*x))/d/x-1/2*c^2*(a+b*arctanh(c*x))^2/d-1/2*(a+b*arctan 
h(c*x))^2/d/x^2+c*(a+b*arctanh(c*x))^2/d/x+b^2*c^2*ln(x)/d-1/2*b^2*c^2*ln( 
-c^2*x^2+1)/d-2*b*c^2*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))/d+c^2*(a+b*arctan 
h(c*x))^2*ln(2-2/(c*x+1))/d+b^2*c^2*polylog(2,-1+2/(c*x+1))/d-b*c^2*(a+b*a 
rctanh(c*x))*polylog(2,-1+2/(c*x+1))/d-1/2*b^2*c^2*polylog(3,-1+2/(c*x+1)) 
/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)} \, dx=\frac {-\frac {a^2}{x^2}+\frac {2 a^2 c}{x}+2 a^2 c^2 \log (x)-2 a^2 c^2 \log (1+c x)+\frac {2 a b \left (\text {arctanh}(c x) \left (-1+2 c x+c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-c x \left (1+2 c x \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )-c^2 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{x^2}+2 b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {\text {arctanh}(c x)}{c x}-\frac {1}{2} \text {arctanh}(c x)^2-\frac {\text {arctanh}(c x)^2}{2 c^2 x^2}+\frac {\text {arctanh}(c x)^2}{c x}-\frac {2}{3} \text {arctanh}(c x)^3-2 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{2 d} \] Input:

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

Output:

(-(a^2/x^2) + (2*a^2*c)/x + 2*a^2*c^2*Log[x] - 2*a^2*c^2*Log[1 + c*x] + (2 
*a*b*(ArcTanh[c*x]*(-1 + 2*c*x + c^2*x^2 + 2*c^2*x^2*Log[1 - E^(-2*ArcTanh 
[c*x])]) - c*x*(1 + 2*c*x*Log[(c*x)/Sqrt[1 - c^2*x^2]]) - c^2*x^2*PolyLog[ 
2, E^(-2*ArcTanh[c*x])]))/x^2 + 2*b^2*c^2*((I/24)*Pi^3 - ArcTanh[c*x]/(c*x 
) - ArcTanh[c*x]^2/2 - ArcTanh[c*x]^2/(2*c^2*x^2) + ArcTanh[c*x]^2/(c*x) - 
 (2*ArcTanh[c*x]^3)/3 - 2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + ArcT 
anh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + Log[(c*x)/Sqrt[1 - c^2*x^2]] + Po 
lyLog[2, E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x]) 
] - PolyLog[3, E^(2*ArcTanh[c*x])]/2))/(2*d)
 

Rubi [A] (verified)

Time = 3.92 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {6496, 27, 6452, 6496, 6452, 6494, 6544, 6452, 243, 47, 14, 16, 6510, 6550, 6494, 2897, 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.

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

\(\Big \downarrow \) 6496

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6496

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

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

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 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, 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 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 6496

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

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 6544

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

rule 6550

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ 
d   Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && GtQ[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 = 5.06 (sec) , antiderivative size = 1492, normalized size of antiderivative = 5.97

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1492\)
default	\(\text {Expression too large to display}\)	\(1492\)
parts	\(\text {Expression too large to display}\)	\(1494\)

Input:

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

Output:

c^2*(a^2/d*(-1/2/c^2/x^2+ln(c*x)+1/c/x-ln(c*x+1))+b^2/d*(arctanh(c*x)^2*ln 
(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+ 
1)^(1/2))+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*p 
olylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^ 
2+1)-1)-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+1/2*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)-1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*arcta 
nh(c*x)^2+arctanh(c*x)^2*ln(c*x)-arctanh(c*x)^2*ln(c*x+1)-2*arctanh(c*x)*l 
n(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+ln((c*x+1)/(-c^2*x^2+1)^(1/2)-1)-1/2*arcta 
nh(c*x)^2/c^2/x^2-2/3*arctanh(c*x)^3+2*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))-2 
*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+3/2*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)))^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+ln(1+(c*x+1) 
/(-c^2*x^2+1)^(1/2))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3, 
(c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)^2/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+...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (d+c d x)} \, dx=\frac {\int \frac {a^{2}}{c x^{4} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx}{d} \] Input:

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

Output:

(Integral(a**2/(c*x**4 + x**3), x) + Integral(b**2*atanh(c*x)**2/(c*x**4 + 
 x**3), x) + Integral(2*a*b*atanh(c*x)/(c*x**4 + x**3), x))/d
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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