3.1.28 \(\int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^4} \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 23, antiderivative size = 269 \[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^4} \, dx=-\frac {b^2 (a+b \text {arctanh}(c+d x))}{d e^4 (c+d x)}+\frac {b (a+b \text {arctanh}(c+d x))^2}{2 d e^4}-\frac {b (a+b \text {arctanh}(c+d x))^2}{2 d e^4 (c+d x)^2}+\frac {(a+b \text {arctanh}(c+d x))^3}{3 d e^4}-\frac {(a+b \text {arctanh}(c+d x))^3}{3 d e^4 (c+d x)^3}+\frac {b^3 \log (c+d x)}{d e^4}-\frac {b^3 \log \left (1-(c+d x)^2\right )}{2 d e^4}+\frac {b (a+b \text {arctanh}(c+d x))^2 \log \left (2-\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^2 (a+b \text {arctanh}(c+d x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c+d x}\right )}{d e^4}-\frac {b^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c+d x}\right )}{2 d e^4} \]

output

-b^2*(a+b*arctanh(d*x+c))/d/e^4/(d*x+c)+1/2*b*(a+b*arctanh(d*x+c))^2/d/e^4 
-1/2*b*(a+b*arctanh(d*x+c))^2/d/e^4/(d*x+c)^2+1/3*(a+b*arctanh(d*x+c))^3/d 
/e^4-1/3*(a+b*arctanh(d*x+c))^3/d/e^4/(d*x+c)^3+b^3*ln(d*x+c)/d/e^4-1/2*b^ 
3*ln(1-(d*x+c)^2)/d/e^4+b*(a+b*arctanh(d*x+c))^2*ln(2-2/(d*x+c+1))/d/e^4-b 
^2*(a+b*arctanh(d*x+c))*polylog(2,-1+2/(d*x+c+1))/d/e^4-1/2*b^3*polylog(3, 
-1+2/(d*x+c+1))/d/e^4
 

3.1.28.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.34 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {-\frac {2 a^3}{(c+d x)^3}-\frac {3 a^2 b}{(c+d x)^2}-\frac {6 a^2 b \text {arctanh}(c+d x)}{(c+d x)^3}+6 a^2 b \log (c+d x)-3 a^2 b \log \left (1-c^2-2 c d x-d^2 x^2\right )+6 a b^2 \left (-\frac {(c+d x)^2+\text {arctanh}(c+d x)^2}{(c+d x)^3}+\text {arctanh}(c+d x) \left (-\frac {1-(c+d x)^2}{(c+d x)^2}+\text {arctanh}(c+d x)+2 \log \left (1-e^{-2 \text {arctanh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c+d x)}\right )\right )+6 b^3 \left (\frac {i \pi ^3}{24}-\frac {\text {arctanh}(c+d x)}{c+d x}-\frac {\left (1+c^3+3 c^2 d x+3 c d^2 x^2+d^3 x^3\right ) \text {arctanh}(c+d x)^3}{3 (c+d x)^3}+\frac {\text {arctanh}(c+d x)^2 \left (-1+c^2+2 c d x+d^2 x^2+2 (c+d x)^2 \log \left (1-e^{2 \text {arctanh}(c+d x)}\right )\right )}{2 (c+d x)^2}+\log (c+d x)+\log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\text {arctanh}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c+d x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c+d x)}\right )\right )}{6 d e^4} \]

input

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

output

((-2*a^3)/(c + d*x)^3 - (3*a^2*b)/(c + d*x)^2 - (6*a^2*b*ArcTanh[c + d*x]) 
/(c + d*x)^3 + 6*a^2*b*Log[c + d*x] - 3*a^2*b*Log[1 - c^2 - 2*c*d*x - d^2* 
x^2] + 6*a*b^2*(-(((c + d*x)^2 + ArcTanh[c + d*x]^2)/(c + d*x)^3) + ArcTan 
h[c + d*x]*(-((1 - (c + d*x)^2)/(c + d*x)^2) + ArcTanh[c + d*x] + 2*Log[1 
- E^(-2*ArcTanh[c + d*x])]) - PolyLog[2, E^(-2*ArcTanh[c + d*x])]) + 6*b^3 
*((I/24)*Pi^3 - ArcTanh[c + d*x]/(c + d*x) - ((1 + c^3 + 3*c^2*d*x + 3*c*d 
^2*x^2 + d^3*x^3)*ArcTanh[c + d*x]^3)/(3*(c + d*x)^3) + (ArcTanh[c + d*x]^ 
2*(-1 + c^2 + 2*c*d*x + d^2*x^2 + 2*(c + d*x)^2*Log[1 - E^(2*ArcTanh[c + d 
*x])]))/(2*(c + d*x)^2) + Log[c + d*x] + Log[1/Sqrt[1 - (c + d*x)^2]] + Ar 
cTanh[c + d*x]*PolyLog[2, E^(2*ArcTanh[c + d*x])] - PolyLog[3, E^(2*ArcTan 
h[c + d*x])]/2))/(6*d*e^4)
 

3.1.28.3 Rubi [A] (warning: unable to verify)

Time = 1.84 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.82, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {6657, 27, 6452, 6544, 6452, 6544, 6452, 243, 47, 14, 16, 6510, 6550, 6494, 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.

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

\(\Big \downarrow \) 6657

\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^3}{e^4 (c+d x)^4}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^3}{(c+d x)^4}d(c+d x)}{d e^4}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

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

input

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

output

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

3.1.28.3.1 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 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 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 6657

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

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

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

Time = 3.94 (sec) , antiderivative size = 1776, normalized size of antiderivative = 6.60

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1776\)
default	\(\text {Expression too large to display}\)	\(1776\)
parts	\(\text {Expression too large to display}\)	\(1784\)

input

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

output

1/d*(-1/3*a^3/e^4/(d*x+c)^3+b^3/e^4*(-1/3/(d*x+c)^3*arctanh(d*x+c)^3-1/2*a 
rctanh(d*x+c)^2*ln(d*x+c-1)-1/2/(d*x+c)^2*arctanh(d*x+c)^2+ln(d*x+c)*arcta 
nh(d*x+c)^2-1/2*arctanh(d*x+c)^2*ln(d*x+c+1)+arctanh(d*x+c)^2*ln((d*x+c+1) 
/(1-(d*x+c)^2)^(1/2))-arctanh(d*x+c)^2*ln((d*x+c+1)^2/(1-(d*x+c)^2)-1)+arc 
tanh(d*x+c)^2*ln(1-(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+2*arctanh(d*x+c)*polylog 
(2,(d*x+c+1)/(1-(d*x+c)^2)^(1/2))-2*polylog(3,(d*x+c+1)/(1-(d*x+c)^2)^(1/2 
))+arctanh(d*x+c)^2*ln(1+(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+2*arctanh(d*x+c)*p 
olylog(2,-(d*x+c+1)/(1-(d*x+c)^2)^(1/2))-2*polylog(3,-(d*x+c+1)/(1-(d*x+c) 
^2)^(1/2))+1/12*arctanh(d*x+c)*(6*I*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))^ 
3*Pi*arctanh(d*x+c)*(d*x+c)+6*I*csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1))*csg 
n(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1)/( 
1-(d*x+c+1)^2/((d*x+c)^2-1)))*Pi*arctanh(d*x+c)*(d*x+c)-6*I*csgn(I/(1-(d*x 
+c+1)^2/((d*x+c)^2-1)))^2*Pi*arctanh(d*x+c)*(d*x+c)-6*I*csgn(I/(1-(d*x+c+1 
)^2/((d*x+c)^2-1)))*csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1)/(1-(d*x+c+1)^2/( 
(d*x+c)^2-1)))^2*Pi*arctanh(d*x+c)*(d*x+c)+3*I*Pi*arctanh(d*x+c)*csgn(I*(d 
*x+c+1)^2/((d*x+c)^2-1))^3*(d*x+c)+3*I*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1 
-(d*x+c+1)^2/((d*x+c)^2-1)))^3*Pi*arctanh(d*x+c)*(d*x+c)+6*I*csgn(I*(d*x+c 
+1)/(1-(d*x+c)^2)^(1/2))*Pi*arctanh(d*x+c)*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1 
))^2*(d*x+c)+6*I*csgn(I*(-(d*x+c+1)^2/((d*x+c)^2-1)-1)/(1-(d*x+c+1)^2/((d* 
x+c)^2-1)))^3*Pi*arctanh(d*x+c)*(d*x+c)+6*I*Pi*arctanh(d*x+c)*(d*x+c)-3...
 

3.1.28.5 Fricas [F]

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

input

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

output

integral((b^3*arctanh(d*x + c)^3 + 3*a*b^2*arctanh(d*x + c)^2 + 3*a^2*b*ar 
ctanh(d*x + c) + a^3)/(d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 
 4*c^3*d*e^4*x + c^4*e^4), x)
 

3.1.28.6 Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{3}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{3} \operatorname {atanh}^{3}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {3 a^{2} b \operatorname {atanh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]

input

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

output

(Integral(a**3/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 
4*x**4), x) + Integral(b**3*atanh(c + d*x)**3/(c**4 + 4*c**3*d*x + 6*c**2* 
d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(3*a*b**2*atanh(c + d 
*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), 
 x) + Integral(3*a**2*b*atanh(c + d*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x* 
*2 + 4*c*d**3*x**3 + d**4*x**4), x))/e**4
 

3.1.28.7 Maxima [F]

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

input

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

output

-1/2*(d*(1/(d^4*e^4*x^2 + 2*c*d^3*e^4*x + c^2*d^2*e^4) + log(d*x + c + 1)/ 
(d^2*e^4) - 2*log(d*x + c)/(d^2*e^4) + log(d*x + c - 1)/(d^2*e^4)) + 2*arc 
tanh(d*x + c)/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4 
))*a^2*b - 1/3*a^3/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3* 
d*e^4) - 1/24*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + (c^3 - 1)* 
b^3)*log(-d*x - c + 1)^3 + 3*(b^3*d*x + b^3*c + 2*a*b^2 + (b^3*d^3*x^3 + 3 
*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + (c^3 + 1)*b^3)*log(d*x + c + 1))*log(-d*x 
 - c + 1)^2)/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) 
 - integrate(-1/8*((b^3*d*x + b^3*(c - 1))*log(d*x + c + 1)^3 + 6*(a*b^2*d 
*x + a*b^2*(c - 1))*log(d*x + c + 1)^2 + (2*b^3*d^2*x^2 + 2*b^3*c^2 + 4*a* 
b^2*c - 3*(b^3*d*x + b^3*(c - 1))*log(d*x + c + 1)^2 + 4*(b^3*c*d + a*b^2* 
d)*x + 2*(b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + (c^4 + c)*b^ 
3 - 6*a*b^2*(c - 1) + ((4*c^3*d + d)*b^3 - 6*a*b^2*d)*x)*log(d*x + c + 1)) 
*log(-d*x - c + 1))/(d^5*e^4*x^5 + c^5*e^4 - c^4*e^4 + (5*c*d^4*e^4 - d^4* 
e^4)*x^4 + 2*(5*c^2*d^3*e^4 - 2*c*d^3*e^4)*x^3 + 2*(5*c^3*d^2*e^4 - 3*c^2* 
d^2*e^4)*x^2 + (5*c^4*d*e^4 - 4*c^3*d*e^4)*x), x)
 

3.1.28.8 Giac [F]

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

input

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

output

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

3.1.28.9 Mupad [F(-1)]

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

input

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

output

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