\(\int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx\) [126]

Optimal result

Integrand size = 18, antiderivative size = 275 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx=-\frac {b^3}{108 c (1+c x)^3}-\frac {19 b^3}{576 c (1+c x)^2}-\frac {85 b^3}{576 c (1+c x)}+\frac {85 b^3 \text {arctanh}(c x)}{576 c}-\frac {b^2 (a+b \text {arctanh}(c x))}{18 c (1+c x)^3}-\frac {5 b^2 (a+b \text {arctanh}(c x))}{48 c (1+c x)^2}-\frac {11 b^2 (a+b \text {arctanh}(c x))}{48 c (1+c x)}+\frac {11 b (a+b \text {arctanh}(c x))^2}{96 c}-\frac {b (a+b \text {arctanh}(c x))^2}{6 c (1+c x)^3}-\frac {b (a+b \text {arctanh}(c x))^2}{8 c (1+c x)^2}-\frac {b (a+b \text {arctanh}(c x))^2}{8 c (1+c x)}+\frac {(a+b \text {arctanh}(c x))^3}{24 c}-\frac {(a+b \text {arctanh}(c x))^3}{3 c (1+c x)^3} \] Output:

-1/108*b^3/c/(c*x+1)^3-19/576*b^3/c/(c*x+1)^2-85/576*b^3/c/(c*x+1)+85/576* 
b^3*arctanh(c*x)/c-1/18*b^2*(a+b*arctanh(c*x))/c/(c*x+1)^3-5/48*b^2*(a+b*a 
rctanh(c*x))/c/(c*x+1)^2-11/48*b^2*(a+b*arctanh(c*x))/c/(c*x+1)+11/96*b*(a 
+b*arctanh(c*x))^2/c-1/6*b*(a+b*arctanh(c*x))^2/c/(c*x+1)^3-1/8*b*(a+b*arc 
tanh(c*x))^2/c/(c*x+1)^2-1/8*b*(a+b*arctanh(c*x))^2/c/(c*x+1)+1/24*(a+b*ar 
ctanh(c*x))^3/c-1/3*(a+b*arctanh(c*x))^3/c/(c*x+1)^3
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx=-\frac {32 \left (36 a^3+18 a^2 b+6 a b^2+b^3\right )+6 b \left (72 a^2+60 a b+19 b^2\right ) (1+c x)+6 b \left (72 a^2+132 a b+85 b^2\right ) (1+c x)^2+24 b \left (144 a^2+12 a b \left (10+9 c x+3 c^2 x^2\right )+b^2 \left (56+81 c x+33 c^2 x^2\right )\right ) \text {arctanh}(c x)-36 b^2 (-1+c x) \left (12 a \left (7+4 c x+c^2 x^2\right )+b \left (29+32 c x+11 c^2 x^2\right )\right ) \text {arctanh}(c x)^2-144 b^3 \left (-7+3 c x+3 c^2 x^2+c^3 x^3\right ) \text {arctanh}(c x)^3+3 b \left (72 a^2+132 a b+85 b^2\right ) (1+c x)^3 \log (1-c x)-3 b \left (72 a^2+132 a b+85 b^2\right ) (1+c x)^3 \log (1+c x)}{3456 c (1+c x)^3} \] Input:

Integrate[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]
 

Output:

-1/3456*(32*(36*a^3 + 18*a^2*b + 6*a*b^2 + b^3) + 6*b*(72*a^2 + 60*a*b + 1 
9*b^2)*(1 + c*x) + 6*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^2 + 24*b*(144 
*a^2 + 12*a*b*(10 + 9*c*x + 3*c^2*x^2) + b^2*(56 + 81*c*x + 33*c^2*x^2))*A 
rcTanh[c*x] - 36*b^2*(-1 + c*x)*(12*a*(7 + 4*c*x + c^2*x^2) + b*(29 + 32*c 
*x + 11*c^2*x^2))*ArcTanh[c*x]^2 - 144*b^3*(-7 + 3*c*x + 3*c^2*x^2 + c^3*x 
^3)*ArcTanh[c*x]^3 + 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 - c 
*x] - 3*b*(72*a^2 + 132*a*b + 85*b^2)*(1 + c*x)^3*Log[1 + c*x])/(c*(1 + c* 
x)^3)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6480, 2009}

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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{(c x+1)^4} \, dx\)

\(\Big \downarrow \) 6480

\(\displaystyle b \int \left (\frac {(a+b \text {arctanh}(c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {(a+b \text {arctanh}(c x))^2}{8 (c x+1)^2}+\frac {(a+b \text {arctanh}(c x))^2}{4 (c x+1)^3}+\frac {(a+b \text {arctanh}(c x))^2}{2 (c x+1)^4}\right )dx-\frac {(a+b \text {arctanh}(c x))^3}{3 c (c x+1)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle b \left (\frac {(a+b \text {arctanh}(c x))^3}{24 b c}+\frac {11 (a+b \text {arctanh}(c x))^2}{96 c}-\frac {(a+b \text {arctanh}(c x))^2}{8 c (c x+1)}-\frac {(a+b \text {arctanh}(c x))^2}{8 c (c x+1)^2}-\frac {(a+b \text {arctanh}(c x))^2}{6 c (c x+1)^3}-\frac {11 b (a+b \text {arctanh}(c x))}{48 c (c x+1)}-\frac {5 b (a+b \text {arctanh}(c x))}{48 c (c x+1)^2}-\frac {b (a+b \text {arctanh}(c x))}{18 c (c x+1)^3}+\frac {85 b^2 \text {arctanh}(c x)}{576 c}-\frac {85 b^2}{576 c (c x+1)}-\frac {19 b^2}{576 c (c x+1)^2}-\frac {b^2}{108 c (c x+1)^3}\right )-\frac {(a+b \text {arctanh}(c x))^3}{3 c (c x+1)^3}\)

Input:

Int[(a + b*ArcTanh[c*x])^3/(1 + c*x)^4,x]
 

Output:

-1/3*(a + b*ArcTanh[c*x])^3/(c*(1 + c*x)^3) + b*(-1/108*b^2/(c*(1 + c*x)^3 
) - (19*b^2)/(576*c*(1 + c*x)^2) - (85*b^2)/(576*c*(1 + c*x)) + (85*b^2*Ar 
cTanh[c*x])/(576*c) - (b*(a + b*ArcTanh[c*x]))/(18*c*(1 + c*x)^3) - (5*b*( 
a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)^2) - (11*b*(a + b*ArcTanh[c*x]))/(48* 
c*(1 + c*x)) + (11*(a + b*ArcTanh[c*x])^2)/(96*c) - (a + b*ArcTanh[c*x])^2 
/(6*c*(1 + c*x)^3) - (a + b*ArcTanh[c*x])^2/(8*c*(1 + c*x)^2) - (a + b*Arc 
Tanh[c*x])^2/(8*c*(1 + c*x)) + (a + b*ArcTanh[c*x])^3/(24*b*c))
 

Defintions of rubi rules used

rule 2009

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

rule 6480

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S 
ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - 
 Simp[b*c*(p/(e*(q + 1)))   Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 
), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] 
 && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
 

Maple [A] (verified)

Time = 1.57 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.72

method	result	size
parallelrisch	\(-\frac {-720 a^{2} b \,x^{3} c^{3}-1944 a^{2} b \,c^{2} x^{2}-1512 a^{2} b c x +54 b^{3} \operatorname {arctanh}\left (c x \right )^{2} c x -378 b^{3} \operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}-198 b^{3} \operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}-729 b^{3} c^{2} x^{2}-576 a^{3} c^{3} x^{3}-1728 a^{3} c^{2} x^{2}-1728 a^{3} c x +1512 \,\operatorname {arctanh}\left (c x \right ) a^{2} b -369 \,\operatorname {arctanh}\left (c x \right ) b^{3} c^{2} x^{2}+207 \,\operatorname {arctanh}\left (c x \right ) b^{3} c x -417 b^{3} c x -1620 a \,b^{2} c^{2} x^{2}-1044 a \,b^{2} c x +504 \operatorname {arctanh}\left (c x \right )^{3} b^{3}+522 b^{3} \operatorname {arctanh}\left (c x \right )^{2}+417 \,\operatorname {arctanh}\left (c x \right ) b^{3}-328 b^{3} c^{3} x^{3}+1512 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2}-672 a \,b^{2} c^{3} x^{3}-216 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2} c^{3} x^{3}-216 \,\operatorname {arctanh}\left (c x \right ) a^{2} b \,c^{3} x^{3}-648 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2} c^{2} x^{2}-648 \,\operatorname {arctanh}\left (c x \right ) a^{2} b \,c^{2} x^{2}-648 \operatorname {arctanh}\left (c x \right )^{2} a \,b^{2} c x -648 \,\operatorname {arctanh}\left (c x \right ) a^{2} b c x -396 a \,b^{2} \operatorname {arctanh}\left (c x \right ) c^{3} x^{3}-756 a \,b^{2} \operatorname {arctanh}\left (c x \right ) c^{2} x^{2}+108 a \,b^{2} \operatorname {arctanh}\left (c x \right ) c x -255 x^{3} \operatorname {arctanh}\left (c x \right ) b^{3} c^{3}+1044 a \,b^{2} \operatorname {arctanh}\left (c x \right )-72 \operatorname {arctanh}\left (c x \right )^{3} b^{3} c^{3} x^{3}-216 \operatorname {arctanh}\left (c x \right )^{3} b^{3} c^{2} x^{2}-216 \operatorname {arctanh}\left (c x \right )^{3} b^{3} c x}{1728 \left (c x +1\right )^{3} c}\)	\(472\)
orering	\(-\frac {\left (9840 c^{6} x^{6}+915 c^{5} x^{5}-19487 c^{4} x^{4}+1928 x^{3} c^{3}+9396 c^{2} x^{2}-5643 c x +3051\right ) \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3}}{2592 \left (c x +1\right )^{3} c}-\frac {\left (c x -1\right ) \left (c x +1\right )^{2} \left (7380 c^{5} x^{5}+5925 c^{4} x^{4}-10013 x^{3} c^{3}-4869 c^{2} x^{2}+4146 c x -1095\right ) \left (\frac {3 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b c}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )}-\frac {4 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3} c}{\left (c x +1\right )^{5}}\right )}{2592 c^{2}}-\frac {\left (c x +1\right )^{3} \left (c x -1\right )^{2} \left (1968 c^{4} x^{4}+2977 x^{3} c^{3}-537 c^{2} x^{2}-1599 c x +139\right ) \left (\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b^{2} c^{2}}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{2}}-\frac {24 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{2}}{\left (c x +1\right )^{5} \left (-c^{2} x^{2}+1\right )}+\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{3} x}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {20 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3} c^{2}}{\left (c x +1\right )^{6}}\right )}{3456 c^{3}}-\frac {x \left (328 c^{2} x^{2}+729 c x +417\right ) \left (c x +1\right )^{4} \left (c x -1\right )^{3} \left (\frac {6 b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{3} \left (c x +1\right )^{4}}-\frac {72 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b^{2} c^{3}}{\left (c x +1\right )^{5} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {36 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b^{2} c^{4} x}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{3}}+\frac {180 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{3}}{\left (c x +1\right )^{6} \left (-c^{2} x^{2}+1\right )}-\frac {72 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{4} x}{\left (c x +1\right )^{5} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {24 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{3}}+\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} b \,c^{3}}{\left (c x +1\right )^{4} \left (-c^{2} x^{2}+1\right )^{2}}-\frac {120 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{3} c^{3}}{\left (c x +1\right )^{7}}\right )}{10368 c^{3}}\)	\(656\)
derivativedivides	\(\text {Expression too large to display}\)	\(1073\)
default	\(\text {Expression too large to display}\)	\(1073\)
parts	\(\text {Expression too large to display}\)	\(1081\)
risch	\(\text {Expression too large to display}\)	\(1221\)

Input:

int((a+b*arctanh(c*x))^3/(c*x+1)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/1728*(-720*a^2*b*x^3*c^3-1944*a^2*b*c^2*x^2-1512*a^2*b*c*x+54*b^3*arcta 
nh(c*x)^2*c*x-378*b^3*arctanh(c*x)^2*c^2*x^2-198*b^3*arctanh(c*x)^2*c^3*x^ 
3-729*b^3*c^2*x^2-576*a^3*c^3*x^3-1728*a^3*c^2*x^2-1728*a^3*c*x+1512*arcta 
nh(c*x)*a^2*b-369*arctanh(c*x)*b^3*c^2*x^2+207*arctanh(c*x)*b^3*c*x-417*b^ 
3*c*x-1620*a*b^2*c^2*x^2-1044*a*b^2*c*x+504*arctanh(c*x)^3*b^3+522*b^3*arc 
tanh(c*x)^2+417*arctanh(c*x)*b^3-328*b^3*c^3*x^3+1512*arctanh(c*x)^2*a*b^2 
-672*a*b^2*c^3*x^3-216*arctanh(c*x)^2*a*b^2*c^3*x^3-216*arctanh(c*x)*a^2*b 
*c^3*x^3-648*arctanh(c*x)^2*a*b^2*c^2*x^2-648*arctanh(c*x)*a^2*b*c^2*x^2-6 
48*arctanh(c*x)^2*a*b^2*c*x-648*arctanh(c*x)*a^2*b*c*x-396*a*b^2*arctanh(c 
*x)*c^3*x^3-756*a*b^2*arctanh(c*x)*c^2*x^2+108*a*b^2*arctanh(c*x)*c*x-255* 
x^3*arctanh(c*x)*b^3*c^3+1044*a*b^2*arctanh(c*x)-72*arctanh(c*x)^3*b^3*c^3 
*x^3-216*arctanh(c*x)^3*b^3*c^2*x^2-216*arctanh(c*x)^3*b^3*c*x)/(c*x+1)^3/ 
c
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx=-\frac {6 \, {\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{2} x^{2} - 18 \, {\left (b^{3} c^{3} x^{3} + 3 \, b^{3} c^{2} x^{2} + 3 \, b^{3} c x - 7 \, b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} + 1152 \, a^{3} + 1440 \, a^{2} b + 1344 \, a b^{2} + 656 \, b^{3} + 162 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 7 \, b^{3}\right )} c x - 9 \, {\left ({\left (12 \, a b^{2} + 11 \, b^{3}\right )} c^{3} x^{3} + 3 \, {\left (12 \, a b^{2} + 7 \, b^{3}\right )} c^{2} x^{2} - 84 \, a b^{2} - 29 \, b^{3} + 3 \, {\left (12 \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 3 \, {\left ({\left (72 \, a^{2} b + 132 \, a b^{2} + 85 \, b^{3}\right )} c^{3} x^{3} + 3 \, {\left (72 \, a^{2} b + 84 \, a b^{2} + 41 \, b^{3}\right )} c^{2} x^{2} - 504 \, a^{2} b - 348 \, a b^{2} - 139 \, b^{3} + 3 \, {\left (72 \, a^{2} b - 12 \, a b^{2} - 23 \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{3456 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \] Input:

integrate((a+b*arctanh(c*x))^3/(c*x+1)^4,x, algorithm="fricas")
 

Output:

-1/3456*(6*(72*a^2*b + 132*a*b^2 + 85*b^3)*c^2*x^2 - 18*(b^3*c^3*x^3 + 3*b 
^3*c^2*x^2 + 3*b^3*c*x - 7*b^3)*log(-(c*x + 1)/(c*x - 1))^3 + 1152*a^3 + 1 
440*a^2*b + 1344*a*b^2 + 656*b^3 + 162*(8*a^2*b + 12*a*b^2 + 7*b^3)*c*x - 
9*((12*a*b^2 + 11*b^3)*c^3*x^3 + 3*(12*a*b^2 + 7*b^3)*c^2*x^2 - 84*a*b^2 - 
 29*b^3 + 3*(12*a*b^2 - b^3)*c*x)*log(-(c*x + 1)/(c*x - 1))^2 - 3*((72*a^2 
*b + 132*a*b^2 + 85*b^3)*c^3*x^3 + 3*(72*a^2*b + 84*a*b^2 + 41*b^3)*c^2*x^ 
2 - 504*a^2*b - 348*a*b^2 - 139*b^3 + 3*(72*a^2*b - 12*a*b^2 - 23*b^3)*c*x 
)*log(-(c*x + 1)/(c*x - 1)))/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c)
 

Sympy [F]

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

integrate((a+b*atanh(c*x))**3/(c*x+1)**4,x)
 

Output:

Integral((a + b*atanh(c*x))**3/(c*x + 1)**4, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (249) = 498\).

Time = 0.07 (sec) , antiderivative size = 1085, normalized size of antiderivative = 3.95 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arctanh(c*x))^3/(c*x+1)^4,x, algorithm="maxima")
 

Output:

-1/3*b^3*arctanh(c*x)^3/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c) - 1/48*(c*(2*( 
3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x 
+ 1)/c^2 + 3*log(c*x - 1)/c^2) + 48*arctanh(c*x)/(c^4*x^3 + 3*c^3*x^2 + 3* 
c^2*x + c))*a^2*b - 1/288*(12*c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c 
^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2)*arctanh 
(c*x) + (66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)^2 + 
 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 162*c*x - 3*(11*c^3* 
x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 
1) + 11)*log(c*x + 1) + 33*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) 
+ 112)*c^2/(c^6*x^3 + 3*c^5*x^2 + 3*c^4*x + c^3))*a*b^2 - 1/3456*(72*c*(2* 
(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2) - 3*log(c*x 
 + 1)/c^2 + 3*log(c*x - 1)/c^2)*arctanh(c*x)^2 + ((510*c^2*x^2 - 18*(c^3*x 
^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x + 1)^3 + 18*(c^3*x^3 + 3*c^2*x^2 + 3*c 
*x + 1)*log(c*x - 1)^3 + 9*(11*c^3*x^3 + 33*c^2*x^2 + 33*c*x + 6*(c^3*x^3 
+ 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 11)*log(c*x + 1)^2 + 99*(c^3*x^3 + 
 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 1134*c*x - 3*(85*c^3*x^3 + 255*c^ 
2*x^2 + 18*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1)^2 + 255*c*x + 66 
*(c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 85)*log(c*x + 1) + 255*( 
c^3*x^3 + 3*c^2*x^2 + 3*c*x + 1)*log(c*x - 1) + 656)*c^2/(c^7*x^3 + 3*c^6* 
x^2 + 3*c^5*x + c^4) + 12*(66*c^2*x^2 + 9*(c^3*x^3 + 3*c^2*x^2 + 3*c*x ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (249) = 498\).

Time = 0.14 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx=\frac {1}{6912} \, {\left (\frac {36 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {3 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {18 \, {\left (\frac {36 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {36 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 12 \, a b^{2} + \frac {18 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {9 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 2 \, b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {6 \, {\left (\frac {216 \, {\left (c x + 1\right )}^{2} a^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {216 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} + 72 \, a^{2} b + \frac {216 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {108 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 24 \, a b^{2} + \frac {108 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {27 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 4 \, b^{3}\right )} {\left (c x - 1\right )}^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {{\left (\frac {864 \, {\left (c x + 1\right )}^{2} a^{3}}{{\left (c x - 1\right )}^{2}} - \frac {864 \, {\left (c x + 1\right )} a^{3}}{c x - 1} + 288 \, a^{3} + \frac {1296 \, {\left (c x + 1\right )}^{2} a^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {648 \, {\left (c x + 1\right )} a^{2} b}{c x - 1} + 144 \, a^{2} b + \frac {1296 \, {\left (c x + 1\right )}^{2} a b^{2}}{{\left (c x - 1\right )}^{2}} - \frac {324 \, {\left (c x + 1\right )} a b^{2}}{c x - 1} + 48 \, a b^{2} + \frac {648 \, {\left (c x + 1\right )}^{2} b^{3}}{{\left (c x - 1\right )}^{2}} - \frac {81 \, {\left (c x + 1\right )} b^{3}}{c x - 1} + 8 \, b^{3}\right )} {\left (c x - 1\right )}^{3}}{{\left (c x + 1\right )}^{3} c^{2}}\right )} c \] Input:

integrate((a+b*arctanh(c*x))^3/(c*x+1)^4,x, algorithm="giac")
 

Output:

1/6912*(36*(3*(c*x + 1)^2*b^3/(c*x - 1)^2 - 3*(c*x + 1)*b^3/(c*x - 1) + b^ 
3)*(c*x - 1)^3*log(-(c*x + 1)/(c*x - 1))^3/((c*x + 1)^3*c^2) + 18*(36*(c*x 
 + 1)^2*a*b^2/(c*x - 1)^2 - 36*(c*x + 1)*a*b^2/(c*x - 1) + 12*a*b^2 + 18*( 
c*x + 1)^2*b^3/(c*x - 1)^2 - 9*(c*x + 1)*b^3/(c*x - 1) + 2*b^3)*(c*x - 1)^ 
3*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1)^3*c^2) + 6*(216*(c*x + 1)^2*a^2*b 
/(c*x - 1)^2 - 216*(c*x + 1)*a^2*b/(c*x - 1) + 72*a^2*b + 216*(c*x + 1)^2* 
a*b^2/(c*x - 1)^2 - 108*(c*x + 1)*a*b^2/(c*x - 1) + 24*a*b^2 + 108*(c*x + 
1)^2*b^3/(c*x - 1)^2 - 27*(c*x + 1)*b^3/(c*x - 1) + 4*b^3)*(c*x - 1)^3*log 
(-(c*x + 1)/(c*x - 1))/((c*x + 1)^3*c^2) + (864*(c*x + 1)^2*a^3/(c*x - 1)^ 
2 - 864*(c*x + 1)*a^3/(c*x - 1) + 288*a^3 + 1296*(c*x + 1)^2*a^2*b/(c*x - 
1)^2 - 648*(c*x + 1)*a^2*b/(c*x - 1) + 144*a^2*b + 1296*(c*x + 1)^2*a*b^2/ 
(c*x - 1)^2 - 324*(c*x + 1)*a*b^2/(c*x - 1) + 48*a*b^2 + 648*(c*x + 1)^2*b 
^3/(c*x - 1)^2 - 81*(c*x + 1)*b^3/(c*x - 1) + 8*b^3)*(c*x - 1)^3/((c*x + 1 
)^3*c^2))*c
 

Mupad [B] (verification not implemented)

Time = 7.11 (sec) , antiderivative size = 1304, normalized size of antiderivative = 4.74 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx=\text {Too large to display} \] Input:

int((a + b*atanh(c*x))^3/(c*x + 1)^4,x)
 

Output:

(1398*b^3*log(1 - c*x) - 1398*b^3*log(c*x + 1) - 1344*a*b^2 - 1440*a^2*b - 
 261*b^3*log(c*x + 1)^2 - 126*b^3*log(c*x + 1)^3 - 261*b^3*log(1 - c*x)^2 
+ 126*b^3*log(1 - c*x)^3 + 1962*b^3*atanh(c*x) - 1152*a^3 - 656*b^3 + 1584 
*a*b^2*atanh(c*x) + 432*a^2*b*atanh(c*x) + 522*b^3*log(c*x + 1)*log(1 - c* 
x) - 1836*a*b^2*log(c*x + 1) - 1728*a^2*b*log(c*x + 1) + 1836*a*b^2*log(1 
- c*x) + 1728*a^2*b*log(1 - c*x) - 378*b^3*log(c*x + 1)*log(1 - c*x)^2 + 3 
78*b^3*log(c*x + 1)^2*log(1 - c*x) - 510*b^3*c^2*x^2 - 756*a*b^2*log(c*x + 
 1)^2 - 756*a*b^2*log(1 - c*x)^2 - 1134*b^3*c*x - 3150*b^3*c*x*log(c*x + 1 
) + 3150*b^3*c*x*log(1 - c*x) - 792*a*b^2*c^2*x^2 - 432*a^2*b*c^2*x^2 + 18 
9*b^3*c^2*x^2*log(c*x + 1)^2 + 54*b^3*c^2*x^2*log(c*x + 1)^3 + 189*b^3*c^2 
*x^2*log(1 - c*x)^2 - 54*b^3*c^2*x^2*log(1 - c*x)^3 + 99*b^3*c^3*x^3*log(c 
*x + 1)^2 + 18*b^3*c^3*x^3*log(c*x + 1)^3 + 99*b^3*c^3*x^3*log(1 - c*x)^2 
- 18*b^3*c^3*x^3*log(1 - c*x)^3 + 5886*b^3*c^2*x^2*atanh(c*x) + 1962*b^3*c 
^3*x^3*atanh(c*x) - 1944*a*b^2*c*x - 1296*a^2*b*c*x - 27*b^3*c*x*log(c*x + 
 1)^2 + 54*b^3*c*x*log(c*x + 1)^3 - 27*b^3*c*x*log(1 - c*x)^2 - 54*b^3*c*x 
*log(1 - c*x)^3 + 1512*a*b^2*log(c*x + 1)*log(1 - c*x) + 5886*b^3*c*x*atan 
h(c*x) - 2574*b^3*c^2*x^2*log(c*x + 1) + 2574*b^3*c^2*x^2*log(1 - c*x) - 7 
26*b^3*c^3*x^3*log(c*x + 1) + 726*b^3*c^3*x^3*log(1 - c*x) + 54*b^3*c*x*lo 
g(c*x + 1)*log(1 - c*x) - 1620*a*b^2*c^2*x^2*log(c*x + 1) + 1620*a*b^2*c^2 
*x^2*log(1 - c*x) - 396*a*b^2*c^3*x^3*log(c*x + 1) + 396*a*b^2*c^3*x^3*...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.67 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(1+c x)^4} \, dx =\text {Too large to display} \] Input:

int((a+b*atanh(c*x))^3/(c*x+1)^4,x)
                                                                                    
                                                                                    
 

Output:

(144*atanh(c*x)**3*b**3*c**3*x**3 + 432*atanh(c*x)**3*b**3*c**2*x**2 + 432 
*atanh(c*x)**3*b**3*c*x - 1008*atanh(c*x)**3*b**3 + 432*atanh(c*x)**2*a*b* 
*2*c**3*x**3 + 1296*atanh(c*x)**2*a*b**2*c**2*x**2 + 1296*atanh(c*x)**2*a* 
b**2*c*x - 3024*atanh(c*x)**2*a*b**2 + 396*atanh(c*x)**2*b**3*c**3*x**3 + 
756*atanh(c*x)**2*b**3*c**2*x**2 - 108*atanh(c*x)**2*b**3*c*x - 1044*atanh 
(c*x)**2*b**3 - 3456*atanh(c*x)*a**2*b + 288*atanh(c*x)*a*b**2*c**3*x**3 - 
 1728*atanh(c*x)*a*b**2*c*x - 2592*atanh(c*x)*a*b**2 + 264*atanh(c*x)*b**3 
*c**3*x**3 - 1152*atanh(c*x)*b**3*c*x - 1080*atanh(c*x)*b**3 - 216*log(c*x 
 - 1)*a**2*b*c**3*x**3 - 648*log(c*x - 1)*a**2*b*c**2*x**2 - 648*log(c*x - 
 1)*a**2*b*c*x - 216*log(c*x - 1)*a**2*b - 252*log(c*x - 1)*a*b**2*c**3*x* 
*3 - 756*log(c*x - 1)*a*b**2*c**2*x**2 - 756*log(c*x - 1)*a*b**2*c*x - 252 
*log(c*x - 1)*a*b**2 - 123*log(c*x - 1)*b**3*c**3*x**3 - 369*log(c*x - 1)* 
b**3*c**2*x**2 - 369*log(c*x - 1)*b**3*c*x - 123*log(c*x - 1)*b**3 + 216*l 
og(c*x + 1)*a**2*b*c**3*x**3 + 648*log(c*x + 1)*a**2*b*c**2*x**2 + 648*log 
(c*x + 1)*a**2*b*c*x + 216*log(c*x + 1)*a**2*b + 252*log(c*x + 1)*a*b**2*c 
**3*x**3 + 756*log(c*x + 1)*a*b**2*c**2*x**2 + 756*log(c*x + 1)*a*b**2*c*x 
 + 252*log(c*x + 1)*a*b**2 + 123*log(c*x + 1)*b**3*c**3*x**3 + 369*log(c*x 
 + 1)*b**3*c**2*x**2 + 369*log(c*x + 1)*b**3*c*x + 123*log(c*x + 1)*b**3 - 
 1152*a**3 + 144*a**2*b*c**3*x**3 - 864*a**2*b*c*x - 1296*a**2*b + 264*a*b 
**2*c**3*x**3 - 1152*a*b**2*c*x - 1080*a*b**2 + 170*b**3*c**3*x**3 - 62...