\(\int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx\) [22]

Optimal result

Integrand size = 23, antiderivative size = 172 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b (a+b \text {arctanh}(c+d x))}{6 d e^5 (c+d x)^3}-\frac {b (a+b \text {arctanh}(c+d x))}{2 d e^5 (c+d x)}+\frac {(a+b \text {arctanh}(c+d x))^2}{4 d e^5}-\frac {(a+b \text {arctanh}(c+d x))^2}{4 d e^5 (c+d x)^4}+\frac {2 b^2 \log (c+d x)}{3 d e^5}-\frac {b^2 \log \left (1-(c+d x)^2\right )}{3 d e^5} \] Output:

-1/12*b^2/d/e^5/(d*x+c)^2-1/6*b*(a+b*arctanh(d*x+c))/d/e^5/(d*x+c)^3-1/2*b 
*(a+b*arctanh(d*x+c))/d/e^5/(d*x+c)+1/4*(a+b*arctanh(d*x+c))^2/d/e^5-1/4*( 
a+b*arctanh(d*x+c))^2/d/e^5/(d*x+c)^4+2/3*b^2*ln(d*x+c)/d/e^5-1/3*b^2*ln(1 
-(d*x+c)^2)/d/e^5
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {\frac {3 a^2}{(c+d x)^4}+\frac {2 a b}{(c+d x)^3}+\frac {b^2}{(c+d x)^2}+\frac {6 a b}{c+d x}+\frac {2 b \left (3 a+b \left (c+3 c^3+d x+9 c^2 d x+9 c d^2 x^2+3 d^3 x^3\right )\right ) \text {arctanh}(c+d x)}{(c+d x)^4}-\frac {3 b^2 \left (-1+c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right ) \text {arctanh}(c+d x)^2}{(c+d x)^4}+b (3 a+4 b) \log (1-c-d x)-8 b^2 \log (c+d x)-(3 a-4 b) b \log (1+c+d x)}{12 d e^5} \] Input:

Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^5,x]
 

Output:

-1/12*((3*a^2)/(c + d*x)^4 + (2*a*b)/(c + d*x)^3 + b^2/(c + d*x)^2 + (6*a* 
b)/(c + d*x) + (2*b*(3*a + b*(c + 3*c^3 + d*x + 9*c^2*d*x + 9*c*d^2*x^2 + 
3*d^3*x^3))*ArcTanh[c + d*x])/(c + d*x)^4 - (3*b^2*(-1 + c^4 + 4*c^3*d*x + 
 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4)*ArcTanh[c + d*x]^2)/(c + d*x)^4 + 
b*(3*a + 4*b)*Log[1 - c - d*x] - 8*b^2*Log[c + d*x] - (3*a - 4*b)*b*Log[1 
+ c + d*x])/(d*e^5)
 

Rubi [A] (warning: unable to verify)

Time = 1.02 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6657, 27, 6452, 6544, 6452, 243, 54, 2009, 6544, 6452, 243, 47, 14, 16, 6510}

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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx\)

\(\Big \downarrow \) 6657

\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{e^5 (c+d x)^5}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{(c+d x)^5}d(c+d x)}{d e^5}\)

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 6510

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

Input:

Int[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^5,x]
 

Output:

(-1/4*(a + b*ArcTanh[c + d*x])^2/(c + d*x)^4 + (b*(-1/3*(a + b*ArcTanh[c + 
 d*x])/(c + d*x)^3 - (a + b*ArcTanh[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 + (b*(-(c 
+ d*x)^(-2) - Log[1 - c - d*x] + Log[(c + d*x)^2]))/6))/2)/(d*e^5)
 

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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

rule 6452

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

rule 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 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]
 

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right )}{6 \left (d x +c \right )^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )^{2}}{16}-\frac {\ln \left (d x +c +1\right )^{2}}{16}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )}{3}-\frac {1}{12 \left (d x +c \right )^{2}}+\frac {2 \ln \left (d x +c \right )}{3}-\frac {\ln \left (d x +c +1\right )}{3}\right )}{e^{5}}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\ln \left (d x +c -1\right )}{8}-\frac {1}{12 \left (d x +c \right )^{3}}-\frac {1}{4 \left (d x +c \right )}+\frac {\ln \left (d x +c +1\right )}{8}\right )}{e^{5}}}{d}\)	\(273\)
default	\(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right )}{6 \left (d x +c \right )^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )^{2}}{16}-\frac {\ln \left (d x +c +1\right )^{2}}{16}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )}{3}-\frac {1}{12 \left (d x +c \right )^{2}}+\frac {2 \ln \left (d x +c \right )}{3}-\frac {\ln \left (d x +c +1\right )}{3}\right )}{e^{5}}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\ln \left (d x +c -1\right )}{8}-\frac {1}{12 \left (d x +c \right )^{3}}-\frac {1}{4 \left (d x +c \right )}+\frac {\ln \left (d x +c +1\right )}{8}\right )}{e^{5}}}{d}\)	\(273\)
parts	\(-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4} d}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right )}{6 \left (d x +c \right )^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )^{2}}{16}-\frac {\ln \left (d x +c +1\right )^{2}}{16}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )}{3}-\frac {1}{12 \left (d x +c \right )^{2}}+\frac {2 \ln \left (d x +c \right )}{3}-\frac {\ln \left (d x +c +1\right )}{3}\right )}{e^{5} d}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\ln \left (d x +c -1\right )}{8}-\frac {1}{12 \left (d x +c \right )^{3}}-\frac {1}{4 \left (d x +c \right )}+\frac {\ln \left (d x +c +1\right )}{8}\right )}{e^{5} d}\)	\(278\)
parallelrisch	\(-\frac {3 a^{2} d^{5}+x^{2} b^{2} d^{7}+6 a b \,c^{3} d^{5}+b^{2} c^{2} d^{5}+2 c b a \,d^{5}+3 b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} d^{5}+32 \ln \left (d x +c -1\right ) x^{3} b^{2} c \,d^{8}-32 \ln \left (d x +c \right ) x^{3} b^{2} c \,d^{8}+48 \ln \left (d x +c -1\right ) x^{2} b^{2} c^{2} d^{7}-48 \ln \left (d x +c \right ) x^{2} b^{2} c^{2} d^{7}+32 \ln \left (d x +c -1\right ) x \,b^{2} c^{3} d^{6}-32 \ln \left (d x +c \right ) x \,b^{2} c^{3} d^{6}-18 x^{2} \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{2} d^{7}+48 x^{2} \operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{7}+18 x^{2} \operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{7}+32 x^{3} \operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{8}-6 x^{4} \operatorname {arctanh}\left (d x +c \right ) a b \,d^{9}+32 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{3} d^{6}+18 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{6}-6 \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{4} d^{5}+18 x a b \,c^{2} d^{6}+18 x^{2} a b c \,d^{7}-12 x \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{3} d^{6}-12 d^{8} c \,b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} x^{3}+2 x \,b^{2} c \,d^{6}+2 x a b \,d^{6}+6 x^{3} a b \,d^{8}-3 b^{2} d^{9} \operatorname {arctanh}\left (d x +c \right )^{2} x^{4}+8 \ln \left (d x +c -1\right ) x^{4} b^{2} d^{9}-8 \ln \left (d x +c \right ) x^{4} b^{2} d^{9}+8 \ln \left (d x +c -1\right ) b^{2} c^{4} d^{5}-8 \ln \left (d x +c \right ) b^{2} c^{4} d^{5}-3 \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{4} d^{5}+6 x^{3} \operatorname {arctanh}\left (d x +c \right ) b^{2} d^{8}+8 x^{4} \operatorname {arctanh}\left (d x +c \right ) b^{2} d^{9}+2 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} d^{6}+8 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{4} d^{5}+6 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{3} d^{5}+2 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{5}+6 \,\operatorname {arctanh}\left (d x +c \right ) a b \,d^{5}-36 x^{2} \operatorname {arctanh}\left (d x +c \right ) a b \,c^{2} d^{7}-24 x^{3} \operatorname {arctanh}\left (d x +c \right ) a b c \,d^{8}-24 x \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{3} d^{6}}{12 \left (d x +c \right )^{4} d^{6} e^{5}}\)	\(702\)
risch	\(\text {Expression too large to display}\)	\(1096\)

Input:

int((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/4*a^2/e^5/(d*x+c)^4+b^2/e^5*(-1/4/(d*x+c)^4*arctanh(d*x+c)^2-1/4*a 
rctanh(d*x+c)*ln(d*x+c-1)-1/6/(d*x+c)^3*arctanh(d*x+c)-1/2/(d*x+c)*arctanh 
(d*x+c)+1/4*arctanh(d*x+c)*ln(d*x+c+1)+1/8*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/ 
2)-1/16*ln(d*x+c-1)^2-1/16*ln(d*x+c+1)^2+1/8*(ln(d*x+c+1)-ln(1/2*d*x+1/2*c 
+1/2))*ln(-1/2*d*x-1/2*c+1/2)-1/3*ln(d*x+c-1)-1/12/(d*x+c)^2+2/3*ln(d*x+c) 
-1/3*ln(d*x+c+1))+2*b*a/e^5*(-1/4/(d*x+c)^4*arctanh(d*x+c)-1/8*ln(d*x+c-1) 
-1/12/(d*x+c)^3-1/4/(d*x+c)+1/8*ln(d*x+c+1)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (158) = 316\).

Time = 0.11 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.18 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {24 \, a b d^{3} x^{3} + 24 \, a b c^{3} + 4 \, {\left (18 \, a b c + b^{2}\right )} d^{2} x^{2} + 4 \, b^{2} c^{2} + 8 \, a b c + 8 \, {\left (9 \, a b c^{2} + b^{2} c + a b\right )} d x - 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 12 \, a^{2} - 4 \, {\left ({\left (3 \, a b - 4 \, b^{2}\right )} d^{4} x^{4} + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c d^{3} x^{3} + 6 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{2} d^{2} x^{2} + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{3} d x + {\left (3 \, a b - 4 \, b^{2}\right )} c^{4}\right )} \log \left (d x + c + 1\right ) - 32 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right ) + 4 \, {\left ({\left (3 \, a b + 4 \, b^{2}\right )} d^{4} x^{4} + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c d^{3} x^{3} + 6 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{2} d^{2} x^{2} + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{3} d x + {\left (3 \, a b + 4 \, b^{2}\right )} c^{4}\right )} \log \left (d x + c - 1\right ) + 4 \, {\left (3 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{3} + b^{2} c + {\left (9 \, b^{2} c^{2} + b^{2}\right )} d x + 3 \, a b\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{48 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \] Input:

integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="fricas")
 

Output:

-1/48*(24*a*b*d^3*x^3 + 24*a*b*c^3 + 4*(18*a*b*c + b^2)*d^2*x^2 + 4*b^2*c^ 
2 + 8*a*b*c + 8*(9*a*b*c^2 + b^2*c + a*b)*d*x - 3*(b^2*d^4*x^4 + 4*b^2*c*d 
^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 - b^2)*log(-(d*x + c 
+ 1)/(d*x + c - 1))^2 + 12*a^2 - 4*((3*a*b - 4*b^2)*d^4*x^4 + 4*(3*a*b - 4 
*b^2)*c*d^3*x^3 + 6*(3*a*b - 4*b^2)*c^2*d^2*x^2 + 4*(3*a*b - 4*b^2)*c^3*d* 
x + (3*a*b - 4*b^2)*c^4)*log(d*x + c + 1) - 32*(b^2*d^4*x^4 + 4*b^2*c*d^3* 
x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(d*x + c) + 4*((3*a* 
b + 4*b^2)*d^4*x^4 + 4*(3*a*b + 4*b^2)*c*d^3*x^3 + 6*(3*a*b + 4*b^2)*c^2*d 
^2*x^2 + 4*(3*a*b + 4*b^2)*c^3*d*x + (3*a*b + 4*b^2)*c^4)*log(d*x + c - 1) 
 + 4*(3*b^2*d^3*x^3 + 9*b^2*c*d^2*x^2 + 3*b^2*c^3 + b^2*c + (9*b^2*c^2 + b 
^2)*d*x + 3*a*b)*log(-(d*x + c + 1)/(d*x + c - 1)))/(d^5*e^5*x^4 + 4*c*d^4 
*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3516 vs. \(2 (148) = 296\).

Time = 3.46 (sec) , antiderivative size = 3516, normalized size of antiderivative = 20.44 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((a+b*atanh(d*x+c))**2/(d*e*x+c*e)**5,x)
 

Output:

Piecewise((-3*a**2/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e* 
*5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 6*a*b*c**4*atanh(c + 
d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c 
*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 24*a*b*c**3*d*x*atanh(c + d*x)/(12* 
c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e** 
5*x**3 + 12*d**5*e**5*x**4) - 6*a*b*c**3/(12*c**4*d*e**5 + 48*c**3*d**2*e* 
*5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 
 36*a*b*c**2*d**2*x**2*atanh(c + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5* 
x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 18 
*a*b*c**2*d*x/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x* 
*2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 24*a*b*c*d**3*x**3*atanh(c 
 + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 4 
8*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 18*a*b*c*d**2*x**2/(12*c**4*d*e* 
*5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 
12*d**5*e**5*x**4) - 2*a*b*c/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c* 
*2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 6*a*b*d**4* 
x**4*atanh(c + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e 
**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 6*a*b*d**3*x**3/(12* 
c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e** 
5*x**3 + 12*d**5*e**5*x**4) - 2*a*b*d*x/(12*c**4*d*e**5 + 48*c**3*d**2*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (158) = 316\).

Time = 0.07 (sec) , antiderivative size = 613, normalized size of antiderivative = 3.56 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {1}{12} \, {\left (d {\left (\frac {2 \, {\left (3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} + 1\right )}}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} - \frac {3 \, \log \left (d x + c + 1\right )}{d^{2} e^{5}} + \frac {3 \, \log \left (d x + c - 1\right )}{d^{2} e^{5}}\right )} + \frac {6 \, \operatorname {artanh}\left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} a b - \frac {1}{48} \, {\left (d^{2} {\left (\frac {3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (8 \, d^{2} x^{2} + 16 \, c d x + 8 \, c^{2} - 3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) + 16 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right ) + 4}{d^{5} e^{5} x^{2} + 2 \, c d^{4} e^{5} x + c^{2} d^{3} e^{5}} - \frac {32 \, \log \left (d x + c\right )}{d^{3} e^{5}}\right )} + 4 \, d {\left (\frac {2 \, {\left (3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} + 1\right )}}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} - \frac {3 \, \log \left (d x + c + 1\right )}{d^{2} e^{5}} + \frac {3 \, \log \left (d x + c - 1\right )}{d^{2} e^{5}}\right )} \operatorname {artanh}\left (d x + c\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} - \frac {a^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \] Input:

integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="maxima")
 

Output:

-1/12*(d*(2*(3*d^2*x^2 + 6*c*d*x + 3*c^2 + 1)/(d^5*e^5*x^3 + 3*c*d^4*e^5*x 
^2 + 3*c^2*d^3*e^5*x + c^3*d^2*e^5) - 3*log(d*x + c + 1)/(d^2*e^5) + 3*log 
(d*x + c - 1)/(d^2*e^5)) + 6*arctanh(d*x + c)/(d^5*e^5*x^4 + 4*c*d^4*e^5*x 
^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5))*a*b - 1/48*(d^2*((3 
*(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c + 1)^2 + 3*(d^2*x^2 + 2*c*d*x + c^2 
)*log(d*x + c - 1)^2 + 2*(8*d^2*x^2 + 16*c*d*x + 8*c^2 - 3*(d^2*x^2 + 2*c* 
d*x + c^2)*log(d*x + c - 1))*log(d*x + c + 1) + 16*(d^2*x^2 + 2*c*d*x + c^ 
2)*log(d*x + c - 1) + 4)/(d^5*e^5*x^2 + 2*c*d^4*e^5*x + c^2*d^3*e^5) - 32* 
log(d*x + c)/(d^3*e^5)) + 4*d*(2*(3*d^2*x^2 + 6*c*d*x + 3*c^2 + 1)/(d^5*e^ 
5*x^3 + 3*c*d^4*e^5*x^2 + 3*c^2*d^3*e^5*x + c^3*d^2*e^5) - 3*log(d*x + c + 
 1)/(d^2*e^5) + 3*log(d*x + c - 1)/(d^2*e^5))*arctanh(d*x + c))*b^2 - 1/4* 
b^2*arctanh(d*x + c)^2/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 
+ 4*c^3*d^2*e^5*x + c^4*d*e^5) - 1/4*a^2/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 
6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (158) = 316\).

Time = 0.14 (sec) , antiderivative size = 730, normalized size of antiderivative = 4.24 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx =\text {Too large to display} \] Input:

integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="giac")
 

Output:

1/12*((c + 1)*d - (c - 1)*d)*(3*((d*x + c + 1)^3*b^2/(d*x + c - 1)^3 + (d* 
x + c + 1)*b^2/(d*x + c - 1))*log(-(d*x + c + 1)/(d*x + c - 1))^2/((d*x + 
c + 1)^4*d^2*e^5/(d*x + c - 1)^4 + 4*(d*x + c + 1)^3*d^2*e^5/(d*x + c - 1) 
^3 + 6*(d*x + c + 1)^2*d^2*e^5/(d*x + c - 1)^2 + 4*(d*x + c + 1)*d^2*e^5/( 
d*x + c - 1) + d^2*e^5) + 2*(6*(d*x + c + 1)^3*a*b/(d*x + c - 1)^3 + 6*(d* 
x + c + 1)*a*b/(d*x + c - 1) + 3*(d*x + c + 1)^3*b^2/(d*x + c - 1)^3 + 6*( 
d*x + c + 1)^2*b^2/(d*x + c - 1)^2 + 5*(d*x + c + 1)*b^2/(d*x + c - 1) + 2 
*b^2)*log(-(d*x + c + 1)/(d*x + c - 1))/((d*x + c + 1)^4*d^2*e^5/(d*x + c 
- 1)^4 + 4*(d*x + c + 1)^3*d^2*e^5/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^2 
*e^5/(d*x + c - 1)^2 + 4*(d*x + c + 1)*d^2*e^5/(d*x + c - 1) + d^2*e^5) + 
2*(6*(d*x + c + 1)^3*a^2/(d*x + c - 1)^3 + 6*(d*x + c + 1)*a^2/(d*x + c - 
1) + 6*(d*x + c + 1)^3*a*b/(d*x + c - 1)^3 + 12*(d*x + c + 1)^2*a*b/(d*x + 
 c - 1)^2 + 10*(d*x + c + 1)*a*b/(d*x + c - 1) + 4*a*b + (d*x + c + 1)^3*b 
^2/(d*x + c - 1)^3 + 2*(d*x + c + 1)^2*b^2/(d*x + c - 1)^2 + (d*x + c + 1) 
*b^2/(d*x + c - 1))/((d*x + c + 1)^4*d^2*e^5/(d*x + c - 1)^4 + 4*(d*x + c 
+ 1)^3*d^2*e^5/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^2*e^5/(d*x + c - 1)^2 
 + 4*(d*x + c + 1)*d^2*e^5/(d*x + c - 1) + d^2*e^5) + 4*b^2*log(-(d*x + c 
+ 1)/(d*x + c - 1) - 1)/(d^2*e^5) - 4*b^2*log(-(d*x + c + 1)/(d*x + c - 1) 
)/(d^2*e^5))
 

Mupad [B] (verification not implemented)

Time = 7.05 (sec) , antiderivative size = 2746, normalized size of antiderivative = 15.97 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=\text {Too large to display} \] Input:

int((a + b*atanh(c + d*x))^2/(c*e + d*e*x)^5,x)
 

Output:

log(1 - d*x - c)^2*(b^2/(16*d*e^5) - b^2/(4*d*(4*c^4*e^5 + 4*d^4*e^5*x^4 + 
 16*c*d^3*e^5*x^3 + 24*c^2*d^2*e^5*x^2 + 16*c^3*d*e^5*x))) + log(c + d*x + 
 1)^2*(b^2/(16*d*e^5) - b^2/(16*d^2*e^5*(4*c^3*x + c^4/d + d^3*x^4 + 6*c^2 
*d*x^2 + 4*c*d^2*x^3))) + log(1 - d*x - c)*(log(c + d*x + 1)*(b^2/(4*d*(2* 
c^4*e^5 + 2*d^4*e^5*x^4 + 8*c*d^3*e^5*x^3 + 12*c^2*d^2*e^5*x^2 + 8*c^3*d*e 
^5*x)) - (b^2*(c^4 + d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x))/( 
4*d*(2*c^4*e^5 + 2*d^4*e^5*x^4 + 8*c*d^3*e^5*x^3 + 12*c^2*d^2*e^5*x^2 + 8* 
c^3*d*e^5*x))) + (3*b^2)/(4*d*(24*c^4*e^5 + 24*d^4*e^5*x^4 + 96*c*d^3*e^5* 
x^3 + 144*c^2*d^2*e^5*x^2 + 96*c^3*d*e^5*x)) + (3*b*(8*a - b))/(4*d*(24*c^ 
4*e^5 + 24*d^4*e^5*x^4 + 96*c*d^3*e^5*x^3 + 144*c^2*d^2*e^5*x^2 + 96*c^3*d 
*e^5*x)) - (b^2*(c*(2*c - 3*c^2 + 4*c^3 + c*(6*c^2 - 3*c + c*(12*c - 3) + 
1) - 1) - 3*c + x^2*(d*(2*d - 6*c*d + 12*c^2*d + d*(6*c^2 - 3*c + c*(12*c 
- 3) + 1) + c*(24*c*d - 3*d + d*(12*c - 3))) - 9*c*d^2 + c*(30*c*d^2 - 3*d 
^2 + d*(24*c*d - 3*d + d*(12*c - 3))) + 3*d^2 + 18*c^2*d^2) + x*(d*(2*c - 
3*c^2 + 4*c^3 + c*(6*c^2 - 3*c + c*(12*c - 3) + 1) - 1) - 3*d + 6*c*d + c* 
(2*d - 6*c*d + 12*c^2*d + d*(6*c^2 - 3*c + c*(12*c - 3) + 1) + c*(24*c*d - 
 3*d + d*(12*c - 3))) - 9*c^2*d + 12*c^3*d) + 3*c^2 - 3*c^3 + 3*c^4 + 25*d 
^4*x^4 + x^3*(34*c*d^3 + d*(30*c*d^2 - 3*d^2 + d*(24*c*d - 3*d + d*(12*c - 
 3))) - 3*d^3) + 3))/(4*d*(24*c^4*e^5 + 24*d^4*e^5*x^4 + 96*c*d^3*e^5*x^3 
+ 144*c^2*d^2*e^5*x^2 + 96*c^3*d*e^5*x)) + (b^2*(c*(c*(6*c*e^5 + 2*e^5 ...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 989, normalized size of antiderivative = 5.75 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx =\text {Too large to display} \] Input:

int((a+b*atanh(d*x+c))^2/(d*e*x+c*e)^5,x)
 

Output:

(12*atanh(c + d*x)**2*b**2*c**5 + 48*atanh(c + d*x)**2*b**2*c**4*d*x + 72* 
atanh(c + d*x)**2*b**2*c**3*d**2*x**2 + 48*atanh(c + d*x)**2*b**2*c**2*d** 
3*x**3 + 12*atanh(c + d*x)**2*b**2*c*d**4*x**4 - 12*atanh(c + d*x)**2*b**2 
*c - 24*atanh(c + d*x)*a*b*c - 18*atanh(c + d*x)*b**2*c**4 - 48*atanh(c + 
d*x)*b**2*c**3*d*x - 36*atanh(c + d*x)*b**2*c**2*d**2*x**2 - 8*atanh(c + d 
*x)*b**2*c**2 - 8*atanh(c + d*x)*b**2*c*d*x + 6*atanh(c + d*x)*b**2*d**4*x 
**4 - 12*log(c + d*x - 1)*a*b*c**5 - 48*log(c + d*x - 1)*a*b*c**4*d*x - 72 
*log(c + d*x - 1)*a*b*c**3*d**2*x**2 - 48*log(c + d*x - 1)*a*b*c**2*d**3*x 
**3 - 12*log(c + d*x - 1)*a*b*c*d**4*x**4 - 16*log(c + d*x - 1)*b**2*c**5 
- 64*log(c + d*x - 1)*b**2*c**4*d*x + 3*log(c + d*x - 1)*b**2*c**4 - 96*lo 
g(c + d*x - 1)*b**2*c**3*d**2*x**2 + 12*log(c + d*x - 1)*b**2*c**3*d*x - 6 
4*log(c + d*x - 1)*b**2*c**2*d**3*x**3 + 18*log(c + d*x - 1)*b**2*c**2*d** 
2*x**2 - 16*log(c + d*x - 1)*b**2*c*d**4*x**4 + 12*log(c + d*x - 1)*b**2*c 
*d**3*x**3 + 3*log(c + d*x - 1)*b**2*d**4*x**4 + 12*log(c + d*x + 1)*a*b*c 
**5 + 48*log(c + d*x + 1)*a*b*c**4*d*x + 72*log(c + d*x + 1)*a*b*c**3*d**2 
*x**2 + 48*log(c + d*x + 1)*a*b*c**2*d**3*x**3 + 12*log(c + d*x + 1)*a*b*c 
*d**4*x**4 - 16*log(c + d*x + 1)*b**2*c**5 - 64*log(c + d*x + 1)*b**2*c**4 
*d*x - 3*log(c + d*x + 1)*b**2*c**4 - 96*log(c + d*x + 1)*b**2*c**3*d**2*x 
**2 - 12*log(c + d*x + 1)*b**2*c**3*d*x - 64*log(c + d*x + 1)*b**2*c**2*d* 
*3*x**3 - 18*log(c + d*x + 1)*b**2*c**2*d**2*x**2 - 16*log(c + d*x + 1)...