\(\int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^2 \, dx\) [16]

Optimal result

Integrand size = 23, antiderivative size = 179 \[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \text {arctanh}(c+d x)}{3 d}+\frac {b e^2 (c+d x)^2 (a+b \text {arctanh}(c+d x))}{3 d}+\frac {e^2 (a+b \text {arctanh}(c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \text {arctanh}(c+d x))^2}{3 d}-\frac {2 b e^2 (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1-c-d x}\right )}{3 d}-\frac {b^2 e^2 \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.84 \[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {e^2 \left (a^2 (c+d x)^3+a b \left ((c+d x)^2+2 (c+d x)^3 \text {arctanh}(c+d x)+\log \left (-1+(c+d x)^2\right )\right )+b^2 \left (c+d x-\text {arctanh}(c+d x)+(c+d x)^2 \text {arctanh}(c+d x)-\text {arctanh}(c+d x)^2+(c+d x)^3 \text {arctanh}(c+d x)^2-2 \text {arctanh}(c+d x) \log \left (1+e^{-2 \text {arctanh}(c+d x)}\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c+d x)}\right )\right )\right )}{3 d} \] Input:

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

Output:

(e^2*(a^2*(c + d*x)^3 + a*b*((c + d*x)^2 + 2*(c + d*x)^3*ArcTanh[c + d*x] 
+ Log[-1 + (c + d*x)^2]) + b^2*(c + d*x - ArcTanh[c + d*x] + (c + d*x)^2*A 
rcTanh[c + d*x] - ArcTanh[c + d*x]^2 + (c + d*x)^3*ArcTanh[c + d*x]^2 - 2* 
ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])] + PolyLog[2, -E^(-2*ArcT 
anh[c + d*x])])))/(3*d)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.80, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6657, 27, 6452, 6542, 6452, 262, 219, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 6657

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

(e^2*(((c + d*x)^3*(a + b*ArcTanh[c + d*x])^2)/3 - (2*b*((b*(-c - d*x + Ar 
cTanh[c + d*x]))/2 - ((c + d*x)^2*(a + b*ArcTanh[c + d*x]))/2 - (a + b*Arc 
Tanh[c + d*x])^2/(2*b) + (a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)] + ( 
b*PolyLog[2, 1 - 2/(1 - c - d*x)])/2))/3))/d
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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

rule 262

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

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6452

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

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6542

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

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 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 = 1.99 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.40

method	result	size
derivativedivides	\(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )^{2}}{3}+\frac {\left (d x +c \right )^{2} \operatorname {arctanh}\left (d x +c \right )}{3}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{3}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{3}+\frac {d x}{3}+\frac {c}{3}+\frac {\ln \left (d x +c -1\right )}{6}-\frac {\ln \left (d x +c +1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (d x +c -1\right )^{2}}{12}-\frac {\ln \left (d x +c +1\right )^{2}}{12}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{6}\right )+2 a b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (d x +c -1\right )}{6}+\frac {\ln \left (d x +c +1\right )}{6}\right )}{d}\)	\(251\)
default	\(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )^{2}}{3}+\frac {\left (d x +c \right )^{2} \operatorname {arctanh}\left (d x +c \right )}{3}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{3}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{3}+\frac {d x}{3}+\frac {c}{3}+\frac {\ln \left (d x +c -1\right )}{6}-\frac {\ln \left (d x +c +1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (d x +c -1\right )^{2}}{12}-\frac {\ln \left (d x +c +1\right )^{2}}{12}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{6}\right )+2 a b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (d x +c -1\right )}{6}+\frac {\ln \left (d x +c +1\right )}{6}\right )}{d}\)	\(251\)
parts	\(\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3 d}+\frac {b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )^{2}}{3}+\frac {\left (d x +c \right )^{2} \operatorname {arctanh}\left (d x +c \right )}{3}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{3}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{3}+\frac {d x}{3}+\frac {c}{3}+\frac {\ln \left (d x +c -1\right )}{6}-\frac {\ln \left (d x +c +1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (d x +c -1\right )^{2}}{12}-\frac {\ln \left (d x +c +1\right )^{2}}{12}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{6}\right )}{d}+\frac {2 a b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arctanh}\left (d x +c \right )}{3}+\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (d x +c -1\right )}{6}+\frac {\ln \left (d x +c +1\right )}{6}\right )}{d}\)	\(256\)
risch	\(\frac {2 a b x \,e^{2} c}{3}+\frac {b^{2} e^{2} x}{3}-\frac {17 b^{2} e^{2}}{54 d}-\frac {11 e^{2} b a}{9 d}+\frac {e^{2} b^{2} c}{3 d}+\frac {e^{2} c^{3} a^{2}}{3 d}-e^{2} d b a \ln \left (-d x -c +1\right ) c \,x^{2}-\frac {b^{2} e^{2} c d \ln \left (d x +c +1\right ) \ln \left (-d x -c +1\right ) x^{2}}{2}+b \,e^{2} a d c \ln \left (d x +c +1\right ) x^{2}+\frac {e^{2} a b \,c^{2}}{3 d}+\frac {b^{2} e^{2} d \ln \left (d x +c +1\right )^{2} c \,x^{2}}{4}+\frac {b \,e^{2} a \,c^{3} \ln \left (d x +c +1\right )}{3 d}+\frac {b \,e^{2} a \,d^{2} \ln \left (d x +c +1\right ) x^{3}}{3}+b \,e^{2} a \,c^{2} \ln \left (d x +c +1\right ) x -\frac {b^{2} e^{2} c^{2} \ln \left (-d x -c +1\right ) \ln \left (d x +c +1\right ) x}{2}-\frac {b^{2} e^{2} c^{3} \ln \left (-d x -c +1\right ) \ln \left (d x +c +1\right )}{6 d}-\frac {b^{2} e^{2} d^{2} \ln \left (-d x -c +1\right ) \ln \left (d x +c +1\right ) x^{3}}{6}-\frac {e^{2} a^{2}}{3 d}-e^{2} b a \ln \left (-d x -c +1\right ) c^{2} x +\frac {e^{2} d \,b^{2} \ln \left (-d x -c +1\right )^{2} c \,x^{2}}{4}-\frac {e^{2} d^{2} b a \ln \left (-d x -c +1\right ) x^{3}}{3}-\frac {e^{2} b a \ln \left (-d x -c +1\right ) c^{3}}{3 d}+e^{2} d c \,a^{2} x^{2}+e^{2} a^{2} c^{2} x +\frac {e^{2} b^{2} \ln \left (-d x -c +1\right )^{2} c^{2} x}{4}+\frac {e^{2} b a \ln \left (-d x -c +1\right )}{3 d}+\frac {e^{2} b^{2} \ln \left (-d x -c +1\right )^{2} c^{3}}{12 d}+\frac {e^{2} d^{2} b^{2} \ln \left (-d x -c +1\right )^{2} x^{3}}{12}+\frac {b^{2} e^{2} \ln \left (d x +c +1\right )^{2} c^{3}}{12 d}+\frac {b^{2} e^{2} \ln \left (d x +c +1\right )^{2} c^{2} x}{4}+\frac {b^{2} e^{2} d^{2} \ln \left (d x +c +1\right )^{2} x^{3}}{12}+\frac {e^{2} d^{2} x^{3} a^{2}}{3}-\frac {e^{2} b^{2} \ln \left (-d x -c +1\right )^{2}}{12 d}+\frac {b^{2} e^{2} \ln \left (d x +c +1\right )^{2}}{12 d}-\frac {b^{2} e^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{3 d}-\frac {4 b^{2} e^{2} \ln \left (d x +c -1\right )}{9 d}-\frac {b^{2} e^{2} \ln \left (d x +c +1\right )}{6 d}+\frac {11 e^{2} b^{2} \ln \left (-d x -c +1\right )}{18 d}+\frac {b^{2} e^{2} \ln \left (d x +c +1\right ) c^{2}}{6 d}+\frac {b^{2} e^{2} d \ln \left (d x +c +1\right ) x^{2}}{6}+\frac {b^{2} e^{2} \ln \left (d x +c +1\right ) x c}{3}-\frac {e^{2} b^{2} \ln \left (-d x -c +1\right ) c^{2}}{6 d}-\frac {e^{2} b^{2} \ln \left (-d x -c +1\right ) x c}{3}+\frac {e^{2} d b a \,x^{2}}{3}-\frac {e^{2} d \,b^{2} \ln \left (-d x -c +1\right ) x^{2}}{6}+\frac {b \,e^{2} a \ln \left (d x +c +1\right )}{3 d}-\frac {b^{2} e^{2} \ln \left (-d x -c +1\right ) \ln \left (d x +c +1\right )}{6 d}+\frac {b^{2} e^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{3 d}-\frac {b^{2} e^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{3 d}\)	\(982\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

e**2*(Integral(a**2*c**2, x) + Integral(a**2*d**2*x**2, x) + Integral(b**2 
*c**2*atanh(c + d*x)**2, x) + Integral(2*a*b*c**2*atanh(c + d*x), x) + Int 
egral(2*a**2*c*d*x, x) + Integral(b**2*d**2*x**2*atanh(c + d*x)**2, x) + I 
ntegral(2*a*b*d**2*x**2*atanh(c + d*x), x) + Integral(2*b**2*c*d*x*atanh(c 
 + d*x)**2, x) + Integral(4*a*b*c*d*x*atanh(c + d*x), x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (157) = 314\).

Time = 0.24 (sec) , antiderivative size = 619, normalized size of antiderivative = 3.46 \[ \int (c e+d e x)^2 (a+b \text {arctanh}(c+d x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/3*a^2*d^2*e^2*x^3 + a^2*c*d*e^2*x^2 + (2*x^2*arctanh(d*x + c) + d*(2*x/d 
^2 - (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 + (c^2 - 2*c + 1)*log(d*x + c - 
1)/d^3))*a*b*c*d*e^2 + 1/3*(2*x^3*arctanh(d*x + c) + d*((d*x^2 - 4*c*x)/d^ 
3 + (c^3 + 3*c^2 + 3*c + 1)*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1) 
*log(d*x + c - 1)/d^4))*a*b*d^2*e^2 + a^2*c^2*e^2*x + (2*(d*x + c)*arctanh 
(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*c^2*e^2/d + 1/3*(log(d*x + c + 1)*l 
og(-1/2*d*x - 1/2*c + 1/2) + dilog(1/2*d*x + 1/2*c + 1/2))*b^2*e^2/d + 1/6 
*(c^2*e^2 - e^2)*b^2*log(d*x + c + 1)/d - 1/6*(c^2*e^2 - e^2)*b^2*log(d*x 
+ c - 1)/d + 1/12*(4*b^2*d*e^2*x + (b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e^2*x^2 
+ 3*b^2*c^2*d*e^2*x + (c^3*e^2 + e^2)*b^2)*log(d*x + c + 1)^2 + (b^2*d^3*e 
^2*x^3 + 3*b^2*c*d^2*e^2*x^2 + 3*b^2*c^2*d*e^2*x + (c^3*e^2 - e^2)*b^2)*lo 
g(-d*x - c + 1)^2 + 2*(b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x)*log(d*x + c + 1) 
 - 2*(b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x + (b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e 
^2*x^2 + 3*b^2*c^2*d*e^2*x + (c^3*e^2 + e^2)*b^2)*log(d*x + c + 1))*log(-d 
*x - c + 1))/d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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