\(\int (e+f x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx\) [38]

Optimal result

Integrand size = 20, antiderivative size = 562 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {a b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{2 d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b^2 f^2 (d e-c f) \text {arctanh}(c+d x)}{d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) (c+d x) \text {arctanh}(c+d x)}{2 d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 (a+b \text {arctanh}(c+d x))}{d^4}+\frac {b f^3 (c+d x)^3 (a+b \text {arctanh}(c+d x))}{6 d^4}+\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (a+b \text {arctanh}(c+d x))^2}{d^4}-\frac {\left (d^4 e^4-4 c d^3 e^3 f+6 \left (1+c^2\right ) d^2 e^2 f^2-4 c \left (3+c^2\right ) d e f^3+\left (1+6 c^2+c^4\right ) f^4\right ) (a+b \text {arctanh}(c+d x))^2}{4 d^4 f}+\frac {(e+f x)^4 (a+b \text {arctanh}(c+d x))^2}{4 f}-\frac {2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1-c-d x}\right )}{d^4}+\frac {b^2 f^3 \log \left (1-(c+d x)^2\right )}{12 d^4}+\frac {b^2 f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) \log \left (1-(c+d x)^2\right )}{4 d^4}-\frac {b^2 (d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^4} \] Output:

b^2*f^2*(-c*f+d*e)*x/d^3+1/2*a*b*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*x/ 
d^3+1/12*b^2*f^3*(d*x+c)^2/d^4-b^2*f^2*(-c*f+d*e)*arctanh(d*x+c)/d^4+1/2*b 
^2*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*(d*x+c)*arctanh(d*x+c)/d^4+b*f^2 
*(-c*f+d*e)*(d*x+c)^2*(a+b*arctanh(d*x+c))/d^4+1/6*b*f^3*(d*x+c)^3*(a+b*ar 
ctanh(d*x+c))/d^4+(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)*(a+b*arctanh( 
d*x+c))^2/d^4-1/4*(d^4*e^4-4*c*d^3*e^3*f+6*(c^2+1)*d^2*e^2*f^2-4*c*(c^2+3) 
*d*e*f^3+(c^4+6*c^2+1)*f^4)*(a+b*arctanh(d*x+c))^2/d^4/f+1/4*(f*x+e)^4*(a+ 
b*arctanh(d*x+c))^2/f-2*b*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)*(a+b* 
arctanh(d*x+c))*ln(2/(-d*x-c+1))/d^4+1/12*b^2*f^3*ln(1-(d*x+c)^2)/d^4+1/4* 
b^2*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*ln(1-(d*x+c)^2)/d^4-b^2*(-c*f+d 
*e)*(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)*polylog(2,-(d*x+c+1)/(-d*x-c+1))/d^4
 

Mathematica [A] (warning: unable to verify)

Time = 4.81 (sec) , antiderivative size = 1082, normalized size of antiderivative = 1.93 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(12*a^2*e^3*x + 18*a^2*e^2*f*x^2 + 12*a^2*e*f^2*x^3 + 3*a^2*f^3*x^4 + a*b* 
(6*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcTanh[c + d*x] - (-2*d* 
f*x*(3*(1 + 3*c^2)*f^2 - 3*c*d*f*(8*e + f*x) + d^2*(18*e^2 + 6*e*f*x + f^2 
*x^2)) + 3*(-1 + c)*(4*d^3*e^3 - 6*(-1 + c)*d^2*e^2*f + 4*(-1 + c)^2*d*e*f 
^2 - (-1 + c)^3*f^3)*Log[1 - c - d*x] + 3*(1 + c)*(-4*d^3*e^3 + 6*(1 + c)* 
d^2*e^2*f - 4*(1 + c)^2*d*e*f^2 + (1 + c)^3*f^3)*Log[1 + c + d*x])/d^4) + 
(12*b^2*e^3*(ArcTanh[c + d*x]*((-1 + c + d*x)*ArcTanh[c + d*x] - 2*Log[1 + 
 E^(-2*ArcTanh[c + d*x])]) + PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d - (1 
8*b^2*e^2*f*((1 - 2*c + c^2 - d^2*x^2)*ArcTanh[c + d*x]^2 - 2*ArcTanh[c + 
d*x]*(c + d*x + 2*c*Log[1 + E^(-2*ArcTanh[c + d*x])]) + 2*Log[1/Sqrt[1 - ( 
c + d*x)^2]] + 2*c*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^2 + (b^2*f^3*( 
-1 - 11*c^2 - 10*c*d*x + d^2*x^2 - 3*(1 - 4*c + 6*c^2 - 4*c^3 + c^4 - d^4* 
x^4)*ArcTanh[c + d*x]^2 + 2*ArcTanh[c + d*x]*(9*c + 13*c^3 + 3*d*x + 9*c^2 
*d*x - 3*c*d^2*x^2 + d^3*x^3 + 12*(c + c^3)*Log[1 + E^(-2*ArcTanh[c + d*x] 
)]) - 8*Log[1/Sqrt[1 - (c + d*x)^2]] - 36*c^2*Log[1/Sqrt[1 - (c + d*x)^2]] 
 - 12*(c + c^3)*PolyLog[2, -E^(-2*ArcTanh[c + d*x])]))/d^4 - (3*b^2*e*f^2* 
(1 - (c + d*x)^2)^(3/2)*(-((c + d*x)/Sqrt[1 - (c + d*x)^2]) + (6*c*(c + d* 
x)*ArcTanh[c + d*x])/Sqrt[1 - (c + d*x)^2] + (3*(c + d*x)*ArcTanh[c + d*x] 
^2)/Sqrt[1 - (c + d*x)^2] - (3*c^2*(c + d*x)*ArcTanh[c + d*x]^2)/Sqrt[1 - 
(c + d*x)^2] + ArcTanh[c + d*x]^2*Cosh[3*ArcTanh[c + d*x]] + 3*c^2*ArcT...
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6661, 27, 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.

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

\(\Big \downarrow \) 6661

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6480

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^4 (a+b \text {arctanh}(c+d x))^2}{4 f}-\frac {b \int \left (-(c+d x)^2 (a+b \text {arctanh}(c+d x)) f^4-4 (d e-c f) (c+d x) (a+b \text {arctanh}(c+d x)) f^3-\left (6 d^2 e^2-12 c d f e+\left (6 c^2+1\right ) f^2\right ) (a+b \text {arctanh}(c+d x)) f^2+\frac {\left (d^4 e^4-4 c d^3 f e^3+6 \left (c^2+1\right ) d^2 f^2 e^2-4 c \left (c^2+3\right ) d f^3 e+\left (c^4+6 c^2+1\right ) f^4+4 f (d e-c f) \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right ) (c+d x)\right ) (a+b \text {arctanh}(c+d x))}{1-(c+d x)^2}\right )d(c+d x)}{2 f}}{d^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^4 (a+b \text {arctanh}(c+d x))^2}{4 f}-\frac {b \left (-\frac {2 f (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) (a+b \text {arctanh}(c+d x))^2}{b}+4 f (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) (a+b \text {arctanh}(c+d x))+\frac {\left (6 \left (c^2+1\right ) d^2 e^2 f^2-4 c \left (c^2+3\right ) d e f^3+\left (c^4+6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right ) (a+b \text {arctanh}(c+d x))^2}{2 b}-2 f^3 (c+d x)^2 (d e-c f) (a+b \text {arctanh}(c+d x))-\frac {1}{3} f^4 (c+d x)^3 (a+b \text {arctanh}(c+d x))-a f^2 (c+d x) \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )-b f^2 (c+d x) \text {arctanh}(c+d x) \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )+2 b f^3 \text {arctanh}(c+d x) (d e-c f)+2 b f (d e-c f) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )-\frac {1}{2} b f^2 \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right ) \log \left (1-(c+d x)^2\right )-2 b f^3 (c+d x) (d e-c f)-\frac {1}{6} b f^4 (c+d x)^2-\frac {1}{6} b f^4 \log \left (1-(c+d x)^2\right )\right )}{2 f}}{d^4}\)

Input:

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

Output:

(((d*e - c*f + f*(c + d*x))^4*(a + b*ArcTanh[c + d*x])^2)/(4*f) - (b*(-2*b 
*f^3*(d*e - c*f)*(c + d*x) - a*f^2*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f 
^2)*(c + d*x) - (b*f^4*(c + d*x)^2)/6 + 2*b*f^3*(d*e - c*f)*ArcTanh[c + d* 
x] - b*f^2*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*(c + d*x)*ArcTanh[c 
+ d*x] - 2*f^3*(d*e - c*f)*(c + d*x)^2*(a + b*ArcTanh[c + d*x]) - (f^4*(c 
+ d*x)^3*(a + b*ArcTanh[c + d*x]))/3 - (2*f*(d*e - c*f)*(d^2*e^2 - 2*c*d*e 
*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])^2)/b + ((d^4*e^4 - 4*c*d^3*e^ 
3*f + 6*(1 + c^2)*d^2*e^2*f^2 - 4*c*(3 + c^2)*d*e*f^3 + (1 + 6*c^2 + c^4)* 
f^4)*(a + b*ArcTanh[c + d*x])^2)/(2*b) + 4*f*(d*e - c*f)*(d^2*e^2 - 2*c*d* 
e*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)] - (b*f^ 
4*Log[1 - (c + d*x)^2])/6 - (b*f^2*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f 
^2)*Log[1 - (c + d*x)^2])/2 + 2*b*f*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 
+ c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))]))/(2*f))/d^4
 

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 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]
 

rule 6661

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2317\) vs. \(2(546)=1092\).

Time = 1.19 (sec) , antiderivative size = 2318, normalized size of antiderivative = 4.12

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2318\)
default	\(\text {Expression too large to display}\)	\(2318\)
parts	\(\text {Expression too large to display}\)	\(2339\)
risch	\(\text {Expression too large to display}\)	\(3412\)

Input:

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

Output:

1/d*(1/4*a^2/d^3*(c*f-d*e-f*(d*x+c))^4/f-b^2/d^3*(-1/4*f^3*arctanh(d*x+c)^ 
2*c^4+f^2*arctanh(d*x+c)^2*c^3*d*e+f^3*arctanh(d*x+c)^2*c^3*(d*x+c)-3/2*f* 
arctanh(d*x+c)^2*c^2*d^2*e^2-3*f^2*arctanh(d*x+c)^2*c^2*d*e*(d*x+c)-3/2*f^ 
3*arctanh(d*x+c)^2*c^2*(d*x+c)^2+arctanh(d*x+c)^2*c*d^3*e^3+3*f*arctanh(d* 
x+c)^2*c*d^2*e^2*(d*x+c)+3*f^2*arctanh(d*x+c)^2*c*d*e*(d*x+c)^2+f^3*arctan 
h(d*x+c)^2*c*(d*x+c)^3-1/4/f*arctanh(d*x+c)^2*d^4*e^4-arctanh(d*x+c)^2*d^3 
*e^3*(d*x+c)-3/2*f*arctanh(d*x+c)^2*d^2*e^2*(d*x+c)^2-f^2*arctanh(d*x+c)^2 
*d*e*(d*x+c)^3-1/4*f^3*arctanh(d*x+c)^2*(d*x+c)^4+1/2/f*(1/2*arctanh(d*x+c 
)*ln(d*x+c+1)*f^4-1/2*arctanh(d*x+c)*ln(d*x+c-1)*f^4-arctanh(d*x+c)*f^4*(d 
*x+c)-1/3*arctanh(d*x+c)*f^4*(d*x+c)^3+2*arctanh(d*x+c)*ln(d*x+c-1)*c^3*d* 
e*f^3-3*arctanh(d*x+c)*ln(d*x+c-1)*c^2*d^2*e^2*f^2+2*arctanh(d*x+c)*ln(d*x 
+c-1)*c*d^3*e^3*f-6*arctanh(d*x+c)*ln(d*x+c-1)*c^2*d*e*f^3+6*arctanh(d*x+c 
)*ln(d*x+c-1)*c*d^2*e^2*f^2+6*arctanh(d*x+c)*ln(d*x+c-1)*c*d*e*f^3-2*arcta 
nh(d*x+c)*ln(d*x+c+1)*c^3*d*e*f^3+3*arctanh(d*x+c)*ln(d*x+c+1)*c^2*d^2*e^2 
*f^2-2*arctanh(d*x+c)*ln(d*x+c+1)*c*d^3*e^3*f-6*arctanh(d*x+c)*ln(d*x+c+1) 
*c^2*d*e*f^3+6*arctanh(d*x+c)*ln(d*x+c+1)*c*d^2*e^2*f^2-6*arctanh(d*x+c)*l 
n(d*x+c+1)*c*d*e*f^3+12*arctanh(d*x+c)*c*d*e*f^3*(d*x+c)-2*arctanh(d*x+c)* 
ln(d*x+c-1)*d*e*f^3-6*arctanh(d*x+c)*d^2*e^2*f^2*(d*x+c)-2*arctanh(d*x+c)* 
d*e*f^3*(d*x+c)^2-2*arctanh(d*x+c)*ln(d*x+c+1)*d^3*e^3*f+3*arctanh(d*x+c)* 
ln(d*x+c+1)*d^2*e^2*f^2-2*arctanh(d*x+c)*ln(d*x+c+1)*d*e*f^3-2*arctanh(...
 

Fricas [F]

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

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

Output:

integral(a^2*f^3*x^3 + 3*a^2*e*f^2*x^2 + 3*a^2*e^2*f*x + a^2*e^3 + (b^2*f^ 
3*x^3 + 3*b^2*e*f^2*x^2 + 3*b^2*e^2*f*x + b^2*e^3)*arctanh(d*x + c)^2 + 2* 
(a*b*f^3*x^3 + 3*a*b*e*f^2*x^2 + 3*a*b*e^2*f*x + a*b*e^3)*arctanh(d*x + c) 
, x)
 

Sympy [F]

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

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

Output:

Integral((a + b*atanh(c + d*x))**2*(e + f*x)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1363 vs. \(2 (538) = 1076\).

Time = 0.25 (sec) , antiderivative size = 1363, normalized size of antiderivative = 2.43 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*a^2*f^3*x^4 + a^2*e*f^2*x^3 + 3/2*a^2*e^2*f*x^2 + 3/2*(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*e^2*f + (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*e*f^2 + 1/12*(6*x^4*arctanh(d*x + 
c) + d*(2*(d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 + 1)*x)/d^4 - 3*(c^4 + 4*c^3 + 6 
*c^2 + 4*c + 1)*log(d*x + c + 1)/d^5 + 3*(c^4 - 4*c^3 + 6*c^2 - 4*c + 1)*l 
og(d*x + c - 1)/d^5))*a*b*f^3 + a^2*e^3*x + (2*(d*x + c)*arctanh(d*x + c) 
+ log(-(d*x + c)^2 + 1))*a*b*e^3/d + (d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 + 
d*e*f^2 - (3*d^2*e^2*f + f^3)*c)*(log(d*x + c + 1)*log(-1/2*d*x - 1/2*c + 
1/2) + dilog(1/2*d*x + 1/2*c + 1/2))*b^2/d^4 + 1/12*(13*c^3*f^3 + 18*d^2*e 
^2*f - 6*d*e*f^2 - 6*(5*d*e*f^2 - 3*f^3)*c^2 + 4*f^3 + 9*(2*d^2*e^2*f - 4* 
d*e*f^2 + f^3)*c)*b^2*log(d*x + c + 1)/d^4 - 1/12*(13*c^3*f^3 - 18*d^2*e^2 
*f - 6*d*e*f^2 - 6*(5*d*e*f^2 + 3*f^3)*c^2 - 4*f^3 + 9*(2*d^2*e^2*f + 4*d* 
e*f^2 + f^3)*c)*b^2*log(d*x + c - 1)/d^4 + 1/48*(4*b^2*d^2*f^3*x^2 + 8*(6* 
d^2*e*f^2 - 5*c*d*f^3)*b^2*x + 3*(b^2*d^4*f^3*x^4 + 4*b^2*d^4*e*f^2*x^3 + 
6*b^2*d^4*e^2*f*x^2 + 4*b^2*d^4*e^3*x - (c^4*f^3 - 4*d^3*e^3 + 6*d^2*e^2*f 
 - 4*(d*e*f^2 - f^3)*c^3 - 4*d*e*f^2 + 6*(d^2*e^2*f - 2*d*e*f^2 + f^3)*c^2 
 + f^3 - 4*(d^3*e^3 - 3*d^2*e^2*f + 3*d*e*f^2 - f^3)*c)*b^2)*log(d*x + c + 
 1)^2 + 3*(b^2*d^4*f^3*x^4 + 4*b^2*d^4*e*f^2*x^3 + 6*b^2*d^4*e^2*f*x^2 ...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(9*atanh(c + d*x)**2*b**2*c**4*f**3 - 24*atanh(c + d*x)**2*b**2*c**3*d*e*f 
**2 + 18*atanh(c + d*x)**2*b**2*c**2*d**2*e**2*f - 6*atanh(c + d*x)**2*b** 
2*c**2*f**3 + 24*atanh(c + d*x)**2*b**2*c*d*e*f**2 + 12*atanh(c + d*x)**2* 
b**2*d**4*e**3*x + 18*atanh(c + d*x)**2*b**2*d**4*e**2*f*x**2 + 12*atanh(c 
 + d*x)**2*b**2*d**4*e*f**2*x**3 + 3*atanh(c + d*x)**2*b**2*d**4*f**3*x**4 
 - 18*atanh(c + d*x)**2*b**2*d**2*e**2*f - 3*atanh(c + d*x)**2*b**2*f**3 - 
 6*atanh(c + d*x)*a*b*c**4*f**3 + 24*atanh(c + d*x)*a*b*c**3*d*e*f**2 - 24 
*atanh(c + d*x)*a*b*c**3*f**3 - 36*atanh(c + d*x)*a*b*c**2*d**2*e**2*f + 7 
2*atanh(c + d*x)*a*b*c**2*d*e*f**2 - 36*atanh(c + d*x)*a*b*c**2*f**3 + 24* 
atanh(c + d*x)*a*b*c*d**3*e**3 - 72*atanh(c + d*x)*a*b*c*d**2*e**2*f + 72* 
atanh(c + d*x)*a*b*c*d*e*f**2 - 24*atanh(c + d*x)*a*b*c*f**3 + 24*atanh(c 
+ d*x)*a*b*d**4*e**3*x + 36*atanh(c + d*x)*a*b*d**4*e**2*f*x**2 + 24*atanh 
(c + d*x)*a*b*d**4*e*f**2*x**3 + 6*atanh(c + d*x)*a*b*d**4*f**3*x**4 + 24* 
atanh(c + d*x)*a*b*d**3*e**3 - 36*atanh(c + d*x)*a*b*d**2*e**2*f + 24*atan 
h(c + d*x)*a*b*d*e*f**2 - 6*atanh(c + d*x)*a*b*f**3 + 26*atanh(c + d*x)*b* 
*2*c**3*f**3 - 60*atanh(c + d*x)*b**2*c**2*d*e*f**2 + 18*atanh(c + d*x)*b* 
*2*c**2*d*f**3*x + 36*atanh(c + d*x)*b**2*c**2*f**3 + 36*atanh(c + d*x)*b* 
*2*c*d**2*e**2*f - 48*atanh(c + d*x)*b**2*c*d**2*e*f**2*x - 6*atanh(c + d* 
x)*b**2*c*d**2*f**3*x**2 - 72*atanh(c + d*x)*b**2*c*d*e*f**2 + 18*atanh(c 
+ d*x)*b**2*c*f**3 + 36*atanh(c + d*x)*b**2*d**3*e**2*f*x + 12*atanh(c ...