\(\int (d+e x)^3 (a+b \text {arctanh}(c x))^2 \, dx\) [9]

Optimal result

Integrand size = 18, antiderivative size = 359 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {b^2 d e^2 x}{c^2}+\frac {a b e \left (6 c^2 d^2+e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \text {arctanh}(c x)}{c^3}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) x \text {arctanh}(c x)}{2 c^3}+\frac {b d e^2 x^2 (a+b \text {arctanh}(c x))}{c}+\frac {b e^3 x^3 (a+b \text {arctanh}(c x))}{6 c}+\frac {d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{c^3}-\frac {\left (c^4 d^4+6 c^2 d^2 e^2+e^4\right ) (a+b \text {arctanh}(c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \text {arctanh}(c x))^2}{4 e}-\frac {2 b d \left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 e^3 \log \left (1-c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2+e^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}-\frac {b^2 d \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.41 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {-b^2 e^3+12 a^2 c^4 d^3 x+36 a b c^3 d^2 e x+12 b^2 c^2 d e^2 x+6 a b c e^3 x+18 a^2 c^4 d^2 e x^2+12 a b c^3 d e^2 x^2+b^2 c^2 e^3 x^2+12 a^2 c^4 d e^2 x^3+2 a b c^3 e^3 x^3+3 a^2 c^4 e^3 x^4+3 b^2 \left (-4 c^3 d^3-6 c^2 d^2 e-4 c d e^2-e^3+c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arctanh}(c x)^2+2 b c \text {arctanh}(c x) \left (3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b e \left (18 c^2 d^2 x+6 d e \left (-1+c^2 x^2\right )+e^2 x \left (3+c^2 x^2\right )\right )-12 b d \left (c^2 d^2+e^2\right ) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+18 a b c^2 d^2 e \log (1-c x)+3 a b e^3 \log (1-c x)-18 a b c^2 d^2 e \log (1+c x)-3 a b e^3 \log (1+c x)+12 a b c^3 d^3 \log \left (1-c^2 x^2\right )+18 b^2 c^2 d^2 e \log \left (1-c^2 x^2\right )+4 b^2 e^3 \log \left (1-c^2 x^2\right )+12 a b c d e^2 \log \left (-1+c^2 x^2\right )+12 b^2 c d \left (c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{12 c^4} \] Input:

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

Output:

(-(b^2*e^3) + 12*a^2*c^4*d^3*x + 36*a*b*c^3*d^2*e*x + 12*b^2*c^2*d*e^2*x + 
 6*a*b*c*e^3*x + 18*a^2*c^4*d^2*e*x^2 + 12*a*b*c^3*d*e^2*x^2 + b^2*c^2*e^3 
*x^2 + 12*a^2*c^4*d*e^2*x^3 + 2*a*b*c^3*e^3*x^3 + 3*a^2*c^4*e^3*x^4 + 3*b^ 
2*(-4*c^3*d^3 - 6*c^2*d^2*e - 4*c*d*e^2 - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 
 4*d*e^2*x^2 + e^3*x^3))*ArcTanh[c*x]^2 + 2*b*c*ArcTanh[c*x]*(3*a*c^3*x*(4 
*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*e*(18*c^2*d^2*x + 6*d*e*(-1 
+ c^2*x^2) + e^2*x*(3 + c^2*x^2)) - 12*b*d*(c^2*d^2 + e^2)*Log[1 + E^(-2*A 
rcTanh[c*x])]) + 18*a*b*c^2*d^2*e*Log[1 - c*x] + 3*a*b*e^3*Log[1 - c*x] - 
18*a*b*c^2*d^2*e*Log[1 + c*x] - 3*a*b*e^3*Log[1 + c*x] + 12*a*b*c^3*d^3*Lo 
g[1 - c^2*x^2] + 18*b^2*c^2*d^2*e*Log[1 - c^2*x^2] + 4*b^2*e^3*Log[1 - c^2 
*x^2] + 12*a*b*c*d*e^2*Log[-1 + c^2*x^2] + 12*b^2*c*d*(c^2*d^2 + e^2)*Poly 
Log[2, -E^(-2*ArcTanh[c*x])])/(12*c^4)
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(803\) vs. \(2(343)=686\).

Time = 0.29 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.24

method	result	size
parts	\(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (\frac {c \,e^{3} \operatorname {arctanh}\left (c x \right )^{2} x^{4}}{4}+c \,e^{2} \operatorname {arctanh}\left (c x \right )^{2} x^{3} d +\frac {3 c e \operatorname {arctanh}\left (c x \right )^{2} x^{2} d^{2}}{2}+\operatorname {arctanh}\left (c x \right )^{2} c x \,d^{3}+\frac {c \operatorname {arctanh}\left (c x \right )^{2} d^{4}}{4 e}-\frac {-6 \,\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} e^{2} x -2 \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,e^{3} x^{2}-\frac {\operatorname {arctanh}\left (c x \right ) e^{4} c^{3} x^{3}}{3}-\operatorname {arctanh}\left (c x \right ) e^{4} c x -\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{4} d^{4}}{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{3} d^{3} e -3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{2} d^{2} e^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d \,e^{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{4} d^{4}}{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{3} d^{3} e +3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{2} d^{2} e^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d \,e^{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e^{4}}{2}-\frac {\left (-3 c^{4} d^{4}+12 c^{3} d^{3} e -18 c^{2} d^{2} e^{2}+12 c d \,e^{3}-3 e^{4}\right ) \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {\left (3 c^{4} d^{4}+12 c^{3} d^{3} e +18 c^{2} d^{2} e^{2}+12 c d \,e^{3}+3 e^{4}\right ) \left (\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {e^{2} \left (6 c^{2} d e x +\frac {e^{2} c^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}+6 c d e +4 e^{2}\right ) \ln \left (c x -1\right )}{2}-\frac {\left (-18 c^{2} d^{2}+6 c d e -4 e^{2}\right ) \ln \left (c x +1\right )}{2}\right )}{3}}{2 c^{3} e}\right )}{c}+\frac {2 a b \left (\frac {c \,e^{3} \operatorname {arctanh}\left (c x \right ) x^{4}}{4}+c \,e^{2} \operatorname {arctanh}\left (c x \right ) x^{3} d +\frac {3 c e \,\operatorname {arctanh}\left (c x \right ) x^{2} d^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x \,d^{3}+\frac {c \,\operatorname {arctanh}\left (c x \right ) d^{4}}{4 e}-\frac {-6 c^{3} d^{2} e^{2} x -2 c^{3} d \,e^{3} x^{2}-\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x -\frac {\left (c^{4} d^{4}+4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}+4 c d \,e^{3}+e^{4}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{4} d^{4}-4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}-4 c d \,e^{3}+e^{4}\right ) \ln \left (c x +1\right )}{2}}{4 c^{3} e}\right )}{c}\)	\(804\)
derivativedivides	\(\frac {\frac {a^{2} \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} d^{4}}{4 e}+\operatorname {arctanh}\left (c x \right )^{2} c^{4} d^{3} x +\frac {3 e \operatorname {arctanh}\left (c x \right )^{2} c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arctanh}\left (c x \right )^{2} c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {-6 \,\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} e^{2} x -2 \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,e^{3} x^{2}-\frac {\operatorname {arctanh}\left (c x \right ) e^{4} c^{3} x^{3}}{3}-\operatorname {arctanh}\left (c x \right ) e^{4} c x -\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{4} d^{4}}{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{3} d^{3} e -3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{2} d^{2} e^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d \,e^{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{4} d^{4}}{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{3} d^{3} e +3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{2} d^{2} e^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d \,e^{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e^{4}}{2}-\frac {\left (-3 c^{4} d^{4}+12 c^{3} d^{3} e -18 c^{2} d^{2} e^{2}+12 c d \,e^{3}-3 e^{4}\right ) \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {\left (3 c^{4} d^{4}+12 c^{3} d^{3} e +18 c^{2} d^{2} e^{2}+12 c d \,e^{3}+3 e^{4}\right ) \left (\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {e^{2} \left (6 c^{2} d e x +\frac {e^{2} c^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}+6 c d e +4 e^{2}\right ) \ln \left (c x -1\right )}{2}-\frac {\left (-18 c^{2} d^{2}+6 c d e -4 e^{2}\right ) \ln \left (c x +1\right )}{2}\right )}{3}}{2 e}\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arctanh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arctanh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arctanh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {-6 c^{3} d^{2} e^{2} x -2 c^{3} d \,e^{3} x^{2}-\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x -\frac {\left (c^{4} d^{4}+4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}+4 c d \,e^{3}+e^{4}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{4} d^{4}-4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}-4 c d \,e^{3}+e^{4}\right ) \ln \left (c x +1\right )}{2}}{4 e}\right )}{c^{3}}}{c}\)	\(828\)
default	\(\frac {\frac {a^{2} \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{4} d^{4}}{4 e}+\operatorname {arctanh}\left (c x \right )^{2} c^{4} d^{3} x +\frac {3 e \operatorname {arctanh}\left (c x \right )^{2} c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arctanh}\left (c x \right )^{2} c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arctanh}\left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {-6 \,\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} e^{2} x -2 \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,e^{3} x^{2}-\frac {\operatorname {arctanh}\left (c x \right ) e^{4} c^{3} x^{3}}{3}-\operatorname {arctanh}\left (c x \right ) e^{4} c x -\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{4} d^{4}}{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{3} d^{3} e -3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{2} d^{2} e^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d \,e^{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e^{4}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{4} d^{4}}{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{3} d^{3} e +3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{2} d^{2} e^{2}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d \,e^{3}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e^{4}}{2}-\frac {\left (-3 c^{4} d^{4}+12 c^{3} d^{3} e -18 c^{2} d^{2} e^{2}+12 c d \,e^{3}-3 e^{4}\right ) \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {\left (3 c^{4} d^{4}+12 c^{3} d^{3} e +18 c^{2} d^{2} e^{2}+12 c d \,e^{3}+3 e^{4}\right ) \left (\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{6}-\frac {e^{2} \left (6 c^{2} d e x +\frac {e^{2} c^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}+6 c d e +4 e^{2}\right ) \ln \left (c x -1\right )}{2}-\frac {\left (-18 c^{2} d^{2}+6 c d e -4 e^{2}\right ) \ln \left (c x +1\right )}{2}\right )}{3}}{2 e}\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arctanh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arctanh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arctanh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {-6 c^{3} d^{2} e^{2} x -2 c^{3} d \,e^{3} x^{2}-\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x -\frac {\left (c^{4} d^{4}+4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}+4 c d \,e^{3}+e^{4}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{4} d^{4}-4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}-4 c d \,e^{3}+e^{4}\right ) \ln \left (c x +1\right )}{2}}{4 e}\right )}{c^{3}}}{c}\)	\(828\)
risch	\(\text {Expression too large to display}\)	\(1221\)

Input:

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

Output:

1/4*a^2*(e*x+d)^4/e+b^2/c*(1/4*c*e^3*arctanh(c*x)^2*x^4+c*e^2*arctanh(c*x) 
^2*x^3*d+3/2*c*e*arctanh(c*x)^2*x^2*d^2+arctanh(c*x)^2*c*x*d^3+1/4*c/e*arc 
tanh(c*x)^2*d^4-1/2/c^3/e*(-6*arctanh(c*x)*c^3*d^2*e^2*x-2*arctanh(c*x)*c^ 
3*d*e^3*x^2-1/3*arctanh(c*x)*e^4*c^3*x^3-arctanh(c*x)*e^4*c*x-1/2*arctanh( 
c*x)*ln(c*x-1)*c^4*d^4-2*arctanh(c*x)*ln(c*x-1)*c^3*d^3*e-3*arctanh(c*x)*l 
n(c*x-1)*c^2*d^2*e^2-2*arctanh(c*x)*ln(c*x-1)*c*d*e^3-1/2*arctanh(c*x)*ln( 
c*x-1)*e^4+1/2*arctanh(c*x)*ln(c*x+1)*c^4*d^4-2*arctanh(c*x)*ln(c*x+1)*c^3 
*d^3*e+3*arctanh(c*x)*ln(c*x+1)*c^2*d^2*e^2-2*arctanh(c*x)*ln(c*x+1)*c*d*e 
^3+1/2*arctanh(c*x)*ln(c*x+1)*e^4-1/6*(-3*c^4*d^4+12*c^3*d^3*e-18*c^2*d^2* 
e^2+12*c*d*e^3-3*e^4)*(-1/4*ln(c*x+1)^2+1/2*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln 
(-1/2*c*x+1/2)-1/2*dilog(1/2*c*x+1/2))-1/6*(3*c^4*d^4+12*c^3*d^3*e+18*c^2* 
d^2*e^2+12*c*d*e^3+3*e^4)*(1/4*ln(c*x-1)^2-1/2*dilog(1/2*c*x+1/2)-1/2*ln(c 
*x-1)*ln(1/2*c*x+1/2))-1/3*e^2*(6*c^2*d*e*x+1/2*e^2*c^2*x^2+1/2*(18*c^2*d^ 
2+6*c*d*e+4*e^2)*ln(c*x-1)-1/2*(-18*c^2*d^2+6*c*d*e-4*e^2)*ln(c*x+1))))+2* 
a*b/c*(1/4*c*e^3*arctanh(c*x)*x^4+c*e^2*arctanh(c*x)*x^3*d+3/2*c*e*arctanh 
(c*x)*x^2*d^2+arctanh(c*x)*c*x*d^3+1/4*c/e*arctanh(c*x)*d^4-1/4/c^3/e*(-6* 
c^3*d^2*e^2*x-2*c^3*d*e^3*x^2-1/3*e^4*c^3*x^3-e^4*c*x-1/2*(c^4*d^4+4*c^3*d 
^3*e+6*c^2*d^2*e^2+4*c*d*e^3+e^4)*ln(c*x-1)+1/2*(c^4*d^4-4*c^3*d^3*e+6*c^2 
*d^2*e^2-4*c*d*e^3+e^4)*ln(c*x+1)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (340) = 680\).

Time = 0.20 (sec) , antiderivative size = 782, normalized size of antiderivative = 2.18 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + 3/2*a^2*d^2*e*x^2 + 3/2*(2*x^2*arctanh(c 
*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a*b*d^2*e + (2*x^ 
3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*d*e^2 + 1/12*(6*x 
^4*arctanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c* 
x - 1)/c^5))*a*b*e^3 + a^2*d^3*x + (2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1) 
)*a*b*d^3/c + (c^2*d^3 + d*e^2)*(log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog( 
1/2*c*x + 1/2))*b^2/c^3 + 1/6*(9*c^2*d^2*e - 3*c*d*e^2 + 2*e^3)*b^2*log(c* 
x + 1)/c^4 + 1/6*(9*c^2*d^2*e + 3*c*d*e^2 + 2*e^3)*b^2*log(c*x - 1)/c^4 + 
1/48*(4*b^2*c^2*e^3*x^2 + 48*b^2*c^2*d*e^2*x + 3*(b^2*c^4*e^3*x^4 + 4*b^2* 
c^4*d*e^2*x^3 + 6*b^2*c^4*d^2*e*x^2 + 4*b^2*c^4*d^3*x + (4*c^3*d^3 - 6*c^2 
*d^2*e + 4*c*d*e^2 - e^3)*b^2)*log(c*x + 1)^2 + 3*(b^2*c^4*e^3*x^4 + 4*b^2 
*c^4*d*e^2*x^3 + 6*b^2*c^4*d^2*e*x^2 + 4*b^2*c^4*d^3*x - (4*c^3*d^3 + 6*c^ 
2*d^2*e + 4*c*d*e^2 + e^3)*b^2)*log(-c*x + 1)^2 + 4*(b^2*c^3*e^3*x^3 + 6*b 
^2*c^3*d*e^2*x^2 + 3*(6*c^3*d^2*e + c*e^3)*b^2*x)*log(c*x + 1) - 2*(2*b^2* 
c^3*e^3*x^3 + 12*b^2*c^3*d*e^2*x^2 + 6*(6*c^3*d^2*e + c*e^3)*b^2*x + 3*(b^ 
2*c^4*e^3*x^4 + 4*b^2*c^4*d*e^2*x^3 + 6*b^2*c^4*d^2*e*x^2 + 4*b^2*c^4*d^3* 
x + (4*c^3*d^3 - 6*c^2*d^2*e + 4*c*d*e^2 - e^3)*b^2)*log(c*x + 1))*log(-c* 
x + 1))/c^4
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x)^3 (a+b \text {arctanh}(c x))^2 \, dx=\frac {8 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} e^{3}+8 \mathit {atanh} \left (c x \right ) b^{2} e^{3}-3 \mathit {atanh} \left (c x \right )^{2} b^{2} e^{3}+24 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{2} c^{3} d \,e^{2}+12 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} d^{3} x +3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} e^{3} x^{4}-18 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} d^{2} e +24 \mathit {atanh} \left (c x \right ) a b \,c^{3} d^{3}+2 \mathit {atanh} \left (c x \right ) b^{2} c^{3} e^{3} x^{3}+36 \mathit {atanh} \left (c x \right ) b^{2} c^{2} d^{2} e -12 \mathit {atanh} \left (c x \right ) b^{2} c d \,e^{2}+6 \mathit {atanh} \left (c x \right ) b^{2} c \,e^{3} x +24 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{3} d^{3}+36 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{2} d^{2} e +18 a^{2} c^{4} d^{2} e \,x^{2}+12 a^{2} c^{4} d \,e^{2} x^{3}+2 a b \,c^{3} e^{3} x^{3}+6 a b c \,e^{3} x +12 b^{2} c^{2} d \,e^{2} x +36 \mathit {atanh} \left (c x \right ) a b \,c^{4} d^{2} e \,x^{2}+b^{2} c^{2} e^{3} x^{2}-6 \mathit {atanh} \left (c x \right ) a b \,e^{3}+12 a^{2} c^{4} d^{3} x +3 a^{2} c^{4} e^{3} x^{4}+18 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} d^{2} e \,x^{2}+12 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{4} d \,e^{2} x^{3}+24 \mathit {atanh} \left (c x \right ) a b \,c^{4} d^{3} x +6 \mathit {atanh} \left (c x \right ) a b \,c^{4} e^{3} x^{4}-36 \mathit {atanh} \left (c x \right ) a b \,c^{2} d^{2} e +24 \mathit {atanh} \left (c x \right ) a b c d \,e^{2}+36 \mathit {atanh} \left (c x \right ) b^{2} c^{3} d^{2} e x +12 \mathit {atanh} \left (c x \right ) b^{2} c^{3} d \,e^{2} x^{2}+24 \,\mathrm {log}\left (c^{2} x -c \right ) a b c d \,e^{2}+36 a b \,c^{3} d^{2} e x +12 a b \,c^{3} d \,e^{2} x^{2}+24 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{2} c^{5} d^{3}+24 \mathit {atanh} \left (c x \right ) a b \,c^{4} d \,e^{2} x^{3}}{12 c^{4}} \] Input:

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

Output:

(12*atanh(c*x)**2*b**2*c**4*d**3*x + 18*atanh(c*x)**2*b**2*c**4*d**2*e*x** 
2 + 12*atanh(c*x)**2*b**2*c**4*d*e**2*x**3 + 3*atanh(c*x)**2*b**2*c**4*e** 
3*x**4 - 18*atanh(c*x)**2*b**2*c**2*d**2*e - 3*atanh(c*x)**2*b**2*e**3 + 2 
4*atanh(c*x)*a*b*c**4*d**3*x + 36*atanh(c*x)*a*b*c**4*d**2*e*x**2 + 24*ata 
nh(c*x)*a*b*c**4*d*e**2*x**3 + 6*atanh(c*x)*a*b*c**4*e**3*x**4 + 24*atanh( 
c*x)*a*b*c**3*d**3 - 36*atanh(c*x)*a*b*c**2*d**2*e + 24*atanh(c*x)*a*b*c*d 
*e**2 - 6*atanh(c*x)*a*b*e**3 + 36*atanh(c*x)*b**2*c**3*d**2*e*x + 12*atan 
h(c*x)*b**2*c**3*d*e**2*x**2 + 2*atanh(c*x)*b**2*c**3*e**3*x**3 + 36*atanh 
(c*x)*b**2*c**2*d**2*e - 12*atanh(c*x)*b**2*c*d*e**2 + 6*atanh(c*x)*b**2*c 
*e**3*x + 8*atanh(c*x)*b**2*e**3 + 24*int((atanh(c*x)*x)/(c**2*x**2 - 1),x 
)*b**2*c**5*d**3 + 24*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b**2*c**3*d*e* 
*2 + 24*log(c**2*x - c)*a*b*c**3*d**3 + 24*log(c**2*x - c)*a*b*c*d*e**2 + 
36*log(c**2*x - c)*b**2*c**2*d**2*e + 8*log(c**2*x - c)*b**2*e**3 + 12*a** 
2*c**4*d**3*x + 18*a**2*c**4*d**2*e*x**2 + 12*a**2*c**4*d*e**2*x**3 + 3*a* 
*2*c**4*e**3*x**4 + 36*a*b*c**3*d**2*e*x + 12*a*b*c**3*d*e**2*x**2 + 2*a*b 
*c**3*e**3*x**3 + 6*a*b*c*e**3*x + 12*b**2*c**2*d*e**2*x + b**2*c**2*e**3* 
x**2)/(12*c**4)