\(\int (d+e x)^2 (a+b \text {arctanh}(c x))^2 \, dx\) [10]

Optimal result

Integrand size = 18, antiderivative size = 257 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \text {arctanh}(c x)}{3 c^3}+\frac {2 b^2 d e x \text {arctanh}(c x)}{c}+\frac {b e^2 x^2 (a+b \text {arctanh}(c x))}{3 c}+\frac {\left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) (a+b \text {arctanh}(c x))^2}{3 e}+\frac {(d+e x)^3 (a+b \text {arctanh}(c x))^2}{3 e}-\frac {2 b \left (3 c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^3}+\frac {b^2 d e \log \left (1-c^2 x^2\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.24 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {3 a^2 c^3 d^2 x+6 a b c^2 d e x+b^2 c e^2 x+3 a^2 c^3 d e x^2+a b c^2 e^2 x^2+a^2 c^3 e^2 x^3+b^2 (-1+c x) \left (e^2+c e (3 d+e x)+c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (b e \left (-e+6 c^2 d x+c^2 e x^2\right )+2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-2 b \left (3 c^2 d^2+e^2\right ) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 a b c d e \log (1-c x)-3 a b c d e \log (1+c x)+3 a b c^2 d^2 \log \left (1-c^2 x^2\right )+3 b^2 c d e \log \left (1-c^2 x^2\right )+a b e^2 \log \left (-1+c^2 x^2\right )+b^2 \left (3 c^2 d^2+e^2\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{3 c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05, 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.

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

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((d + e*x)^3*(a + b*ArcTanh[c*x])^2)/(3*e) - (2*b*c*((-3*a*d*e^2*x)/c^2 - 
(b*e^3*x)/(2*c^3) + (b*e^3*ArcTanh[c*x])/(2*c^4) - (3*b*d*e^2*x*ArcTanh[c* 
x])/c^2 - (e^3*x^2*(a + b*ArcTanh[c*x]))/(2*c^2) - (e*(3*c^2*d^2 + e^2)*(a 
 + b*ArcTanh[c*x])^2)/(2*b*c^4) + (d*(c^2*d^2 + 3*e^2)*(a + b*ArcTanh[c*x] 
)^2)/(2*b*c^3) + (e*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x) 
])/c^4 - (3*b*d*e^2*Log[1 - c^2*x^2])/(2*c^3) + (b*e*(3*c^2*d^2 + e^2)*Pol 
yLog[2, 1 - 2/(1 - c*x)])/(2*c^4)))/(3*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. \(589\) vs. \(2(241)=482\).

Time = 0.20 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.30

method	result	size
parts	\(\frac {a^{2} \left (e x +d \right )^{3}}{3 e}+\frac {b^{2} \left (\frac {c \,e^{2} \operatorname {arctanh}\left (c x \right )^{2} x^{3}}{3}+c e \operatorname {arctanh}\left (c x \right )^{2} x^{2} d +\operatorname {arctanh}\left (c x \right )^{2} c x \,d^{2}+\frac {c \operatorname {arctanh}\left (c x \right )^{2} d^{3}}{3 e}-\frac {2 \left (-3 \,\operatorname {arctanh}\left (c x \right ) c^{2} d \,e^{2} x -\frac {\operatorname {arctanh}\left (c x \right ) e^{3} c^{2} x^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{3} d^{3}}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{2} d^{2} e}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d \,e^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e^{3}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{3} d^{3}}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{2} d^{2} e}{2}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d \,e^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e^{3}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\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 )}{2}-\frac {e^{2} \left (c e x +\frac {\left (6 c d +e \right ) \ln \left (c x -1\right )}{2}-\frac {\left (-6 c d +e \right ) \ln \left (c x +1\right )}{2}\right )}{2}-\frac {\left (-c^{3} d^{3}+3 c^{2} d^{2} e -3 c d \,e^{2}+e^{3}\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 )}{2}\right )}{3 c^{2} e}\right )}{c}+\frac {2 a b \left (\frac {c \,e^{2} \operatorname {arctanh}\left (c x \right ) x^{3}}{3}+c e \,\operatorname {arctanh}\left (c x \right ) x^{2} d +\operatorname {arctanh}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arctanh}\left (c x \right ) d^{3}}{3 e}-\frac {-3 c^{2} d \,e^{2} x -\frac {e^{3} c^{2} x^{2}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \ln \left (c x +1\right )}{2}}{3 c^{2} e}\right )}{c}\)	\(590\)
derivativedivides	\(\frac {\frac {a^{2} \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} d^{3}}{3 e}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} d^{2} x +e \operatorname {arctanh}\left (c x \right )^{2} c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \left (-3 \,\operatorname {arctanh}\left (c x \right ) c^{2} d \,e^{2} x -\frac {\operatorname {arctanh}\left (c x \right ) e^{3} c^{2} x^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{3} d^{3}}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{2} d^{2} e}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d \,e^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e^{3}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{3} d^{3}}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{2} d^{2} e}{2}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d \,e^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e^{3}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\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 )}{2}-\frac {e^{2} \left (c e x +\frac {\left (6 c d +e \right ) \ln \left (c x -1\right )}{2}-\frac {\left (-6 c d +e \right ) \ln \left (c x +1\right )}{2}\right )}{2}-\frac {\left (-c^{3} d^{3}+3 c^{2} d^{2} e -3 c d \,e^{2}+e^{3}\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 )}{2}\right )}{3 e}\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {-3 c^{2} d \,e^{2} x -\frac {e^{3} c^{2} x^{2}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \ln \left (c x +1\right )}{2}}{3 e}\right )}{c^{2}}}{c}\)	\(610\)
default	\(\frac {\frac {a^{2} \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} d^{3}}{3 e}+\operatorname {arctanh}\left (c x \right )^{2} c^{3} d^{2} x +e \operatorname {arctanh}\left (c x \right )^{2} c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \left (-3 \,\operatorname {arctanh}\left (c x \right ) c^{2} d \,e^{2} x -\frac {\operatorname {arctanh}\left (c x \right ) e^{3} c^{2} x^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{3} d^{3}}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c^{2} d^{2} e}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d \,e^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e^{3}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{3} d^{3}}{2}-\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c^{2} d^{2} e}{2}+\frac {3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d \,e^{2}}{2}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e^{3}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\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 )}{2}-\frac {e^{2} \left (c e x +\frac {\left (6 c d +e \right ) \ln \left (c x -1\right )}{2}-\frac {\left (-6 c d +e \right ) \ln \left (c x +1\right )}{2}\right )}{2}-\frac {\left (-c^{3} d^{3}+3 c^{2} d^{2} e -3 c d \,e^{2}+e^{3}\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 )}{2}\right )}{3 e}\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {-3 c^{2} d \,e^{2} x -\frac {e^{3} c^{2} x^{2}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \ln \left (c x +1\right )}{2}}{3 e}\right )}{c^{2}}}{c}\)	\(610\)
risch	\(\frac {2 a b d e x}{c}+\frac {b^{2} e^{2} x}{3 c^{2}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) e^{2}}{3 c^{3}}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right ) e^{2}}{3 c^{3}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{c}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right ) d^{2}}{c}-\frac {2 b a d e}{c^{2}}-\frac {b^{2} d e \ln \left (-c x +1\right )^{2}}{4 c^{2}}+\frac {b^{2} d e \ln \left (-c x +1\right )}{c^{2}}+\frac {\ln \left (-c x +1\right ) a b \,d^{2}}{c}+\frac {b a \,e^{2} \ln \left (-c x +1\right )}{3 c^{3}}-\frac {b a \,e^{2} \ln \left (-c x +1\right ) x^{3}}{3}+\frac {b^{2} d e \ln \left (-c x +1\right )^{2} x^{2}}{4}-\ln \left (-c x +1\right ) x a b \,d^{2}+\frac {b a \,e^{2} x^{2}}{3 c}-\frac {b^{2} e^{2} \ln \left (-c x +1\right ) x^{2}}{6 c}+\left (-\frac {\left (e x +d \right )^{3} b^{2} \ln \left (-c x +1\right )}{6 e}+\frac {b \left (2 a \,c^{3} e^{3} x^{3}+6 a \,c^{3} d \,e^{2} x^{2}+6 a \,c^{3} d^{2} e x +\ln \left (-c x +1\right ) b \,c^{3} d^{3}+b \,c^{2} e^{3} x^{2}+3 \ln \left (-c x +1\right ) b \,c^{2} d^{2} e +6 b \,c^{2} d \,e^{2} x +3 \ln \left (-c x +1\right ) b c d \,e^{2}+\ln \left (-c x +1\right ) b \,e^{3}\right )}{6 c^{3} e}\right ) \ln \left (c x +1\right )-\frac {d^{2} a^{2}}{c}-\frac {e^{2} a^{2}}{3 c^{3}}-\frac {b^{2} e^{2}}{3 c^{3}}+d^{2} a^{2} x +\frac {e^{2} a^{2} x^{3}}{3}-\frac {b^{2} \ln \left (-c x -1\right ) e^{2}}{6 c^{3}}+a^{2} d e \,x^{2}+\frac {\ln \left (-c x +1\right )^{2} x \,b^{2} d^{2}}{4}+\frac {b^{2} e^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}-\frac {b^{2} e^{2} \ln \left (-c x +1\right )^{2}}{12 c^{3}}+\frac {b^{2} e^{2} \ln \left (-c x +1\right )}{6 c^{3}}-\frac {\ln \left (-c x +1\right )^{2} b^{2} d^{2}}{4 c}-\frac {a^{2} d e}{c^{2}}+\frac {b^{2} \left (c^{3} e^{2} x^{3}+3 c^{3} d e \,x^{2}+3 c^{3} d^{2} x +3 c^{2} d^{2}-3 c d e +e^{2}\right ) \ln \left (c x +1\right )^{2}}{12 c^{3}}-\frac {b a \,e^{2}}{3 c^{3}}+\frac {b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right ) e^{2}}{3 c^{3}}+\frac {b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right ) d^{2}}{c}-\frac {b \ln \left (-c x -1\right ) a d e}{c^{2}}-b a d e \ln \left (-c x +1\right ) x^{2}+\frac {b a d e \ln \left (-c x +1\right )}{c^{2}}-\frac {b^{2} d e \ln \left (-c x +1\right ) x}{c}+\frac {b \ln \left (-c x -1\right ) a \,d^{2}}{c}+\frac {b^{2} \ln \left (-c x -1\right ) d e}{c^{2}}+\frac {b \ln \left (-c x -1\right ) a \,e^{2}}{3 c^{3}}\)	\(848\)

Input:

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

Output:

1/3*a^2*(e*x+d)^3/e+b^2/c*(1/3*c*e^2*arctanh(c*x)^2*x^3+c*e*arctanh(c*x)^2 
*x^2*d+arctanh(c*x)^2*c*x*d^2+1/3*c/e*arctanh(c*x)^2*d^3-2/3/c^2/e*(-3*arc 
tanh(c*x)*c^2*d*e^2*x-1/2*arctanh(c*x)*e^3*c^2*x^2-1/2*arctanh(c*x)*ln(c*x 
-1)*c^3*d^3-3/2*arctanh(c*x)*ln(c*x-1)*c^2*d^2*e-3/2*arctanh(c*x)*ln(c*x-1 
)*c*d*e^2-1/2*arctanh(c*x)*ln(c*x-1)*e^3+1/2*arctanh(c*x)*ln(c*x+1)*c^3*d^ 
3-3/2*arctanh(c*x)*ln(c*x+1)*c^2*d^2*e+3/2*arctanh(c*x)*ln(c*x+1)*c*d*e^2- 
1/2*arctanh(c*x)*ln(c*x+1)*e^3-1/2*(c^3*d^3+3*c^2*d^2*e+3*c*d*e^2+e^3)*(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))-1/2*e^ 
2*(c*e*x+1/2*(6*c*d+e)*ln(c*x-1)-1/2*(-6*c*d+e)*ln(c*x+1))-1/2*(-c^3*d^3+3 
*c^2*d^2*e-3*c*d*e^2+e^3)*(-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))))+2*a*b/c*(1/3*c*e^2*arctanh(c* 
x)*x^3+c*e*arctanh(c*x)*x^2*d+arctanh(c*x)*c*x*d^2+1/3*c/e*arctanh(c*x)*d^ 
3-1/3/c^2/e*(-3*c^2*d*e^2*x-1/2*e^3*c^2*x^2-1/2*(c^3*d^3+3*c^2*d^2*e+3*c*d 
*e^2+e^3)*ln(c*x-1)+1/2*(c^3*d^3-3*c^2*d^2*e+3*c*d*e^2-e^3)*ln(c*x+1)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (238) = 476\).

Time = 0.19 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.04 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} + {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b d e + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b e^{2} + a^{2} d^{2} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d^{2}}{c} + \frac {{\left (3 \, c^{2} d^{2} + e^{2}\right )} {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2}}{3 \, c^{3}} + \frac {{\left (6 \, c d e - e^{2}\right )} b^{2} \log \left (c x + 1\right )}{6 \, c^{3}} + \frac {{\left (6 \, c d e + e^{2}\right )} b^{2} \log \left (c x - 1\right )}{6 \, c^{3}} + \frac {4 \, b^{2} c e^{2} x + {\left (b^{2} c^{3} e^{2} x^{3} + 3 \, b^{2} c^{3} d e x^{2} + 3 \, b^{2} c^{3} d^{2} x + {\left (3 \, c^{2} d^{2} - 3 \, c d e + e^{2}\right )} b^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (b^{2} c^{3} e^{2} x^{3} + 3 \, b^{2} c^{3} d e x^{2} + 3 \, b^{2} c^{3} d^{2} x - {\left (3 \, c^{2} d^{2} + 3 \, c d e + e^{2}\right )} b^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (b^{2} c^{2} e^{2} x^{2} + 6 \, b^{2} c^{2} d e x\right )} \log \left (c x + 1\right ) - 2 \, {\left (b^{2} c^{2} e^{2} x^{2} + 6 \, b^{2} c^{2} d e x + {\left (b^{2} c^{3} e^{2} x^{3} + 3 \, b^{2} c^{3} d e x^{2} + 3 \, b^{2} c^{3} d^{2} x + {\left (3 \, c^{2} d^{2} - 3 \, c d e + e^{2}\right )} b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{12 \, c^{3}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*atanh(c*x)**2*b**2*c**3*d**2*x + 3*atanh(c*x)**2*b**2*c**3*d*e*x**2 + a 
tanh(c*x)**2*b**2*c**3*e**2*x**3 - 3*atanh(c*x)**2*b**2*c*d*e + 6*atanh(c* 
x)*a*b*c**3*d**2*x + 6*atanh(c*x)*a*b*c**3*d*e*x**2 + 2*atanh(c*x)*a*b*c** 
3*e**2*x**3 + 6*atanh(c*x)*a*b*c**2*d**2 - 6*atanh(c*x)*a*b*c*d*e + 2*atan 
h(c*x)*a*b*e**2 + 6*atanh(c*x)*b**2*c**2*d*e*x + atanh(c*x)*b**2*c**2*e**2 
*x**2 + 6*atanh(c*x)*b**2*c*d*e - atanh(c*x)*b**2*e**2 + 6*int((atanh(c*x) 
*x)/(c**2*x**2 - 1),x)*b**2*c**4*d**2 + 2*int((atanh(c*x)*x)/(c**2*x**2 - 
1),x)*b**2*c**2*e**2 + 6*log(c**2*x - c)*a*b*c**2*d**2 + 2*log(c**2*x - c) 
*a*b*e**2 + 6*log(c**2*x - c)*b**2*c*d*e + 3*a**2*c**3*d**2*x + 3*a**2*c** 
3*d*e*x**2 + a**2*c**3*e**2*x**3 + 6*a*b*c**2*d*e*x + a*b*c**2*e**2*x**2 + 
 b**2*c*e**2*x)/(3*c**3)