\(\int (d+e x) (a+b \text {arctanh}(c x))^3 \, dx\) [17]

Optimal result

Integrand size = 16, antiderivative size = 244 \[ \int (d+e x) (a+b \text {arctanh}(c x))^3 \, dx=\frac {3 b e (a+b \text {arctanh}(c x))^2}{2 c^2}+\frac {3 b e x (a+b \text {arctanh}(c x))^2}{2 c}+\frac {d (a+b \text {arctanh}(c x))^3}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) (a+b \text {arctanh}(c x))^3}{2 e}+\frac {(d+e x)^2 (a+b \text {arctanh}(c x))^3}{2 e}-\frac {3 b^2 e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b d (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c^2}-\frac {3 b^2 d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.36 \[ \int (d+e x) (a+b \text {arctanh}(c x))^3 \, dx=\frac {2 a^2 c (2 a c d+3 b e) x+2 a^3 c^2 e x^2+6 a^2 b c^2 x (2 d+e x) \text {arctanh}(c x)+3 a^2 b (2 c d+e) \log (1-c x)+3 a^2 b (2 c d-e) \log (1+c x)+6 a b^2 e \left (2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+\log \left (1-c^2 x^2\right )\right )-2 b^3 e \left (\text {arctanh}(c x) \left ((3-3 c x) \text {arctanh}(c x)+\left (1-c^2 x^2\right ) \text {arctanh}(c x)^2+6 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )-3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+12 a b^2 c d \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+4 b^3 c d \left (\text {arctanh}(c x)^2 \left ((-1+c x) \text {arctanh}(c x)-3 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )\right )}{4 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((d + e*x)^2*(a + b*ArcTanh[c*x])^3)/(2*e) - (3*b*c*(-((e^2*(a + b*ArcTanh 
[c*x])^2)/c^3) - (e^2*x*(a + b*ArcTanh[c*x])^2)/c^2 - (2*d*e*(a + b*ArcTan 
h[c*x])^3)/(3*b*c^2) + ((c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^3)/(3*b*c^3) 
+ (2*b*e^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^3 + (2*d*e*(a + b*ArcT 
anh[c*x])^2*Log[2/(1 - c*x)])/c^2 + (b^2*e^2*PolyLog[2, 1 - 2/(1 - c*x)])/ 
c^3 + (2*b*d*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^2 - (b^ 
2*d*e*PolyLog[3, 1 - 2/(1 - c*x)])/c^2))/(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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.79 (sec) , antiderivative size = 6347, normalized size of antiderivative = 26.01

method	result	size
parts	\(\text {Expression too large to display}\)	\(6347\)
derivativedivides	\(\text {Expression too large to display}\)	\(6355\)
default	\(\text {Expression too large to display}\)	\(6355\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*a^3*e*x^2 + 3/4*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + 
log(c*x - 1)/c^3))*a^2*b*e + a^3*d*x + 3/2*(2*c*x*arctanh(c*x) + log(-c^2* 
x^2 + 1))*a^2*b*d/c - 1/16*((b^3*c^2*e*x^2 + 2*b^3*c^2*d*x - (2*c*d + e)*b 
^3)*log(-c*x + 1)^3 - 3*(2*a*b^2*c^2*e*x^2 + 2*(2*a*b^2*c^2*d + b^3*c*e)*x 
 + (b^3*c^2*e*x^2 + 2*b^3*c^2*d*x + (2*c*d - e)*b^3)*log(c*x + 1))*log(-c* 
x + 1)^2)/c^2 - integrate(-1/8*((b^3*c^2*e*x^2 - b^3*c*d + (c^2*d - c*e)*b 
^3*x)*log(c*x + 1)^3 + 6*(a*b^2*c^2*e*x^2 - a*b^2*c*d + (c^2*d - c*e)*a*b^ 
2*x)*log(c*x + 1)^2 - 3*(2*a*b^2*c^2*e*x^2 + (b^3*c^2*e*x^2 - b^3*c*d + (c 
^2*d - c*e)*b^3*x)*log(c*x + 1)^2 + 2*(2*a*b^2*c^2*d + b^3*c*e)*x - (4*a*b 
^2*c*d - (2*c*d - e)*b^3 - (4*a*b^2*c^2*e + b^3*c^2*e)*x^2 - 2*(b^3*c^2*d 
+ 2*(c^2*d - c*e)*a*b^2)*x)*log(c*x + 1))*log(-c*x + 1))/(c^2*x - c), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*atanh(c*x)**3*b**3*c**2*d*x + atanh(c*x)**3*b**3*c**2*e*x**2 - atanh(c* 
x)**3*b**3*e + 6*atanh(c*x)**2*a*b**2*c**2*d*x + 3*atanh(c*x)**2*a*b**2*c* 
*2*e*x**2 - 3*atanh(c*x)**2*a*b**2*e + 3*atanh(c*x)**2*b**3*c*e*x + 6*atan 
h(c*x)*a**2*b*c**2*d*x + 3*atanh(c*x)*a**2*b*c**2*e*x**2 + 6*atanh(c*x)*a* 
*2*b*c*d - 3*atanh(c*x)*a**2*b*e + 6*atanh(c*x)*a*b**2*c*e*x + 6*atanh(c*x 
)*a*b**2*e + 12*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*a*b**2*c**3*d + 6*in 
t((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b**3*c**2*e + 6*int((atanh(c*x)**2*x)/ 
(c**2*x**2 - 1),x)*b**3*c**3*d + 6*log(c**2*x - c)*a**2*b*c*d + 6*log(c**2 
*x - c)*a*b**2*e + 2*a**3*c**2*d*x + a**3*c**2*e*x**2 + 3*a**2*b*c*e*x)/(2 
*c**2)