\(\int x (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx\) [70]

Optimal result

Integrand size = 18, antiderivative size = 196 \[ \int x (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}-\frac {b^2 d \text {arctanh}(c x)}{3 c^2}+\frac {b^2 d x \text {arctanh}(c x)}{c}+\frac {1}{3} b d x^2 (a+b \text {arctanh}(c x))-\frac {d (a+b \text {arctanh}(c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \text {arctanh}(c x))^2+\frac {1}{3} c d x^3 (a+b \text {arctanh}(c x))^2-\frac {2 b d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03 \[ \int x (d+c d x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {d \left (6 a b c x+2 b^2 c x+3 a^2 c^2 x^2+2 a b c^2 x^2+2 a^2 c^3 x^3+b^2 \left (-5+3 c^2 x^2+2 c^3 x^3\right ) \text {arctanh}(c x)^2+2 b \text {arctanh}(c x) \left (a c^2 x^2 (3+2 c x)+b \left (-1+3 c x+c^2 x^2\right )-2 b \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+3 a b \log (1-c x)-3 a b \log (1+c x)+3 b^2 \log \left (1-c^2 x^2\right )+2 a b \log \left (-1+c^2 x^2\right )+2 b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )}{6 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6502, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.27

method	result	size
parts	\(d \,a^{2} \left (\frac {1}{3} c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\operatorname {arctanh}\left (c x \right ) c x +\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}+\frac {5 \ln \left (c x -1\right )^{2}}{24}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x +1\right )^{2}}{24}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {c x}{3}+\frac {2 \ln \left (c x -1\right )}{3}+\frac {\ln \left (c x +1\right )}{3}\right )}{c^{2}}+\frac {2 d a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {c^{2} x^{2}}{6}+\frac {c x}{2}+\frac {5 \ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{2}}\)	\(249\)
derivativedivides	\(\frac {d \,a^{2} \left (\frac {1}{3} x^{3} c^{3}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\operatorname {arctanh}\left (c x \right ) c x +\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}+\frac {5 \ln \left (c x -1\right )^{2}}{24}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x +1\right )^{2}}{24}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {c x}{3}+\frac {2 \ln \left (c x -1\right )}{3}+\frac {\ln \left (c x +1\right )}{3}\right )+2 d a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {c^{2} x^{2}}{6}+\frac {c x}{2}+\frac {5 \ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{2}}\)	\(252\)
default	\(\frac {d \,a^{2} \left (\frac {1}{3} x^{3} c^{3}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{3}+\operatorname {arctanh}\left (c x \right ) c x +\frac {5 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{6}+\frac {5 \ln \left (c x -1\right )^{2}}{24}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {5 \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (c x +1\right )^{2}}{24}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {c x}{3}+\frac {2 \ln \left (c x -1\right )}{3}+\frac {\ln \left (c x +1\right )}{3}\right )+2 d a b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {c^{2} x^{2}}{6}+\frac {c x}{2}+\frac {5 \ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{2}}\)	\(252\)
risch	\(\frac {d \,b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{2}}+\left (-\frac {d \,b^{2} x^{2} \left (2 c x +3\right ) \ln \left (-c x +1\right )}{12}-\frac {d b \left (-4 a \,c^{3} x^{3}-6 a \,c^{2} x^{2}-2 b \,c^{2} x^{2}-6 b c x -5 b \ln \left (-c x +1\right )\right )}{12 c^{2}}\right ) \ln \left (c x +1\right )-\frac {d \,b^{2}}{3 c^{2}}+\frac {2 d \,b^{2} \ln \left (-c x +1\right )}{3 c^{2}}-\frac {d b \ln \left (-c x -1\right ) a}{6 c^{2}}+\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{2}}-\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{3 c^{2}}-\frac {4 d b a}{3 c^{2}}+\frac {d c \,b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}-\frac {d c a b \ln \left (-c x +1\right ) x^{3}}{3}+\frac {d \,b^{2} \ln \left (-c x -1\right )}{3 c^{2}}+\frac {d \,b^{2} \ln \left (-c x +1\right )^{2} x^{2}}{8}-\frac {d \,b^{2} \ln \left (-c x +1\right ) x^{2}}{6}+\frac {d b a \,x^{2}}{3}-\frac {5 d \,b^{2} \ln \left (-c x +1\right )^{2}}{24 c^{2}}+\frac {d c \,x^{3} a^{2}}{3}-\frac {d \,b^{2} \ln \left (-c x +1\right ) x}{2 c}-\frac {d a b \ln \left (-c x +1\right ) x^{2}}{2}+\frac {5 d a b \ln \left (-c x +1\right )}{6 c^{2}}+\frac {d \,b^{2} \left (2 x^{3} c^{3}+3 c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{2}}{24 c^{2}}-\frac {5 d \,a^{2}}{6 c^{2}}+\frac {d \,x^{2} a^{2}}{2}+\frac {b^{2} d x}{3 c}+\frac {a b d x}{c}\)	\(431\)

Input:

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

Output:

d*a^2*(1/3*c*x^3+1/2*x^2)+d*b^2/c^2*(1/3*arctanh(c*x)^2*c^3*x^3+1/2*arctan 
h(c*x)^2*c^2*x^2+1/3*arctanh(c*x)*c^2*x^2+arctanh(c*x)*c*x+5/6*arctanh(c*x 
)*ln(c*x-1)-1/6*arctanh(c*x)*ln(c*x+1)+5/24*ln(c*x-1)^2-1/3*dilog(1/2*c*x+ 
1/2)-5/12*ln(c*x-1)*ln(1/2*c*x+1/2)+1/24*ln(c*x+1)^2-1/12*(ln(c*x+1)-ln(1/ 
2*c*x+1/2))*ln(-1/2*c*x+1/2)+1/3*c*x+2/3*ln(c*x-1)+1/3*ln(c*x+1))+2*d*a*b/ 
c^2*(1/3*arctanh(c*x)*c^3*x^3+1/2*arctanh(c*x)*c^2*x^2+1/6*c^2*x^2+1/2*c*x 
+5/12*ln(c*x-1)-1/12*ln(c*x+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*a^2*c*d*x^3 + 1/2*b^2*d*x^2*arctanh(c*x)^2 + 1/3*(2*x^3*arctanh(c*x) + 
 c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*a*b*c*d - 1/216*(2*c^4*(2*(c^2*x^3 + 
3*x)/c^6 - 3*log(c*x + 1)/c^7 + 3*log(c*x - 1)/c^7) - 3*c^3*(x^2/c^4 + log 
(c^2*x^2 - 1)/c^6) - 648*c^3*integrate(1/9*x^3*log(c*x + 1)/(c^4*x^2 - c^2 
), x) + 9*c^2*(2*x/c^4 - log(c*x + 1)/c^5 + log(c*x - 1)/c^5) - 324*c*inte 
grate(1/9*x*log(c*x + 1)/(c^4*x^2 - c^2), x) - 6*(3*c^3*x^3*log(c*x + 1)^2 
 + (2*c^3*x^3 - 3*c^2*x^2 + 6*c*x - 6*(c^3*x^3 + 1)*log(c*x + 1))*log(-c*x 
 + 1))/c^3 - (2*(c*x - 1)^3*(9*log(-c*x + 1)^2 - 6*log(-c*x + 1) + 2) + 27 
*(c*x - 1)^2*(2*log(-c*x + 1)^2 - 2*log(-c*x + 1) + 1) + 54*(c*x - 1)*(log 
(-c*x + 1)^2 - 2*log(-c*x + 1) + 2))/c^3 + 18*log(9*c^4*x^2 - 9*c^2)/c^3 - 
 324*integrate(1/9*log(c*x + 1)/(c^4*x^2 - c^2), x))*b^2*c*d + 1/2*a^2*d*x 
^2 + 1/2*(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 + 1/8*(4*c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*a 
rctanh(c*x) - (2*(log(c*x - 1) - 2)*log(c*x + 1) - log(c*x + 1)^2 - log(c* 
x - 1)^2 - 4*log(c*x - 1))/c^2)*b^2*d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d*(2*atanh(c*x)**2*b**2*c**3*x**3 + 3*atanh(c*x)**2*b**2*c**2*x**2 - 2*at 
anh(c*x)**2*b**2*c*x - 3*atanh(c*x)**2*b**2 + 4*atanh(c*x)*a*b*c**3*x**3 + 
 6*atanh(c*x)*a*b*c**2*x**2 - 2*atanh(c*x)*a*b + 2*atanh(c*x)*b**2*c**2*x* 
*2 + 6*atanh(c*x)*b**2*c*x + 4*atanh(c*x)*b**2 + 2*int(atanh(c*x)**2,x)*b* 
*2*c + 4*log(c**2*x - c)*a*b + 6*log(c**2*x - c)*b**2 + 2*a**2*c**3*x**3 + 
 3*a**2*c**2*x**2 + 2*a*b*c**2*x**2 + 6*a*b*c*x + 2*b**2*c*x))/(6*c**2)