\(\int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx\) [80]

Optimal result

Integrand size = 22, antiderivative size = 278 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=a b c d^2 x+b^2 c d^2 x \text {arctanh}(c x)+\frac {3}{2} d^2 (a+b \text {arctanh}(c x))^2+2 c d^2 x (a+b \text {arctanh}(c x))^2+\frac {1}{2} c^2 d^2 x^2 (a+b \text {arctanh}(c x))^2+2 d^2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-4 b d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^2 \log \left (1-c^2 x^2\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.17 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=\frac {1}{2} d^2 \left (4 a^2 c x+a^2 c^2 x^2+2 a^2 \log (c x)+a b \left (2 c x+2 c^2 x^2 \text {arctanh}(c x)+\log (1-c x)-\log (1+c x)\right )+4 a b \left (2 c x \text {arctanh}(c x)+\log \left (1-c^2 x^2\right )\right )+b^2 \left (2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+\log \left (1-c^2 x^2\right )\right )+4 b^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+2 a b (-\operatorname {PolyLog}(2,-c x)+\operatorname {PolyLog}(2,c x))+2 b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right ) \] Input:

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

Output:

(d^2*(4*a^2*c*x + a^2*c^2*x^2 + 2*a^2*Log[c*x] + a*b*(2*c*x + 2*c^2*x^2*Ar 
cTanh[c*x] + Log[1 - c*x] - Log[1 + c*x]) + 4*a*b*(2*c*x*ArcTanh[c*x] + Lo 
g[1 - c^2*x^2]) + b^2*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 
+ Log[1 - c^2*x^2]) + 4*b^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log 
[1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 2*a*b*(-P 
olyLog[2, -(c*x)] + PolyLog[2, c*x]) + 2*b^2*((I/24)*Pi^3 - (2*ArcTanh[c*x 
]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[ 
1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 
ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c* 
x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2)))/2
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

a*b*c*d^2*x + b^2*c*d^2*x*ArcTanh[c*x] + (3*d^2*(a + b*ArcTanh[c*x])^2)/2 
+ 2*c*d^2*x*(a + b*ArcTanh[c*x])^2 + (c^2*d^2*x^2*(a + b*ArcTanh[c*x])^2)/ 
2 + 2*d^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 4*b*d^2*(a + b 
*ArcTanh[c*x])*Log[2/(1 - c*x)] + (b^2*d^2*Log[1 - c^2*x^2])/2 - 2*b^2*d^2 
*PolyLog[2, 1 - 2/(1 - c*x)] - b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2 
/(1 - c*x)] + b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b 
^2*d^2*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*d^2*PolyLog[3, -1 + 2/(1 - c* 
x)])/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 [C] (warning: unable to verify)

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

Time = 2.45 (sec) , antiderivative size = 895, normalized size of antiderivative = 3.22

method	result	size
parts	\(\text {Expression too large to display}\)	\(895\)
derivativedivides	\(\text {Expression too large to display}\)	\(897\)
default	\(\text {Expression too large to display}\)	\(897\)

Input:

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

Output:

d^2*a^2*(1/2*c^2*x^2+2*c*x+ln(x))+d^2*b^2*(1/2*arctanh(c*x)^2*c^2*x^2+2*ar 
ctanh(c*x)^2*c*x+arctanh(c*x)^2*ln(c*x)-arctanh(c*x)^2*ln((c*x+1)^2/(-c^2* 
x^2+1)-1)+arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*p 
olylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2 
))+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylog( 
2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-ar 
ctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/2*polylog(3,-(c*x+1)^2/(-c 
^2*x^2+1))+1/2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/( 
c^2*x^2-1)-1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)) 
)*arctanh(c*x)^2-4*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-4*dilog(1-I*(c*x+ 
1)/(-c^2*x^2+1)^(1/2))+3/2*arctanh(c*x)^2+(c*x+1)*arctanh(c*x)-1/2*I*Pi*cs 
gn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x 
+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2-ln(1+(c*x+1)^2/(-c^2*x^2+1))-4*arctan 
h(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-4*arctanh(c*x)*ln(1-I*(c*x+1)/(- 
c^2*x^2+1)^(1/2))-1/2*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(-(c*x 
+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2+1/2*I*Pi* 
csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*x 
)^2)+2*d^2*a*b*(1/2*arctanh(c*x)*c^2*x^2+2*arctanh(c*x)*c*x+arctanh(c*x)*l 
n(c*x)-1/2*dilog(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1)+1/2*c*x+5/4*l 
n(c*x-1)+3/4*ln(c*x+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

d**2*(Integral(2*a**2*c, x) + Integral(a**2/x, x) + Integral(a**2*c**2*x, 
x) + Integral(2*b**2*c*atanh(c*x)**2, x) + Integral(b**2*atanh(c*x)**2/x, 
x) + Integral(4*a*b*c*atanh(c*x), x) + Integral(2*a*b*atanh(c*x)/x, x) + I 
ntegral(b**2*c**2*x*atanh(c*x)**2, x) + Integral(2*a*b*c**2*x*atanh(c*x), 
x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(d**2*(atanh(c*x)**2*b**2*c**2*x**2 - atanh(c*x)**2*b**2 + 2*atanh(c*x)*a* 
b*c**2*x**2 + 8*atanh(c*x)*a*b*c*x + 6*atanh(c*x)*a*b + 2*atanh(c*x)*b**2* 
c*x + 2*atanh(c*x)*b**2 + 4*int(atanh(c*x)**2,x)*b**2*c + 4*int(atanh(c*x) 
/x,x)*a*b + 2*int(atanh(c*x)**2/x,x)*b**2 + 8*log(c**2*x - c)*a*b + 2*log( 
c**2*x - c)*b**2 + 2*log(x)*a**2 + a**2*c**2*x**2 + 4*a**2*c*x + 2*a*b*c*x 
))/2