\(\int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^2} \, dx\) [81]

Optimal result

Integrand size = 22, antiderivative size = 283 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=2 c d^2 (a+b \text {arctanh}(c x))^2-\frac {d^2 (a+b \text {arctanh}(c x))^2}{x}+c^2 d^2 x (a+b \text {arctanh}(c x))^2+4 c d^2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-2 b c d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+2 b c d^2 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b^2 c d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-2 b c d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+2 b c d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-b^2 c d^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+b^2 c d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-b^2 c d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.20 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\frac {d^2 \left (-12 a^2+i b^2 c \pi ^3 x+12 a^2 c^2 x^2-24 a b \text {arctanh}(c x)+24 a b c^2 x^2 \text {arctanh}(c x)-12 b^2 \text {arctanh}(c x)^2+12 b^2 c^2 x^2 \text {arctanh}(c x)^2-16 b^2 c x \text {arctanh}(c x)^3+24 b^2 c x \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-24 b^2 c x \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-24 b^2 c x \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c x \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+24 a^2 c x \log (x)+24 a b c x \log (c x)+12 b^2 c x (1+2 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+24 b^2 c x \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-24 a b c x \operatorname {PolyLog}(2,-c x)+24 a b c x \operatorname {PolyLog}(2,c x)+12 b^2 c x \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-12 b^2 c x \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{12 x} \] Input:

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

Output:

(d^2*(-12*a^2 + I*b^2*c*Pi^3*x + 12*a^2*c^2*x^2 - 24*a*b*ArcTanh[c*x] + 24 
*a*b*c^2*x^2*ArcTanh[c*x] - 12*b^2*ArcTanh[c*x]^2 + 12*b^2*c^2*x^2*ArcTanh 
[c*x]^2 - 16*b^2*c*x*ArcTanh[c*x]^3 + 24*b^2*c*x*ArcTanh[c*x]*Log[1 - E^(- 
2*ArcTanh[c*x])] - 24*b^2*c*x*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 
24*b^2*c*x*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTan 
h[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*a^2*c*x*Log[x] + 24*a*b*c*x*Log[ 
c*x] + 12*b^2*c*x*(1 + 2*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 
12*b^2*c*x*PolyLog[2, E^(-2*ArcTanh[c*x])] + 24*b^2*c*x*ArcTanh[c*x]*PolyL 
og[2, E^(2*ArcTanh[c*x])] - 24*a*b*c*x*PolyLog[2, -(c*x)] + 24*a*b*c*x*Pol 
yLog[2, c*x] + 12*b^2*c*x*PolyLog[3, -E^(-2*ArcTanh[c*x])] - 12*b^2*c*x*Po 
lyLog[3, E^(2*ArcTanh[c*x])]))/(12*x)
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 283, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 2.18 (sec) , antiderivative size = 2519, normalized size of antiderivative = 8.90

method	result	size
parts	\(\text {Expression too large to display}\)	\(2519\)
derivativedivides	\(\text {Expression too large to display}\)	\(2521\)
default	\(\text {Expression too large to display}\)	\(2521\)

Input:

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

Output:

d^2*a^2*(c^2*x+2*c*ln(x)-1/x)+d^2*b^2*c*(2*arctanh(c*x)^2*ln(1-(c*x+1)/(-c 
^2*x^2+1)^(1/2))+4*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+2*ar 
ctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+4*arctanh(c*x)*polylog(2,-(c 
*x+1)/(-c^2*x^2+1)^(1/2))-2*arctanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-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)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+ 
1)^(1/2))+arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+I*(c*x+1 
)/(-c^2*x^2+1)^(1/2))+dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-dilog((c*x+1)/ 
(-c^2*x^2+1)^(1/2))+dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2)))+2*arctanh(c*x)^2* 
ln(c*x)+2*arctanh(c*x)*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*ln( 
1+(c*x+1)/(-c^2*x^2+1)^(1/2))-2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))+ 
arctanh(c*x)^2*c*x-polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+polylog(3,-(c*x+1)^2 
/(-c^2*x^2+1))-2*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+2*polylog 
(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+2*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-4* 
polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2))-4*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1 
/2))-arctanh(c*x)^2/c/x-1/4*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)))*(2*arctanh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-4*arctanh( 
c*x)^2+2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))+2*arctanh(c*x)*ln(1-...
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a^2*c^2*d^2*x - 1/2*b^2*c^2*d^2*integrate(log(c*x + 1)*log(-c*x + 1), x) + 
 1/4*b^2*c^2*d^2*integrate(log(c*x + 1)^2/(c^2*x^2), x) + (2*c*x*arctanh(c 
*x) + log(-c^2*x^2 + 1))*a*b*c*d^2 + 1/2*(c*x - (c*x - 1)*log(-c*x + 1) - 
1)*b^2*c*d^2 + 1/4*b^2*c*d^2*gamma(3, -log(c*x + 1)) + 1/2*b^2*c*d^2*integ 
rate(log(c*x + 1)^2/x, x) - b^2*c*d^2*integrate(log(c*x + 1)*log(-c*x + 1) 
/x, x) + 2*a*b*c*d^2*integrate(log(c*x + 1)/x, x) - 2*a*b*c*d^2*integrate( 
log(-c*x + 1)/x, x) - 1/2*b^2*c*d^2*integrate(log(-c*x + 1)/x, x) + 2*a^2* 
c*d^2*log(x) - (c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b*d^ 
2 - 1/2*b^2*d^2*integrate(log(c*x + 1)*log(-c*x + 1)/x^2, x) - a^2*d^2/x + 
 1/4*(b^2*c^2*d^2*x^2 - b^2*d^2)*log(-c*x + 1)^2/x
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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