\(\int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx\) [82]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {b c d^2 (a+b \text {arctanh}(c x))}{x}+\frac {5}{2} c^2 d^2 (a+b \text {arctanh}(c x))^2-\frac {d^2 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {2 c d^2 (a+b \text {arctanh}(c x))^2}{x}+2 c^2 d^2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )+b^2 c^2 d^2 \log (x)-\frac {1}{2} b^2 c^2 d^2 \log \left (1-c^2 x^2\right )+4 b c^2 d^2 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b c^2 d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b c^2 d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-2 b^2 c^2 d^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:

-b*c*d^2*(a+b*arctanh(c*x))/x+5/2*c^2*d^2*(a+b*arctanh(c*x))^2-1/2*d^2*(a+ 
b*arctanh(c*x))^2/x^2-2*c*d^2*(a+b*arctanh(c*x))^2/x-2*c^2*d^2*(a+b*arctan 
h(c*x))^2*arctanh(-1+2/(-c*x+1))+b^2*c^2*d^2*ln(x)-1/2*b^2*c^2*d^2*ln(-c^2 
*x^2+1)+4*b*c^2*d^2*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b*c^2*d^2*(a+b*arct 
anh(c*x))*polylog(2,1-2/(-c*x+1))+b*c^2*d^2*(a+b*arctanh(c*x))*polylog(2,- 
1+2/(-c*x+1))-2*b^2*c^2*d^2*polylog(2,-1+2/(c*x+1))+1/2*b^2*c^2*d^2*polylo 
g(3,1-2/(-c*x+1))-1/2*b^2*c^2*d^2*polylog(3,-1+2/(-c*x+1))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.18 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=-\frac {d^2 \left (a^2+4 a^2 c x-2 a^2 c^2 x^2 \log (x)+a b (2 \text {arctanh}(c x)+c x (2+c x \log (1-c x)-c x \log (1+c x)))+b^2 \left (2 c x \text {arctanh}(c x)+\left (1-c^2 x^2\right ) \text {arctanh}(c x)^2-2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )+4 a b c x \left (2 \text {arctanh}(c x)+c x \left (-2 \log (c x)+\log \left (1-c^2 x^2\right )\right )\right )+4 b^2 c x \left (\text {arctanh}(c x) \left ((1-c x) \text {arctanh}(c x)-2 c x \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )+c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )+2 a b c^2 x^2 (\operatorname {PolyLog}(2,-c x)-\operatorname {PolyLog}(2,c x))-2 b^2 c^2 x^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right )}{2 x^2} \] Input:

Integrate[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^3,x]
 

Output:

-1/2*(d^2*(a^2 + 4*a^2*c*x - 2*a^2*c^2*x^2*Log[x] + a*b*(2*ArcTanh[c*x] + 
c*x*(2 + c*x*Log[1 - c*x] - c*x*Log[1 + c*x])) + b^2*(2*c*x*ArcTanh[c*x] + 
 (1 - c^2*x^2)*ArcTanh[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]) + 
4*a*b*c*x*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[1 - c^2*x^2])) + 4*b^2* 
c*x*(ArcTanh[c*x]*((1 - c*x)*ArcTanh[c*x] - 2*c*x*Log[1 - E^(-2*ArcTanh[c* 
x])]) + c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]) + 2*a*b*c^2*x^2*(PolyLog[2, - 
(c*x)] - PolyLog[2, c*x]) - 2*b^2*c^2*x^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])] + Arc 
Tanh[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)))/x^2
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 313, 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.

\(\displaystyle \int \frac {(c d x+d)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx\)

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

\(\displaystyle -b c^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b c^2 d^2 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+\frac {5}{2} c^2 d^2 (a+b \text {arctanh}(c x))^2+2 c^2 d^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2+4 b c^2 d^2 \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {d^2 (a+b \text {arctanh}(c x))^2}{2 x^2}-\frac {2 c d^2 (a+b \text {arctanh}(c x))^2}{x}-\frac {b c d^2 (a+b \text {arctanh}(c x))}{x}-2 b^2 c^2 d^2 \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^2 d^2 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )-\frac {1}{2} b^2 c^2 d^2 \log \left (1-c^2 x^2\right )+b^2 c^2 d^2 \log (x)\)

Input:

Int[((d + c*d*x)^2*(a + b*ArcTanh[c*x])^2)/x^3,x]
 

Output:

-((b*c*d^2*(a + b*ArcTanh[c*x]))/x) + (5*c^2*d^2*(a + b*ArcTanh[c*x])^2)/2 
 - (d^2*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (2*c*d^2*(a + b*ArcTanh[c*x])^2) 
/x + 2*c^2*d^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + b^2*c^2*d 
^2*Log[x] - (b^2*c^2*d^2*Log[1 - c^2*x^2])/2 + 4*b*c^2*d^2*(a + b*ArcTanh[ 
c*x])*Log[2 - 2/(1 + c*x)] - b*c^2*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 
 2/(1 - c*x)] + b*c^2*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x) 
] - 2*b^2*c^2*d^2*PolyLog[2, -1 + 2/(1 + c*x)] + (b^2*c^2*d^2*PolyLog[3, 1 
 - 2/(1 - c*x)])/2 - (b^2*c^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 = 4.34 (sec) , antiderivative size = 952, normalized size of antiderivative = 3.04

method	result	size
parts	\(\text {Expression too large to display}\)	\(952\)
derivativedivides	\(\text {Expression too large to display}\)	\(953\)
default	\(\text {Expression too large to display}\)	\(953\)

Input:

int((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

d^2*a^2*(c^2*ln(x)-2*c/x-1/2/x^2)+d^2*b^2*c^2*(-1/2*arctanh(c*x)^2/c^2/x^2 
+arctanh(c*x)^2*ln(c*x)-2*arctanh(c*x)^2/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)*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))+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*pol 
ylog(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)*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)^2-1/2*(c*x-(-c^2*x^2+1)^(1/2)+1)/c/x*arctanh(c*x)-1/2* 
arctanh(c*x)*(c*x+(-c^2*x^2+1)^(1/2)+1)/c/x+4*arctanh(c*x)*ln(1+(c*x+1)/(- 
c^2*x^2+1)^(1/2))-3/2*arctanh(c*x)^2-4*dilog((c*x+1)/(-c^2*x^2+1)^(1/2))+4 
*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+ln((c*x+1)/(-c^2*x^2+1)^(1/2)-1)+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*arctan 
h(c*x)^2-1/2*I*Pi*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)^2+ln(1+(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)+2*d^2*a*b* 
c^2*(-1/2*arctanh(c*x)/c^2/x^2+arctanh(c*x)*ln(c*x)-2*arctanh(c*x)/c/x-5/4 
*ln(c*x-1)-1/2/c/x+2*ln(c*x)-3/4*ln(c*x+1)-1/2*dilog(c*x)-1/2*dilog(c*x...
 

Fricas [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^3,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^3, x)
 

Sympy [F]

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

integrate((c*d*x+d)**2*(a+b*atanh(c*x))**2/x**3,x)
 

Output:

d**2*(Integral(a**2/x**3, x) + Integral(2*a**2*c/x**2, x) + Integral(a**2* 
c**2/x, x) + Integral(b**2*atanh(c*x)**2/x**3, x) + Integral(2*a*b*atanh(c 
*x)/x**3, x) + Integral(2*b**2*c*atanh(c*x)**2/x**2, x) + Integral(b**2*c* 
*2*atanh(c*x)**2/x, x) + Integral(4*a*b*c*atanh(c*x)/x**2, x) + Integral(2 
*a*b*c**2*atanh(c*x)/x, x))
 

Maxima [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:

integrate((c*d*x+d)^2*(a+b*arctanh(c*x))^2/x^3,x, algorithm="giac")
 

Output:

integrate((c*d*x + d)^2*(b*arctanh(c*x) + a)^2/x^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(((a + b*atanh(c*x))^2*(d + c*d*x)^2)/x^3,x)
 

Output:

int(((a + b*atanh(c*x))^2*(d + c*d*x)^2)/x^3, x)
 

Reduce [F]

\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x^3} \, dx=\frac {d^{2} \left (\mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-4 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -\mathit {atanh} \left (c x \right )^{2} b^{2}-6 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-8 \mathit {atanh} \left (c x \right ) a b c x -2 \mathit {atanh} \left (c x \right ) a b -2 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}-2 \mathit {atanh} \left (c x \right ) b^{2} c x -8 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c^{2} x^{2}+4 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+2 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}-8 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}+8 \,\mathrm {log}\left (x \right ) a b \,c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{2}-4 a^{2} c x -a^{2}-2 a b c x \right )}{2 x^{2}} \] Input:

int((c*d*x+d)^2*(a+b*atanh(c*x))^2/x^3,x)
 

Output:

(d**2*(atanh(c*x)**2*b**2*c**2*x**2 - 4*atanh(c*x)**2*b**2*c*x - atanh(c*x 
)**2*b**2 - 6*atanh(c*x)*a*b*c**2*x**2 - 8*atanh(c*x)*a*b*c*x - 2*atanh(c* 
x)*a*b - 2*atanh(c*x)*b**2*c**2*x**2 - 2*atanh(c*x)*b**2*c*x - 8*int(atanh 
(c*x)/(c**2*x**3 - x),x)*b**2*c**2*x**2 + 4*int(atanh(c*x)/x,x)*a*b*c**2*x 
**2 + 2*int(atanh(c*x)**2/x,x)*b**2*c**2*x**2 - 8*log(c**2*x - c)*a*b*c**2 
*x**2 - 2*log(c**2*x - c)*b**2*c**2*x**2 + 2*log(x)*a**2*c**2*x**2 + 8*log 
(x)*a*b*c**2*x**2 + 2*log(x)*b**2*c**2*x**2 - 4*a**2*c*x - a**2 - 2*a*b*c* 
x))/(2*x**2)