\(\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx\) [105]

Optimal result

Integrand size = 22, antiderivative size = 260 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=-\frac {b^2}{2 c^3 d^2 (1+c x)}+\frac {b^2 \text {arctanh}(c x)}{2 c^3 d^2}-\frac {b (a+b \text {arctanh}(c x))}{c^3 d^2 (1+c x)}+\frac {3 (a+b \text {arctanh}(c x))^2}{2 c^3 d^2}+\frac {x (a+b \text {arctanh}(c x))^2}{c^2 d^2}-\frac {(a+b \text {arctanh}(c x))^2}{c^3 d^2 (1+c x)}-\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^3 d^2}+\frac {2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3 d^2}-\frac {2 b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^3 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{c^3 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.13 \[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {4 a^2 c x-\frac {4 a^2}{1+c x}-8 a^2 \log (1+c x)+b^2 \left (-4 \text {arctanh}(c x)^2+4 c x \text {arctanh}(c x)^2-\cosh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))-8 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+8 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+(4-8 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-4 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )+2 a b \left (-\cosh (2 \text {arctanh}(c x))+2 \log \left (1-c^2 x^2\right )-4 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \left (2 c x-\cosh (2 \text {arctanh}(c x))+4 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))\right )\right )}{4 c^3 d^2} \] Input:

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

Output:

(4*a^2*c*x - (4*a^2)/(1 + c*x) - 8*a^2*Log[1 + c*x] + b^2*(-4*ArcTanh[c*x] 
^2 + 4*c*x*ArcTanh[c*x]^2 - Cosh[2*ArcTanh[c*x]] - 2*ArcTanh[c*x]*Cosh[2*A 
rcTanh[c*x]] - 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] - 8*ArcTanh[c*x]*Log[ 
1 + E^(-2*ArcTanh[c*x])] + 8*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 
 (4 - 8*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 4*PolyLog[3, -E^( 
-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]*Sinh[2*ArcTanh[c 
*x]] + 2*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 2*a*b*(-Cosh[2*ArcTanh[c*x 
]] + 2*Log[1 - c^2*x^2] - 4*PolyLog[2, -E^(-2*ArcTanh[c*x])] + Sinh[2*ArcT 
anh[c*x]] + 2*ArcTanh[c*x]*(2*c*x - Cosh[2*ArcTanh[c*x]] + 4*Log[1 + E^(-2 
*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]])))/(4*c^3*d^2)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 260, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*b^2/(c^3*d^2*(1 + c*x)) + (b^2*ArcTanh[c*x])/(2*c^3*d^2) - (b*(a + b* 
ArcTanh[c*x]))/(c^3*d^2*(1 + c*x)) + (3*(a + b*ArcTanh[c*x])^2)/(2*c^3*d^2 
) + (x*(a + b*ArcTanh[c*x])^2)/(c^2*d^2) - (a + b*ArcTanh[c*x])^2/(c^3*d^2 
*(1 + c*x)) - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^3*d^2) + (2*( 
a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d^2) - (b^2*PolyLog[2, 1 - 2/ 
(1 - c*x)])/(c^3*d^2) - (2*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c* 
x)])/(c^3*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^3*d^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 = 1.63 (sec) , antiderivative size = 2674, normalized size of antiderivative = 10.28

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2674\)
default	\(\text {Expression too large to display}\)	\(2674\)
parts	\(\text {Expression too large to display}\)	\(2684\)

Input:

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

Output:

1/c^3*(a^2/d^2*(c*x-1/(c*x+1)-2*ln(c*x+1))+b^2/d^2*(1/2*I*Pi*csgn(I*(c*x+1 
)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*c 
sgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*(2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+ 
1))-2*arctanh(c*x)^2+polylog(2,-(c*x+1)^2/(-c^2*x^2+1)))-2*arctanh(c*x)^2* 
ln(c*x+1)-3/2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-1/2*arctanh(c*x)*l 
n(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^ 
2+1)^(1/2))-ln(2)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+2*ln(2)*dilog(1+I*(c* 
x+1)/(-c^2*x^2+1)^(1/2))+2*ln(2)*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2 
*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^ 
(1/2))-1/2*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2 
*x^2-1))*(2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-2*arctanh(c*x)^2+pol 
ylog(2,-(c*x+1)^2/(-c^2*x^2+1)))+1/2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*cs 
gn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*(2*arctanh(c*x)*ln 
(1+(c*x+1)^2/(-c^2*x^2+1))-2*arctanh(c*x)^2+polylog(2,-(c*x+1)^2/(-c^2*x^2 
+1)))-I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1) 
)^2*(2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-2*arctanh(c*x)^2+polylog( 
2,-(c*x+1)^2/(-c^2*x^2+1)))+I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2 
/(c^2*x^2-1)))^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*(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...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a**2*x**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*x**2*atanh 
(c*x)**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(2*a*b*x**2*atanh(c*x)/(c** 
2*x**2 + 2*c*x + 1), x))/d**2
 

Maxima [F]

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

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

Output:

-a^2*(1/(c^4*d^2*x + c^3*d^2) - x/(c^2*d^2) + 2*log(c*x + 1)/(c^3*d^2)) + 
1/4*(b^2*c^2*x^2 + b^2*c*x - b^2 - 2*(b^2*c*x + b^2)*log(c*x + 1))*log(-c* 
x + 1)^2/(c^4*d^2*x + c^3*d^2) - integrate(-1/4*((b^2*c^3*x^3 - b^2*c^2*x^ 
2)*log(c*x + 1)^2 + 4*(a*b*c^3*x^3 - a*b*c^2*x^2)*log(c*x + 1) - 2*((2*a*b 
*c^3 + b^2*c^3)*x^3 - 2*(a*b*c^2 - b^2*c^2)*x^2 - b^2 + (b^2*c^3*x^3 - 3*b 
^2*c^2*x^2 - 4*b^2*c*x - 2*b^2)*log(c*x + 1))*log(-c*x + 1))/(c^5*d^2*x^3 
+ c^4*d^2*x^2 - c^3*d^2*x - c^2*d^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{2}}{c^{2} x^{2}+2 c x +1}d x \right ) a b \,c^{4} x +2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{2}}{c^{2} x^{2}+2 c x +1}d x \right ) a b \,c^{3}+\left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{2}}{c^{2} x^{2}+2 c x +1}d x \right ) b^{2} c^{4} x +\left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{2}}{c^{2} x^{2}+2 c x +1}d x \right ) b^{2} c^{3}-2 \,\mathrm {log}\left (c x +1\right ) a^{2} c x -2 \,\mathrm {log}\left (c x +1\right ) a^{2}+a^{2} c^{2} x^{2}+2 a^{2} c x}{c^{3} d^{2} \left (c x +1\right )} \] Input:

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

Output:

(2*int((atanh(c*x)*x**2)/(c**2*x**2 + 2*c*x + 1),x)*a*b*c**4*x + 2*int((at 
anh(c*x)*x**2)/(c**2*x**2 + 2*c*x + 1),x)*a*b*c**3 + int((atanh(c*x)**2*x* 
*2)/(c**2*x**2 + 2*c*x + 1),x)*b**2*c**4*x + int((atanh(c*x)**2*x**2)/(c** 
2*x**2 + 2*c*x + 1),x)*b**2*c**3 - 2*log(c*x + 1)*a**2*c*x - 2*log(c*x + 1 
)*a**2 + a**2*c**2*x**2 + 2*a**2*c*x)/(c**3*d**2*(c*x + 1))