\(\int \frac {x^3 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx\) [104]

Optimal result

Integrand size = 22, antiderivative size = 331 \[ \int \frac {x^3 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {a b x}{c^3 d^2}+\frac {b^2}{2 c^4 d^2 (1+c x)}-\frac {b^2 \text {arctanh}(c x)}{2 c^4 d^2}+\frac {b^2 x \text {arctanh}(c x)}{c^3 d^2}+\frac {b (a+b \text {arctanh}(c x))}{c^4 d^2 (1+c x)}-\frac {3 (a+b \text {arctanh}(c x))^2}{c^4 d^2}-\frac {2 x (a+b \text {arctanh}(c x))^2}{c^3 d^2}+\frac {x^2 (a+b \text {arctanh}(c x))^2}{2 c^2 d^2}+\frac {(a+b \text {arctanh}(c x))^2}{c^4 d^2 (1+c x)}+\frac {4 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^4 d^2}-\frac {3 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^4 d^2}+\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^4 d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^4 d^2}+\frac {3 b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^4 d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^4 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.07 \[ \int \frac {x^3 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {-8 a^2 c x+2 a^2 c^2 x^2+\frac {4 a^2}{1+c x}+12 a^2 \log (1+c x)+2 a b \left (2 c x+\cosh (2 \text {arctanh}(c x))-4 \log \left (1-c^2 x^2\right )+6 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+2 \text {arctanh}(c x) \left (-1-4 c x+c^2 x^2+\cosh (2 \text {arctanh}(c x))-6 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-\sinh (2 \text {arctanh}(c x))\right )-\sinh (2 \text {arctanh}(c x))\right )+b^2 \left (4 c x \text {arctanh}(c x)+6 \text {arctanh}(c x)^2-8 c x \text {arctanh}(c x)^2+2 c^2 x^2 \text {arctanh}(c x)^2+\cosh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))+2 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))+16 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-12 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+2 \log \left (1-c^2 x^2\right )+4 (-2+3 \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\sinh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))-2 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))\right )}{4 c^4 d^2} \] Input:

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

Output:

(-8*a^2*c*x + 2*a^2*c^2*x^2 + (4*a^2)/(1 + c*x) + 12*a^2*Log[1 + c*x] + 2* 
a*b*(2*c*x + Cosh[2*ArcTanh[c*x]] - 4*Log[1 - c^2*x^2] + 6*PolyLog[2, -E^( 
-2*ArcTanh[c*x])] + 2*ArcTanh[c*x]*(-1 - 4*c*x + c^2*x^2 + Cosh[2*ArcTanh[ 
c*x]] - 6*Log[1 + E^(-2*ArcTanh[c*x])] - Sinh[2*ArcTanh[c*x]]) - Sinh[2*Ar 
cTanh[c*x]]) + b^2*(4*c*x*ArcTanh[c*x] + 6*ArcTanh[c*x]^2 - 8*c*x*ArcTanh[ 
c*x]^2 + 2*c^2*x^2*ArcTanh[c*x]^2 + Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]* 
Cosh[2*ArcTanh[c*x]] + 2*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 16*ArcTanh[ 
c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - 12*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTan 
h[c*x])] + 2*Log[1 - c^2*x^2] + 4*(-2 + 3*ArcTanh[c*x])*PolyLog[2, -E^(-2* 
ArcTanh[c*x])] + 6*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]]))/(4*c^4*d^2)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 331, 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.

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

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (\frac {3 (a+b \text {arctanh}(c x))^2}{c^3 d^2 (c x+1)}-\frac {2 (a+b \text {arctanh}(c x))^2}{c^3 d^2}-\frac {(a+b \text {arctanh}(c x))^2}{c^3 d^2 (c x+1)^2}+\frac {x (a+b \text {arctanh}(c x))^2}{c^2 d^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a*b*x)/(c^3*d^2) + b^2/(2*c^4*d^2*(1 + c*x)) - (b^2*ArcTanh[c*x])/(2*c^4* 
d^2) + (b^2*x*ArcTanh[c*x])/(c^3*d^2) + (b*(a + b*ArcTanh[c*x]))/(c^4*d^2* 
(1 + c*x)) - (3*(a + b*ArcTanh[c*x])^2)/(c^4*d^2) - (2*x*(a + b*ArcTanh[c* 
x])^2)/(c^3*d^2) + (x^2*(a + b*ArcTanh[c*x])^2)/(2*c^2*d^2) + (a + b*ArcTa 
nh[c*x])^2/(c^4*d^2*(1 + c*x)) + (4*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x) 
])/(c^4*d^2) - (3*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^4*d^2) + (b^ 
2*Log[1 - c^2*x^2])/(2*c^4*d^2) + (2*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^4 
*d^2) + (3*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^4*d^2) + 
 (3*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^4*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 = 4.28 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.97

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(983\)
default	\(\text {Expression too large to display}\)	\(983\)
parts	\(\text {Expression too large to display}\)	\(994\)

Input:

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

Output:

1/c^4*(a^2/d^2*(1/2*c^2*x^2-2*c*x+1/(c*x+1)+3*ln(c*x+1))+b^2/d^2*(3/2*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)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*arctanh(c*x)^2+3*arctanh(c* 
x)^2*ln(c*x+1)+4*arctanh(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))+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))+2*arctanh(c*x)^3+1/2*arctan 
h(c*x)^2*c^2*x^2-2*arctanh(c*x)^2*c*x-1/4/(c*x+1)*(c*x-1)-3*arctanh(c*x)^2 
+3/2*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+(c*x+1)*arctanh(c*x)-ln(1+(c*x+1)^ 
2/(-c^2*x^2+1))-3*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-3/2*I*Pi 
*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*x)^2- 
3/2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2-6*arctanh(c*x)^2*l 
n((c*x+1)/(-c^2*x^2+1)^(1/2))-3*ln(2)*arctanh(c*x)^2-1/2*arctanh(c*x)*(c*x 
-1)/(c*x+1)+1/(c*x+1)*arctanh(c*x)^2+3/2*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)))^2*arctanh(c*x)^2 
-3*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 
*arctanh(c*x)^2-3/2*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))*arctanh(c*x)^2-3/2*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)^2 
)+2/d^2*a*b*(1/2*arctanh(c*x)*c^2*x^2-2*arctanh(c*x)*c*x+1/(c*x+1)*arctanh 
(c*x)+3*arctanh(c*x)*ln(c*x+1)-3/4*ln(c*x+1)^2+3/2*(ln(c*x+1)-ln(1/2*c*...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

(Integral(a**2*x**3/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*x**3*atanh 
(c*x)**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(2*a*b*x**3*atanh(c*x)/(c** 
2*x**2 + 2*c*x + 1), x))/d**2
 

Maxima [F]

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

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

Output:

1/2*a^2*(2/(c^5*d^2*x + c^4*d^2) + (c*x^2 - 4*x)/(c^3*d^2) + 6*log(c*x + 1 
)/(c^4*d^2)) + 1/8*(b^2*c^3*x^3 - 3*b^2*c^2*x^2 - 4*b^2*c*x + 2*b^2 + 6*(b 
^2*c*x + b^2)*log(c*x + 1))*log(-c*x + 1)^2/(c^5*d^2*x + c^4*d^2) - integr 
ate(-1/4*((b^2*c^4*x^4 - b^2*c^3*x^3)*log(c*x + 1)^2 + 4*(a*b*c^4*x^4 - a* 
b*c^3*x^3)*log(c*x + 1) + (7*b^2*c^2*x^2 - (4*a*b*c^4 + b^2*c^4)*x^4 + 2*b 
^2*c*x + 2*(2*a*b*c^3 + b^2*c^3)*x^3 - 2*b^2 - 2*(b^2*c^4*x^4 - b^2*c^3*x^ 
3 + 3*b^2*c^2*x^2 + 6*b^2*c*x + 3*b^2)*log(c*x + 1))*log(-c*x + 1))/(c^6*d 
^2*x^3 + c^5*d^2*x^2 - c^4*d^2*x - c^3*d^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{3}}{c^{2} x^{2}+2 c x +1}d x \right ) a b \,c^{5} x +4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{3}}{c^{2} x^{2}+2 c x +1}d x \right ) a b \,c^{4}+2 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{3}}{c^{2} x^{2}+2 c x +1}d x \right ) b^{2} c^{5} x +2 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x^{3}}{c^{2} x^{2}+2 c x +1}d x \right ) b^{2} c^{4}+6 \,\mathrm {log}\left (c x +1\right ) a^{2} c x +6 \,\mathrm {log}\left (c x +1\right ) a^{2}+a^{2} c^{3} x^{3}-3 a^{2} c^{2} x^{2}-6 a^{2} c x}{2 c^{4} d^{2} \left (c x +1\right )} \] Input:

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

Output:

(4*int((atanh(c*x)*x**3)/(c**2*x**2 + 2*c*x + 1),x)*a*b*c**5*x + 4*int((at 
anh(c*x)*x**3)/(c**2*x**2 + 2*c*x + 1),x)*a*b*c**4 + 2*int((atanh(c*x)**2* 
x**3)/(c**2*x**2 + 2*c*x + 1),x)*b**2*c**5*x + 2*int((atanh(c*x)**2*x**3)/ 
(c**2*x**2 + 2*c*x + 1),x)*b**2*c**4 + 6*log(c*x + 1)*a**2*c*x + 6*log(c*x 
 + 1)*a**2 + a**2*c**3*x**3 - 3*a**2*c**2*x**2 - 6*a**2*c*x)/(2*c**4*d**2* 
(c*x + 1))