\(\int \frac {\text {arctanh}(c x)^2}{x (d+e x)} \, dx\) [159]

Optimal result

Integrand size = 17, antiderivative size = 275 \[ \int \frac {\text {arctanh}(c x)^2}{x (d+e x)} \, dx=\frac {2 \text {arctanh}(c x)^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d}+\frac {\text {arctanh}(c x)^2 \log \left (\frac {2}{1+c x}\right )}{d}-\frac {\text {arctanh}(c x)^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {\text {arctanh}(c x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d}+\frac {\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d}-\frac {\text {arctanh}(c x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d}+\frac {\text {arctanh}(c x) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.65 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.09 \[ \int \frac {\text {arctanh}(c x)^2}{x (d+e x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(I*c*d*Pi^3 - 8*c*d*ArcTanh[c*x]^3 - 8*e*ArcTanh[c*x]^3 + 24*c*d*ArcTanh[c 
*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + 24*c*d*ArcTanh[c*x]*PolyLog[2, E^(2*Ar 
cTanh[c*x])] - 12*c*d*PolyLog[3, E^(2*ArcTanh[c*x])] - (24*(c*d - e)*(c*d 
+ e)*(-6*c*d*ArcTanh[c*x]^3 + 2*e*ArcTanh[c*x]^3 - (4*Sqrt[1 - (c^2*d^2)/e 
^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[( 
E^(-ArcTanh[c*x]) + E^ArcTanh[c*x])/2] - 6*c*d*ArcTanh[c*x]^2*Log[1 - (Sqr 
t[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] - 6*c*d*ArcTanh[c*x]^2*Log[1 
+ (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] + 6*c*d*ArcTanh[c*x]^2* 
Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 
+ E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(2 
*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x 
]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d) 
/e] + ArcTanh[c*x])))] + 6*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c* 
x])) + c*d*(1 + E^(2*ArcTanh[c*x])))/(2*E^ArcTanh[c*x])] - 6*c*d*ArcTanh[c 
*x]^2*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]] - (3*I)*c*d*Pi*ArcTanh[c*x]*Log 
[1 - c^2*x^2] - 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[I*Sinh[ArcTanh[(c 
*d)/e] + ArcTanh[c*x]]] - 12*c*d*ArcTanh[c*x]*PolyLog[2, -((Sqrt[c*d + e]* 
E^ArcTanh[c*x])/Sqrt[-(c*d) + e])] - 12*c*d*ArcTanh[c*x]*PolyLog[2, (Sqrt[ 
c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] + 12*c*d*ArcTanh[c*x]*PolyLog[2 
, -E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 12*c*d*ArcTanh[c*x]*PolyLog[2...
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.

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

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*ArcTanh[c*x]^2*ArcTanh[1 - 2/(1 - c*x)])/d + (ArcTanh[c*x]^2*Log[2/(1 + 
 c*x)])/d - (ArcTanh[c*x]^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d 
- (ArcTanh[c*x]*PolyLog[2, 1 - 2/(1 - c*x)])/d + (ArcTanh[c*x]*PolyLog[2, 
-1 + 2/(1 - c*x)])/d - (ArcTanh[c*x]*PolyLog[2, 1 - 2/(1 + c*x)])/d + (Arc 
Tanh[c*x]*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d + PolyL 
og[3, 1 - 2/(1 - c*x)]/(2*d) - PolyLog[3, -1 + 2/(1 - c*x)]/(2*d) - PolyLo 
g[3, 1 - 2/(1 + c*x)]/(2*d) + PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 
 + c*x))]/(2*d)
 

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 = 2.32 (sec) , antiderivative size = 1466, normalized size of antiderivative = 5.33

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1466\)
default	\(\text {Expression too large to display}\)	\(1466\)
parts	\(\text {Expression too large to display}\)	\(2304\)

Input:

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

Output:

-arctanh(c*x)^2/d*ln(c*e*x+c*d)+arctanh(c*x)^2/d*ln(c*x)-2*c*(-1/2/d/c*arc 
tanh(c*x)^2*ln(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1)-1) 
)+1/4*I/d/c*Pi*(csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(d*c*(1-(c*x+1)^2 
/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1)))*csgn(I*(d*c*(1-(c*x+1)^2/(c^2 
*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))-csgn(I/( 
1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1 
)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))^2-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)))+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-csgn(I*(d*c*(1-(c*x+ 
1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1)))*csgn(I*(d*c*(1-(c*x+1)^2/ 
(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))^2+cs 
gn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+ 
1)^2/(c^2*x^2-1)))^3+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-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+1/2/d/c*arctanh(c*x)^2*l 
n((c*x+1)^2/(-c^2*x^2+1)-1)-1/2/d/c*arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+ 
1)^(1/2))-1/d/c*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))+1/d/c*p 
olylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2/d/c*arctanh(c*x)^2*ln(1+(c*x+1)/( 
-c^2*x^2+1)^(1/2))-1/d/c*arctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(atanh(c*x)**2/(x*(d + e*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - atanh(c*x)**3*c - 3*int(atanh(c*x)**2/(c**2*d*x**3 + c**2*e*x**4 - d*x 
 - e*x**2),x)*e - 3*int(atanh(c*x)**2/(c**2*d*x**2 + c**2*e*x**3 - d - e*x 
),x)*c**2*d)/(3*e)