\(\int \frac {(a+b \text {arctanh}(c x))^2}{d+e x} \, dx\) [12]

Optimal result

Integrand size = 18, antiderivative size = 188 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+e x} \, dx=-\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{e}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.04 (sec) , antiderivative size = 1055, normalized size of antiderivative = 5.61 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*a^2*Log[d + e*x] + 6*a*b*ArcTanh[c*x]*(Log[1 - c^2*x^2] + 2*Log[I*Sinh[ 
ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) - (6*I)*a*b*((-1/4*I)*(Pi - (2*I)*ArcTa 
nh[c*x])^2 + I*(ArcTanh[(c*d)/e] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh[c 
*x])*Log[1 + E^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x]) 
*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - (Pi - (2*I)*ArcTanh[c 
*x])*Log[2/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[(c*d)/e] + ArcTanh[c*x])*Lo 
g[(2*I)*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTan 
h[c*x])] - I*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]) + (b^2* 
(-8*c*d*ArcTanh[c*x]^3 + 4*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*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*A 
rcTanh[c*x])] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[c*x]) + E^ArcTa 
nh[c*x])/2] - 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 + (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] + Arc 
Tanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + ArcTan 
h[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*ArcTan 
h[c*x])))/(2*E^ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[(c*(d + e*x))/...
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6474}

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.

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

\(\Big \downarrow \) 6474

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

Input:

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

Output:

-(((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])^2*L 
og[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b*(a + b*ArcTanh[c*x])*Pol 
yLog[2, 1 - 2/(1 + c*x)])/e - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c* 
(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/( 
2*e) - (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e)
 

Defintions of rubi rules used

rule 6474

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> 
 Simp[(-(a + b*ArcTanh[c*x])^2)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc 
Tanh[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[b*(a 
 + b*ArcTanh[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcT 
anh[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + S 
imp[b^2*(PolyLog[3, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[b^2*(PolyLog[3, 1 - 
2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e} 
, x] && NeQ[c^2*d^2 - e^2, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.88 (sec) , antiderivative size = 1087, normalized size of antiderivative = 5.78

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1087\)
default	\(\text {Expression too large to display}\)	\(1087\)
parts	\(\text {Expression too large to display}\)	\(1090\)

Input:

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

Output:

1/c*(a^2*c*ln(c*e*x+c*d)/e+b^2*c*(ln(c*e*x+c*d)/e*arctanh(c*x)^2-2/e*(1/2* 
arctanh(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*Pi*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*(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/(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)))*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*(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)*arctanh(c*x)^2+1/2*arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2 
+1))-1/4*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-1/2/(c*d+e)*e*arctanh(c*x)^2*l 
n(1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2/(c*d+e)*e*arctanh(c*x)*po 
lylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+1/4/(c*d+e)*e*polylog(3,( 
c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2/(c*d+e)*d*c*arctanh(c*x)^2*ln( 
1-(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2/(c*d+e)*d*c*arctanh(c*x)*po 
lylog(2,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+1/4/(c*d+e)*d*c*polylog(3 
,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))))+2*a*b*c*(ln(c*e*x+c*d)/e*arcta 
nh(c*x)-1/e^2*(-1/2*e*(dilog((c*e*x-e)/(-c*d-e))+ln(c*e*x+c*d)*ln((c*e*x-e 
)/(-c*d-e)))+1/2*e*(dilog((c*e*x+e)/(-c*d+e))+ln(c*e*x+c*d)*ln((c*e*x+e...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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