3.42 \(\int \frac{(a+b \tanh ^{-1}(c+d x))^2}{e+f x} \, dx\)

Optimal. Leaf size=214 \[ -\frac{b \left (a+b \tanh ^{-1}(c+d x)\right ) \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c+d x+1}\right )}{2 f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{f} \]

[Out]

-(((a + b*ArcTanh[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*Ar
cTanh[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1
+ c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)

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Rubi [A]  time = 0.154244, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6111, 5922} \[ -\frac{b \left (a+b \tanh ^{-1}(c+d x)\right ) \text{PolyLog}\left (2,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac{b \text{PolyLog}\left (2,1-\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{c+d x+1}\right )}{2 f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac{2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac{\log \left (\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b*ArcTanh[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*Ar
cTanh[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1
+ c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)

Rule 6111

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rule 5922

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*ArcTanh[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*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(
d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2}{e+f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2}{\frac{d e-c f}{d}+\frac{f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1+c+d x}\right )}{f}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac{b \left (a+b \tanh ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1+c+d x}\right )}{f}-\frac{b \left (a+b \tanh ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+c+d x}\right )}{2 f}-\frac{b^2 \text{Li}_3\left (1-\frac{2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}\\ \end{align*}

Mathematica [C]  time = 23.5301, size = 2404, normalized size = 11.23 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*Log[e + f*x])/f - ((2*I)*a*b*(I*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[I*Sinh[ArcTanh[(d*e
- c*f)/f] + ArcTanh[c + d*x]]]) + ((-I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])^2 - (I/4)*(Pi - (2*I)
*ArcTanh[c + d*x])^2 + 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])*Log[1 - E^((2*I)*(I*ArcTanh[(d*e - c*
f)/f] + I*ArcTanh[c + d*x]))] + (Pi - (2*I)*ArcTanh[c + d*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))] - (
Pi - (2*I)*ArcTanh[c + d*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c + d*x])/2]] - 2*(I*ArcTanh[(d*e - c*f)/f] + I*Arc
Tanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] - I*PolyLog[2, E^((2*I)*(I*ArcTanh[(
d*e - c*f)/f] + I*ArcTanh[c + d*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))])/2))/f + (b^2*(d*e -
c*f + f*(c + d*x))*((2*ArcTanh[c + d*x]^2*(d*e*ArcTanh[c + d*x] - (1 + c)*f*ArcTanh[c + d*x] + 3*(d*e - c*f)*
Log[1 + E^(-2*ArcTanh[c + d*x])]) + (-6*d*e*ArcTanh[c + d*x] + 6*c*f*ArcTanh[c + d*x])*PolyLog[2, -E^(-2*ArcTa
nh[c + d*x])] + (-3*d*e + 3*c*f)*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/(6*f*(-(d*e) + c*f)) + ((-(d*e) - f + c
*f)*(-(d*e) + f + c*f)*((ArcTanh[c + d*x]^2*(f*ArcTanh[c + d*x] + (-(d*e) + c*f)*Log[(d*e)/Sqrt[1 - (c + d*x)^
2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]]))/((d*e + f - c*f)*(d*e - (1 + c)*f))
- (ArcTanh[c + d*x]*(I*d*e*Pi*ArcTanh[c + d*x] - I*c*f*Pi*ArcTanh[c + d*x] + 2*f*ArcTanh[c + d*x]^2 - (Sqrt[1
- c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x]^2)/E^ArcTanh[(d*e - c*f)/f] - I*d*e*Pi*Log[1 + E^(2*Ar
cTanh[c + d*x])] + I*c*f*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] + 2*d*e*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d
*e - c*f)/f] + ArcTanh[c + d*x]))] - 2*c*f*ArcTanh[c + d*x]*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x]))] + I*d*e*Pi*Log[1/Sqrt[1 - (c + d*x)^2]] - I*c*f*Pi*Log[1/Sqrt[1 - (c + d*x)^2]] - 2*d*e*ArcTanh[c +
d*x]*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] + 2*
c*f*ArcTanh[c + d*x]*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c
+ d*x)^2]] + 2*(d*e - c*f)*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] +
ArcTanh[c + d*x]))] - Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]]) + (-(d*e) + c*f)*PolyLog[2, E^(
-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (((-2*d*e + (2 + 2*c
- Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]/E^ArcTanh[(d*e - c*f)/f])*f)*ArcTanh[c + d*x]^3)/3 + (d*e - c*f)
*ArcTanh[c + d*x]^2*Log[-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + (d*e - c*f)*ArcTanh[c + d*x]
*(I*Pi*(ArcTanh[c + d*x] - Log[1 + E^(2*ArcTanh[c + d*x])] + Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c +
d*x])]) + 2*ArcTanh[(d*e - c*f)/f]*(ArcTanh[c + d*x] + Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*
x]))] - Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[
c + d*x])))])) - (d*e - c*f)*ArcTanh[c + d*x]^2*Log[d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c
+ d*x]) + c*E^(2*ArcTanh[c + d*x]))*f] + (d*e - c*f)*ArcTanh[c + d*x]^2*(ArcTanh[c + d*x] + Log[1 - E^(-2*(Ar
cTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - Log[-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + Log
[d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f] - Log[(d*e*
(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c +
d*x])]) + ((d*e - c*f)*(2*ArcTanh[c + d*x]^3 + 3*ArcTanh[c + d*x]^2*Log[1 - E^(-2*(ArcTanh[(d*e - c*f)/f] + A
rcTanh[c + d*x]))] - 3*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*ArcTanh[c
+ d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*ArcTanh[c + d*x]*PolyLog[2, E^(-2*(ArcTan
h[(d*e - c*f)/f] + ArcTanh[c + d*x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c
+ d*x])] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*PolyLog[3, -E^(Arc
Tanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])]))/3 + (
(d*e - c*f)*(4*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*Log[1 - E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*
x]))] - 6*ArcTanh[c + d*x]*PolyLog[2, E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))] + 3*PolyLog[3, E^(2*(
ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]))]))/6 - ((d*e - c*f)*(4*ArcTanh[c + d*x]^3 - 6*ArcTanh[c + d*x]^2*L
og[1 + (E^(2*ArcTanh[c + d*x])*(d*e + f - c*f))/(d*e - (1 + c)*f)] - 6*ArcTanh[c + d*x]*PolyLog[2, -((E^(2*Arc
Tanh[c + d*x])*(d*e + f - c*f))/(d*e - (1 + c)*f))] + 3*PolyLog[3, -((E^(2*ArcTanh[c + d*x])*(d*e + f - c*f))/
(d*e - (1 + c)*f))]))/6)/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)))/(f*(-(d*e) + c*f))))/(d*(e + f*x))

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Maple [C]  time = 0.878, size = 1984, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*b^2/f*csgn(I/((d*x+c+1)^2/(1-(d*x+c)^2)+1))*csgn(I*(c*f*((d*x+c+1)^2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-(
d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f))*csgn(I*(c*f*((d*x+c+1)^2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-
(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/((d*x+c+1)^2/(1-(d*x+c)^2)+1))*arctanh(d*x+c)^2*Pi+2*a*b*ln
((d*x+c)*f-c*f+d*e)/f*arctanh(d*x+c)-a*b/f*ln((d*x+c)*f-c*f+d*e)*ln(((d*x+c)*f+f)/(c*f-d*e+f))+a*b/f*ln((d*x+c
)*f-c*f+d*e)*ln(((d*x+c)*f-f)/(c*f-d*e-f))+b^2*c/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-
(d*x+c)^2)/(-c*f+d*e-f))+b^2*c/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*
f+d*e-f))-I*b^2/f*Pi*arctanh(d*x+c)^2+a^2*ln((d*x+c)*f-c*f+d*e)/f+1/2*b^2/f*polylog(3,-(d*x+c+1)^2/(1-(d*x+c)^
2))+1/2*b^2/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+1/2*d*b^2/f*e/(c*f-d*e-f
)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+I*b^2/f*csgn(I*(c*f*((d*x+c+1)^2/(1-(d*x+c)^2)
+1)+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/((d*x+c+1)^2/(1-(d*x+c)^2)+1))^2*arcta
nh(d*x+c)^2*Pi-1/2*I*b^2/f*csgn(I*(c*f*((d*x+c+1)^2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*
x+c+1)^2/(1-(d*x+c)^2))*f)/((d*x+c+1)^2/(1-(d*x+c)^2)+1))^3*arctanh(d*x+c)^2*Pi-b^2/(c*f-d*e-f)*arctanh(d*x+c)
^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-b^2/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-
f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+b^2*ln((d*x+c)*f-c*f+d*e)/f*arctanh(d*x+c)^2-1/2*b^2*c/(c*f-d*e-f)*
polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-a*b/f*dilog(((d*x+c)*f+f)/(c*f-d*e+f))+a*b/f*dil
og(((d*x+c)*f-f)/(c*f-d*e-f))-b^2/f*arctanh(d*x+c)^2*ln(c*f*((d*x+c+1)^2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-(d*
x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)-b^2/f*arctanh(d*x+c)*polylog(2,-(d*x+c+1)^2/(1-(d*x+c)^2))-1/2
*I*b^2/f*csgn(I/((d*x+c+1)^2/(1-(d*x+c)^2)+1))*csgn(I*(c*f*((d*x+c+1)^2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-(d*x
+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/((d*x+c+1)^2/(1-(d*x+c)^2)+1))^2*arctanh(d*x+c)^2*Pi-d*b^2/f*e/
(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-d*b^2/f*e/(c*f-d*e-f)*ar
ctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-1/2*I*b^2/f*csgn(I*(c*f*((d*x+c+1)^
2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f))*csgn(I*(c*f*((d*x+c+1)
^2/(1-(d*x+c)^2)+1)+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/((d*x+c+1)^2/(1-(d*x+c
)^2)+1))^2*arctanh(d*x+c)^2*Pi

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \log \left (f x + e\right )}{f} + \int \frac{b^{2}{\left (\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )\right )}^{2}}{4 \,{\left (f x + e\right )}} + \frac{a b{\left (\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )\right )}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{artanh}\left (d x + c\right ) + a^{2}}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (d x + c\right ) + a\right )}^{2}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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