\(\int \frac {(a+b \coth ^{-1}(c+d x))^2}{e+f x} \, dx\) [112]

Optimal result

Integrand size = 20, antiderivative size = 214 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6247, 6060} \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=-\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right )}{2 f} \]

[In]

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

[Out]

-(((a + b*ArcCoth[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*Ar
cCoth[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 6060

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^2)*(Lo
g[2/(1 + c*x)]/e), x] + (Simp[(a + b*ArcCoth[c*x])^2*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp
[b*(a + b*ArcCoth[c*x])*(PolyLog[2, 1 - 2/(1 + c*x)]/e), x] - Simp[b*(a + b*ArcCoth[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*(PolyL
og[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]

Rule 6247

Int[((a_.) + ArcCoth[(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*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b \coth ^{-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 \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \coth ^{-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 \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 13.43 (sec) , antiderivative size = 1767, normalized size of antiderivative = 8.26 \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\frac {a^2 \log (e+f x)}{f}+2 a b \left (\frac {\left (\coth ^{-1}(c+d x)-\text {arctanh}(c+d x)\right ) \log (e+f x)}{f}-\frac {i \left (i \text {arctanh}(c+d x) \left (-\log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\log \left (i \sinh \left (\text {arctanh}\left (\frac {d e-c f}{f}\right )+\text {arctanh}(c+d x)\right )\right )\right )+\frac {1}{2} \left (-i \left (i \text {arctanh}\left (\frac {d e-c f}{f}\right )+i \text {arctanh}(c+d x)\right )^2-\frac {1}{4} i (\pi -2 i \text {arctanh}(c+d x))^2+2 \left (i \text {arctanh}\left (\frac {d e-c f}{f}\right )+i \text {arctanh}(c+d x)\right ) \log \left (1-e^{2 i \left (i \text {arctanh}\left (\frac {d e-c f}{f}\right )+i \text {arctanh}(c+d x)\right )}\right )+(\pi -2 i \text {arctanh}(c+d x)) \log \left (1-e^{i (\pi -2 i \text {arctanh}(c+d x))}\right )-(\pi -2 i \text {arctanh}(c+d x)) \log \left (2 \sin \left (\frac {1}{2} (\pi -2 i \text {arctanh}(c+d x))\right )\right )-2 \left (i \text {arctanh}\left (\frac {d e-c f}{f}\right )+i \text {arctanh}(c+d x)\right ) \log \left (2 i \sinh \left (\text {arctanh}\left (\frac {d e-c f}{f}\right )+\text {arctanh}(c+d x)\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \left (i \text {arctanh}\left (\frac {d e-c f}{f}\right )+i \text {arctanh}(c+d x)\right )}\right )-i \operatorname {PolyLog}\left (2,e^{i (\pi -2 i \text {arctanh}(c+d x))}\right )\right )\right )}{f}\right )-\frac {b^2 (d e-c f+f (c+d x)) \left (1-(c+d x)^2\right ) \left (-\frac {i f \pi ^3-8 d e \coth ^{-1}(c+d x)^3-8 f \coth ^{-1}(c+d x)^3+8 c f \coth ^{-1}(c+d x)^3+24 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )+24 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )-12 f \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )}{24 f^2}+\frac {(-d e-f+c f) (-d e+f+c f) \left (2 d e \coth ^{-1}(c+d x)^3-6 f \coth ^{-1}(c+d x)^3-2 c f \coth ^{-1}(c+d x)^3-4 d e e^{-\text {arctanh}\left (\frac {f}{d e-c f}\right )} \sqrt {\frac {d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}{(d e-c f)^2}} \coth ^{-1}(c+d x)^3+4 c e^{-\text {arctanh}\left (\frac {f}{d e-c f}\right )} f \sqrt {\frac {d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}{(d e-c f)^2}} \coth ^{-1}(c+d x)^3+6 i f \pi \coth ^{-1}(c+d x) \log (2)-f \coth ^{-1}(c+d x)^2 \log (64)-6 i f \pi \coth ^{-1}(c+d x) \log \left (e^{-\coth ^{-1}(c+d x)}+e^{\coth ^{-1}(c+d x)}\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (1+e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )+12 f \coth ^{-1}(c+d x) \text {arctanh}\left (\frac {f}{d e-c f}\right ) \log \left (\frac {1}{2} i e^{-\coth ^{-1}(c+d x)-\text {arctanh}\left (\frac {f}{d e-c f}\right )} \left (-1+e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )\right )+6 f \coth ^{-1}(c+d x)^2 \log \left (-e^{-\coth ^{-1}(c+d x)} \left (d e \left (-1+e^{2 \coth ^{-1}(c+d x)}\right )+\left (1+c+e^{2 \coth ^{-1}(c+d x)}-c e^{2 \coth ^{-1}(c+d x)}\right ) f\right )\right )-6 f \coth ^{-1}(c+d x)^2 \log \left (1-\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )-6 f \coth ^{-1}(c+d x)^2 \log \left (1+\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )+6 i f \pi \coth ^{-1}(c+d x) \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right )-6 f \coth ^{-1}(c+d x)^2 \log \left (-\frac {f}{\sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {d e}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {c f}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-12 f \coth ^{-1}(c+d x) \text {arctanh}\left (\frac {f}{d e-c f}\right ) \log \left (i \sinh \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )\right )+12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,-e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )+6 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )-12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,-\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )-12 f \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )-12 f \operatorname {PolyLog}\left (3,-e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )-12 f \operatorname {PolyLog}\left (3,e^{\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )}\right )-3 f \operatorname {PolyLog}\left (3,e^{2 \left (\coth ^{-1}(c+d x)+\text {arctanh}\left (\frac {f}{d e-c f}\right )\right )}\right )+12 f \operatorname {PolyLog}\left (3,-\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )+12 f \operatorname {PolyLog}\left (3,\frac {e^{\coth ^{-1}(c+d x)} \sqrt {d e+f-c f}}{\sqrt {d e-(1+c) f}}\right )\right )}{6 f^2 (d e+f-c f) (d e-(1+c) f)}\right )}{d (c+d x)^2 (e+f x) \left (1-\frac {1}{(c+d x)^2}\right )} \]

[In]

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

[Out]

(a^2*Log[e + f*x])/f + 2*a*b*(((ArcCoth[c + d*x] - ArcTanh[c + d*x])*Log[e + f*x])/f - (I*(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)*Arc
Tanh[c + d*x])/2]] - 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[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))*(1 - (c + d*x)^2)*(-1/24*(I*f*Pi^
3 - 8*d*e*ArcCoth[c + d*x]^3 - 8*f*ArcCoth[c + d*x]^3 + 8*c*f*ArcCoth[c + d*x]^3 + 24*f*ArcCoth[c + d*x]^2*Log
[1 - E^(2*ArcCoth[c + d*x])] + 24*f*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] - 12*f*PolyLog[3, E^(2
*ArcCoth[c + d*x])])/f^2 + ((-(d*e) - f + c*f)*(-(d*e) + f + c*f)*(2*d*e*ArcCoth[c + d*x]^3 - 6*f*ArcCoth[c +
d*x]^3 - 2*c*f*ArcCoth[c + d*x]^3 - (4*d*e*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCoth[
c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (4*c*f*Sqrt[(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)/(d*e - c*f)^2]*ArcCo
th[c + d*x]^3)/E^ArcTanh[f/(d*e - c*f)] + (6*I)*f*Pi*ArcCoth[c + d*x]*Log[2] - f*ArcCoth[c + d*x]^2*Log[64] -
(6*I)*f*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + 6*f*ArcCoth[c + d*x]^2*Log[1 - E
^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] + 6*f*ArcCoth[c + d*x]^2*Log[1 + E^(ArcCoth[c + d*x] + ArcTanh[f
/(d*e - c*f)])] + 6*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*Arc
Coth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCoth[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcC
oth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 6*f*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) +
(1 + c + E^(2*ArcCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] - 6*f*ArcCoth[c + d*x]^2*L
og[1 - (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*f*ArcCoth[c + d*x]^2*Log[1 + (E^Arc
Coth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + (6*I)*f*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*
x)^(-2)]] - 6*f*ArcCoth[c + d*x]^2*Log[-(f/Sqrt[1 - (c + d*x)^(-2)]) - (d*e)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2
)]) + (c*f)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] - 12*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[Ar
cCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 12*f*ArcCoth[c + d*x]*PolyLog[2, -E^(ArcCoth[c + d*x] + ArcTanh[f/
(d*e - c*f)])] + 12*f*ArcCoth[c + d*x]*PolyLog[2, E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] + 6*f*ArcCoth
[c + d*x]*PolyLog[2, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 12*f*ArcCoth[c + d*x]*PolyLog[2, -((
E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*f*ArcCoth[c + d*x]*PolyLog[2, (E^ArcCoth[
c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*f*PolyLog[3, -E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e -
c*f)])] - 12*f*PolyLog[3, E^(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])] - 3*f*PolyLog[3, E^(2*(ArcCoth[c + d
*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c
)*f])] + 12*f*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]]))/(6*f^2*(d*e + f - c
*f)*(d*e - (1 + c)*f))))/(d*(c + d*x)^2*(e + f*x)*(1 - (c + d*x)^(-2)))

Maple [C] (warning: unable to verify)

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

Time = 9.72 (sec) , antiderivative size = 1603, normalized size of antiderivative = 7.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1603\)
default	\(\text {Expression too large to display}\)	\(1603\)
parts	\(\text {Expression too large to display}\)	\(1684\)

[In]

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

[Out]

1/d*(a^2*d*ln(c*f-d*e-f*(d*x+c))/f-b^2*d*(-ln(c*f-d*e-f*(d*x+c))/f*arccoth(d*x+c)^2-2/f*(-1/2*arccoth(d*x+c)^2
*ln(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)+1/4*I*Pi*csgn(I*(f*c*(
(d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))*(csgn(
I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I/((d*x+c+1)/(d*x
+c-1)-1))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)/((d*x+c+
1)/(d*x+c-1)-1))*csgn(I/((d*x+c+1)/(d*x+c-1)-1))-csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e
*d+(-(d*x+c+1)/(d*x+c-1)-1)*f))*csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+c+1)/(d
*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))+csgn(I*(f*c*((d*x+c+1)/(d*x+c-1)-1)+(1-(d*x+c+1)/(d*x+c-1))*e*d+(-(d*x+
c+1)/(d*x+c-1)-1)*f)/((d*x+c+1)/(d*x+c-1)-1))^2)*arccoth(d*x+c)^2+1/2*arccoth(d*x+c)^2*ln((d*x+c+1)/(d*x+c-1)-
1)-1/2*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-arccoth(d*x+c)*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(
1/2))+polylog(3,1/((d*x+c-1)/(d*x+c+1))^(1/2))-1/2*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-arccot
h(d*x+c)*polylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/2*f*c/(c*f-d*e-
f)*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/2*f*c/(c*f-d*e-f)*arccoth(d*x+c)*polyl
og(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/4*f*c/(c*f-d*e-f)*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e
+f)/(d*x+c-1))-1/2*f/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*f/(c*f
-d*e-f)*arccoth(d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/4*f/(c*f-d*e-f)*polylog(3,(d*x
+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))-1/2*e*d/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(d*x+c+1)*(c*f-d*e-f)/(c*f-
d*e+f)/(d*x+c-1))-1/2*e*d/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))+1/
4*e*d/(c*f-d*e-f)*polylog(3,(d*x+c+1)*(c*f-d*e-f)/(c*f-d*e+f)/(d*x+c-1))))-2*a*b*d*(-ln(c*f-d*e-f*(d*x+c))/f*a
rccoth(d*x+c)+1/f^2*(-1/2*f*(dilog((-f*(d*x+c)+f)/(-c*f+d*e+f))+ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)+f)/(-c*f+
d*e+f)))+1/2*f*(dilog((-f*(d*x+c)-f)/(-c*f+d*e-f))+ln(c*f-d*e-f*(d*x+c))*ln((-f*(d*x+c)-f)/(-c*f+d*e-f))))))

Fricas [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx=\int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2}{e+f\,x} \,d x \]

[In]

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

[Out]

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