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

Optimal. Leaf size=214 \[ -\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{f}-\frac {b^2 \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c+d x+1}\right )}{2 f} \]

[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

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Rubi [A]  time = 0.15, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6112, 5923} \[ -\frac {b \left (a+b \coth ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \text {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{f}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right )}{2 f}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{f} \]

Antiderivative was successfully verified.

[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 5923

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCoth[c*x])^2*Log[
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*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]

Rule 6112

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*} \int \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{e+f x} \, dx &=\frac {\operatorname {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 ) \text {Li}_2\left (1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \coth ^{-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*}

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Mathematica [C]  time = 31.06, size = 3759, normalized size = 17.57 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[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^2*e^2*ArcCoth[c + d*x]^3 - 8*d*e*f*ArcC
oth[c + d*x]^3 - 4*c*d*e*f*ArcCoth[c + d*x]^3 + 4*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*ArcCoth[c + d*x]^3 - 10*f
^2*ArcCoth[c + d*x]^3 + 8*c*f^2*ArcCoth[c + d*x]^3 + 2*c^2*f^2*ArcCoth[c + d*x]^3 - 4*E^(2*ArcTanh[f/(d*e - c*
f)])*f^2*ArcCoth[c + d*x]^3 - 4*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*ArcCoth[c + d*x]^3 - (4*d^2*e^2*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*d*e*f*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)] + (8*c*d*e*f
*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^2*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^2*f^2*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)] + (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[2] + (6*I)*f^2*Pi*ArcCoth[c + d*x]*Log[2] - (6*I)*c*f^2*Pi*ArcCot
h[c + d*x]*Log[2] - d*e*f*ArcCoth[c + d*x]^2*Log[64] - f^2*ArcCoth[c + d*x]^2*Log[64] + c*f^2*ArcCoth[c + d*x]
^2*Log[64] - (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] - (6*I)*f^2*Pi*Ar
cCoth[c + d*x]*Log[E^(-ArcCoth[c + d*x]) + E^ArcCoth[c + d*x]] + (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[E^(-ArcCo
th[c + d*x]) + E^ArcCoth[c + d*x]] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*
e - c*f)]))] - 6*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTan
h[f/(d*e - c*f)]))] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 6*
c*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*E^(2*ArcTanh[f/(d*e -
c*f)])*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 6*c*E^(2*ArcTanh[f/
(d*e - c*f)])*f^2*ArcCoth[c + d*x]^2*Log[1 - E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*d*e*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)])))] + 12*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[(I/2)*E^(-ArcCo
th[c + d*x] - ArcTanh[f/(d*e - c*f)])*(-1 + E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] - 12*c*f^2*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*d*e*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^2*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*c*f^2*ArcCoth[c + d*x]^2*Log[-((d*e*(-1 + E^(2*ArcCoth[c + d*x])) + (1 + c + E^(2*Ar
cCoth[c + d*x]) - c*E^(2*ArcCoth[c + d*x]))*f)/E^ArcCoth[c + d*x])] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 - (E^Ar
cCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d
*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 - (E^ArcCoth[c + d*x]*Sqrt[
d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] + 6*d*e*f*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f -
c*f])/Sqrt[d*e - (1 + c)*f]] + 6*f^2*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt
[d*e - (1 + c)*f]] - 6*c*f^2*ArcCoth[c + d*x]^2*Log[1 + (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1
+ c)*f]] + (6*I)*d*e*f*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] + (6*I)*f^2*Pi*ArcCoth[c + d*x]*Lo
g[1/Sqrt[1 - (c + d*x)^(-2)]] - (6*I)*c*f^2*Pi*ArcCoth[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^(-2)]] - 6*d*e*f*ArcC
oth[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)])] - 6*f^2*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)])] + 6*c*f^2*ArcCoth[c + d*x]^2*Log[-(f/S
qrt[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*d*e*f*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]
] - 12*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 12
*c*f^2*ArcCoth[c + d*x]*ArcTanh[f/(d*e - c*f)]*Log[I*Sinh[ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]]] + 6*f*(-
(d*e*(-1 + E^(2*ArcTanh[f/(d*e - c*f)]))) + (1 + E^(2*ArcTanh[f/(d*e - c*f)]) + c*(-1 + E^(2*ArcTanh[f/(d*e -
c*f)])))*f)*ArcCoth[c + d*x]*PolyLog[2, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*(d*e + f - c
*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*d*e*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^2*ArcCoth[c
+ d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*c*f^2*ArcCoth[c + d*x]
*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 3*d*e*f*PolyLog[3, E^(2*(ArcCoth
[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*d*e*E^(2*ArcTanh[f/(d*e - c*f)])*f*PolyLog[3, E^(2*(ArcCoth[c + d*x]
+ ArcTanh[f/(d*e - c*f)]))] - 3*f^2*PolyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*c*f^2*P
olyLog[3, E^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*PolyLog[3, E
^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 3*c*E^(2*ArcTanh[f/(d*e - c*f)])*f^2*PolyLog[3, E^(2*(ArcC
oth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] - 12*d*e*f*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt
[d*e - (1 + c)*f])] - 12*f^2*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] + 1
2*c*f^2*PolyLog[3, -((E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f])] - 12*d*e*f*PolyLog[3, (E
^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 12*f^2*PolyLog[3, (E^ArcCoth[c + d*x]*Sqrt[d*e
+ f - c*f])/Sqrt[d*e - (1 + c)*f]] + 12*c*f^2*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)^2*(d*e - (1 + c)*f))))/(d*(c + d*x)^2*(e + f*x)*(1 - (c + d*x)^(-2)))

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcoth}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arcoth}\left (d x + c\right ) + a^{2}}{f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 1.50, size = 1845, normalized size = 8.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^2*c/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))+b^2*ln((d*x+c)*f-c*f+d*e)/f*arcc
oth(d*x+c)^2-b^2/f*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*b^2/f*arccoth(d*x+c)*polylog(2,-1/((
d*x+c-1)/(d*x+c+1))^(1/2))-b^2/f*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*b^2/f*arccoth(d*x+c)*p
olylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))+b^2/f*arccoth(d*x+c)^2*ln((d*x+c+1)/(d*x+c-1)-1)-b^2/(c*f-d*e-f)*arcco
th(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-b^2/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(c*f-d*
e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-b^2/f*arccoth(d*x+c)^2*ln(((d*x+c+1)/(d*x+c-1)-1)*c*f+(1-(d*x+c+1)/(d*x+
c-1))*e*d+(-(d*x+c+1)/(d*x+c-1)-1)*f)-a*b/f*dilog(((d*x+c)*f+f)/(c*f-d*e+f))+a*b/f*dilog(((d*x+c)*f-f)/(c*f-d*
e-f))+2*b^2/f*polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/2*b^2/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d
*x+c-1)/(c*f-d*e+f))+a^2*ln((d*x+c)*f-c*f+d*e)/f+2*b^2/f*polylog(3,1/((d*x+c-1)/(d*x+c+1))^(1/2))+1/2*I*b^2/f*
Pi*arccoth(d*x+c)^2*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(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)*c*f+(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))-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)*arccoth(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)/
(d*x+c-1)/(c*f-d*e+f))+b^2*c/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))
+2*a*b*ln((d*x+c)*f-c*f+d*e)/f*arccoth(d*x+c)-I*b^2/f*Pi*arccoth(d*x+c)^2+I*b^2/f*Pi*arccoth(d*x+c)^2*csgn(I*(
((d*x+c+1)/(d*x+c-1)-1)*c*f+(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
+1/2*d*b^2/f*e/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-1/2*I*b^2/f*Pi*arccoth(d*x+c
)^2*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(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))^3-d*b^2/f*e/(c*f-d*e-f)*arccoth(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-d*b^2/f*
e/(c*f-d*e-f)*arccoth(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)/(d*x+c-1)/(c*f-d*e+f))-1/2*I*b^2/f*Pi*arccoth(d*x
+c)^2*csgn(I/((d*x+c+1)/(d*x+c-1)-1))*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)*c*f+(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))^2-1/2*I*b^2/f*Pi*arccoth(d*x+c)^2*csgn(I*(((d*x+c+1)/(d*x+c-1)-1)
*c*f+(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)*c*f+(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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \log \left (f x + e\right )}{f} + \int \frac {b^{2} {\left (\log \left (\frac {1}{d x + c} + 1\right ) - \log \left (-\frac {1}{d x + c} + 1\right )\right )}^{2}}{4 \, {\left (f x + e\right )}} + \frac {a b {\left (\log \left (\frac {1}{d x + c} + 1\right ) - \log \left (-\frac {1}{d x + c} + 1\right )\right )}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2}{e+f\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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