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

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

[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)

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Rubi [A]  time = 0.153347, 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 = {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 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]

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]

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*}

Mathematica [C]  time = 16.6836, size = 1721, normalized size = 8.04 \[ \text{result too large to display} \]

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)*(-(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*ArcC
oth[c + d*x])])/(24*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
^(2*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 12*f*ArcCoth[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*(ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)])))] + 6*f*Ar
cCoth[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*Log[(-(d*e*(-1 + E^(2*ArcCoth[c + d*x]))) + (-1 - E
^(2*ArcCoth[c + d*x]) + c*(-1 + E^(2*ArcCoth[c + d*x])))*f)/(d*e - (1 + c)*f)] + 6*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*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)*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[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)]))] - 6*f*ArcCoth[c + d*x]*PolyLog[2, (E^(2*ArcCoth[c + d*x])*(d*e + f - c*f))/(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*ArcCo
th[c + d*x]*PolyLog[2, (E^ArcCoth[c + d*x]*Sqrt[d*e + f - c*f])/Sqrt[d*e - (1 + c)*f]] - 3*f*PolyLog[3, E^(2*(
ArcCoth[c + d*x] + ArcTanh[f/(d*e - c*f)]))] + 3*f*PolyLog[3, (E^(2*ArcCoth[c + d*x])*(d*e + f - c*f))/(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])] - 12*f*PolyLo
g[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)))

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Maple [C]  time = 0.855, size = 1845, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

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*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*csgn(I/((d*x+c+1)/(d*x+c-1)-1))-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)*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))-b^2/(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*f-d*e-f)*arccoth(d*x+c)*polyl
og(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)-b^2/f*arcco
th(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)+b^2*ln((d*x
+c)*f-c*f+d*e)/f*arccoth(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)/(d*x+c-1)/(c*f-d*e+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))+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))*csgn(I/((
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))+2*a*b*ln((d*x+c)*f-c*f+d*e)/f
*arccoth(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))+b^2*c/(c*f-d*e-f)*arccoth(d*x+c)^2*l
n(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))-I*b^2/f*Pi*arccoth(d*x+c)^2+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))+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))+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))+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*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)*ar
ccoth(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)
*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., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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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*acoth(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{arcoth}\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*arccoth(d*x+c))^2/(f*x+e),x, algorithm="giac")

[Out]

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