3.34 \(\int \frac{\coth ^{-1}(c x)^2}{d+e x} \, dx\)

Optimal. Leaf size=164 \[ -\frac{\text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e}-\frac{\coth ^{-1}(c x) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac{\text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 e}+\frac{\coth ^{-1}(c x) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{e}+\frac{\coth ^{-1}(c x)^2 \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac{\log \left (\frac{2}{c x+1}\right ) \coth ^{-1}(c x)^2}{e} \]

[Out]

-((ArcCoth[c*x]^2*Log[2/(1 + c*x)])/e) + (ArcCoth[c*x]^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (ArcC
oth[c*x]*PolyLog[2, 1 - 2/(1 + c*x)])/e - (ArcCoth[c*x]*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])
/e + PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) - PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))]/(2*e)

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Rubi [A]  time = 0.0337196, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5923} \[ -\frac{\text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e}-\frac{\coth ^{-1}(c x) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac{\text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{2 e}+\frac{\coth ^{-1}(c x) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{e}+\frac{\coth ^{-1}(c x)^2 \log \left (\frac{2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}-\frac{\log \left (\frac{2}{c x+1}\right ) \coth ^{-1}(c x)^2}{e} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[c*x]^2/(d + e*x),x]

[Out]

-((ArcCoth[c*x]^2*Log[2/(1 + c*x)])/e) + (ArcCoth[c*x]^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (ArcC
oth[c*x]*PolyLog[2, 1 - 2/(1 + c*x)])/e - (ArcCoth[c*x]*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])
/e + PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) - PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))]/(2*e)

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{\coth ^{-1}(c x)^2}{d+e x} \, dx &=-\frac{\coth ^{-1}(c x)^2 \log \left (\frac{2}{1+c x}\right )}{e}+\frac{\coth ^{-1}(c x)^2 \log \left (\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac{\coth ^{-1}(c x) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{e}-\frac{\coth ^{-1}(c x) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac{\text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 e}-\frac{\text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}\\ \end{align*}

Mathematica [C]  time = 7.68431, size = 741, normalized size = 4.52 \[ \frac{\frac{24 (e-c d) (c d+e) \left (6 e \coth ^{-1}(c x) \text{PolyLog}\left (2,\frac{(c d+e) e^{2 \coth ^{-1}(c x)}}{c d-e}\right )-3 e \text{PolyLog}\left (3,\frac{(c d+e) e^{2 \coth ^{-1}(c x)}}{c d-e}\right )-12 e \coth ^{-1}(c x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)}\right )-12 e \coth ^{-1}(c x) \text{PolyLog}\left (2,e^{\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)}\right )-6 e \coth ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \left (\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)\right )}\right )+12 e \text{PolyLog}\left (3,-e^{\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)}\right )+12 e \text{PolyLog}\left (3,e^{\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)}\right )+3 e \text{PolyLog}\left (3,e^{2 \left (\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)\right )}\right )+4 c d \sqrt{1-\frac{e^2}{c^2 d^2}} \coth ^{-1}(c x)^3 e^{-\tanh ^{-1}\left (\frac{e}{c d}\right )}+6 e \coth ^{-1}(c x)^2 \log \left (\frac{d+e x}{x \sqrt{1-\frac{1}{c^2 x^2}}}\right )-6 i \pi e \log \left (\frac{1}{\sqrt{1-\frac{1}{c^2 x^2}}}\right ) \coth ^{-1}(c x)+6 e \coth ^{-1}(c x)^2 \log \left (\frac{(c d+e) e^{2 \coth ^{-1}(c x)}}{e-c d}+1\right )-6 e \coth ^{-1}(c x)^2 \log \left (\frac{1}{2} e^{-\coth ^{-1}(c x)} \left (c d \left (e^{2 \coth ^{-1}(c x)}-1\right )+e \left (e^{2 \coth ^{-1}(c x)}+1\right )\right )\right )-6 e \coth ^{-1}(c x)^2 \log \left (1-e^{\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)}\right )-6 e \coth ^{-1}(c x)^2 \log \left (e^{\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)}+1\right )-6 e \coth ^{-1}(c x)^2 \log \left (1-e^{2 \left (\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)\right )}\right )-12 e \coth ^{-1}(c x) \tanh ^{-1}\left (\frac{e}{c d}\right ) \log \left (\frac{1}{2} i e^{-\tanh ^{-1}\left (\frac{e}{c d}\right )-\coth ^{-1}(c x)} \left (e^{2 \left (\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)\right )}-1\right )\right )+12 e \coth ^{-1}(c x) \tanh ^{-1}\left (\frac{e}{c d}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac{e}{c d}\right )+\coth ^{-1}(c x)\right )\right )-2 c d \coth ^{-1}(c x)^3+6 e \coth ^{-1}(c x)^3+6 i \pi e \coth ^{-1}(c x) \log \left (\frac{1}{2} \left (e^{-\coth ^{-1}(c x)}+e^{\coth ^{-1}(c x)}\right )\right )\right )}{6 c^2 d^2-6 e^2}-24 e \coth ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(c x)}\right )+12 e \text{PolyLog}\left (3,e^{2 \coth ^{-1}(c x)}\right )+8 c d \coth ^{-1}(c x)^3+8 e \coth ^{-1}(c x)^3-24 e \coth ^{-1}(c x)^2 \log \left (1-e^{2 \coth ^{-1}(c x)}\right )-i \pi ^3 e}{24 e^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCoth[c*x]^2/(d + e*x),x]

[Out]

((-I)*e*Pi^3 + 8*c*d*ArcCoth[c*x]^3 + 8*e*ArcCoth[c*x]^3 - 24*e*ArcCoth[c*x]^2*Log[1 - E^(2*ArcCoth[c*x])] - 2
4*e*ArcCoth[c*x]*PolyLog[2, E^(2*ArcCoth[c*x])] + 12*e*PolyLog[3, E^(2*ArcCoth[c*x])] + (24*(-(c*d) + e)*(c*d
+ e)*(-2*c*d*ArcCoth[c*x]^3 + 6*e*ArcCoth[c*x]^3 + (4*c*d*Sqrt[1 - e^2/(c^2*d^2)]*ArcCoth[c*x]^3)/E^ArcTanh[e/
(c*d)] + (6*I)*e*Pi*ArcCoth[c*x]*Log[(E^(-ArcCoth[c*x]) + E^ArcCoth[c*x])/2] + 6*e*ArcCoth[c*x]^2*Log[1 + ((c*
d + e)*E^(2*ArcCoth[c*x]))/(-(c*d) + e)] - 6*e*ArcCoth[c*x]^2*Log[1 - E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 6
*e*ArcCoth[c*x]^2*Log[1 + E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 6*e*ArcCoth[c*x]^2*Log[1 - E^(2*(ArcCoth[c*x]
+ ArcTanh[e/(c*d)]))] - 12*e*ArcCoth[c*x]*ArcTanh[e/(c*d)]*Log[(I/2)*E^(-ArcCoth[c*x] - ArcTanh[e/(c*d)])*(-1
+ E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)])))] - 6*e*ArcCoth[c*x]^2*Log[(c*d*(-1 + E^(2*ArcCoth[c*x])) + e*(1 +
E^(2*ArcCoth[c*x])))/(2*E^ArcCoth[c*x])] - (6*I)*e*Pi*ArcCoth[c*x]*Log[1/Sqrt[1 - 1/(c^2*x^2)]] + 6*e*ArcCoth[
c*x]^2*Log[(d + e*x)/(Sqrt[1 - 1/(c^2*x^2)]*x)] + 12*e*ArcCoth[c*x]*ArcTanh[e/(c*d)]*Log[I*Sinh[ArcCoth[c*x] +
ArcTanh[e/(c*d)]]] + 6*e*ArcCoth[c*x]*PolyLog[2, ((c*d + e)*E^(2*ArcCoth[c*x]))/(c*d - e)] - 12*e*ArcCoth[c*x
]*PolyLog[2, -E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] - 12*e*ArcCoth[c*x]*PolyLog[2, E^(ArcCoth[c*x] + ArcTanh[e/
(c*d)])] - 6*e*ArcCoth[c*x]*PolyLog[2, E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)]))] - 3*e*PolyLog[3, ((c*d + e)*E^
(2*ArcCoth[c*x]))/(c*d - e)] + 12*e*PolyLog[3, -E^(ArcCoth[c*x] + ArcTanh[e/(c*d)])] + 12*e*PolyLog[3, E^(ArcC
oth[c*x] + ArcTanh[e/(c*d)])] + 3*e*PolyLog[3, E^(2*(ArcCoth[c*x] + ArcTanh[e/(c*d)]))]))/(6*c^2*d^2 - 6*e^2))
/(24*e^2)

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Maple [C]  time = 0.625, size = 926, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(c*e*x+c*d)/e*arccoth(c*x)^2-1/e*arccoth(c*x)^2*ln(c*d*((c*x+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)*e)+1/e*arccot
h(c*x)^2*ln((c*x+1)/(c*x-1)-1)-1/e*arccoth(c*x)^2*ln(1-1/((c*x-1)/(c*x+1))^(1/2))-2/e*arccoth(c*x)*polylog(2,1
/((c*x-1)/(c*x+1))^(1/2))+2/e*polylog(3,1/((c*x-1)/(c*x+1))^(1/2))-1/e*arccoth(c*x)^2*ln(1+1/((c*x-1)/(c*x+1))
^(1/2))-2/e*arccoth(c*x)*polylog(2,-1/((c*x-1)/(c*x+1))^(1/2))+2/e*polylog(3,-1/((c*x-1)/(c*x+1))^(1/2))-1/2*I
/e*Pi*arccoth(c*x)^2*csgn(I*(c*d*((c*x+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)*e)/((c*x+1)/(c*x-1)-1))^2*csgn(I/((c*
x+1)/(c*x-1)-1))+1/2*I/e*Pi*arccoth(c*x)^2*csgn(I*(c*d*((c*x+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)*e)/((c*x+1)/(c*
x-1)-1))^3+1/2*I/e*Pi*arccoth(c*x)^2*csgn(I*(c*d*((c*x+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)*e))*csgn(I*(c*d*((c*x
+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)*e)/((c*x+1)/(c*x-1)-1))*csgn(I/((c*x+1)/(c*x-1)-1))-1/2*I/e*Pi*arccoth(c*x)
^2*csgn(I*(c*d*((c*x+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)*e))*csgn(I*(c*d*((c*x+1)/(c*x-1)-1)+((c*x+1)/(c*x-1)+1)
*e)/((c*x+1)/(c*x-1)-1))^2+1/(c*d+e)*arccoth(c*x)^2*ln(1-(c*d+e)/(c*d-e)*(c*x+1)/(c*x-1))+1/(c*d+e)*arccoth(c*
x)*polylog(2,(c*d+e)/(c*d-e)*(c*x+1)/(c*x-1))-1/2/(c*d+e)*polylog(3,(c*d+e)/(c*d-e)*(c*x+1)/(c*x-1))+c/e*d/(c*
d+e)*arccoth(c*x)^2*ln(1-(c*d+e)/(c*d-e)*(c*x+1)/(c*x-1))+c/e*d/(c*d+e)*arccoth(c*x)*polylog(2,(c*d+e)/(c*d-e)
*(c*x+1)/(c*x-1))-1/2*c/e*d/(c*d+e)*polylog(3,(c*d+e)/(c*d-e)*(c*x+1)/(c*x-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccoth(c*x)^2/(e*x + d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}^{2}{\left (c x \right )}}{d + e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(c*x)**2/(d + e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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