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

Optimal. Leaf size=335 \[ -\frac{\text{PolyLog}\left (2,\frac{2 \left (-e \left (b-\sqrt{b^2-4 a c}\right )-2 c (d+e x)+2 c d\right )}{(d+e x+1) \left (-e \sqrt{b^2-4 a c}+b e-2 c d+2 c\right )}+1\right )}{2 \sqrt{b^2-4 a c}}+\frac{\text{PolyLog}\left (2,\frac{2 \left (-e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d+e x)+2 c d\right )}{(d+e x+1) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (1-d)\right )}+1\right )}{2 \sqrt{b^2-4 a c}}+\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (b-\sqrt{b^2-4 a c}\right )+2 c (1-d)\right )}\right )}{\sqrt{b^2-4 a c}}-\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (1-d)\right )}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

(ArcCoth[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 + d
+ e*x))])/Sqrt[b^2 - 4*a*c] - (ArcCoth[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b +
Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])
*e - 2*c*(d + e*x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLo
g[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1
+ d + e*x))]/(2*Sqrt[b^2 - 4*a*c])

________________________________________________________________________________________

Rubi [A]  time = 0.748577, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {618, 206, 6728, 6112, 5921, 2402, 2315, 2447} \[ -\frac{\text{PolyLog}\left (2,\frac{2 \left (-e \left (b-\sqrt{b^2-4 a c}\right )-2 c (d+e x)+2 c d\right )}{(d+e x+1) \left (-e \sqrt{b^2-4 a c}+b e-2 c d+2 c\right )}+1\right )}{2 \sqrt{b^2-4 a c}}+\frac{\text{PolyLog}\left (2,\frac{2 \left (-e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d+e x)+2 c d\right )}{(d+e x+1) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (1-d)\right )}+1\right )}{2 \sqrt{b^2-4 a c}}+\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (b-\sqrt{b^2-4 a c}\right )+2 c (1-d)\right )}\right )}{\sqrt{b^2-4 a c}}-\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (1-d)\right )}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcCoth[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 + d
+ e*x))])/Sqrt[b^2 - 4*a*c] - (ArcCoth[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b +
Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])
*e - 2*c*(d + e*x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLo
g[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1
+ d + e*x))]/(2*Sqrt[b^2 - 4*a*c])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 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]

Rule 5921

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCoth[c*x])*Log[2/(1
+ c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d +
e*x))/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x], x] + Simp[((a + b*ArcCoth[c*x])*Log[(2*c*(d + e*x))/((c*d + e)
*(1 + c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx &=\int \left (\frac{2 c \coth ^{-1}(d+e x)}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}-\frac{2 c \coth ^{-1}(d+e x)}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}\right ) \, dx\\ &=\frac{(2 c) \int \frac{\coth ^{-1}(d+e x)}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{\coth ^{-1}(d+e x)}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{(2 c) \operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{\frac{-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e}{e}+\frac{2 c x}{e}} \, dx,x,d+e x\right )}{\sqrt{b^2-4 a c} e}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{\frac{-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e}{e}+\frac{2 c x}{e}} \, dx,x,d+e x\right )}{\sqrt{b^2-4 a c} e}\\ &=\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c-2 c d+b e-\sqrt{b^2-4 a c} e\right ) (1+d+e x)}\right )}{\sqrt{b^2-4 a c}}-\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt{b^2-4 a c}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (\frac{-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e}{e}+\frac{2 c x}{e}\right )}{\left (\frac{2 c}{e}+\frac{-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e}{e}\right ) (1+x)}\right )}{1-x^2} \, dx,x,d+e x\right )}{\sqrt{b^2-4 a c}}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (\frac{-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e}{e}+\frac{2 c x}{e}\right )}{\left (\frac{2 c}{e}+\frac{-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e}{e}\right ) (1+x)}\right )}{1-x^2} \, dx,x,d+e x\right )}{\sqrt{b^2-4 a c}}\\ &=\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c-2 c d+b e-\sqrt{b^2-4 a c} e\right ) (1+d+e x)}\right )}{\sqrt{b^2-4 a c}}-\frac{\coth ^{-1}(d+e x) \log \left (\frac{2 e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt{b^2-4 a c}}-\frac{\text{Li}_2\left (1-\frac{2 e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c-2 c d+b e-\sqrt{b^2-4 a c} e\right ) (1+d+e x)}\right )}{2 \sqrt{b^2-4 a c}}+\frac{\text{Li}_2\left (1-\frac{2 e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{2 \sqrt{b^2-4 a c}}\\ \end{align*}

Mathematica [A]  time = 0.898204, size = 596, normalized size = 1.78 \[ \frac{-\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d+1)}\right )+\text{PolyLog}\left (2,\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{-e \sqrt{b^2-4 a c}+b e-2 c d+2 c}\right )-\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d-1)}\right )+\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d+1)}\right )+\log \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x-1)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d-1)}\right )-\log \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x-1)}{2 c (d-1)-e \left (\sqrt{b^2-4 a c}+b\right )}\right )-\log \left (\frac{d+e x-1}{d+e x}\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x\right )+\log \left (\frac{d+e x-1}{d+e x}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )-\log \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x+1)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d+1)}\right )+\log \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x+1)}{2 c (d+1)-e \left (\sqrt{b^2-4 a c}+b\right )}\right )+\log \left (\frac{d+e x+1}{d+e x}\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x\right )-\log \left (\frac{d+e x+1}{d+e x}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 \sqrt{b^2-4 a c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-1 + d + e*x))/(2*c*(-1 + d) + (-b + Sqrt[b^2 - 4*a*c])*e)] - Lo
g[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-1 + d + e*x))/(2*c*(-1 + d) - (b + Sqrt[b^2 - 4*a*c])*e)] - Log[b
- Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-1 + d + e*x)/(d + e*x)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-1 + d +
e*x)/(d + e*x)] - Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(1 + d + e*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4
*a*c])*e)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(1 + d + e*x))/(2*c*(1 + d) - (b + Sqrt[b^2 - 4*a*c])
*e)] + Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(1 + d + e*x)/(d + e*x)] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Lo
g[(1 + d + e*x)/(d + e*x)] - PolyLog[2, (e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(1 + d) + (-b + Sqrt[b^2 - 4
*a*c])*e)] + PolyLog[2, (e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)] - PolyL
og[2, (e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(-1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2, (e*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(1 + d) + (b + Sqrt[b^2 - 4*a*c])*e)])/(2*Sqrt[b^2 - 4*a*c])

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Maple [B]  time = 0.586, size = 2098, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-
1)/(a*e^2-b*e*d+c*d^2+(-e^2*(4*a*c-b^2))^(1/2)-c))-1/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)^2+1
/2/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a*e^2
-b*e*d+c*d^2+(-e^2*(4*a*c-b^2))^(1/2)-c))-e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2
*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*arccoth(e*x+d)*a/(a*e^2-b*e*d+c*d^
2-(e^2*(-4*a*c+b^2))^(1/2)-c)+(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d
+1)/(e*x+d-1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*arccoth(e*x+d)*b*d/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a
*c+b^2))^(1/2)-c)-1/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x
+d-1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*arccoth(e*x+d)*c*d^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2
))^(1/2)-c)+e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)^2*a/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1
/2)-c)-(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)^2*b*d/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c
)+1/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*arccoth(e*x+d)^2*c*d^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-
c)-1/2*e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a
*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*a/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)+1/2*(e^2*(-4*a*
c+b^2))^(1/2)/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a*e^2-b*e*d+c*d^2-(-e
^2*(4*a*c-b^2))^(1/2)-c))*b*d/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-1/2/e*(e^2*(-4*a*c+b^2))^(1/2)/(4
*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^
(1/2)-c))*c*d^2/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-e/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c
)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*arc
coth(e*x+d)+1/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*ln(1-(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/
(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*arccoth(e*x+d)*c/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c
)+e/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)*arccoth(e*x+d)^2-1/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*a
rccoth(e*x+d)^2*c/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)-1/2*e/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(
1/2)-c)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1
/2)-c))+1/2/e*(e^2*(-4*a*c+b^2))^(1/2)/(4*a*c-b^2)*polylog(2,(a*e^2-b*d*e+c*d^2+b*e-2*c*d+c)*(e*x+d+1)/(e*x+d-
1)/(a*e^2-b*e*d+c*d^2-(-e^2*(4*a*c-b^2))^(1/2)-c))*c/(a*e^2-b*e*d+c*d^2-(e^2*(-4*a*c+b^2))^(1/2)-c)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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