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

Optimal. Leaf size=335 \[ -\frac {\text {Li}_2\left (\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (-2 d c+2 c+b e-\sqrt {b^2-4 a c} e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {Li}_2\left (\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (b-\sqrt {b^2-4 a c}\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (\sqrt {b^2-4 a c}+b\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.75, antiderivative size = 335, normalized size of antiderivative = 1.00, 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 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 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 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 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 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]]

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 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 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]

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 = 0.96, size = 596, normalized size = 1.78 \[ \frac {-\text {Li}_2\left (\frac {e \left (-b-2 c x+\sqrt {b^2-4 a c}\right )}{2 c (d+1)+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )+\text {Li}_2\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{-2 d c+2 c+b e-\sqrt {b^2-4 a c} e}\right )-\text {Li}_2\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d-1)}\right )+\text {Li}_2\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+1)}\right )+\log \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (\frac {2 c (d+e x-1)}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c (d-1)}\right )-\log \left (\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (\frac {2 c (d+e x-1)}{2 c (d-1)-e \left (\sqrt {b^2-4 a c}+b\right )}\right )-\log \left (\frac {d+e x-1}{d+e x}\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x\right )+\log \left (\frac {d+e x-1}{d+e x}\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x\right )-\log \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (\frac {2 c (d+e x+1)}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c (d+1)}\right )+\log \left (\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (\frac {2 c (d+e x+1)}{2 c (d+1)-e \left (\sqrt {b^2-4 a c}+b\right )}\right )+\log \left (\frac {d+e x+1}{d+e x}\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x\right )-\log \left (\frac {d+e x+1}{d+e x}\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 \sqrt {b^2-4 a c}} \]

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])

________________________________________________________________________________________

fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (e x + d\right )}{c x^{2} + b x + a}, x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [B]  time = 1.11, size = 2098, normalized size = 6.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))*arccoth(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)*arccoth(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)+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))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________