\(\int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx\) [82]

Optimal result

Integrand size = 19, antiderivative size = 335 \[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c-2 c d+b e-\sqrt {b^2-4 a c} e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+d+e x)}\right )}{2 \sqrt {b^2-4 a c}} \]

[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] (verified)

Time = 0.53 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {632, 212, 6860, 6247, 6058, 2449, 2352, 2497} \[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (-2 d c+2 c+b e-\sqrt {b^2-4 a c} e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\operatorname {PolyLog}\left (2,\frac {2 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-2 c (d+e x)\right )}{\left (2 c (1-d)+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (d+e x+1)}+1\right )}{2 \sqrt {b^2-4 a c}}+\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (b-\sqrt {b^2-4 a c}\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\coth ^{-1}(d+e x) \log \left (\frac {2 e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(d+e x+1) \left (e \left (\sqrt {b^2-4 a c}+b\right )+2 c (1-d)\right )}\right )}{\sqrt {b^2-4 a c}} \]

[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 212

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

Rule 632

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 2352

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

Rule 2449

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 2497

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 6058

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 6247

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 6860

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*} \text {integral}& = \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) \text {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) \text {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 {\text {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 {\text {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 {\operatorname {PolyLog}\left (2,1-\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 {\operatorname {PolyLog}\left (2,1-\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] (verified)

Time = 0.58 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.78 \[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\frac {\log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {2 c (-1+d+e x)}{2 c (-1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )-\log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {2 c (-1+d+e x)}{2 c (-1+d)-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )-\log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {-1+d+e x}{d+e x}\right )+\log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {-1+d+e x}{d+e x}\right )-\log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {2 c (1+d+e x)}{2 c (1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )+\log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {2 c (1+d+e x)}{2 c (1+d)-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )+\log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {1+d+e x}{d+e x}\right )-\log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {1+d+e x}{d+e x}\right )-\operatorname {PolyLog}\left (2,\frac {e \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c (1+d)+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )+\operatorname {PolyLog}\left (2,\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c-2 c d+b e-\sqrt {b^2-4 a c} e}\right )-\operatorname {PolyLog}\left (2,\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c (-1+d)+\left (b+\sqrt {b^2-4 a c}\right ) e}\right )+\operatorname {PolyLog}\left (2,\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c (1+d)+\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {b^2-4 a c}} \]

[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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(749\) vs. \(2(307)=614\).

Time = 1.38 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.24

method	result	size
risch	\(-\frac {e \ln \left (e x +d -1\right ) \ln \left (\frac {-2 \left (e x +d -1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \ln \left (e x +d -1\right ) \ln \left (\frac {2 \left (e x +d -1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{b e -2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \operatorname {dilog}\left (\frac {-2 \left (e x +d -1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \operatorname {dilog}\left (\frac {2 \left (e x +d -1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{b e -2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \ln \left (e x +d +1\right ) \ln \left (\frac {-2 \left (e x +d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{-b e +2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \ln \left (e x +d +1\right ) \ln \left (\frac {2 \left (e x +d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \operatorname {dilog}\left (\frac {-2 \left (e x +d +1\right ) c -b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 c}{-b e +2 c d +2 c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \operatorname {dilog}\left (\frac {2 \left (e x +d +1\right ) c +b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}{b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 c}\right )}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(750\)
derivativedivides	\(\text {Expression too large to display}\)	\(2090\)
default	\(\text {Expression too large to display}\)	\(2090\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (e x + d\right )}{c x^{2} + b x + a} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]

[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 deta

Giac [F]

\[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (e x + d\right )}{c x^{2} + b x + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^{-1}(d+e x)}{a+b x+c x^2} \, dx=\int \frac {\mathrm {acoth}\left (d+e\,x\right )}{c\,x^2+b\,x+a} \,d x \]

[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)