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

Optimal. Leaf size=367 \[ \frac{i \text{PolyLog}\left (2,1+\frac{2 \left (-e \left (b-\sqrt{b^2-4 a c}\right )-2 c (d+e x)+2 c d\right )}{(1-i (d+e x)) \left (-e \sqrt{b^2-4 a c}+b e-2 c d+2 i c\right )}\right )}{2 \sqrt{b^2-4 a c}}-\frac{i \text{PolyLog}\left (2,1+\frac{2 \left (-e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d+e x)+2 c d\right )}{(1-i (d+e x)) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (-d+i)\right )}\right )}{2 \sqrt{b^2-4 a c}}+\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{(1-i (d+e x)) \left (e \left (b-\sqrt{b^2-4 a c}\right )+2 c (-d+i)\right )}\right )}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{(1-i (d+e x)) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (-d+i)\right )}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.686065, antiderivative size = 367, 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, 5048, 4857, 2402, 2315, 2447} \[ \frac{i \text{PolyLog}\left (2,1+\frac{2 \left (-e \left (b-\sqrt{b^2-4 a c}\right )-2 c (d+e x)+2 c d\right )}{(1-i (d+e x)) \left (-e \sqrt{b^2-4 a c}+b e-2 c d+2 i c\right )}\right )}{2 \sqrt{b^2-4 a c}}-\frac{i \text{PolyLog}\left (2,1+\frac{2 \left (-e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d+e x)+2 c d\right )}{(1-i (d+e x)) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (-d+i)\right )}\right )}{2 \sqrt{b^2-4 a c}}+\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{(1-i (d+e x)) \left (e \left (b-\sqrt{b^2-4 a c}\right )+2 c (-d+i)\right )}\right )}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{(1-i (d+e x)) \left (e \left (\sqrt{b^2-4 a c}+b\right )+2 c (-d+i)\right )}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcCot[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 - I*(
d + e*x)))])/Sqrt[b^2 - 4*a*c] - (ArcCot[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b
+ Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] + ((I/2)*PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[
b^2 - 4*a*c])*e - 2*c*(d + e*x)))/(((2*I)*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 - I*(d + e*x)))])/Sqrt[b^2
- 4*a*c] - ((I/2)*PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(I - d) + (b +
Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/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 5048

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 4857

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (-Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcCot[c*x])*Log[(2*c*(d + e*x))/((c*
d + I*e)*(1 - I*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{\cot ^{-1}(d+e x)}{a+b x+c x^2} \, dx &=\int \left (\frac{2 c \cot ^{-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 \cot ^{-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{\cot ^{-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{\cot ^{-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{\cot ^{-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{\cot ^{-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{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 i c-2 c d+b e-\sqrt{b^2-4 a c} e\right ) (1-i (d+e x))}\right )}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1-i (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 i c}{e}+\frac{-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e}{e}\right ) (1-i 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 i c}{e}+\frac{-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e}{e}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,d+e x\right )}{\sqrt{b^2-4 a c}}\\ &=\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 i c-2 c d+b e-\sqrt{b^2-4 a c} e\right ) (1-i (d+e x))}\right )}{\sqrt{b^2-4 a c}}-\frac{\cot ^{-1}(d+e x) \log \left (\frac{2 e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{\sqrt{b^2-4 a c}}+\frac{i \text{Li}_2\left (1-\frac{2 e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 i c-2 c d+b e-\sqrt{b^2-4 a c} e\right ) (1-i (d+e x))}\right )}{2 \sqrt{b^2-4 a c}}-\frac{i \text{Li}_2\left (1-\frac{2 e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c (i-d)+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1-i (d+e x))}\right )}{2 \sqrt{b^2-4 a c}}\\ \end{align*}

Mathematica [A]  time = 0.454859, size = 629, normalized size = 1.71 \[ -\frac{i \left (\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d-i)}\right )-\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d+i)}\right )-\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d-i)}\right )+\text{PolyLog}\left (2,\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c (d+i)}\right )+\log \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x-i)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d-i)}\right )-\log \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x-i)}{-e \left (\sqrt{b^2-4 a c}+b\right )+2 c (d-i)}\right )-\log \left (\frac{d+e x-i}{d+e x}\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x\right )+\log \left (\frac{d+e x-i}{d+e x}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )-\log \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x+i)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c (d+i)}\right )+\log \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (\frac{2 c (d+e x+i)}{-e \left (\sqrt{b^2-4 a c}+b\right )+2 c (d+i)}\right )+\log \left (\frac{d+e x+i}{d+e x}\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x\right )-\log \left (\frac{d+e x+i}{d+e x}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )\right )}{2 \sqrt{b^2-4 a c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/2)*(Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-I + d + e*x))/(2*c*(-I + d) + (-b + Sqrt[b^2 - 4*a*c])*
e)] - Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(-I + d + e*x))/(2*c*(-I + d) - (b + Sqrt[b^2 - 4*a*c])*e)]
- Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-I + d + e*x)/(d + e*x)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(-
I + d + e*x)/(d + e*x)] - Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(I + d + e*x))/(2*c*(I + d) + (-b + Sqrt
[b^2 - 4*a*c])*e)] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(2*c*(I + d + e*x))/(2*c*(I + d) - (b + Sqrt[b^2 -
4*a*c])*e)] + Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(I + d + e*x)/(d + e*x)] - Log[b + Sqrt[b^2 - 4*a*c] + 2
*c*x]*Log[(I + d + e*x)/(d + e*x)] + PolyLog[2, (e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(-I + d) + (-b + Sqr
t[b^2 - 4*a*c])*e)] - PolyLog[2, (e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*(I + d) + (-b + Sqrt[b^2 - 4*a*c])*
e)] - PolyLog[2, (e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(-I + d) + (b + Sqrt[b^2 - 4*a*c])*e)] + PolyLog[2,
(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*(I + d) + (b + Sqrt[b^2 - 4*a*c])*e)]))/Sqrt[b^2 - 4*a*c]

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Maple [B]  time = 1.072, size = 4601, normalized size = 12.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*arccot(e*x+d)^2-1/2*e/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1
/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(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)*arccot(e*x+d)^2-1/2*e/(a*e^2-b*e*d+c*d^2-(e^2*
(4*a*c-b^2))^(1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d
^2-(e^2*(4*a*c-b^2))^(1/2)+c))+1/4*I*e*(e^2*(4*a*c-b^2))^(1/2)/c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^
2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c
-b^2))^(1/2)+c))*arccot(e*x+d)*b^2-1/4*I*e*(e^2*(4*a*c-b^2))^(1/2)/c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*
c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2*(4
*a*c-b^2))^(1/2)+c))*arccot(e*x+d)*b^2+1/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*
c-b^2))^(1/2)+c)*arccot(e*x+d)^2*d^2-1/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-
b^2))^(1/2)+c)*arccot(e*x+d)^2*d^2-1/2/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-
b^2))^(1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2
*(4*a*c-b^2))^(1/2)+c))*d^2+1/2/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(
1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c
-b^2))^(1/2)+c))*d^2+1/8*e*(e^2*(4*a*c-b^2))^(1/2)/c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)
*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^
(1/2)+c))*b^2+1/4*e*(e^2*(4*a*c-b^2))^(1/2)/c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*arccot
(e*x+d)^2*b^2-1/4*e*(e^2*(4*a*c-b^2))^(1/2)/c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*arccot
(e*x+d)^2*b^2-1/8*e*(e^2*(4*a*c-b^2))^(1/2)/c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*polylo
g(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c
))*b^2+1/4*I*e*(e^2*(4*a*c-b^2))^(1/2)/c/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e
^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)-1/4*I
*e*(e^2*(4*a*c-b^2))^(1/2)/c/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d
^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)-I*(e^2*(4*a*c-b^2
))^(1/2)/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e
*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)*b*d+I/e*(e^2*(4*a*c-b^2))
^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e
*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)+I*(e^2*(4*a*c-b^2))^(1/2)
/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^
2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)*b*d-I/e*(e^2*(4*a*c-b^2))^(1/2)*c
/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^
2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)+1/8*e*(e^2*(4*a*c-b^2))^(1/2)/c/(
a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d
)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))+I/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c
*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*
d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)*d^2-I/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c
*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*
d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)*d^2-1/4*e*(e^2*(4*a*c-b^2))^(1/2)/c/(a*e^2-b*e*d+c*d^2+(e^2*
(4*a*c-b^2))^(1/2)+c)*arccot(e*x+d)^2-I*e/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*
e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)-I*e/
(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*ln(1-(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+
1)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))*arccot(e*x+d)+(e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(a*e^2-b*e
*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*arccot(e*x+d)^2*b*d+1/2*(e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(a*e^2-b*e*d+c
*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2
-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))*b*d-(e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*
c-b^2))^(1/2)+c)*arccot(e*x+d)^2*b*d-1/2*(e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^
2))^(1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(
4*a*c-b^2))^(1/2)+c))*b*d-1/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)
+c)*arccot(e*x+d)^2-1/2/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*
polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+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)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*arccot(e*x+d)
^2+1/2/e*(e^2*(4*a*c-b^2))^(1/2)*c/(4*a*c-b^2)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*polylog(2,(I*b*e-
2*I*d*c+a*e^2-b*e*d+c*d^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c))-1/8*e*(e
^2*(4*a*c-b^2))^(1/2)/c/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c)*polylog(2,(I*b*e-2*I*d*c+a*e^2-b*e*d+c*d
^2-c)*(e*x+d+I)^2/((e*x+d)^2+1)/(a*e^2-b*e*d+c*d^2+(e^2*(4*a*c-b^2))^(1/2)+c))+1/4*e*(e^2*(4*a*c-b^2))^(1/2)/c
/(a*e^2-b*e*d+c*d^2-(e^2*(4*a*c-b^2))^(1/2)+c)*arccot(e*x+d)^2

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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(arccot(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{arccot}\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(arccot(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

integral(arccot(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(acot(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{arccot}\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(arccot(e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

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