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

Optimal. Leaf size=403 \[ -\frac{\text{PolyLog}\left (2,1-\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left (a \sqrt{c}-\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,1+\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left (a \sqrt{c}+\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{i \log \left (1-\frac{i}{a x}\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{i \log \left (1+\frac{i}{a x}\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{i \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left (a \sqrt{c}-\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{i \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left (a \sqrt{c}+\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}} \]

[Out]

((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - I/(a*x)])/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Lo
g[1 + I/(a*x)])/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/(
(a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((
-2*I)*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) - PolyLog
[2, 1 - ((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c]*Sqrt[d])
+ PolyLog[2, 1 + ((2*I)*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c
]*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.919218, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {4909, 205, 2470, 12, 260, 6688, 4876, 4848, 2391, 4856, 2402, 2315, 2447} \[ -\frac{\text{PolyLog}\left (2,1-\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left (a \sqrt{c}-\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,1+\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left (a \sqrt{c}+\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{i \log \left (1-\frac{i}{a x}\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{i \log \left (1+\frac{i}{a x}\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{i \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (\frac{2 i \sqrt{c} \sqrt{d} (-a x+i)}{\left (a \sqrt{c}-\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}}+\frac{i \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ) \log \left (-\frac{2 i \sqrt{c} \sqrt{d} (a x+i)}{\left (a \sqrt{c}+\sqrt{d}\right ) \left (\sqrt{c}-i \sqrt{d} x\right )}\right )}{2 \sqrt{c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/(c + d*x^2),x]

[Out]

((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - I/(a*x)])/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Lo
g[1 + I/(a*x)])/(Sqrt[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/(
(a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) + ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[((
-2*I)*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(Sqrt[c]*Sqrt[d]) - PolyLog
[2, 1 - ((2*I)*Sqrt[c]*Sqrt[d]*(I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c]*Sqrt[d])
+ PolyLog[2, 1 + ((2*I)*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c
]*Sqrt[d])

Rule 4909

Int[ArcCot[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I/(c*x)]/(d + e*x^2), x], x]
- Dist[I/2, Int[Log[1 + I/(c*x)]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 2391

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

Rule 4856

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

Mathematica [A]  time = 0.293219, size = 523, normalized size = 1.3 \[ \frac{i \left (-\text{PolyLog}\left (2,\frac{a \left (\sqrt{-c}-\sqrt{d} x\right )}{a \sqrt{-c}-i \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{a \left (\sqrt{-c}-\sqrt{d} x\right )}{a \sqrt{-c}+i \sqrt{d}}\right )-\text{PolyLog}\left (2,\frac{a \left (\sqrt{-c}+\sqrt{d} x\right )}{a \sqrt{-c}-i \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{a \left (\sqrt{-c}+\sqrt{d} x\right )}{a \sqrt{-c}+i \sqrt{d}}\right )+\log \left (1-\frac{i}{a x}\right ) \log \left (\sqrt{-c}-\sqrt{d} x\right )-\log \left (1+\frac{i}{a x}\right ) \log \left (\sqrt{-c}-\sqrt{d} x\right )-\log \left (\sqrt{-c}-\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (a x-i)}{a \sqrt{-c}-i \sqrt{d}}\right )+\log \left (\sqrt{-c}-\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (a x+i)}{a \sqrt{-c}+i \sqrt{d}}\right )-\log \left (1-\frac{i}{a x}\right ) \log \left (\sqrt{-c}+\sqrt{d} x\right )+\log \left (1+\frac{i}{a x}\right ) \log \left (\sqrt{-c}+\sqrt{d} x\right )+\log \left (\sqrt{-c}+\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (-a x+i)}{a \sqrt{-c}+i \sqrt{d}}\right )-\log \left (\sqrt{-c}+\sqrt{d} x\right ) \log \left (-\frac{\sqrt{d} (a x+i)}{a \sqrt{-c}-i \sqrt{d}}\right )\right )}{4 \sqrt{-c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[a*x]/(c + d*x^2),x]

[Out]

((I/4)*(Log[1 - I/(a*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[1 + I/(a*x)]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[(Sqrt[d]
*(-I + a*x))/(a*Sqrt[-c] - I*Sqrt[d])]*Log[Sqrt[-c] - Sqrt[d]*x] + Log[(Sqrt[d]*(I + a*x))/(a*Sqrt[-c] + I*Sqr
t[d])]*Log[Sqrt[-c] - Sqrt[d]*x] - Log[1 - I/(a*x)]*Log[Sqrt[-c] + Sqrt[d]*x] + Log[1 + I/(a*x)]*Log[Sqrt[-c]
+ Sqrt[d]*x] + Log[(Sqrt[d]*(I - a*x))/(a*Sqrt[-c] + I*Sqrt[d])]*Log[Sqrt[-c] + Sqrt[d]*x] - Log[-((Sqrt[d]*(I
+ a*x))/(a*Sqrt[-c] - I*Sqrt[d]))]*Log[Sqrt[-c] + Sqrt[d]*x] - PolyLog[2, (a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[
-c] - I*Sqrt[d])] + PolyLog[2, (a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] + I*Sqrt[d])] - PolyLog[2, (a*(Sqrt[-c]
+ Sqrt[d]*x))/(a*Sqrt[-c] - I*Sqrt[d])] + PolyLog[2, (a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqrt[-c] + I*Sqrt[d])]))/(S
qrt[-c]*Sqrt[d])

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Maple [B]  time = 0.264, size = 826, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I/a*(a^2*c*d)^(1/2)/c/d*arccot(a*x)*ln(1-(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c+2*(a^2*c*d)^(1/2)+d))+1/2
*a^3*arccot(a*x)^2/d/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)*c-a*arccot(a*x)^2/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*
d)^(1/2)+1/2*I*a^3*ln(1-(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c-2*(a^2*c*d)^(1/2)+d))*arccot(a*x)/d/(a^4*c^2-2*
a^2*c*d+d^2)*(a^2*c*d)^(1/2)*c-I*a*ln(1-(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c-2*(a^2*c*d)^(1/2)+d))*arccot(a*
x)/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)+1/4*a^3*polylog(2,(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c-2*(a^2*c*d
)^(1/2)+d))/d/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)*c-1/2*a*polylog(2,(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c
-2*(a^2*c*d)^(1/2)+d))/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)+1/2/a*arccot(a*x)^2/c/(a^4*c^2-2*a^2*c*d+d^2)*(
a^2*c*d)^(1/2)*d+1/4/a*polylog(2,(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c-2*(a^2*c*d)^(1/2)+d))/c/(a^4*c^2-2*a^2
*c*d+d^2)*(a^2*c*d)^(1/2)*d+1/2*I/a*ln(1-(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c-2*(a^2*c*d)^(1/2)+d))*arccot(a
*x)/c/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)*d-1/2/a*(a^2*c*d)^(1/2)/c/d*arccot(a*x)^2-1/4/a*(a^2*c*d)^(1/2)/
c/d*polylog(2,(a^2*c-d)*(a*x+I)^2/(a^2*x^2+1)/(a^2*c+2*(a^2*c*d)^(1/2)+d))

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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(a*x)/(d*x^2+c),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 (a x\right )}{d x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(a*x)/(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/(d*x**2+c),x)

[Out]

Integral(acot(a*x)/(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x)/(d*x^2 + c), x)