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

Optimal. Leaf size=403 \[ -\frac {\text {Li}_2\left (1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (\sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \log \left (1-\frac {i}{a x}\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \log \left (1+\frac {i}{a x}\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (-a x+i)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.92, antiderivative size = 403, normalized size of antiderivative = 1.00, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 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 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 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 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 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 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 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 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 = 0.33, size = 523, normalized size = 1.30 \[ \frac {i \left (-\text {Li}_2\left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-i \sqrt {d}}\right )+\text {Li}_2\left (\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {-c} a+i \sqrt {d}}\right )-\text {Li}_2\left (\frac {a \left (\sqrt {d} x+\sqrt {-c}\right )}{a \sqrt {-c}-i \sqrt {d}}\right )+\text {Li}_2\left (\frac {a \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {-c} a+i \sqrt {d}}\right )+\log \left (1-\frac {i}{a x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (1+\frac {i}{a x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\sqrt {-c}-\sqrt {d} x\right ) \log \left (\frac {\sqrt {d} (a x-i)}{a \sqrt {-c}-i \sqrt {d}}\right )+\log \left (\sqrt {-c}-\sqrt {d} x\right ) \log \left (\frac {\sqrt {d} (a x+i)}{a \sqrt {-c}+i \sqrt {d}}\right )-\log \left (1-\frac {i}{a x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (1+\frac {i}{a x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (\sqrt {-c}+\sqrt {d} x\right ) \log \left (\frac {\sqrt {d} (-a x+i)}{a \sqrt {-c}+i \sqrt {d}}\right )-\log \left (\sqrt {-c}+\sqrt {d} x\right ) \log \left (-\frac {\sqrt {d} (a x+i)}{a \sqrt {-c}-i \sqrt {d}}\right )\right )}{4 \sqrt {-c} \sqrt {d}} \]

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])

________________________________________________________________________________________

fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )}{d x^{2} + c}, x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [B]  time = 0.72, size = 826, normalized size = 2.05 \[ -\frac {i \sqrt {a^{2} c d}\, \mathrm {arccot}\left (a x \right ) \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{2 a c d}-\frac {\sqrt {a^{2} c d}\, \mathrm {arccot}\left (a x \right )^{2}}{2 a c d}-\frac {\sqrt {a^{2} c d}\, \polylog \left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c +2 \sqrt {a^{2} c d}+d \right )}\right )}{4 a c d}+\frac {i a^{3} \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \mathrm {arccot}\left (a x \right ) \sqrt {a^{2} c d}\, c}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i a \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \mathrm {arccot}\left (a x \right ) \sqrt {a^{2} c d}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {a^{3} \mathrm {arccot}\left (a x \right )^{2} \sqrt {a^{2} c d}\, c}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {a \mathrm {arccot}\left (a x \right )^{2} \sqrt {a^{2} c d}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \ln \left (1-\frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \mathrm {arccot}\left (a x \right ) \sqrt {a^{2} c d}\, d}{2 a c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {a^{3} \polylog \left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \sqrt {a^{2} c d}\, c}{4 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {a \polylog \left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \sqrt {a^{2} c d}}{2 \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\mathrm {arccot}\left (a x \right )^{2} \sqrt {a^{2} c d}\, d}{2 a c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\polylog \left (2, \frac {\left (a^{2} c -d \right ) \left (a x +i\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (a^{2} c -2 \sqrt {a^{2} c d}+d \right )}\right ) \sqrt {a^{2} c d}\, d}{4 a c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.65, size = 528, normalized size = 1.31 \[ -\frac {a {\left (\frac {8 \, \arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{a} - \frac {4 \, \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \arctan \left (-\frac {a \sqrt {d} x}{a \sqrt {c} - \sqrt {d}}, -\frac {\sqrt {d}}{a \sqrt {c} - \sqrt {d}}\right ) + \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d + 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) - \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} - 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d - 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x + {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x - {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x - {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x + {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right )}{a}\right )}}{8 \, \sqrt {c d}} + \frac {\operatorname {arccot}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {\arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a*(8*arctan(a*x)*arctan(d*x/sqrt(c*d))/a - (4*arctan(a*x)*arctan(sqrt(d)*x/sqrt(c)) + 4*arctan(sqrt(d)*x/
sqrt(c))*arctan2(-a*sqrt(d)*x/(a*sqrt(c) - sqrt(d)), -sqrt(d)/(a*sqrt(c) - sqrt(d))) + log(d*x^2 + c)*log((a^2
*c*d + (a^4*c*d + a^2*d^2)*x^2 + 2*(a^3*d*x^2 + a*d)*sqrt(c)*sqrt(d) + d^2)/(a^4*c^2 + 6*a^2*c*d + 4*(a^3*c +
a*d)*sqrt(c)*sqrt(d) + d^2)) - log(d*x^2 + c)*log((a^2*c*d + (a^4*c*d + a^2*d^2)*x^2 - 2*(a^3*d*x^2 + a*d)*sqr
t(c)*sqrt(d) + d^2)/(a^4*c^2 + 6*a^2*c*d - 4*(a^3*c + a*d)*sqrt(c)*sqrt(d) + d^2)) + 2*dilog((a^2*c + I*a*d*x
+ (I*a^2*x + a)*sqrt(c)*sqrt(d))/(a^2*c + 2*a*sqrt(c)*sqrt(d) + d)) + 2*dilog((a^2*c - I*a*d*x - (I*a^2*x - a)
*sqrt(c)*sqrt(d))/(a^2*c + 2*a*sqrt(c)*sqrt(d) + d)) - 2*dilog((a^2*c + I*a*d*x - (I*a^2*x + a)*sqrt(c)*sqrt(d
))/(a^2*c - 2*a*sqrt(c)*sqrt(d) + d)) - 2*dilog((a^2*c - I*a*d*x + (I*a^2*x - a)*sqrt(c)*sqrt(d))/(a^2*c - 2*a
*sqrt(c)*sqrt(d) + d)))/a)/sqrt(c*d) + arccot(a*x)*arctan(d*x/sqrt(c*d))/sqrt(c*d) + arctan(a*x)*arctan(d*x/sq
rt(c*d))/sqrt(c*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)/(c + d*x^2),x)

[Out]

int(acot(a*x)/(c + d*x^2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}{\left (a x \right )}}{c + d x^{2}}\, dx \]

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)

________________________________________________________________________________________