\(\int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx\) [57]

Optimal result

Integrand size = 14, antiderivative size = 403 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]

[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] (verified)

Time = 0.70 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5029, 211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {i \log \left (1-\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \log \left (1+\frac {i}{a x}\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (-a x+i)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (\sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}} \]

[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 211

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

Rule 266

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 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 2438

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 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 2520

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 4940

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 4966

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 4996

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 5029

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \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 {\arctan \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \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 {\arctan \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-i+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (i+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-i+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (i+a x)} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {i a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-i+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}+\frac {i a \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{i+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \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 {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-i+a x} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {(i a) \int \frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{i+a x} \, dx}{2 \sqrt {c} \sqrt {d}} \\ & = \frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \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 \arctan \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 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \arctan \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 \arctan \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 {\operatorname {PolyLog}\left (2,1-\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 {\operatorname {PolyLog}\left (2,1+\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] (verified)

Time = 1.22 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.78 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=\frac {a \left (-2 \arccos \left (\frac {a^2 c+d}{a^2 c-d}\right ) \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-4 \cot ^{-1}(a x) \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )-\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )-2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 i d \left (i a^2 c+\sqrt {-a^2 c d}\right ) (i+a x)}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )-\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )+2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {-a^2 c d}\right ) (-i+a x)}{\left (a^2 c-d\right ) \left (-\sqrt {-a^2 c d}+a d x\right )}\right )+\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )+2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )+2 i \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-a^2 c d} e^{-i \cot ^{-1}(a x)}}{\sqrt {a^2 c-d} \sqrt {-a^2 c-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+\left (\arccos \left (\frac {a^2 c+d}{a^2 c-d}\right )-2 i \text {arctanh}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-2 i \text {arctanh}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-a^2 c d} e^{i \cot ^{-1}(a x)}}{\sqrt {a^2 c-d} \sqrt {-a^2 c-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (a^2 c+d-2 i \sqrt {-a^2 c d}\right ) \left (\sqrt {-a^2 c d}+a d x\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (a^2 c+d+2 i \sqrt {-a^2 c d}\right ) \left (\sqrt {-a^2 c d}+a d x\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )\right )\right )}{4 \sqrt {-a^2 c d}} \]

[In]

Integrate[ArcCot[a*x]/(c + d*x^2),x]

[Out]

(a*(-2*ArcCos[(a^2*c + d)/(a^2*c - d)]*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)] - 4*ArcCot[a*x]*ArcTanh[(a*d*x)/Sqr
t[-(a^2*c*d)]] - (ArcCos[(a^2*c + d)/(a^2*c - d)] - (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)])*Log[((2*I)*d*(I
*a^2*c + Sqrt[-(a^2*c*d)])*(I + a*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))] - (ArcCos[(a^2*c + d)/(a^2*c -
d)] + (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)])*Log[(2*d*(a^2*c + I*Sqrt[-(a^2*c*d)])*(-I + a*x))/((a^2*c -
d)*(-Sqrt[-(a^2*c*d)] + a*d*x))] + (ArcCos[(a^2*c + d)/(a^2*c - d)] + (2*I)*ArcTanh[(a*c)/(Sqrt[-(a^2*c*d)]*x)
] + (2*I)*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)]])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)])/(Sqrt[a^2*c - d]*E^(I*ArcCot[a*x])
*Sqrt[-(a^2*c) - d + (a^2*c - d)*Cos[2*ArcCot[a*x]]])] + (ArcCos[(a^2*c + d)/(a^2*c - d)] - (2*I)*ArcTanh[(a*c
)/(Sqrt[-(a^2*c*d)]*x)] - (2*I)*ArcTanh[(a*d*x)/Sqrt[-(a^2*c*d)]])*Log[(Sqrt[2]*Sqrt[-(a^2*c*d)]*E^(I*ArcCot[a
*x]))/(Sqrt[a^2*c - d]*Sqrt[-(a^2*c) - d + (a^2*c - d)*Cos[2*ArcCot[a*x]]])] + I*(-PolyLog[2, ((a^2*c + d - (2
*I)*Sqrt[-(a^2*c*d)])*(Sqrt[-(a^2*c*d)] + a*d*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))] + PolyLog[2, ((a^2
*c + d + (2*I)*Sqrt[-(a^2*c*d)])*(Sqrt[-(a^2*c*d)] + a*d*x))/((a^2*c - d)*(Sqrt[-(a^2*c*d)] - a*d*x))])))/(4*S
qrt[-(a^2*c*d)])

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {i \pi \,\operatorname {arctanh}\left (\frac {2 \left (-i a x +1\right ) d -2 d}{2 a \sqrt {c d}}\right )}{2 \sqrt {c d}}-\frac {\ln \left (-i a x +1\right ) \ln \left (\frac {a \sqrt {c d}-\left (-i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (-i a x +1\right ) \ln \left (\frac {a \sqrt {c d}+\left (-i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}-\left (-i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}+\left (-i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\ln \left (i a x +1\right ) \ln \left (\frac {a \sqrt {c d}-\left (i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (i a x +1\right ) \ln \left (\frac {a \sqrt {c d}+\left (i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}-\left (i a x +1\right ) d +d}{a \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\operatorname {dilog}\left (\frac {a \sqrt {c d}+\left (i a x +1\right ) d -d}{a \sqrt {c d}-d}\right )}{4 \sqrt {c d}}\)	\(392\)
derivativedivides	\(\frac {-\frac {i \sqrt {a^{2} c d}\, \operatorname {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 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right )^{2}}{2 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {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 c d}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \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 ) \operatorname {arccot}\left (a x \right )}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i \left (a^{2} c +2 \sqrt {a^{2} c d}+d \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 ) \operatorname {arccot}\left (a x \right ) a^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \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 ) \operatorname {arccot}\left (a x \right )}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {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 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {arccot}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) \operatorname {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 ) a^{2}}{2 \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {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 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}}{a}\)	\(977\)
default	\(\frac {-\frac {i \sqrt {a^{2} c d}\, \operatorname {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 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {arccot}\left (a x \right )^{2}}{2 c d}-\frac {\sqrt {a^{2} c d}\, \operatorname {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 c d}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \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 ) \operatorname {arccot}\left (a x \right )}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{2 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {i \left (a^{2} c +2 \sqrt {a^{2} c d}+d \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 ) \operatorname {arccot}\left (a x \right ) a^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {i \left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \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 ) \operatorname {arccot}\left (a x \right )}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) a^{2} \operatorname {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 d \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) a^{2} \operatorname {arccot}\left (a x \right )^{2}}{a^{4} c^{2}-2 a^{2} c d +d^{2}}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {arccot}\left (a x \right )^{2}}{2 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}-\frac {\left (a^{2} c +2 \sqrt {a^{2} c d}+d \right ) \operatorname {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 ) a^{2}}{2 \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}+\frac {\left (\sqrt {a^{2} c d}\, a^{2} c +2 a^{2} c d +\sqrt {a^{2} c d}\, d \right ) \operatorname {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 c \left (a^{4} c^{2}-2 a^{2} c d +d^{2}\right )}}{a}\)	\(977\)

[In]

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

[Out]

1/2*I*Pi/(c*d)^(1/2)*arctanh(1/2*(2*(1-I*a*x)*d-2*d)/a/(c*d)^(1/2))-1/4*ln(1-I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1
/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d))+1/4*ln(1-I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1
/2)-d))-1/4/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)-(1-I*a*x)*d+d)/(a*(c*d)^(1/2)+d))+1/4/(c*d)^(1/2)*dilog((a*(c*d)^
(1/2)+(1-I*a*x)*d-d)/(a*(c*d)^(1/2)-d))-1/4*ln(1+I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^
(1/2)+d))+1/4*ln(1+I*a*x)/(c*d)^(1/2)*ln((a*(c*d)^(1/2)+(1+I*a*x)*d-d)/(a*(c*d)^(1/2)-d))-1/4/(c*d)^(1/2)*dilo
g((a*(c*d)^(1/2)-(1+I*a*x)*d+d)/(a*(c*d)^(1/2)+d))+1/4/(c*d)^(1/2)*dilog((a*(c*d)^(1/2)+(1+I*a*x)*d-d)/(a*(c*d
)^(1/2)-d))

Fricas [F]

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

[In]

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

[Out]

integral(arccot(a*x)/(d*x^2 + c), x)

Sympy [F]

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

[In]

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

[Out]

Integral(acot(a*x)/(c + d*x**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx=-\frac {a {\left (\frac {8 \, \arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{a} - \frac {4 \, \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \arctan \left (-\frac {a \sqrt {d} x}{a \sqrt {c} - \sqrt {d}}, -\frac {\sqrt {d}}{a \sqrt {c} - \sqrt {d}}\right ) + \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d + 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) - \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} - 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d - 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x + {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x - {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x - {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x + {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right )}{a}\right )}}{8 \, \sqrt {c d}} + \frac {\operatorname {arccot}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {\arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} \]

[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)

Giac [F]

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

[In]

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

[Out]

integrate(arccot(a*x)/(d*x^2 + c), x)

Mupad [F(-1)]

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

[In]

int(acot(a*x)/(c + d*x^2),x)

[Out]

int(acot(a*x)/(c + d*x^2), x)