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

Optimal. Leaf size=801 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (d x^2+c\right )}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (a^2 x^2+1\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (d x^2+c\right )}{4 c \left (a^2 c-d\right )}-\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \]

[Out]

1/2*x*arccot(a*x)/c/(d*x^2+c)+1/4*a*ln(a^2*x^2+1)/c/(a^2*c-d)-1/4*a*ln(d*x^2+c)/c/(a^2*c-d)+1/2*arccot(a*x)*ar
ctan(x*d^(1/2)/c^(1/2))/c^(3/2)/d^(1/2)+1/8*I*a*ln(-(1+x*(-a^2)^(1/2))*d^(1/2)/(I*(-a^2)^(1/2)*c^(1/2)-d^(1/2)
))*ln(1-I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)-1/8*I*a*ln((1-x*(-a^2)^(1/2))*d^(1/2)/(I*(-a^2)^(1/2
)*c^(1/2)+d^(1/2)))*ln(1-I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)+1/8*I*a*ln(-(1-x*(-a^2)^(1/2))*d^(1
/2)/(I*(-a^2)^(1/2)*c^(1/2)-d^(1/2)))*ln(1+I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)-1/8*I*a*ln((1+x*(
-a^2)^(1/2))*d^(1/2)/(I*(-a^2)^(1/2)*c^(1/2)+d^(1/2)))*ln(1+I*x*d^(1/2)/c^(1/2))/c^(3/2)/(-a^2)^(1/2)/d^(1/2)-
1/8*I*a*polylog(2,(-a^2)^(1/2)*(c^(1/2)-I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)-I*d^(1/2)))/c^(3/2)/(-a^2)^(1/2)/d^
(1/2)+1/8*I*a*polylog(2,(-a^2)^(1/2)*(c^(1/2)-I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)+I*d^(1/2)))/c^(3/2)/(-a^2)^(1
/2)/d^(1/2)-1/8*I*a*polylog(2,(-a^2)^(1/2)*(c^(1/2)+I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)-I*d^(1/2)))/c^(3/2)/(-a
^2)^(1/2)/d^(1/2)+1/8*I*a*polylog(2,(-a^2)^(1/2)*(c^(1/2)+I*x*d^(1/2))/((-a^2)^(1/2)*c^(1/2)+I*d^(1/2)))/c^(3/
2)/(-a^2)^(1/2)/d^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.16, antiderivative size = 801, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {199, 205, 4913, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \cot ^{-1}(a x)}{2 c^{3/2} \sqrt {d}}+\frac {x \cot ^{-1}(a x)}{2 c \left (d x^2+c\right )}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (\sqrt {-a^2} x+1\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (\frac {i \sqrt {d} x}{\sqrt {c}}+1\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (a^2 x^2+1\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (d x^2+c\right )}{4 c \left (a^2 c-d\right )}-\frac {i a \text {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \text {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \text {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \text {PolyLog}\left (2,\frac {\sqrt {-a^2} \left (i \sqrt {d} x+\sqrt {c}\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/(c + d*x^2)^2,x]

[Out]

(x*ArcCot[a*x])/(2*c*(c + d*x^2)) + (ArcCot[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*Sqrt[d]) - ((I/8)*a*L
og[(Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*
c^(3/2)*Sqrt[d]) + ((I/8)*a*Log[-((Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*S
qrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/8)*a*Log[-((Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*S
qrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) - ((I/8)*a*Log[(Sqrt[d]*(1 +
Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d])
+ (a*Log[1 + a^2*x^2])/(4*c*(a^2*c - d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c - d)) - ((I/8)*a*PolyLog[2, (Sqrt[-a
^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + ((I/8)*a*PolyLo
g[2, (Sqrt[-a^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*Sqrt[d]) - ((
I/8)*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sqrt[d])])/(Sqrt[-a^2]*c^(3/2)*
Sqrt[d]) + ((I/8)*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(Sqrt[-
a^2]*c^(3/2)*Sqrt[d])

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 4908

Int[ArcTan[(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 4913

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+a \int \frac {\frac {x}{2 c \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}}{1+a^2 x^2} \, dx\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+a \int \left (\frac {x}{2 c \left (1+a^2 x^2\right ) \left (c+d x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d} \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a \int \frac {x}{\left (1+a^2 x^2\right ) \left (c+d x^2\right )} \, dx}{2 c}+\frac {a \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a^2 x^2} \, dx}{2 c^{3/2} \sqrt {d}}\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\left (1+a^2 x\right ) (c+d x)} \, dx,x,x^2\right )}{4 c}+\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+a^2 x^2} \, dx}{4 c^{3/2} \sqrt {d}}\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{4 c \left (a^2 c-d\right )}+\frac {(i a) \int \left (\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1-\sqrt {-a^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1+\sqrt {-a^2} x\right )}\right ) \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(i a) \int \left (\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1-\sqrt {-a^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{2 \left (1+\sqrt {-a^2} x\right )}\right ) \, dx}{4 c^{3/2} \sqrt {d}}-\frac {(a d) \operatorname {Subst}\left (\int \frac {1}{c+d x} \, dx,x,x^2\right )}{4 c \left (a^2 c-d\right )}\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}+\frac {a \log \left (1+a^2 x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c-d\right )}+\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-\sqrt {-a^2} x} \, dx}{8 c^{3/2} \sqrt {d}}+\frac {(i a) \int \frac {\log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+\sqrt {-a^2} x} \, dx}{8 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1-\sqrt {-a^2} x} \, dx}{8 c^{3/2} \sqrt {d}}-\frac {(i a) \int \frac {\log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{1+\sqrt {-a^2} x} \, dx}{8 c^{3/2} \sqrt {d}}\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (1+a^2 x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c-d\right )}+\frac {a \int \frac {\log \left (-\frac {i \sqrt {d} \left (1-\sqrt {-a^2} x\right )}{\sqrt {c} \left (\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}+\frac {a \int \frac {\log \left (\frac {i \sqrt {d} \left (1-\sqrt {-a^2} x\right )}{\sqrt {c} \left (\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}-\frac {a \int \frac {\log \left (-\frac {i \sqrt {d} \left (1+\sqrt {-a^2} x\right )}{\sqrt {c} \left (-\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1-\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}-\frac {a \int \frac {\log \left (\frac {i \sqrt {d} \left (1+\sqrt {-a^2} x\right )}{\sqrt {c} \left (-\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}\right )}\right )}{1+\frac {i \sqrt {d} x}{\sqrt {c}}} \, dx}{8 \sqrt {-a^2} c^2}\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (1+a^2 x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {(i a) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a^2} x}{-\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {(i a) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a^2} x}{\sqrt {-a^2}-\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {(i a) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a^2} x}{-\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {(i a) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a^2} x}{\sqrt {-a^2}+\frac {i \sqrt {d}}{\sqrt {c}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}\\ &=\frac {x \cot ^{-1}(a x)}{2 c \left (c+d x^2\right )}+\frac {\cot ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \log \left (-\frac {\sqrt {d} \left (1-\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}-\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \log \left (\frac {\sqrt {d} \left (1+\sqrt {-a^2} x\right )}{i \sqrt {-a^2} \sqrt {c}+\sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {a \log \left (1+a^2 x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {a \log \left (c+d x^2\right )}{4 c \left (a^2 c-d\right )}-\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (\sqrt {c}-i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}-\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}-i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}+\frac {i a \text {Li}_2\left (\frac {\sqrt {-a^2} \left (\sqrt {c}+i \sqrt {d} x\right )}{\sqrt {-a^2} \sqrt {c}+i \sqrt {d}}\right )}{8 \sqrt {-a^2} c^{3/2} \sqrt {d}}\\ \end {align*}

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Mathematica [A]  time = 8.26, size = 806, normalized size = 1.01 \[ -\frac {a \left (\frac {2 \log \left (\frac {c a^2+d+\left (d-a^2 c\right ) \cos \left (2 \cot ^{-1}(a x)\right )}{c a^2+d}\right )}{a^2 c-d}+\frac {2 \cos ^{-1}\left (\frac {c a^2+d}{a^2 c-d}\right ) \tanh ^{-1}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )+4 \cot ^{-1}(a x) \tanh ^{-1}\left (\frac {a d x}{\sqrt {-a^2 c d}}\right )+\left (\cos ^{-1}\left (\frac {c a^2+d}{a^2 c-d}\right )-2 i \tanh ^{-1}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 i d \left (i c a^2+\sqrt {-a^2 c d}\right ) (a x+i)}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )+\left (\cos ^{-1}\left (\frac {c a^2+d}{a^2 c-d}\right )+2 i \tanh ^{-1}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (c a^2+i \sqrt {-a^2 c d}\right ) (a x-i)}{\left (a^2 c-d\right ) \left (a d x-\sqrt {-a^2 c d}\right )}\right )-\left (\cos ^{-1}\left (\frac {c a^2+d}{a^2 c-d}\right )+2 i \tanh ^{-1}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )+2 i \tanh ^{-1}\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 {-c a^2-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )-\left (\cos ^{-1}\left (\frac {c a^2+d}{a^2 c-d}\right )-2 i \tanh ^{-1}\left (\frac {a c}{\sqrt {-a^2 c d} x}\right )-2 i \tanh ^{-1}\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 {-c a^2-d+\left (a^2 c-d\right ) \cos \left (2 \cot ^{-1}(a x)\right )}}\right )+i \left (\text {Li}_2\left (\frac {\left (c a^2+d-2 i \sqrt {-a^2 c d}\right ) \left (a d x+\sqrt {-a^2 c d}\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )-\text {Li}_2\left (\frac {\left (c a^2+d+2 i \sqrt {-a^2 c d}\right ) \left (a d x+\sqrt {-a^2 c d}\right )}{\left (a^2 c-d\right ) \left (\sqrt {-a^2 c d}-a d x\right )}\right )\right )}{\sqrt {-a^2 c d}}-\frac {4 \cot ^{-1}(a x) \sin \left (2 \cot ^{-1}(a x)\right )}{c a^2+d+\left (d-a^2 c\right ) \cos \left (2 \cot ^{-1}(a x)\right )}\right )}{8 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/8*(a*((2*Log[(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])/(a^2*c + d)])/(a^2*c - d) + (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)/Sqrt[-(a^2*c*d)]] + (ArcC
os[(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))]))/Sqrt[-(a^2*c*d)] - (4*ArcCot
[a*x]*Sin[2*ArcCot[a*x]])/(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])))/c

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(a*x)/(d^2*x^4 + 2*c*d*x^2 + c^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.80, size = 2177, normalized size = 2.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a*(a^2*c*d)^(1/2)/c^2/(a^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))+3/
8*a^3/(a^4*c^2-2*a^2*c*d+d^2)/(a^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))
*(a^2*c*d)^(1/2)+1/2*a^4*arccot(a*x)/(a^2*c-d)/(a^2*d*x^2+a^2*c)*x+1/4*(d*c)^(1/2)/c^2*d*arctanh(1/4*(2*(a^2*c
-d)*(I+a*x)^2/(a^2*x^2+1)-2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*c-d)^2-1/2*I*a^3*arccot(a*x)/(a^2*c-d)/(a^2*d*x^2+a
^2*c)+1/4*(d*c)^(1/2)/c^2*arctanh(1/4*(2*(a^2*c-d)*(I+a*x)^2/(a^2*x^2+1)-2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*c-d)
+3/4*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)/(a^4*c^2-2*a^2*c*d+d^
2)/(a^2*c-d)*(a^2*c*d)^(1/2)-1/4*I/a*(a^2*c*d)^(1/2)/c^2/(a^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))-1/4*a^3/(a^2*c-d)^2*ln((I+a*x)^4/(a^2*x^2+1)^2*a^2*c-2*a^2*c*(I+a*x)^2/(a^2
*x^2+1)-(I+a*x)^4/(a^2*x^2+1)^2*d+a^2*c-2*(I+a*x)^2/(a^2*x^2+1)*d-d)+1/8*a*(a^2*c*d)^(1/2)/(a^2*c-d)/c/d*polyl
og(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))+1/4*a*(a^2*c*d)^(1/2)/(a^2*c-d)/c/d*arccot(a
*x)^2+1/4*a^2*(d*c)^(1/2)/d/c*arctanh(1/4*(2*(a^2*c-d)*(I+a*x)^2/(a^2*x^2+1)-2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*
c-d)+a^3/(a^2*c-d)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))-1/4*a^4*(d*c)^(1/2)/d*arctanh(1/4*(2*(a^2*c-d)*(I+a*x)^2/(a
^2*x^2+1)-2*a^2*c-2*d)/a/(d*c)^(1/2))/(a^2*c-d)^2-3/4*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)*d/(a^2*c-d)/c/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)-1/4*I*a^5*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^2*c-d)/d/(a^4*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)*
c+1/4*I/a*d^2*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^2/(a^2*c-d)/(a^4
*c^2-2*a^2*c*d+d^2)*(a^2*c*d)^(1/2)-1/8*a^5/(a^2*c-d)/d/(a^4*c^2-2*a^2*c*d+d^2)*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))*(a^2*c*d)^(1/2)*c+1/8/a/c^2*d^2/(a^2*c-d)/(a^4*c^2-2*a^2*c*d+d^2)*pol
ylog(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^2*c*d)^(1/2)-3/8*a/(a^2*c-d)/c/(a^4*c^2
-2*a^2*c*d+d^2)*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))*(a^2*c*d)^(1/2)-1/4*a
^5/(a^2*c-d)/d/(a^4*c^2-2*a^2*c*d+d^2)*arccot(a*x)^2*(a^2*c*d)^(1/2)*c-3/4*a/(a^2*c-d)/c/(a^4*c^2-2*a^2*c*d+d^
2)*d*arccot(a*x)^2*(a^2*c*d)^(1/2)+1/4/a/c^2*d^2/(a^2*c-d)/(a^4*c^2-2*a^2*c*d+d^2)*arccot(a*x)^2*(a^2*c*d)^(1/
2)-1/2*a^2*arccot(a*x)/c/(a^2*c-d)/(a^2*d*x^2+a^2*c)*x*d-a/c/(a^2*c-d)^2*d*ln((I+a*x)/(a^2*x^2+1)^(1/2))+1/4*a
/c/(a^2*c-d)^2*d*ln((I+a*x)^4/(a^2*x^2+1)^2*a^2*c-2*a^2*c*(I+a*x)^2/(a^2*x^2+1)-(I+a*x)^4/(a^2*x^2+1)^2*d+a^2*
c-2*(I+a*x)^2/(a^2*x^2+1)*d-d)+3/4*a^3/(a^2*c-d)/(a^4*c^2-2*a^2*c*d+d^2)*arccot(a*x)^2*(a^2*c*d)^(1/2)-1/4/a*(
a^2*c*d)^(1/2)/c^2/(a^2*c-d)*arccot(a*x)^2-1/2*I*a^3*arccot(a*x)/c/(a^2*c-d)/(a^2*d*x^2+a^2*c)*x^2*d+1/4*I*a*(
a^2*c*d)^(1/2)/(a^2*c-d)/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))

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maxima [A]  time = 0.55, size = 628, normalized size = 0.78 \[ \frac {1}{2} \, {\left (\frac {x}{c d x^{2} + c^{2}} + \frac {\arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c}\right )} \operatorname {arccot}\left (a x\right ) + \frac {{\left (4 \, a c d \log \left (a^{2} x^{2} + 1\right ) - 4 \, a c d \log \left (d x^{2} + c\right ) + {\left (4 \, {\left (a^{2} c - d\right )} \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, {\left (a^{2} c - d\right )} \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 ) + {\left (a^{2} c - 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 ) - {\left (a^{2} c - 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 ) + 2 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 \, {\left (a^{2} c - d\right )} {\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 )\right )} \sqrt {c} \sqrt {d}\right )} a}{16 \, {\left (a^{3} c^{3} d - a c^{2} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x/(c*d*x^2 + c^2) + arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c))*arccot(a*x) + 1/16*(4*a*c*d*log(a^2*x^2 + 1) - 4
*a*c*d*log(d*x^2 + c) + (4*(a^2*c - d)*arctan(a*x)*arctan(sqrt(d)*x/sqrt(c)) + 4*(a^2*c - d)*arctan(sqrt(d)*x/
sqrt(c))*arctan2(-a*sqrt(d)*x/(a*sqrt(c) - sqrt(d)), -sqrt(d)/(a*sqrt(c) - sqrt(d))) + (a^2*c - 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)) - (a^2*c - 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)) + 2
*(a^2*c - d)*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*(a
^2*c - d)*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*(a^2*
c - d)*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*(a^2*c -
d)*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)))*sqrt(c)*sqrt(d
))*a/(a^3*c^3*d - a*c^2*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)/(c + d*x^2)^2,x)

[Out]

int(acot(a*x)/(c + d*x^2)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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