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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.15834, antiderivative size = 801, normalized size of antiderivative = 1., 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 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 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]

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 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 31

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

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 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 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 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 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]

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*}

Mathematica [A]  time = 7.30628, size = 802, normalized size = 1. \[ -\frac{a \left (\frac{2 \log \left (1-\frac{\left (a^2 c-d\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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]

-(a*((2*Log[1 - ((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)]] + (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))] - (A
rcCos[(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)*ArcTa
nh[(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*Arc
Cot[a*x]])/(a^2*c + d + (-(a^2*c) + d)*Cos[2*ArcCot[a*x]])))/(8*c)

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Maple [B]  time = 0.447, size = 2177, normalized size = 2.7 \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)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/(d*x^2+c)^2,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^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

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)