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

Optimal. Leaf size=642 \[ \frac{\text{PolyLog}\left (2,-\frac{(-a-b x+i) \left (b \sqrt{c}-i a \sqrt{d}\right )}{(a+b x) \left (b \sqrt{c}-(1+i a) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,-\frac{(-a-b x+i) \left (b \sqrt{c}+i a \sqrt{d}\right )}{(a+b x) \left (b \sqrt{c}+(1+i a) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}-\frac{\text{PolyLog}\left (2,\frac{(a+b x+i) \left (b \sqrt{c}-i a \sqrt{d}\right )}{(a+b x) \left (b \sqrt{c}+(1-i a) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{\text{PolyLog}\left (2,\frac{(a+b x+i) \left (b \sqrt{c}+i a \sqrt{d}\right )}{(a+b x) \left (b \sqrt{c}+i (a+i) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (-\frac{b \left (-\sqrt{d} x+i \sqrt{c}\right )}{(a+b x) \left (b \sqrt{c}+(1-i a) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{\log \left (-\frac{-a-b x+i}{a+b x}\right ) \log \left (\frac{i b \left (\sqrt{c}+i \sqrt{d} x\right )}{(a+b x) \left (b \sqrt{c}-(1+i a) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}-\frac{\log \left (-\frac{-a-b x+i}{a+b x}\right ) \log \left (\frac{b \left (\sqrt{d} x+i \sqrt{c}\right )}{(a+b x) \left (b \sqrt{c}+(1+i a) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}}+\frac{\log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (-\frac{b \left (\sqrt{d} x+i \sqrt{c}\right )}{(a+b x) \left (b \sqrt{c}+i (a+i) \sqrt{d}\right )}\right )}{4 \sqrt{c} \sqrt{d}} \]

[Out]

-(Log[(I + a + b*x)/(a + b*x)]*Log[-((b*(I*Sqrt[c] - Sqrt[d]*x))/((b*Sqrt[c] + (1 - I*a)*Sqrt[d])*(a + b*x)))]
)/(4*Sqrt[c]*Sqrt[d]) + (Log[-((I - a - b*x)/(a + b*x))]*Log[(I*b*(Sqrt[c] + I*Sqrt[d]*x))/((b*Sqrt[c] - (1 +
I*a)*Sqrt[d])*(a + b*x))])/(4*Sqrt[c]*Sqrt[d]) - (Log[-((I - a - b*x)/(a + b*x))]*Log[(b*(I*Sqrt[c] + Sqrt[d]*
x))/((b*Sqrt[c] + (1 + I*a)*Sqrt[d])*(a + b*x))])/(4*Sqrt[c]*Sqrt[d]) + (Log[(I + a + b*x)/(a + b*x)]*Log[-((b
*(I*Sqrt[c] + Sqrt[d]*x))/((b*Sqrt[c] + I*(I + a)*Sqrt[d])*(a + b*x)))])/(4*Sqrt[c]*Sqrt[d]) + PolyLog[2, -(((
b*Sqrt[c] - I*a*Sqrt[d])*(I - a - b*x))/((b*Sqrt[c] - (1 + I*a)*Sqrt[d])*(a + b*x)))]/(4*Sqrt[c]*Sqrt[d]) - Po
lyLog[2, -(((b*Sqrt[c] + I*a*Sqrt[d])*(I - a - b*x))/((b*Sqrt[c] + (1 + I*a)*Sqrt[d])*(a + b*x)))]/(4*Sqrt[c]*
Sqrt[d]) - PolyLog[2, ((b*Sqrt[c] - I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + (1 - I*a)*Sqrt[d])*(a + b*x))]/(
4*Sqrt[c]*Sqrt[d]) + PolyLog[2, ((b*Sqrt[c] + I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + I*(I + a)*Sqrt[d])*(a
+ b*x))]/(4*Sqrt[c]*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 0.997957, antiderivative size = 655, normalized size of antiderivative = 1.02, number of steps used = 37, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5052, 2513, 2409, 2394, 2393, 2391, 205} \[ \frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} (-a-b x+i)}{b \sqrt{-c}-(-a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} (-a-b x+i)}{b \sqrt{-c}+(-a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{i \log (a+b x-i) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}-(-a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{i \log (a+b x+i) \log \left (\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}+(a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}-\frac{i \log (a+b x-i) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}+(-a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{i \log (a+b x+i) \log \left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+i) \sqrt{d}}\right )}{4 \sqrt{-c} \sqrt{d}}+\frac{i \left (\log \left (-\frac{-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}}-\frac{i \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac{a+b x+i}{a+b x}\right )\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 \sqrt{c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

Int[ArcCot[a + b*x]/(c + d*x^2),x]

[Out]

((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/(Sqrt
[c]*Sqrt[d]) - ((I/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*
x)]))/(Sqrt[c]*Sqrt[d]) + ((I/4)*Log[-I + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (I - a)*Sqrt[d
])])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*Log[I + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d]
)])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*Log[-I + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] + (I - a)*Sqrt[d]
)])/(Sqrt[-c]*Sqrt[d]) + ((I/4)*Log[I + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d])
])/(Sqrt[-c]*Sqrt[d]) + ((I/4)*PolyLog[2, -((Sqrt[d]*(I - a - b*x))/(b*Sqrt[-c] - (I - a)*Sqrt[d]))])/(Sqrt[-c
]*Sqrt[d]) - ((I/4)*PolyLog[2, (Sqrt[d]*(I - a - b*x))/(b*Sqrt[-c] + (I - a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d]) + (
(I/4)*PolyLog[2, -((Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/4)*PolyL
og[2, (Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])])/(Sqrt[-c]*Sqrt[d])

Rule 5052

Int[ArcCot[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Dist[I/2, Int[Log[(-I + a + b*x)/(a + b*
x)]/(c + d*x^n), x], x] - Dist[I/2, Int[Log[(I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}
, x] && RationalQ[n]

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.548539, size = 563, normalized size = 0.88 \[ -\frac{i \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}+(a-i) \sqrt{d}}\right )-\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{b \sqrt{-c}+(a+i) \sqrt{d}}\right )-\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a-i) \sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{b \sqrt{-c}-(a+i) \sqrt{d}}\right )+\log \left (\sqrt{-c}-\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (a+b x-i)}{b \sqrt{-c}+(a-i) \sqrt{d}}\right )-\log \left (\frac{a+b x-i}{a+b x}\right ) \log \left (\sqrt{-c}-\sqrt{d} x\right )-\log \left (\sqrt{-c}-\sqrt{d} x\right ) \log \left (\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}+(a+i) \sqrt{d}}\right )+\log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (\sqrt{-c}-\sqrt{d} x\right )-\log \left (\sqrt{-c}+\sqrt{d} x\right ) \log \left (-\frac{\sqrt{d} (a+b x-i)}{b \sqrt{-c}-(a-i) \sqrt{d}}\right )+\log \left (\frac{a+b x-i}{a+b x}\right ) \log \left (\sqrt{-c}+\sqrt{d} x\right )+\log \left (\sqrt{-c}+\sqrt{d} x\right ) \log \left (-\frac{\sqrt{d} (a+b x+i)}{b \sqrt{-c}-(a+i) \sqrt{d}}\right )-\log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (\sqrt{-c}+\sqrt{d} x\right )\right )}{4 \sqrt{-c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.814, size = 2082, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x
+a)^2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)*a^2-1/2*I*b/d*(b^2*c*d)^(1/2)/(a^2*d+c*b^2+2*(b^2*c*d)
^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d))*arccot(b*
x+a)-I*b/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^
2+2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)-I*b/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a
+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)+1/2*b/d*(b^2*c*d)^(1/2)/(a^2*d+c*b^2-2*
(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-b/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2+1/2/b*(b^2*c*d)^(1/2)/c
/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2+1/2/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*a
rccot(b*x+a)^2*a^2+1/4*b/d*(b^2*c*d)^(1/2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+c*b^2-d
)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d))-1/2*b/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*polylog
(2,(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d))+1/4/b*(b^2*c*d)^(1/2)
/c/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2
-2*(b^2*c*d)^(1/2)+d))+1/4/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+c*b
^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d))*a^2+1/2*I*b/d*(b^2*c*d)^(1/2)/(a^2*d+c*b^2-
2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)
)*arccot(b*x+a)+1/2*I/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a
+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)+1/2*I/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2-
2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2-2*(b^2*c*d)^(1/2)+d)
)*arccot(b*x+a)*a^2-1/2*I/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+c*b^2-d)*
(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)-1/2*b/d*(b^2*c*d)^(1/2)/(a^2*d+c*b^
2+2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-b/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-1/2/b*(b^2*c*d)^(1/
2)/c/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-1/2/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+
d)*arccot(b*x+a)^2*a^2-1/4*b/d*(b^2*c*d)^(1/2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+c*b
^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d))-1/2*b/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*pol
ylog(2,(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d))-1/4/b*(b^2*c*d)^(
1/2)/c/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c
*b^2+2*(b^2*c*d)^(1/2)+d))-1/4/b*(b^2*c*d)^(1/2)/c/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d
+c*b^2-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+c*b^2+2*(b^2*c*d)^(1/2)+d))*a^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(b*x+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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