3.2.7 \(\int \frac {\cot ^{-1}(a+b x)}{c+d x^2} \, dx\) [107]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.64, antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5160, 2576, 2404, 2354, 2438} \begin {gather*} \frac {\text {Li}_2\left (-\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (-a-b x+i)}{\left (b \sqrt {c}-(i a+1) \sqrt {d}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\text {Li}_2\left (-\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (-a-b x+i)}{\left (\sqrt {d} (i a+1)+b \sqrt {c}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}-\frac {\text {Li}_2\left (\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) (a+b x+i)}{\left (\sqrt {d} (1-i a)+b \sqrt {c}\right ) (a+b x)}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {\left (i \sqrt {d} a+b \sqrt {c}\right ) (a+b x+i)}{\left (i \sqrt {d} (a+i)+b \sqrt {c}\right ) (a+b x)}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[a + b*x]/(c + d*x^2),x]

[Out]

-1/4*(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
)))])/(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]) -
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] + (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])

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 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 2576

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol]
:> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Dist[b*c - a*d, Subst[Int[(b^2*f
- a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^2*h)*x^2)^m*((A + B*Log[e*x^n])^p
/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x,
x, 2] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rule 5160

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]

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 \text {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 \text {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 \text {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 \text {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*}

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Mathematica [A]
time = 0.40, size = 563, normalized size = 0.88 \begin {gather*} -\frac {i \left (\log \left (\frac {\sqrt {d} (-i+a+b x)}{b \sqrt {-c}+(-i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}+(i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )+\log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (-\frac {\sqrt {d} (-i+a+b x)}{b \sqrt {-c}-(-i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (-\frac {\sqrt {d} (i+a+b x)}{b \sqrt {-c}-(i+a) \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )-\log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\text {PolyLog}\left (2,\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(-i+a) \sqrt {d}}\right )-\text {PolyLog}\left (2,\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{b \sqrt {-c}+(i+a) \sqrt {d}}\right )-\text {PolyLog}\left (2,\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(-i+a) \sqrt {d}}\right )+\text {PolyLog}\left (2,\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{b \sqrt {-c}-(i+a) \sqrt {d}}\right )\right )}{4 \sqrt {-c} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/4*I)*(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])]*L
og[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)
*Sqrt[d])]))/(Sqrt[-c]*Sqrt[d])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2073 vs. \(2 (495 ) = 990\).
time = 0.70, size = 2074, normalized size = 3.23

method	result	size
risch	\(-\frac {\dilog \left (\frac {i a d -b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d -b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}+\frac {\dilog \left (\frac {i a d +b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d +b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\dilog \left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d +b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\dilog \left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d -b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}-\frac {i b \pi \arctan \left (\frac {2 i a d +2 \left (-i b x -i a +1\right ) d -2 d}{2 \sqrt {-b^{2} c d}}\right )}{2 \sqrt {-b^{2} c d}}-\frac {\ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d -b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d -b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d +b \sqrt {c d}+\left (-i b x -i a +1\right ) d -d}{i a d +b \sqrt {c d}-d}\right )}{4 \sqrt {c d}}-\frac {\ln \left (i b x +i a +1\right ) \ln \left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d +b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}+\frac {\ln \left (i b x +i a +1\right ) \ln \left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a +1\right ) d +d}{i a d -b \sqrt {c d}+d}\right )}{4 \sqrt {c d}}\)	\(543\)
derivativedivides	\(\text {Expression too large to display}\)	\(2074\)
default	\(\text {Expression too large to display}\)	\(2074\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/2*I*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(
b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)+1/2*I*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c-2*(b^2*c*d)^(1
/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d))*a^2*arccot(b
*x+a)+1/2*I*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(
b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)-I*b^2/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*
a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)-1/2*b^2/d*(b^2*c
*d)^(1/2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-b^2/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a
)^2-1/2*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-1/2*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c
+2*(b^2*c*d)^(1/2)+d)*a^2*arccot(b*x+a)^2-1/4*b^2/d*(b^2*c*d)^(1/2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*polylog(
2,(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))-1/2*b^2/(a^2*d+b^2*c+2
*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2
)+d))-1/4*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(
1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))-1/4*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*polylo
g(2,(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))*a^2-I*b^2/(a^2*d+b^2
*c-2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)
+d))*arccot(b*x+a)-1/2*I*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*c-d)*(I+
a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))*a^2*arccot(b*x+a)+1/2*I*b^2/d*(b^2*c*d)^(1/2)/(a^2*d
+b^2*c-2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(
1/2)+d))*arccot(b*x+a)-1/2*I*b^2/d*(b^2*c*d)^(1/2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d)*ln(1-(-2*I*a*d+a^2*d+b^2*
c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c+2*(b^2*c*d)^(1/2)+d))*arccot(b*x+a)+1/2*b^2/d*(b^2*c*d)^(1/2)/(a^2
*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2-b^2/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2+1/2*(b^2*c
*d)^(1/2)/c/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*arccot(b*x+a)^2+1/2*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c-2*(b^2*c*d)^(
1/2)+d)*a^2*arccot(b*x+a)^2+1/4*b^2/d*(b^2*c*d)^(1/2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^
2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d))-1/2*b^2/(a^2*d+b^2*c-2*(b^2*c*d)^(1/
2)+d)*polylog(2,(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d))+1/4*(b^2
*c*d)^(1/2)/c/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(
a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d))+1/4*(b^2*c*d)^(1/2)/c/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d)*polylog(2,(-2*I*a*d+
a^2*d+b^2*c-d)*(I+a+b*x)^2/(1+(b*x+a)^2)/(a^2*d+b^2*c-2*(b^2*c*d)^(1/2)+d))*a^2)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8519 vs. \(2 (456) = 912\).
time = 5.31, size = 8519, normalized size = 13.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*(8*arctan(d*x/sqrt(c*d))*arctan((b^2*x + a*b)/b)/b - (4*arctan(sqrt(d)*x/sqrt(c))*arctan2((2*a*b^2*c*d
+ (a*b^3*c + (a^3 + a)*b*d + (b^4*c + (a^2 + 3)*b^2*d)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*d^2)*x)/(
b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d)), ((a^2 + 3)
*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + (2*a*b^2*d*x + b^3*c + 3*(a^2 + 1)*b*d)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3
+ a)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 + 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqr
t(d))) + 4*arctan(sqrt(d)*x/sqrt(c))*arctan2((2*a*b^2*c*d - (a*b^3*c + (a^3 + a)*b*d + (b^4*c + (a^2 + 3)*b^2*
d)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^
2 - 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*d^2 - (2*a*b^2*d*x + b^
3*c + 3*(a^2 + 1)*b*d)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*d^2)*x)/(b^4*c^2 + 2*(a^2 + 3)*b^2*c*d + (a^
4 + 2*a^2 + 1)*d^2 - 4*(b^3*c + (a^2 + 1)*b*d)*sqrt(c)*sqrt(d))) + log(d*x^2 + c)*log(((a^2 + 1)*b^22*c^11*d +
11*(a^4 + 22*a^2 + 21)*b^20*c^10*d^2 + 55*(a^6 + 39*a^4 + 171*a^2 + 133)*b^18*c^9*d^3 + 33*(5*a^8 + 260*a^6 +
1870*a^4 + 3876*a^2 + 2261)*b^16*c^8*d^4 + 330*(a^10 + 61*a^8 + 570*a^6 + 1802*a^4 + 2261*a^2 + 969)*b^14*c^7
*d^5 + 22*(21*a^12 + 1386*a^10 + 15015*a^8 + 60060*a^6 + 109395*a^4 + 92378*a^2 + 29393)*b^12*c^6*d^6 + 22*(21
*a^14 + 1407*a^12 + 16401*a^10 + 75075*a^8 + 169455*a^6 + 201773*a^4 + 121771*a^2 + 29393)*b^10*c^5*d^7 + 330*
(a^16 + 64*a^14 + 756*a^12 + 3696*a^10 + 9438*a^8 + 13728*a^6 + 11492*a^4 + 5168*a^2 + 969)*b^8*c^4*d^8 + 33*(
5*a^18 + 285*a^16 + 3220*a^14 + 15876*a^12 + 42966*a^10 + 70070*a^8 + 70980*a^6 + 43860*a^4 + 15181*a^2 + 2261
)*b^6*c^3*d^9 + 55*(a^20 + 46*a^18 + 465*a^16 + 2184*a^14 + 5922*a^12 + 10164*a^10 + 11466*a^8 + 8520*a^6 + 40
29*a^4 + 1102*a^2 + 133)*b^4*c^2*d^10 + 11*(a^22 + 31*a^20 + 255*a^18 + 1065*a^16 + 2730*a^14 + 4662*a^12 + 55
02*a^10 + 4530*a^8 + 2565*a^6 + 955*a^4 + 211*a^2 + 21)*b^2*c*d^11 + (a^24 + 12*a^22 + 66*a^20 + 220*a^18 + 49
5*a^16 + 792*a^14 + 924*a^12 + 792*a^10 + 495*a^8 + 220*a^6 + 66*a^4 + 12*a^2 + 1)*d^12 + (b^24*c^11*d + 11*(a
^2 + 21)*b^22*c^10*d^2 + 55*(a^4 + 38*a^2 + 133)*b^20*c^9*d^3 + 33*(5*a^6 + 255*a^4 + 1615*a^2 + 2261)*b^18*c^
8*d^4 + 330*(a^8 + 60*a^6 + 510*a^4 + 1292*a^2 + 969)*b^16*c^7*d^5 + 22*(21*a^10 + 1365*a^8 + 13650*a^6 + 4641
0*a^4 + 62985*a^2 + 29393)*b^14*c^6*d^6 + 22*(21*a^12 + 1386*a^10 + 15015*a^8 + 60060*a^6 + 109395*a^4 + 92378
*a^2 + 29393)*b^12*c^5*d^7 + 330*(a^14 + 63*a^12 + 693*a^10 + 3003*a^8 + 6435*a^6 + 7293*a^4 + 4199*a^2 + 969)
*b^10*c^4*d^8 + 33*(5*a^16 + 280*a^14 + 2940*a^12 + 12936*a^10 + 30030*a^8 + 40040*a^6 + 30940*a^4 + 12920*a^2
+ 2261)*b^8*c^3*d^9 + 55*(a^18 + 45*a^16 + 420*a^14 + 1764*a^12 + 4158*a^10 + 6006*a^8 + 5460*a^6 + 3060*a^4
+ 969*a^2 + 133)*b^6*c^2*d^10 + 11*(a^20 + 30*a^18 + 225*a^16 + 840*a^14 + 1890*a^12 + 2772*a^10 + 2730*a^8 +
1800*a^6 + 765*a^4 + 190*a^2 + 21)*b^4*c*d^11 + (a^22 + 11*a^20 + 55*a^18 + 165*a^16 + 330*a^14 + 462*a^12 + 4
62*a^10 + 330*a^8 + 165*a^6 + 55*a^4 + 11*a^2 + 1)*b^2*d^12)*x^2 + 2*(11*(a^2 + 1)*b^21*c^10*d + 110*(a^4 + 8*
a^2 + 7)*b^19*c^9*d^2 + 33*(15*a^6 + 205*a^4 + 589*a^2 + 399)*b^17*c^8*d^3 + 264*(5*a^8 + 90*a^6 + 408*a^4 + 6
46*a^2 + 323)*b^15*c^7*d^4 + 110*(21*a^10 + 441*a^8 + 2562*a^6 + 6018*a^4 + 6137*a^2 + 2261)*b^13*c^6*d^5 + 4*
(693*a^12 + 15708*a^10 + 105105*a^8 + 308880*a^6 + 449735*a^4 + 319124*a^2 + 88179)*b^11*c^5*d^6 + 110*(21*a^1
4 + 483*a^12 + 3465*a^10 + 11583*a^8 + 20735*a^6 + 20553*a^4 + 10659*a^2 + 2261)*b^9*c^4*d^7 + 264*(5*a^16 + 1
10*a^14 + 798*a^12 + 2838*a^10 + 5720*a^8 + 6890*a^6 + 4930*a^4 + 1938*a^2 + 323)*b^7*c^3*d^8 + 33*(15*a^18 +
295*a^16 + 2044*a^14 + 7308*a^12 + 15554*a^10 + 20930*a^8 + 18060*a^6 + 9724*a^4 + 2983*a^2 + 399)*b^5*c^2*d^9
+ 110*(a^20 + 16*a^18 + 99*a^16 + 336*a^14 + 714*a^12 + 1008*a^10 + 966*a^8 + 624*a^6 + 261*a^4 + 64*a^2 + 7)
*b^3*c*d^10 + 11*(a^22 + 11*a^20 + 55*a^18 + 165*a^16 + 330*a^14 + 462*a^12 + 462*a^10 + 330*a^8 + 165*a^6 + 5
5*a^4 + 11*a^2 + 1)*b*d^11 + (11*b^23*c^10*d + 110*(a^2 + 7)*b^21*c^9*d^2 + 33*(15*a^4 + 190*a^2 + 399)*b^19*c
^8*d^3 + 264*(5*a^6 + 85*a^4 + 323*a^2 + 323)*b^17*c^7*d^4 + 110*(21*a^8 + 420*a^6 + 2142*a^4 + 3876*a^2 + 226
1)*b^15*c^6*d^5 + 4*(693*a^10 + 15015*a^8 + 90090*a^6 + 218790*a^4 + 230945*a^2 + 88179)*b^13*c^5*d^6 + 110*(2
1*a^12 + 462*a^10 + 3003*a^8 + 8580*a^6 + 12155*a^4 + 8398*a^2 + 2261)*b^11*c^4*d^7 + 264*(5*a^14 + 105*a^12 +
693*a^10 + 2145*a^8 + 3575*a^6 + 3315*a^4 + 1615*a^2 + 323)*b^9*c^3*d^8 + 33*(15*a^16 + 280*a^14 + 1764*a^12
+ 5544*a^10 + 10010*a^8 + 10920*a^6 + 7140*a^4 + 2584*a^2 + 399)*b^7*c^2*d^9 + 110*(a^18 + 15*a^16 + 84*a^14 +
252*a^12 + 462*a^10 + 546*a^8 + 420*a^6 + 204*a^4 + 57*a^2 + 7)*b^5*c*d^10 + 11*(a^20 + 10*a^18 + 45*a^16 + 1
20*a^14 + 210*a^12 + 252*a^10 + 210*a^8 + 120*a^6 + 45*a^4 + 10*a^2 + 1)*b^3*d^11)*x^2 + 2*(11*a*b^22*c^10*d +
110*(a^3 + 7*a)*b^20*c^9*d^2 + 33*(15*a^5 + 19...

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a + b*x)/(c + d*x^2),x)

[Out]

int(acot(a + b*x)/(c + d*x^2), x)

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