3.2.10 \(\int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx\) [110]

Optimal. Leaf size=735 \[ \frac {\log (i-a-b x)}{2 b c}+\frac {i (a+b x) \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 b c}-\frac {i \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c^{3/2}}+\frac {\log (i+a+b x)}{2 b c}-\frac {i (a+b x) \log \left (\frac {i+a+b x}{a+b x}\right )}{2 b c}+\frac {i \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c^{3/2}}-\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i-a-b x)}{(i-a) \sqrt {c}+i b \sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i+a+b x)}{(i+a) \sqrt {c}-i b \sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i-a-b x)}{(i-a) \sqrt {c}-i b \sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}-\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i+a+b x)}{(i+a) \sqrt {c}+i b \sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {b \left (\sqrt {d}-i \sqrt {c} x\right )}{(1+i a) \sqrt {c}+b \sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {b \left (\sqrt {d}-i \sqrt {c} x\right )}{i (i+a) \sqrt {c}+b \sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \text {PolyLog}\left (2,-\frac {b \left (\sqrt {d}+i \sqrt {c} x\right )}{(1+i a) \sqrt {c}-b \sqrt {d}}\right )}{4 c^{3/2}}-\frac {\sqrt {d} \text {PolyLog}\left (2,\frac {b \left (\sqrt {d}+i \sqrt {c} x\right )}{(1-i a) \sqrt {c}+b \sqrt {d}}\right )}{4 c^{3/2}} \]

[Out]

1/2*ln(I-a-b*x)/b/c+1/2*I*(b*x+a)*ln((-I+a+b*x)/(b*x+a))/b/c+1/2*ln(I+a+b*x)/b/c-1/2*I*(b*x+a)*ln((I+a+b*x)/(b
*x+a))/b/c-1/2*I*arctan(x*c^(1/2)/d^(1/2))*ln((-I+a+b*x)/(b*x+a))*d^(1/2)/c^(3/2)+1/2*I*arctan(x*c^(1/2)/d^(1/
2))*ln((I+a+b*x)/(b*x+a))*d^(1/2)/c^(3/2)+1/4*ln(1+I*x*c^(1/2)/d^(1/2))*ln((I-a-b*x)*c^(1/2)/((I-a)*c^(1/2)-I*
b*d^(1/2)))*d^(1/2)/c^(3/2)+1/4*ln(1-I*x*c^(1/2)/d^(1/2))*ln((I+a+b*x)*c^(1/2)/((I+a)*c^(1/2)-I*b*d^(1/2)))*d^
(1/2)/c^(3/2)-1/4*ln(1-I*x*c^(1/2)/d^(1/2))*ln((I-a-b*x)*c^(1/2)/((I-a)*c^(1/2)+I*b*d^(1/2)))*d^(1/2)/c^(3/2)-
1/4*ln(1+I*x*c^(1/2)/d^(1/2))*ln((I+a+b*x)*c^(1/2)/((I+a)*c^(1/2)+I*b*d^(1/2)))*d^(1/2)/c^(3/2)+1/4*polylog(2,
-b*(I*x*c^(1/2)+d^(1/2))/((1+I*a)*c^(1/2)-b*d^(1/2)))*d^(1/2)/c^(3/2)-1/4*polylog(2,b*(I*x*c^(1/2)+d^(1/2))/((
1-I*a)*c^(1/2)+b*d^(1/2)))*d^(1/2)/c^(3/2)-1/4*polylog(2,b*(-I*x*c^(1/2)+d^(1/2))/((1+I*a)*c^(1/2)+b*d^(1/2)))
*d^(1/2)/c^(3/2)+1/4*polylog(2,b*(-I*x*c^(1/2)+d^(1/2))/(I*(I+a)*c^(1/2)+b*d^(1/2)))*d^(1/2)/c^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 1.21, antiderivative size = 818, normalized size of antiderivative = 1.11, number of steps used = 57, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {5160, 2593, 2456, 2436, 2332, 2441, 2440, 2438, 199, 327, 211} \begin {gather*} \frac {i x \left (\log \left (-\frac {-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right )}{2 c}-\frac {i \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log \left (-\frac {-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right )}{2 c^{3/2}}-\frac {i (-a-b x+i) \log (a+b x-i)}{2 b c}-\frac {i (a+b x+i) \log (a+b x+i)}{2 b c}-\frac {i x \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right )}{2 c}+\frac {i \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right )}{2 c^{3/2}}-\frac {i \sqrt {d} \log (a+b x-i) \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(i-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (a+b x+i) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{\sqrt {-c} (a+i)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \log (a+b x+i) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{(a+i) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \log (a+b x-i) \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{\sqrt {-c} (i-a)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+i)}{(i-a) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (-a-b x+i)}{\sqrt {-c} (i-a)+b \sqrt {d}}\right )}{4 (-c)^{3/2}}-\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+i)}{(a+i) \sqrt {-c}-b \sqrt {d}}\right )}{4 (-c)^{3/2}}+\frac {i \sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x+i)}{\sqrt {-c} (a+i)+b \sqrt {d}}\right )}{4 (-c)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[a + b*x]/(c + d/x^2),x]

[Out]

((I/2)*x*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/c - ((I/2)*Sqrt[d]*ArcTan[(Sqrt
[c]*x)/Sqrt[d]]*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/c^(3/2) - ((I/2)*(I - a
- b*x)*Log[-I + a + b*x])/(b*c) - ((I/2)*(I + a + b*x)*Log[I + a + b*x])/(b*c) - ((I/2)*x*(Log[a + b*x] - Log[
I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/c + ((I/2)*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - L
og[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/c^(3/2) - ((I/4)*Sqrt[d]*Log[-I + a + b*x]*Log[-((b*(Sqrt[d]
- Sqrt[-c]*x))/((I - a)*Sqrt[-c] - b*Sqrt[d]))])/(-c)^(3/2) + ((I/4)*Sqrt[d]*Log[I + a + b*x]*Log[(b*(Sqrt[d]
- Sqrt[-c]*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*Log[I + a + b*x]*Log[-((b*(Sqrt[d]
+ Sqrt[-c]*x))/((I + a)*Sqrt[-c] - b*Sqrt[d]))])/(-c)^(3/2) + ((I/4)*Sqrt[d]*Log[-I + a + b*x]*Log[(b*(Sqrt[d
] + Sqrt[-c]*x))/((I - a)*Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*
x))/((I - a)*Sqrt[-c] - b*Sqrt[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/((I - a)*
Sqrt[-c] + b*Sqrt[d])])/(-c)^(3/2) - ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/((I + a)*Sqrt[-c] - b*
Sqrt[d])])/(-c)^(3/2) + ((I/4)*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(-
c)^(3/2)

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 211

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

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

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 2440

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 2441

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 2456

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 2593

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5117\) vs. \(2(735)=1470\).
time = 30.05, size = 5117, normalized size = 6.96 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.21, size = 13390, normalized size = 18.22

method	result	size
risch	\(\frac {i \pi }{2 b c}+\frac {\pi a}{2 b c}-\frac {i \ln \left (-i b x -i a +1\right ) x}{2 c}+\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b c}+\frac {i \ln \left (i b x +i a +1\right ) x}{2 c}+\frac {i b \pi d \arctan \left (\frac {2 i a c +2 \left (-i b x -i a +1\right ) c -2 c}{2 \sqrt {-b^{2} c d}}\right )}{2 c \sqrt {-b^{2} c d}}-\frac {1}{b c}+\frac {\pi x}{2 c}-\frac {a \arctan \left (b x +a \right )}{b c}-\frac {d \dilog \left (\frac {i a c +b \sqrt {c d}+\left (-i b x -i a +1\right ) c -c}{i a c +b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}+\frac {d \dilog \left (\frac {i a c +b \sqrt {c d}-\left (i b x +i a +1\right ) c +c}{i a c +b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}-\frac {d \ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a c +b \sqrt {c d}+\left (-i b x -i a +1\right ) c -c}{i a c +b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}+\frac {d \ln \left (i b x +i a +1\right ) \ln \left (\frac {i a c +b \sqrt {c d}-\left (i b x +i a +1\right ) c +c}{i a c +b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}+\frac {d \ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a c -b \sqrt {c d}+\left (-i b x -i a +1\right ) c -c}{i a c -b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}-\frac {d \ln \left (i b x +i a +1\right ) \ln \left (\frac {i a c -b \sqrt {c d}-\left (i b x +i a +1\right ) c +c}{i a c -b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}-\frac {d \dilog \left (\frac {i a c -b \sqrt {c d}-\left (i b x +i a +1\right ) c +c}{i a c -b \sqrt {c d}+c}\right )}{4 c \sqrt {c d}}+\frac {d \dilog \left (\frac {i a c -b \sqrt {c d}+\left (-i b x -i a +1\right ) c -c}{i a c -b \sqrt {c d}-c}\right )}{4 c \sqrt {c d}}\)	\(693\)
derivativedivides	\(\text {Expression too large to display}\)	\(13390\)
default	\(\text {Expression too large to display}\)	\(13390\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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. 8518 vs. \(2 (502) = 1004\).
time = 1.20, size = 8518, normalized size = 11.59 \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)/(c+d/x^2),x, algorithm="maxima")

[Out]

-(d*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c) - x/c)*arccot(b*x + a) - 1/8*(8*a*c*arctan(b*x + a) + (4*b*arctan(sqrt
(c)*x/sqrt(d))*arctan2((2*a*b^2*c*d + (a*b^3*d + (a^3 + a)*b*c + (b^4*d + (a^2 + 3)*b^2*c)*x)*sqrt(c)*sqrt(d)
+ (3*b^3*c*d + (a^2 + 1)*b*c^2)*x)/(b^4*d^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 + 4*(b^3*d + (a^2 +
1)*b*c)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 + (2*a*b^2*c*x + b^3*d + 3*(a^2 + 1)*b*c)
*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*c^2)*x)/(b^4*d^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 + 4
*(b^3*d + (a^2 + 1)*b*c)*sqrt(c)*sqrt(d))) + 4*b*arctan(sqrt(c)*x/sqrt(d))*arctan2((2*a*b^2*c*d - (a*b^3*d + (
a^3 + a)*b*c + (b^4*d + (a^2 + 3)*b^2*c)*x)*sqrt(c)*sqrt(d) + (3*b^3*c*d + (a^2 + 1)*b*c^2)*x)/(b^4*d^2 + 2*(a
^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 - 4*(b^3*d + (a^2 + 1)*b*c)*sqrt(c)*sqrt(d)), ((a^2 + 3)*b^2*c*d + (a^
4 + 2*a^2 + 1)*c^2 - (2*a*b^2*c*x + b^3*d + 3*(a^2 + 1)*b*c)*sqrt(c)*sqrt(d) + (a*b^3*c*d + (a^3 + a)*b*c^2)*x
)/(b^4*d^2 + 2*(a^2 + 3)*b^2*c*d + (a^4 + 2*a^2 + 1)*c^2 - 4*(b^3*d + (a^2 + 1)*b*c)*sqrt(c)*sqrt(d))) + b*log
(c*x^2 + d)*log(((a^2 + 1)*b^22*c*d^11 + 11*(a^4 + 22*a^2 + 21)*b^20*c^2*d^10 + 55*(a^6 + 39*a^4 + 171*a^2 + 1
33)*b^18*c^3*d^9 + 33*(5*a^8 + 260*a^6 + 1870*a^4 + 3876*a^2 + 2261)*b^16*c^4*d^8 + 330*(a^10 + 61*a^8 + 570*a
^6 + 1802*a^4 + 2261*a^2 + 969)*b^14*c^5*d^7 + 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^7*d^5 + 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^8*d^4 + 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^9*d^3 + 55*(a^20 + 46*a^18 + 465*a^16 + 2184*a^14 + 5922*a^12
+ 10164*a^10 + 11466*a^8 + 8520*a^6 + 4029*a^4 + 1102*a^2 + 133)*b^4*c^10*d^2 + 11*(a^22 + 31*a^20 + 255*a^18
+ 1065*a^16 + 2730*a^14 + 4662*a^12 + 5502*a^10 + 4530*a^8 + 2565*a^6 + 955*a^4 + 211*a^2 + 21)*b^2*c^11*d + (
a^24 + 12*a^22 + 66*a^20 + 220*a^18 + 495*a^16 + 792*a^14 + 924*a^12 + 792*a^10 + 495*a^8 + 220*a^6 + 66*a^4 +
12*a^2 + 1)*c^12 + (b^24*c*d^11 + 11*(a^2 + 21)*b^22*c^2*d^10 + 55*(a^4 + 38*a^2 + 133)*b^20*c^3*d^9 + 33*(5*
a^6 + 255*a^4 + 1615*a^2 + 2261)*b^18*c^4*d^8 + 330*(a^8 + 60*a^6 + 510*a^4 + 1292*a^2 + 969)*b^16*c^5*d^7 + 2
2*(21*a^10 + 1365*a^8 + 13650*a^6 + 46410*a^4 + 62985*a^2 + 29393)*b^14*c^6*d^6 + 22*(21*a^12 + 1386*a^10 + 15
015*a^8 + 60060*a^6 + 109395*a^4 + 92378*a^2 + 29393)*b^12*c^7*d^5 + 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^8*d^4 + 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^9*d^3 + 55*(a^18 + 45*a^16 + 420*a^14 + 1764*a^12 + 415
8*a^10 + 6006*a^8 + 5460*a^6 + 3060*a^4 + 969*a^2 + 133)*b^6*c^10*d^2 + 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^11*d + (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 + 55*a^4 + 11*a^2 + 1)*b^2*c^12)*x^2 + 2*(
11*(a^2 + 1)*b^21*c*d^10 + 110*(a^4 + 8*a^2 + 7)*b^19*c^2*d^9 + 33*(15*a^6 + 205*a^4 + 589*a^2 + 399)*b^17*c^3
*d^8 + 264*(5*a^8 + 90*a^6 + 408*a^4 + 646*a^2 + 323)*b^15*c^4*d^7 + 110*(21*a^10 + 441*a^8 + 2562*a^6 + 6018*
a^4 + 6137*a^2 + 2261)*b^13*c^5*d^6 + 4*(693*a^12 + 15708*a^10 + 105105*a^8 + 308880*a^6 + 449735*a^4 + 319124
*a^2 + 88179)*b^11*c^6*d^5 + 110*(21*a^14 + 483*a^12 + 3465*a^10 + 11583*a^8 + 20735*a^6 + 20553*a^4 + 10659*a
^2 + 2261)*b^9*c^7*d^4 + 264*(5*a^16 + 110*a^14 + 798*a^12 + 2838*a^10 + 5720*a^8 + 6890*a^6 + 4930*a^4 + 1938
*a^2 + 323)*b^7*c^8*d^3 + 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^9*d^2 + 110*(a^20 + 16*a^18 + 99*a^16 + 336*a^14 + 714*a^12 + 1008*a^10 + 9
66*a^8 + 624*a^6 + 261*a^4 + 64*a^2 + 7)*b^3*c^10*d + 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 + 55*a^4 + 11*a^2 + 1)*b*c^11 + (11*b^23*c*d^10 + 110*(a^2 + 7)*b^21*c^2*
d^9 + 33*(15*a^4 + 190*a^2 + 399)*b^19*c^3*d^8 + 264*(5*a^6 + 85*a^4 + 323*a^2 + 323)*b^17*c^4*d^7 + 110*(21*a
^8 + 420*a^6 + 2142*a^4 + 3876*a^2 + 2261)*b^15*c^5*d^6 + 4*(693*a^10 + 15015*a^8 + 90090*a^6 + 218790*a^4 + 2
30945*a^2 + 88179)*b^13*c^6*d^5 + 110*(21*a^12 + 462*a^10 + 3003*a^8 + 8580*a^6 + 12155*a^4 + 8398*a^2 + 2261)
*b^11*c^7*d^4 + 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^8*d
^3 + 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
^9*d^2 + 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
^10*d + 11*(a^20 + 10*a^18 + 45*a^16 + 120*a^14 + 210*a^12 + 252*a^10 + 210*a^8 + 120*a^6 + 45*a^4 + 10*a^2 +
1)*b^3*c^11)*x^2 + 2*(11*a*b^22*c*d^10 + 110*(a...

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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)/(c+d/x^2),x, algorithm="fricas")

[Out]

integral(x^2*arccot(b*x + a)/(c*x^2 + d), 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)/(c+d/x**2),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)/(c+d/x^2),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 )}{c+\frac {d}{x^2}} \,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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