\(\int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx\) [39]

Optimal result

Integrand size = 16, antiderivative size = 338 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {\log (i-a-b x)}{2 b c}+\frac {i (a+b x) \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 b c}+\frac {\log (i+a+b x)}{2 b c}-\frac {i (a+b x) \log \left (\frac {i+a+b x}{a+b x}\right )}{2 b c}+\frac {i d \log \left (\frac {c (i-a-b x)}{i c-a c+b d}\right ) \log (d+c x)}{2 c^2}-\frac {i d \log \left (-\frac {i-a-b x}{a+b x}\right ) \log (d+c x)}{2 c^2}-\frac {i d \log \left (\frac {c (i+a+b x)}{(i+a) c-b d}\right ) \log (d+c x)}{2 c^2}+\frac {i d \log \left (\frac {i+a+b x}{a+b x}\right ) \log (d+c x)}{2 c^2}-\frac {i d \operatorname {PolyLog}\left (2,-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}+\frac {i d \operatorname {PolyLog}\left (2,\frac {b (d+c x)}{i c-a c+b d}\right )}{2 c^2} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 7.43 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.51 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {2 a c^2 \cot ^{-1}(a+b x)-i b c d \pi \cot ^{-1}(a+b x)+2 b c^2 x \cot ^{-1}(a+b x)-i b c d \cot ^{-1}(a+b x)^2+a b c d \cot ^{-1}(a+b x)^2-b^2 d^2 \cot ^{-1}(a+b x)^2-a b c d \sqrt {1+\frac {c^2}{(a c-b d)^2}} e^{i \arctan \left (\frac {c}{-a c+b d}\right )} \cot ^{-1}(a+b x)^2+b^2 d^2 \sqrt {1+\frac {c^2}{(a c-b d)^2}} e^{i \arctan \left (\frac {c}{-a c+b d}\right )} \cot ^{-1}(a+b x)^2+2 i b c d \cot ^{-1}(a+b x) \arctan \left (\frac {c}{-a c+b d}\right )-b c d \pi \log \left (1+e^{-2 i \cot ^{-1}(a+b x)}\right )+2 b c d \cot ^{-1}(a+b x) \log \left (1-e^{2 i \cot ^{-1}(a+b x)}\right )-2 b c d \cot ^{-1}(a+b x) \log \left (1-e^{2 i \left (\cot ^{-1}(a+b x)+\arctan \left (\frac {c}{-a c+b d}\right )\right )}\right )-2 b c d \arctan \left (\frac {c}{-a c+b d}\right ) \log \left (1-e^{2 i \left (\cot ^{-1}(a+b x)+\arctan \left (\frac {c}{-a c+b d}\right )\right )}\right )+b c d \pi \log \left (\frac {1}{\sqrt {1+\frac {1}{(a+b x)^2}}}\right )-2 c^2 \log \left (\frac {1}{(a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}\right )+2 b c d \arctan \left (\frac {c}{-a c+b d}\right ) \log \left (\sin \left (\cot ^{-1}(a+b x)+\arctan \left (\frac {c}{-a c+b d}\right )\right )\right )-i b c d \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a+b x)}\right )+i b c d \operatorname {PolyLog}\left (2,e^{2 i \left (\cot ^{-1}(a+b x)+\arctan \left (\frac {c}{-a c+b d}\right )\right )}\right )}{2 b c^3} \] Input:

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

Output:

(2*a*c^2*ArcCot[a + b*x] - I*b*c*d*Pi*ArcCot[a + b*x] + 2*b*c^2*x*ArcCot[a 
 + b*x] - I*b*c*d*ArcCot[a + b*x]^2 + a*b*c*d*ArcCot[a + b*x]^2 - b^2*d^2* 
ArcCot[a + b*x]^2 - a*b*c*d*Sqrt[1 + c^2/(a*c - b*d)^2]*E^(I*ArcTan[c/(-(a 
*c) + b*d)])*ArcCot[a + b*x]^2 + b^2*d^2*Sqrt[1 + c^2/(a*c - b*d)^2]*E^(I* 
ArcTan[c/(-(a*c) + b*d)])*ArcCot[a + b*x]^2 + (2*I)*b*c*d*ArcCot[a + b*x]* 
ArcTan[c/(-(a*c) + b*d)] - b*c*d*Pi*Log[1 + E^((-2*I)*ArcCot[a + b*x])] + 
2*b*c*d*ArcCot[a + b*x]*Log[1 - E^((2*I)*ArcCot[a + b*x])] - 2*b*c*d*ArcCo 
t[a + b*x]*Log[1 - E^((2*I)*(ArcCot[a + b*x] + ArcTan[c/(-(a*c) + b*d)]))] 
 - 2*b*c*d*ArcTan[c/(-(a*c) + b*d)]*Log[1 - E^((2*I)*(ArcCot[a + b*x] + Ar 
cTan[c/(-(a*c) + b*d)]))] + b*c*d*Pi*Log[1/Sqrt[1 + (a + b*x)^(-2)]] - 2*c 
^2*Log[1/((a + b*x)*Sqrt[1 + (a + b*x)^(-2)])] + 2*b*c*d*ArcTan[c/(-(a*c) 
+ b*d)]*Log[Sin[ArcCot[a + b*x] + ArcTan[c/(-(a*c) + b*d)]]] - I*b*c*d*Pol 
yLog[2, E^((2*I)*ArcCot[a + b*x])] + I*b*c*d*PolyLog[2, E^((2*I)*(ArcCot[a 
 + b*x] + ArcTan[c/(-(a*c) + b*d)]))])/(2*b*c^3)
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.42, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5575, 2993, 772, 49, 2009, 2856, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5575

\(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{c+\frac {d}{x}}dx-\frac {1}{2} i \int \frac {\log \left (\frac {a+b x+i}{a+b x}\right )}{c+\frac {d}{x}}dx\)

\(\Big \downarrow \) 2993

\(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \int \frac {1}{c+\frac {d}{x}}dx\right )+\int \frac {\log (-a-b x+i)}{c+\frac {d}{x}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \int \frac {1}{c+\frac {d}{x}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+i)}{c+\frac {d}{x}}dx\right )\)

\(\Big \downarrow \) 772

\(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \int \frac {x}{d+c x}dx\right )+\int \frac {\log (-a-b x+i)}{c+\frac {d}{x}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \int \frac {x}{d+c x}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+i)}{c+\frac {d}{x}}dx\right )\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right )dx\right )+\int \frac {\log (-a-b x+i)}{c+\frac {d}{x}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right )dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+i)}{c+\frac {d}{x}}dx\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} i \left (\int \frac {\log (-a-b x+i)}{c+\frac {d}{x}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )\right )-\frac {1}{2} i \left (-\int \frac {\log (a+b x)}{c+\frac {d}{x}}dx+\int \frac {\log (a+b x+i)}{c+\frac {d}{x}}dx+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )\)

\(\Big \downarrow \) 2856

\(\displaystyle \frac {1}{2} i \left (\int \left (\frac {\log (-a-b x+i)}{c}-\frac {d \log (-a-b x+i)}{c (d+c x)}\right )dx-\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c (d+c x)}\right )dx-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )\right )-\frac {1}{2} i \left (-\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c (d+c x)}\right )dx+\int \left (\frac {\log (a+b x+i)}{c}-\frac {d \log (a+b x+i)}{c (d+c x)}\right )dx+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} i \left (-\frac {d \operatorname {PolyLog}\left (2,\frac {c (-a-b x+i)}{-a c+i c+b d}\right )}{c^2}+\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x)}{a c-b d}\right )}{c^2}-\frac {d \log (-a-b x+i) \log \left (\frac {b (c x+d)}{-a c+b d+i c}\right )}{c^2}-\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )+\frac {d \log (a+b x) \log \left (-\frac {b (c x+d)}{a c-b d}\right )}{c^2}-\frac {(-a-b x+i) \log (-a-b x+i)}{b c}-\frac {(a+b x) \log (a+b x)}{b c}\right )-\frac {1}{2} i \left (\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x)}{a c-b d}\right )}{c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x+i)}{(a+i) c-b d}\right )}{c^2}+\frac {d \log (a+b x) \log \left (-\frac {b (c x+d)}{a c-b d}\right )}{c^2}+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {d \log (c x+d)}{c^2}\right )-\frac {d \log (a+b x+i) \log \left (-\frac {b (c x+d)}{-b d+(a+i) c}\right )}{c^2}-\frac {(a+b x) \log (a+b x)}{b c}+\frac {(a+b x+i) \log (a+b x+i)}{b c}\right )\)

Input:

Int[ArcCot[a + b*x]/(c + d/x),x]
 

Output:

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

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 772

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[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] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 2993

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r   Int[RFx*Log[a + b*x], x], x] + (Si 
mp[q*r   Int[RFx*Log[c + d*x], x], x] - Simp[(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_.) /; IntegersQ[ 
m, n]]
 

rule 5575

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

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arccot}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arccot}\left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}-\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )+c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (-\frac {i \ln \left (a c -b d -c \left (b x +a \right )\right ) \left (\ln \left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\ln \left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}-\frac {i \left (\operatorname {dilog}\left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\operatorname {dilog}\left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}\right )}{c}}{b}\)	\(296\)
default	\(\frac {\frac {\operatorname {arccot}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arccot}\left (b x +a \right ) d b \ln \left (a c -b d -c \left (b x +a \right )\right )}{c^{2}}-\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -b d -c \left (b x +a \right )\right )+2 b d \left (a c -b d -c \left (b x +a \right )\right )+c^{2}+\left (a c -b d -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (-\frac {i \ln \left (a c -b d -c \left (b x +a \right )\right ) \left (\ln \left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\ln \left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}-\frac {i \left (\operatorname {dilog}\left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )-\operatorname {dilog}\left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )\right )}{2 c}\right )}{c}}{b}\)	\(296\)
parts	\(\frac {\operatorname {arccot}\left (b x +a \right ) x}{c}-\frac {\operatorname {arccot}\left (b x +a \right ) d \ln \left (c x +d \right )}{c^{2}}+\frac {b \left (\frac {\ln \left (a^{2} c^{2}-2 a b c d +2 a b c \left (c x +d \right )+b^{2} d^{2}-2 b^{2} d \left (c x +d \right )+b^{2} \left (c x +d \right )^{2}+c^{2}\right )}{2 b^{2}}-\frac {a \arctan \left (\frac {2 a b c -2 b^{2} d +2 b^{2} \left (c x +d \right )}{2 b c}\right )}{b^{2}}-d \left (-\frac {i \ln \left (c x +d \right ) \left (\ln \left (\frac {i c -a c +b d -b \left (c x +d \right )}{-a c +b d +i c}\right )-\ln \left (\frac {i c +a c -b d +b \left (c x +d \right )}{a c -b d +i c}\right )\right )}{2 b c}-\frac {i \left (\operatorname {dilog}\left (\frac {i c -a c +b d -b \left (c x +d \right )}{-a c +b d +i c}\right )-\operatorname {dilog}\left (\frac {i c +a c -b d +b \left (c x +d \right )}{a c -b d +i c}\right )\right )}{2 b c}\right )\right )}{c}\)	\(312\)
risch	\(\frac {i \pi }{2 b c}+\frac {i d \operatorname {dilog}\left (\frac {i a c -i b d +\left (-b x i-a i+1\right ) c -c}{i a c -i b d -c}\right )}{2 c^{2}}+\frac {i d \ln \left (-b x i-a i+1\right ) \ln \left (\frac {i a c -i b d +\left (-b x i-a i+1\right ) c -c}{i a c -i b d -c}\right )}{2 c^{2}}+\frac {i \ln \left (b x i+a i+1\right ) a}{2 b c}+\frac {\ln \left (-b x i-a i+1\right )}{2 b c}-\frac {1}{b c}-\frac {i d \operatorname {dilog}\left (\frac {-i a c +i b d +\left (b x i+a i+1\right ) c -c}{-i a c +i b d -c}\right )}{2 c^{2}}-\frac {i \ln \left (-b x i-a i+1\right ) a}{2 b c}+\frac {\pi x}{2 c}+\frac {\pi a}{2 b c}-\frac {i \ln \left (-b x i-a i+1\right ) x}{2 c}-\frac {\pi d \ln \left (i a c -i b d +\left (-b x i-a i+1\right ) c -c \right )}{2 c^{2}}-\frac {i d \ln \left (b x i+a i+1\right ) \ln \left (\frac {-i a c +i b d +\left (b x i+a i+1\right ) c -c}{-i a c +i b d -c}\right )}{2 c^{2}}+\frac {i \ln \left (b x i+a i+1\right ) x}{2 c}+\frac {\ln \left (b x i+a i+1\right )}{2 b c}\)	\(426\)

Input:

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

Output:

1/b*(arccot(b*x+a)/c*(b*x+a)-arccot(b*x+a)*d*b/c^2*ln(a*c-b*d-c*(b*x+a))-1 
/c*(-1/2*ln(a^2*c^2-2*a*b*c*d+b^2*d^2-2*a*c*(a*c-b*d-c*(b*x+a))+2*b*d*(a*c 
-b*d-c*(b*x+a))+c^2+(a*c-b*d-c*(b*x+a))^2)-b*d*(-1/2*I*ln(a*c-b*d-c*(b*x+a 
))*(ln((I*c+c*(b*x+a))/(a*c-b*d+I*c))-ln((I*c-c*(b*x+a))/(I*c-a*c+b*d)))/c 
-1/2*I*(dilog((I*c+c*(b*x+a))/(a*c-b*d+I*c))-dilog((I*c-c*(b*x+a))/(I*c-a* 
c+b*d)))/c)))
 

Fricas [F]

\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {2 \, b c x \arctan \left (1, b x + a\right ) - b d \arctan \left (1, b x + a\right ) \log \left (-\frac {b^{2} c^{2} x^{2} + 2 \, b^{2} c d x + b^{2} d^{2}}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}\right ) - 2 \, a c \arctan \left (b x + a\right ) + i \, b d {\rm Li}_2\left (\frac {b c x + {\left (a + i\right )} c}{{\left (a + i\right )} c - b d}\right ) - i \, b d {\rm Li}_2\left (\frac {b c x + {\left (a - i\right )} c}{{\left (a - i\right )} c - b d}\right ) - {\left (b d \arctan \left (-\frac {b c^{2} x + b c d}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}, \frac {a b c d - b^{2} d^{2} + {\left (a b c^{2} - b^{2} c d\right )} x}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}\right ) - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b c^{2}} \] Input:

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

Output:

1/2*(2*b*c*x*arctan2(1, b*x + a) - b*d*arctan2(1, b*x + a)*log(-(b^2*c^2*x 
^2 + 2*b^2*c*d*x + b^2*d^2)/(2*a*b*c*d - b^2*d^2 - (a^2 + 1)*c^2)) - 2*a*c 
*arctan(b*x + a) + I*b*d*dilog((b*c*x + (a + I)*c)/((a + I)*c - b*d)) - I* 
b*d*dilog((b*c*x + (a - I)*c)/((a - I)*c - b*d)) - (b*d*arctan2(-(b*c^2*x 
+ b*c*d)/(2*a*b*c*d - b^2*d^2 - (a^2 + 1)*c^2), (a*b*c*d - b^2*d^2 + (a*b* 
c^2 - b^2*c*d)*x)/(2*a*b*c*d - b^2*d^2 - (a^2 + 1)*c^2)) - c)*log(b^2*x^2 
+ 2*a*b*x + a^2 + 1))/(b*c^2)
 

Giac [F]

\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \] Input:

int(acot(a + b*x)/(c + d/x),x)
 

Output:

int(acot(a + b*x)/(c + d/x), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathit {acot} \left (b x +a \right ) x}{c x +d}d x \] Input:

int(acot(b*x+a)/(c+d/x),x)
 

Output:

int((acot(a + b*x)*x)/(c*x + d),x)