\(\int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx\) [40]

Optimal result

Integrand size = 16, antiderivative size = 735 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\frac {\log (i-a-b x)}{2 b c}+\frac {i (a+b x) \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 b c}-\frac {i \sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c^{3/2}}+\frac {\log (i+a+b x)}{2 b c}-\frac {i (a+b x) \log \left (\frac {i+a+b x}{a+b x}\right )}{2 b c}+\frac {i \sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c^{3/2}}-\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i-a-b x)}{(i-a) \sqrt {c}+i b \sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i+a+b x)}{(i+a) \sqrt {c}-i b \sqrt {d}}\right ) \log \left (1-\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i-a-b x)}{(i-a) \sqrt {c}-i b \sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}-\frac {\sqrt {d} \log \left (\frac {\sqrt {c} (i+a+b x)}{(i+a) \sqrt {c}+i b \sqrt {d}}\right ) \log \left (1+\frac {i \sqrt {c} x}{\sqrt {d}}\right )}{4 c^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}-i \sqrt {c} x\right )}{(1+i a) \sqrt {c}+b \sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}-i \sqrt {c} x\right )}{i (i+a) \sqrt {c}+b \sqrt {d}}\right )}{4 c^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,-\frac {b \left (\sqrt {d}+i \sqrt {c} x\right )}{(1+i a) \sqrt {c}-b \sqrt {d}}\right )}{4 c^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}+i \sqrt {c} x\right )}{(1-i a) \sqrt {c}+b \sqrt {d}}\right )}{4 c^{3/2}} \] 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*I*d^(1/2) 
*arctan(c^(1/2)*x/d^(1/2))*ln(-(I-a-b*x)/(b*x+a))/c^(3/2)+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^(1/2)*arctan(c^(1/2)*x 
/d^(1/2))*ln((I+a+b*x)/(b*x+a))/c^(3/2)-1/4*d^(1/2)*ln(c^(1/2)*(I-a-b*x)/( 
(I-a)*c^(1/2)+I*b*d^(1/2)))*ln(1-I*c^(1/2)*x/d^(1/2))/c^(3/2)+1/4*d^(1/2)* 
ln(c^(1/2)*(I+a+b*x)/((I+a)*c^(1/2)-I*b*d^(1/2)))*ln(1-I*c^(1/2)*x/d^(1/2) 
)/c^(3/2)+1/4*d^(1/2)*ln(c^(1/2)*(I-a-b*x)/((I-a)*c^(1/2)-I*b*d^(1/2)))*ln 
(1+I*c^(1/2)*x/d^(1/2))/c^(3/2)-1/4*d^(1/2)*ln(c^(1/2)*(I+a+b*x)/((I+a)*c^ 
(1/2)+I*b*d^(1/2)))*ln(1+I*c^(1/2)*x/d^(1/2))/c^(3/2)-1/4*d^(1/2)*polylog( 
2,b*(d^(1/2)-I*c^(1/2)*x)/((1+I*a)*c^(1/2)+b*d^(1/2)))/c^(3/2)+1/4*d^(1/2) 
*polylog(2,b*(d^(1/2)-I*c^(1/2)*x)/(I*(I+a)*c^(1/2)+b*d^(1/2)))/c^(3/2)+1/ 
4*d^(1/2)*polylog(2,-b*(d^(1/2)+I*c^(1/2)*x)/((1+I*a)*c^(1/2)-b*d^(1/2)))/ 
c^(3/2)-1/4*d^(1/2)*polylog(2,b*(d^(1/2)+I*c^(1/2)*x)/((1-I*a)*c^(1/2)+b*d 
^(1/2)))/c^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.27 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\frac {4 c \log (a+b x)+2 c \log \left (\frac {-i+a+b x}{a+b x}\right )+2 i a c \log \left (\frac {-i+a+b x}{a+b x}\right )+2 i b c x \log \left (\frac {-i+a+b x}{a+b x}\right )+2 c \log \left (\frac {i+a+b x}{a+b x}\right )-2 i a c \log \left (\frac {i+a+b x}{a+b x}\right )-2 i b c x \log \left (\frac {i+a+b x}{a+b x}\right )+i b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (-i+a+b x)}{-i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )-i b \sqrt {-c} \sqrt {d} \log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )-i b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (i+a+b x)}{(i+a) \sqrt {-c}+b \sqrt {d}}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )+i b \sqrt {-c} \sqrt {d} \log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\sqrt {d}-\sqrt {-c} x\right )-i b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (i-a-b x)}{-\left ((-i+a) \sqrt {-c}\right )+b \sqrt {d}}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )+i b \sqrt {-c} \sqrt {d} \log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )+i b \sqrt {-c} \sqrt {d} \log \left (\frac {\sqrt {-c} (i+a+b x)}{(i+a) \sqrt {-c}-b \sqrt {d}}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )-i b \sqrt {-c} \sqrt {d} \log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\sqrt {d}+\sqrt {-c} x\right )+i b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{-i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )-i b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{i \sqrt {-c}+a \sqrt {-c}+b \sqrt {d}}\right )+i b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{-i \sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )-i b \sqrt {-c} \sqrt {d} \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{i \sqrt {-c}-a \sqrt {-c}+b \sqrt {d}}\right )}{4 b c^2} \] Input:

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

Output:

(4*c*Log[a + b*x] + 2*c*Log[(-I + a + b*x)/(a + b*x)] + (2*I)*a*c*Log[(-I 
+ a + b*x)/(a + b*x)] + (2*I)*b*c*x*Log[(-I + a + b*x)/(a + b*x)] + 2*c*Lo 
g[(I + a + b*x)/(a + b*x)] - (2*I)*a*c*Log[(I + a + b*x)/(a + b*x)] - (2*I 
)*b*c*x*Log[(I + a + b*x)/(a + b*x)] + I*b*Sqrt[-c]*Sqrt[d]*Log[(Sqrt[-c]* 
(-I + a + b*x))/((-I)*Sqrt[-c] + a*Sqrt[-c] + b*Sqrt[d])]*Log[Sqrt[d] - Sq 
rt[-c]*x] - I*b*Sqrt[-c]*Sqrt[d]*Log[(-I + a + b*x)/(a + b*x)]*Log[Sqrt[d] 
 - Sqrt[-c]*x] - I*b*Sqrt[-c]*Sqrt[d]*Log[(Sqrt[-c]*(I + a + b*x))/((I + a 
)*Sqrt[-c] + b*Sqrt[d])]*Log[Sqrt[d] - Sqrt[-c]*x] + I*b*Sqrt[-c]*Sqrt[d]* 
Log[(I + a + b*x)/(a + b*x)]*Log[Sqrt[d] - Sqrt[-c]*x] - I*b*Sqrt[-c]*Sqrt 
[d]*Log[(Sqrt[-c]*(I - a - b*x))/(-((-I + a)*Sqrt[-c]) + b*Sqrt[d])]*Log[S 
qrt[d] + Sqrt[-c]*x] + I*b*Sqrt[-c]*Sqrt[d]*Log[(-I + a + b*x)/(a + b*x)]* 
Log[Sqrt[d] + Sqrt[-c]*x] + I*b*Sqrt[-c]*Sqrt[d]*Log[(Sqrt[-c]*(I + a + b* 
x))/((I + a)*Sqrt[-c] - b*Sqrt[d])]*Log[Sqrt[d] + Sqrt[-c]*x] - I*b*Sqrt[- 
c]*Sqrt[d]*Log[(I + a + b*x)/(a + b*x)]*Log[Sqrt[d] + Sqrt[-c]*x] + I*b*Sq 
rt[-c]*Sqrt[d]*PolyLog[2, (b*(Sqrt[d] - Sqrt[-c]*x))/((-I)*Sqrt[-c] + a*Sq 
rt[-c] + b*Sqrt[d])] - I*b*Sqrt[-c]*Sqrt[d]*PolyLog[2, (b*(Sqrt[d] - Sqrt[ 
-c]*x))/(I*Sqrt[-c] + a*Sqrt[-c] + b*Sqrt[d])] + I*b*Sqrt[-c]*Sqrt[d]*Poly 
Log[2, (b*(Sqrt[d] + Sqrt[-c]*x))/((-I)*Sqrt[-c] - a*Sqrt[-c] + b*Sqrt[d]) 
] - I*b*Sqrt[-c]*Sqrt[d]*PolyLog[2, (b*(Sqrt[d] + Sqrt[-c]*x))/(I*Sqrt[-c] 
 - a*Sqrt[-c] + b*Sqrt[d])])/(4*b*c^2)
 

Rubi [A] (verified)

Time = 2.08 (sec) , antiderivative size = 1222, normalized size of antiderivative = 1.66, 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, 262, 218, 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.

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

\(\Big \downarrow \) 5575

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

\(\Big \downarrow \) 2993

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

\(\Big \downarrow \) 772

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} i \left (\int \frac {\log (-a-b x+i)}{c+\frac {d}{x^2}}dx-\int \frac {\log (a+b x)}{c+\frac {d}{x^2}}dx-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{c^{3/2}}\right )\right )\right )-\frac {1}{2} i \left (-\int \frac {\log (a+b x)}{c+\frac {d}{x^2}}dx+\int \frac {\log (a+b x+i)}{c+\frac {d}{x^2}}dx+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{c^{3/2}}\right )\right )\)

\(\Big \downarrow \) 2856

\(\displaystyle \frac {1}{2} i \left (\int \left (\frac {\log (-a-b x+i)}{c}-\frac {d \log (-a-b x+i)}{c \left (c x^2+d\right )}\right )dx-\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c \left (c x^2+d\right )}\right )dx-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {x}{c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{c^{3/2}}\right )\right )\right )-\frac {1}{2} i \left (-\int \left (\frac {\log (a+b x)}{c}-\frac {d \log (a+b x)}{c \left (c x^2+d\right )}\right )dx+\int \left (\frac {\log (a+b x+i)}{c}-\frac {d \log (a+b x+i)}{c \left (c x^2+d\right )}\right )dx+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {x}{c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{c^{3/2}}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} i \left (-\frac {(-a-b x+i) \log (-a-b x+i)}{b c}-\frac {\sqrt {d} \log \left (-\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{(i-a) \sqrt {-c}-b \sqrt {d}}\right ) \log (-a-b x+i)}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log \left (\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{\sqrt {-c} (i-a)+b \sqrt {d}}\right ) \log (-a-b x+i)}{2 (-c)^{3/2}}-\left (\frac {x}{c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{c^{3/2}}\right ) \left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right )-\frac {(a+b x) \log (a+b x)}{b c}+\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+i)}{-\sqrt {-c} a+i \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (-a-b x+i)}{\sqrt {-c} (i-a)+b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}\right )-\frac {1}{2} i \left (-\frac {(a+b x) \log (a+b x)}{b c}+\frac {\sqrt {d} \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{\sqrt {-c} a+b \sqrt {d}}\right ) \log (a+b x)}{2 (-c)^{3/2}}-\frac {\sqrt {d} \log \left (-\frac {b \left (\sqrt {-c} x+\sqrt {d}\right )}{a \sqrt {-c}-b \sqrt {d}}\right ) \log (a+b x)}{2 (-c)^{3/2}}+\frac {(a+b x+i) \log (a+b x+i)}{b c}+\left (\frac {x}{c}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )}{c^{3/2}}\right ) \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right )-\frac {\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 )}{2 (-c)^{3/2}}+\frac {\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 )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x)}{\sqrt {-c} a+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+i)}{(a+i) \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (a+b x+i)}{\sqrt {-c} (a+i)+b \sqrt {d}}\right )}{2 (-c)^{3/2}}\right )\)

Input:

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

Output:

(I/2)*(-(((I - a - b*x)*Log[I - a - b*x])/(b*c)) - (x/c - (Sqrt[d]*ArcTan[ 
(Sqrt[c]*x)/Sqrt[d]])/c^(3/2))*(Log[I - a - b*x] - Log[-((I - a - b*x)/(a 
+ b*x))] - Log[a + b*x]) - ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[I 
 - a - b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((I - a)*Sqrt[-c] - b*Sqrt[d] 
))])/(2*(-c)^(3/2)) + (Sqrt[d]*Log[a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x)) 
/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*Log[a + b*x]*Log[-(( 
b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))])/(2*(-c)^(3/2)) + (Sq 
rt[d]*Log[I - a - b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((I - a)*Sqrt[-c] + 
b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x)) 
/(I*Sqrt[-c] - a*Sqrt[-c] - b*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]*PolyLog 
[2, (Sqrt[-c]*(I - a - b*x))/((I - a)*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2 
)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b*Sqrt[d])])/( 
2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] + b*S 
qrt[d])])/(2*(-c)^(3/2))) - (I/2)*(-(((a + b*x)*Log[a + b*x])/(b*c)) + ((I 
 + a + b*x)*Log[I + a + b*x])/(b*c) + (x/c - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/S 
qrt[d]])/c^(3/2))*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a 
+ b*x)]) + (Sqrt[d]*Log[a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(a*Sqrt[-c 
] + b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*Log[I + a + b*x]*Log[(b*(Sqrt[d 
] - Sqrt[-c]*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d 
]*Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]...
 

Defintions of rubi rules used

rule 218

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

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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 = 2.97 (sec) , antiderivative size = 728, normalized size of antiderivative = 0.99

method	result	size
risch	\(\frac {\pi x}{2 c}+\frac {\pi a}{2 b c}+\frac {i b \pi d \arctan \left (\frac {2 i a c +2 \left (-b x i-a i+1\right ) c -2 c}{2 \sqrt {-b^{2} c d}}\right )}{2 c \sqrt {-b^{2} c d}}+\frac {i \pi }{2 b c}+\frac {i \ln \left (b x i+a i+1\right ) a}{2 b c}-\frac {i \ln \left (-b x i-a i+1\right ) a}{2 b c}+\frac {i \ln \left (b x i+a i+1\right ) x}{2 c}-\frac {i \ln \left (-b x i-a i+1\right ) x}{2 c}+\frac {\ln \left (-b x i-a i+1\right )}{2 b c}-\frac {1}{b c}+\frac {\ln \left (-b x i-a i+1\right ) \ln \left (\frac {i a c -b \sqrt {c d}+\left (-b x i-a i+1\right ) c -c}{i a c -b \sqrt {c d}-c}\right ) \sqrt {c d}}{4 c^{2}}-\frac {\ln \left (-b x i-a i+1\right ) \ln \left (\frac {i a c +b \sqrt {c d}+\left (-b x i-a i+1\right ) c -c}{i a c +b \sqrt {c d}-c}\right ) \sqrt {c d}}{4 c^{2}}+\frac {\operatorname {dilog}\left (\frac {i a c -b \sqrt {c d}+\left (-b x i-a i+1\right ) c -c}{i a c -b \sqrt {c d}-c}\right ) \sqrt {c d}}{4 c^{2}}-\frac {\operatorname {dilog}\left (\frac {i a c +b \sqrt {c d}+\left (-b x i-a i+1\right ) c -c}{i a c +b \sqrt {c d}-c}\right ) \sqrt {c d}}{4 c^{2}}+\frac {\ln \left (b x i+a i+1\right )}{2 b c}+\frac {\ln \left (b x i+a i+1\right ) \sqrt {c d}\, \ln \left (\frac {i a c +b \sqrt {c d}-\left (b x i+a i+1\right ) c +c}{i a c +b \sqrt {c d}+c}\right )}{4 c^{2}}-\frac {\ln \left (b x i+a i+1\right ) \sqrt {c d}\, \ln \left (\frac {i a c -b \sqrt {c d}-\left (b x i+a i+1\right ) c +c}{i a c -b \sqrt {c d}+c}\right )}{4 c^{2}}-\frac {\sqrt {c d}\, \operatorname {dilog}\left (\frac {i a c -b \sqrt {c d}-\left (b x i+a i+1\right ) c +c}{i a c -b \sqrt {c d}+c}\right )}{4 c^{2}}+\frac {\sqrt {c d}\, \operatorname {dilog}\left (\frac {i a c +b \sqrt {c d}-\left (b x i+a i+1\right ) c +c}{i a c +b \sqrt {c d}+c}\right )}{4 c^{2}}\)	\(728\)
derivativedivides	\(\text {Expression too large to display}\)	\(13390\)
default	\(\text {Expression too large to display}\)	\(13390\)

Input:

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

Output:

1/2*Pi/c*x+1/2/b*Pi/c*a+1/2*I*b*Pi*d/c/(-b^2*c*d)^(1/2)*arctan(1/2*(2*I*a* 
c+2*(1-I*a-I*b*x)*c-2*c)/(-b^2*c*d)^(1/2))+1/2*I/b*Pi/c+1/2*I/b/c*ln(1+I*a 
+I*b*x)*a-1/2*I/b/c*ln(1-I*a-I*b*x)*a+1/2*I/c*ln(1+I*a+I*b*x)*x-1/2*I/c*ln 
(1-I*a-I*b*x)*x+1/2/b/c*ln(1-I*a-I*b*x)-1/b/c+1/4/c^2*ln(1-I*a-I*b*x)*ln(( 
I*a*c-b*(c*d)^(1/2)+(1-I*a-I*b*x)*c-c)/(I*a*c-b*(c*d)^(1/2)-c))*(c*d)^(1/2 
)-1/4/c^2*ln(1-I*a-I*b*x)*ln((I*a*c+b*(c*d)^(1/2)+(1-I*a-I*b*x)*c-c)/(I*a* 
c+b*(c*d)^(1/2)-c))*(c*d)^(1/2)+1/4/c^2*dilog((I*a*c-b*(c*d)^(1/2)+(1-I*a- 
I*b*x)*c-c)/(I*a*c-b*(c*d)^(1/2)-c))*(c*d)^(1/2)-1/4/c^2*dilog((I*a*c+b*(c 
*d)^(1/2)+(1-I*a-I*b*x)*c-c)/(I*a*c+b*(c*d)^(1/2)-c))*(c*d)^(1/2)+1/2/b/c* 
ln(1+I*a+I*b*x)+1/4/c^2*ln(1+I*a+I*b*x)*(c*d)^(1/2)*ln((I*a*c+b*(c*d)^(1/2 
)-(1+I*a+I*b*x)*c+c)/(I*a*c+b*(c*d)^(1/2)+c))-1/4/c^2*ln(1+I*a+I*b*x)*(c*d 
)^(1/2)*ln((I*a*c-b*(c*d)^(1/2)-(1+I*a+I*b*x)*c+c)/(I*a*c-b*(c*d)^(1/2)+c) 
)-1/4/c^2*(c*d)^(1/2)*dilog((I*a*c-b*(c*d)^(1/2)-(1+I*a+I*b*x)*c+c)/(I*a*c 
-b*(c*d)^(1/2)+c))+1/4/c^2*(c*d)^(1/2)*dilog((I*a*c+b*(c*d)^(1/2)-(1+I*a+I 
*b*x)*c+c)/(I*a*c+b*(c*d)^(1/2)+c))
 

Fricas [F]

\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{x^{2}}} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

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 = 0.86 (sec) , antiderivative size = 8518, normalized size of antiderivative = 11.59 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-(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)*sqr 
t(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 + 133)*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 + 6 
0060*a^6 + 109395*a^4 + 92378*a^2 + 29393)*b^12*c^6*d^6 + 22*(21*a^14 +...
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x^2}} \, dx=\int \frac {\mathit {acot} \left (b x +a \right ) x^{2}}{c \,x^{2}+d}d x \] Input:

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

Output:

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