\(\int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx\) [42]

Optimal result

Integrand size = 18, antiderivative size = 693 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=-\frac {2 i \sqrt {i+a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}+\frac {2 i \sqrt {i-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}-\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{d^2}-\frac {i \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}+\frac {i c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 618, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=-\frac {i \left (\frac {2 \sqrt {i+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b}}-\frac {2 \sqrt {i-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b}}+c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )+c \log \left (\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log \left (\frac {-i+a+b x}{a+b x}\right )+c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {-i+a+b x}{a+b x}\right )+d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )-c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )+c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )-c \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )\right )}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 1133, normalized size of antiderivative = 1.63, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5575, 7267, 3008, 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.

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

\(\Big \downarrow \) 5575

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 3008

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 3008

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With 
[{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u 
]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFuncti 
onQ[RGx, x] && IGtQ[n, 0]
 

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]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.32 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.53

method	result	size
derivativedivides	\(\frac {2 \,\operatorname {arccot}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\)	\(364\)
default	\(\frac {2 \,\operatorname {arccot}\left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}+\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\)	\(364\)

Input:

int(arccot(b*x+a)/(c+d*x^(1/2)),x,method=_RETURNVERBOSE)
 

Output:

2*arccot(b*x+a)/d*x^(1/2)-2*arccot(b*x+a)*c/d^2*ln(c+d*x^(1/2))+4*b/d^2*(1 
/4*d^2/b*sum((_R^2-2*_R*c+c^2)/(_R^3*b-3*_R^2*b*c+_R*a*d^2+3*_R*b*c^2-a*c* 
d^2-b*c^3)*ln(d*x^(1/2)-_R+c),_R=RootOf(b^2*_Z^4-4*b^2*c*_Z^3+(2*a*b*d^2+6 
*b^2*c^2)*_Z^2+(-4*a*b*c*d^2-4*b^2*c^3)*_Z+a^2*d^4+2*a*b*c^2*d^2+b^2*c^4+d 
^4))-1/4*c*d^2/b*sum(1/(_R1^2*b-2*_R1*b*c+a*d^2+b*c^2)*(ln(c+d*x^(1/2))*ln 
((-d*x^(1/2)+_R1-c)/_R1)+dilog((-d*x^(1/2)+_R1-c)/_R1)),_R1=RootOf(b^2*_Z^ 
4-4*b^2*c*_Z^3+(2*a*b*d^2+6*b^2*c^2)*_Z^2+(-4*a*b*c*d^2-4*b^2*c^3)*_Z+a^2* 
d^4+2*a*b*c^2*d^2+b^2*c^4+d^4)))
 

Fricas [F]

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

integrate(arccot(b*x+a)/(c+d*x^(1/2)),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate(acot(b*x+a)/(c+d*x**(1/2)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(arccot(b*x+a)/(c+d*x^(1/2)),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arccot(b*x+a)/(c+d*x^(1/2)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.The choice 
 was done
 

Mupad [F(-1)]

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

int(acot(a + b*x)/(c + d*x^(1/2)),x)
 

Output:

int(acot(a + b*x)/(c + d*x^(1/2)), x)
 

Reduce [F]

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

int(acot(b*x+a)/(c+d*x^(1/2)),x)
 

Output:

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