Integrand size = 18, antiderivative size = 830 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\frac {2 i \sqrt {i+a} d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} c^2}-\frac {2 i \sqrt {i-a} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} c^2}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {i d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}-\frac {i d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )}{c^3}+\frac {(1+i a) \log (i-a-b x)}{2 b c}-\frac {i d \sqrt {x} \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^2}+\frac {i x \log \left (-\frac {i-a-b x}{a+b x}\right )}{2 c}+\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (-\frac {i-a-b x}{a+b x}\right )}{c^3}+\frac {(1-i a) \log (i+a+b x)}{2 b c}+\frac {i d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )}{c^2}-\frac {i x \log \left (\frac {i+a+b x}{a+b x}\right )}{2 c}-\frac {i d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )}{c^3}+\frac {i d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right )}{c^3}+\frac {i d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )}{c^3}-\frac {i d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{c^3} \]
1/2*(1+I*a)*ln(I-a-b*x)/b/c+I*d^2*polylog(2,-b^(1/2)*(d+c*x^(1/2))/(c*(-I- a)^(1/2)-d*b^(1/2)))/c^3+1/2*(1-I*a)*ln(I+a+b*x)/b/c-I*d*ln((-I+a+b*x)/(b* x+a))*x^(1/2)/c^2+I*d^2*ln(d+c*x^(1/2))*ln(c*((-I-a)^(1/2)-b^(1/2)*x^(1/2) )/(c*(-I-a)^(1/2)+d*b^(1/2)))/c^3+2*I*d*arctan(b^(1/2)*x^(1/2)/(I+a)^(1/2) )*(I+a)^(1/2)/c^2/b^(1/2)-2*I*d*arctanh(b^(1/2)*x^(1/2)/(I-a)^(1/2))*(I-a) ^(1/2)/c^2/b^(1/2)-I*d^2*polylog(2,b^(1/2)*(d+c*x^(1/2))/(c*(I-a)^(1/2)+d* b^(1/2)))/c^3+I*d^2*polylog(2,b^(1/2)*(d+c*x^(1/2))/(c*(-I-a)^(1/2)+d*b^(1 /2)))/c^3-I*d^2*ln(d+c*x^(1/2))*ln(c*((I-a)^(1/2)+b^(1/2)*x^(1/2))/(c*(I-a )^(1/2)-d*b^(1/2)))/c^3-I*d^2*ln((I+a+b*x)/(b*x+a))*ln(d+c*x^(1/2))/c^3-I* d^2*polylog(2,-b^(1/2)*(d+c*x^(1/2))/(c*(I-a)^(1/2)-d*b^(1/2)))/c^3+I*d^2* ln(d+c*x^(1/2))*ln(c*((-I-a)^(1/2)+b^(1/2)*x^(1/2))/(c*(-I-a)^(1/2)-d*b^(1 /2)))/c^3+1/2*I*x*ln((-I+a+b*x)/(b*x+a))/c+I*d^2*ln((-I+a+b*x)/(b*x+a))*ln (d+c*x^(1/2))/c^3-1/2*I*x*ln((I+a+b*x)/(b*x+a))/c+I*d*ln((I+a+b*x)/(b*x+a) )*x^(1/2)/c^2-I*d^2*ln(d+c*x^(1/2))*ln(c*((I-a)^(1/2)-b^(1/2)*x^(1/2))/(c* (I-a)^(1/2)+d*b^(1/2)))/c^3
Time = 0.53 (sec) , antiderivative size = 809, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\frac {4 i \sqrt {i+a} \sqrt {b} c d \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )-4 i \sqrt {i-a} \sqrt {b} c d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )+2 i b d^2 \log \left (\frac {c \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )-2 i b d^2 \log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+2 i b d^2 \log \left (\frac {c \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )-2 i b d^2 \log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (d+c \sqrt {x}\right )+c^2 \log (i-a-b x)+i a c^2 \log (i-a-b x)-2 i b c d \sqrt {x} \log \left (\frac {-i+a+b x}{a+b x}\right )+i b c^2 x \log \left (\frac {-i+a+b x}{a+b x}\right )+2 i b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {-i+a+b x}{a+b x}\right )+c^2 \log (i+a+b x)-i a c^2 \log (i+a+b x)+2 i b c d \sqrt {x} \log \left (\frac {i+a+b x}{a+b x}\right )-i b c^2 x \log \left (\frac {i+a+b x}{a+b x}\right )-2 i b d^2 \log \left (d+c \sqrt {x}\right ) \log \left (\frac {i+a+b x}{a+b x}\right )+2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {-i-a} c+\sqrt {b} d}\right )+2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {-i-a} c+\sqrt {b} d}\right )-2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{-\sqrt {i-a} c+\sqrt {b} d}\right )-2 i b d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (d+c \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right )}{2 b c^3} \]
((4*I)*Sqrt[I + a]*Sqrt[b]*c*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]] - (4* I)*Sqrt[I - a]*Sqrt[b]*c*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]] + (2*I)* b*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d )]*Log[d + c*Sqrt[x]] - (2*I)*b*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]) )/(Sqrt[I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + (2*I)*b*d^2*Log[(c*(Sq rt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqr t[x]] - (2*I)*b*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]] + c^2*Log[I - a - b*x] + I*a*c^2*Log[I - a - b*x] - (2*I)*b*c*d*Sqrt[x]*Log[(-I + a + b*x)/(a + b*x)] + I*b*c^2*x* Log[(-I + a + b*x)/(a + b*x)] + (2*I)*b*d^2*Log[d + c*Sqrt[x]]*Log[(-I + a + b*x)/(a + b*x)] + c^2*Log[I + a + b*x] - I*a*c^2*Log[I + a + b*x] + (2* I)*b*c*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)] - I*b*c^2*x*Log[(I + a + b*x )/(a + b*x)] - (2*I)*b*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)] + (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sqrt[-I - a]*c) + S qrt[b]*d)] + (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a ]*c + Sqrt[b]*d)] - (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(-(Sq rt[I - a]*c) + Sqrt[b]*d)] - (2*I)*b*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x ]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/(2*b*c^3)
Time = 2.68 (sec) , antiderivative size = 1316, normalized size of antiderivative = 1.59, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx\) |
| \(\Big \downarrow \) 5575 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{c+\frac {d}{\sqrt {x}}}dx-\frac {1}{2} i \int \frac {\log \left (\frac {a+b x+i}{a+b x}\right )}{c+\frac {d}{\sqrt {x}}}dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle i \int \frac {x \log \left (-\frac {-a-b x+i}{a+b x}\right )}{\sqrt {x} c+d}d\sqrt {x}-i \int \frac {x \log \left (\frac {a+b x+i}{a+b x}\right )}{\sqrt {x} c+d}d\sqrt {x}\) |
| \(\Big \downarrow \) 3008 |
| \(\displaystyle i \int \left (\frac {\log \left (-\frac {-a-b x+i}{a+b x}\right ) d^2}{c^2 \left (\sqrt {x} c+d\right )}-\frac {\log \left (-\frac {-a-b x+i}{a+b x}\right ) d}{c^2}+\frac {\sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right )}{c}\right )d\sqrt {x}-i \int \left (\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) d^2}{c^2 \left (\sqrt {x} c+d\right )}-\frac {\log \left (\frac {a+b x+i}{a+b x}\right ) d}{c^2}+\frac {\sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right )}{c}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (-\frac {\log \left (\frac {c \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {\log \left (\frac {c \left (\sqrt {-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {\log \left (\frac {c \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {\log \left (\frac {c \left (\sqrt {-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {\log \left (\sqrt {x} c+d\right ) \log \left (-\frac {-a-b x+i}{a+b x}\right ) d^2}{c^3}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c-\sqrt {b} d}\right ) d^2}{c^3}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a} c-\sqrt {b} d}\right ) d^2}{c^3}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {i-a} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ) d}{\sqrt {b} c^2}-\frac {2 \sqrt {i-a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right ) d}{\sqrt {b} c^2}-\frac {\sqrt {x} \log \left (-\frac {-a-b x+i}{a+b x}\right ) d}{c^2}-\frac {(i-a) \log (-a-b x+i)}{2 b c}+\frac {x \log \left (-\frac {-a-b x+i}{a+b x}\right )}{2 c}-\frac {a \log (a+b x)}{2 b c}\right )-i \left (-\frac {\log \left (\frac {c \left (\sqrt {-a-i}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {\log \left (\frac {c \left (\sqrt {-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {-a} c+\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}-\frac {\log \left (\frac {c \left (\sqrt {-a-i}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {\log \left (\frac {c \left (\sqrt {-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {-a} c-\sqrt {b} d}\right ) \log \left (\sqrt {x} c+d\right ) d^2}{c^3}+\frac {\log \left (\sqrt {x} c+d\right ) \log \left (\frac {a+b x+i}{a+b x}\right ) d^2}{c^3}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c-\sqrt {b} d}\right ) d^2}{c^3}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a} c-\sqrt {b} d}\right ) d^2}{c^3}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a-i} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {b} \left (\sqrt {x} c+d\right )}{\sqrt {-a} c+\sqrt {b} d}\right ) d^2}{c^3}+\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ) d}{\sqrt {b} c^2}-\frac {2 \sqrt {a+i} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right ) d}{\sqrt {b} c^2}-\frac {\sqrt {x} \log \left (\frac {a+b x+i}{a+b x}\right ) d}{c^2}-\frac {a \log (a+b x)}{2 b c}+\frac {(a+i) \log (a+b x+i)}{2 b c}+\frac {x \log \left (\frac {a+b x+i}{a+b x}\right )}{2 c}\right )\) |
(-I)*((2*Sqrt[a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*c^2) - (2*S qrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*c^2) - (d^2*L og[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]*Log[ d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]* c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-I - a] + Sqrt [b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d^2 *Log[(c*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*c - Sqrt[b]*d)]*Log[d + c* Sqrt[x]])/c^3 - (a*Log[a + b*x])/(2*b*c) + ((I + a)*Log[I + a + b*x])/(2*b *c) - (d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/c^2 + (x*Log[(I + a + b*x)/ (a + b*x)])/(2*c) + (d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/ c^3 - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b ]*d))])/c^3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-a]*c - Sq rt[b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]* c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-a] *c + Sqrt[b]*d)])/c^3) + I*((2*Sqrt[a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]] )/(Sqrt[b]*c^2) - (2*Sqrt[I - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]]) /(Sqrt[b]*c^2) - (d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a] *c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[-a] - Sqrt[b] *Sqrt[x]))/(Sqrt[-a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (d^2*Log[(c *(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + ...
3.2.12.3.1 Defintions of rubi rules used
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]
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]
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]]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.47 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.47
| method | result | size |
| derivativedivides | \(\frac {\operatorname {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+5 \textit {\_R}^{2} d -7 \textit {\_R} \,d^{2}+3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b d +\textit {\_R} a \,c^{2}+3 \textit {\_R} b \,d^{2}-a \,c^{2} d -b \,d^{3}}\right )}{8 b}+\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+b \,d^{2}}\right )}{4 b}\right )}{c^{2}}\) | \(388\) |
| default | \(\frac {\operatorname {arccot}\left (b x +a \right ) x}{c}-\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d \sqrt {x}}{c^{2}}+\frac {2 \,\operatorname {arccot}\left (b x +a \right ) d^{2} \ln \left (d +c \sqrt {x}\right )}{c^{3}}+\frac {4 b \left (-\frac {c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+5 \textit {\_R}^{2} d -7 \textit {\_R} \,d^{2}+3 d^{3}\right ) \ln \left (c \sqrt {x}-\textit {\_R} +d \right )}{\textit {\_R}^{3} b -3 \textit {\_R}^{2} b d +\textit {\_R} a \,c^{2}+3 \textit {\_R} b \,d^{2}-a \,c^{2} d -b \,d^{3}}\right )}{8 b}+\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{4}-4 b^{2} d \,\textit {\_Z}^{3}+\left (2 a \,c^{2} b +6 b^{2} d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b \,c^{2} d -4 b^{2} d^{3}\right ) \textit {\_Z} +a^{2} c^{4}+2 a b \,c^{2} d^{2}+b^{2} d^{4}+c^{4}\right )}{\sum }\frac {\ln \left (d +c \sqrt {x}\right ) \ln \left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-c \sqrt {x}+\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b d +a \,c^{2}+b \,d^{2}}\right )}{4 b}\right )}{c^{2}}\) | \(388\) |
arccot(b*x+a)/c*x-2*arccot(b*x+a)/c^2*d*x^(1/2)+2*arccot(b*x+a)*d^2/c^3*ln (d+c*x^(1/2))+4*b/c^2*(-1/8*c/b*sum((-_R^3+5*_R^2*d-7*_R*d^2+3*d^3)/(_R^3* b-3*_R^2*b*d+_R*a*c^2+3*_R*b*d^2-a*c^2*d-b*d^3)*ln(c*x^(1/2)-_R+d),_R=Root Of(b^2*_Z^4-4*b^2*d*_Z^3+(2*a*b*c^2+6*b^2*d^2)*_Z^2+(-4*a*b*c^2*d-4*b^2*d^ 3)*_Z+a^2*c^4+2*a*b*c^2*d^2+b^2*d^4+c^4))+1/4*c*d^2/b*sum(1/(_R1^2*b-2*_R1 *b*d+a*c^2+b*d^2)*(ln(d+c*x^(1/2))*ln((-c*x^(1/2)+_R1-d)/_R1)+dilog((-c*x^ (1/2)+_R1-d)/_R1)),_R1=RootOf(b^2*_Z^4-4*b^2*d*_Z^3+(2*a*b*c^2+6*b^2*d^2)* _Z^2+(-4*a*b*c^2*d-4*b^2*d^3)*_Z+a^2*c^4+2*a*b*c^2*d^2+b^2*d^4+c^4)))
\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\text {Timed out} \]
\[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{\sqrt {x}}} \,d x } \]
Exception generated. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0]W arning, replacing 0 by -24, a substitution variable should perhaps be purg ed.Warnin
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{\sqrt {x}}} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+\frac {d}{\sqrt {x}}} \,d x \]