3.1.14 \(\int \frac {\cot (d+e x)}{(a+b \cot (d+e x)+c \cot ^2(d+e x))^{3/2}} \, dx\) [14]

3.1.14.1 Optimal result

Integrand size = 31, antiderivative size = 635 \[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \]

output

-2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(e*x+d))/(b^2+(a-c)^2)/(-4*a*c+b^2)/e/( 
a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)+1/2*arctanh(1/2*(b^2-(a-c)*(a-c-(a^2- 
2*a*c+b^2+c^2)^(1/2))-b*cot(e*x+d)*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))*2^ 
(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(2*a-2*c+(a^2-2*a*c+b^2+c^2)^( 
1/2))^(1/2)/(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2))*(2* 
a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c 
+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-1/2*arctanh(1/2 
*(b^2-b*cot(e*x+d)*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))-(a-c)*(a-c+(a^2-2*a 
*c+b^2+c^2)^(1/2)))*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(2*a-2*c 
-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+ 
c^2)^(1/2))^(1/2))*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a* 
c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2 
^(1/2)
 

3.1.14.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.40 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.72 \[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {2 \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (-\frac {\frac {i \left (4 a^2 c+b^2 (i b+c)-a \left (b^2+4 i b c+4 c^2\right )\right ) \arctan \left (\frac {-i b+2 c+(-2 i a+b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}+\frac {\left (-b^2 (b+i c)-4 i a^2 c+a \left (i b^2+4 b c+4 i c^2\right )\right ) \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}}{4 \left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}+\frac {b+2 a \tan (d+e x)}{\left (-b^2+4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {b^3+a b (a-3 c)+a \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \]

input

Integrate[Cot[d + e*x]/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
 

output

(2*Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*(-1/4*((I*(4*a 
^2*c + b^2*(I*b + c) - a*(b^2 + (4*I)*b*c + 4*c^2))*ArcTan[((-I)*b + 2*c + 
 ((-2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b*Tan[d + e*x] 
 + a*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] + ((-(b^2*(b + I*c)) - (4*I)*a^2 
*c + a*(I*b^2 + 4*b*c + (4*I)*c^2))*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[ 
d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2] 
)])/Sqrt[a - I*b - c])/((b^2 + (a - c)^2)*(b^2 - 4*a*c)) + (b + 2*a*Tan[d 
+ e*x])/((-b^2 + 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) + (b^ 
3 + a*b*(a - 3*c) + a*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x])/((b^2 + (a - c)^ 
2)*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])))/(e*Sqrt[a 
+ b*Cot[d + e*x] + c*Cot[d + e*x]^2])
 

3.1.14.3 Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3042, 4184, 1351, 27, 27, 1369, 25, 1363, 221}

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.

\(\displaystyle \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot (d+e x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 1351

\(\displaystyle -\frac {\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \int -\frac {b \left (b^2-4 a c\right )+(a-c) \cot (d+e x) \left (b^2-4 a c\right )}{2 \left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right )}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {\left (b^2-4 a c\right ) (b+(a-c) \cot (d+e x))}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right )}+\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {b+(a-c) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{(a-c)^2+b^2}+\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

\(\Big \downarrow \) 1369

\(\displaystyle -\frac {\frac {\frac {\int -\frac {b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int -\frac {b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}}{(a-c)^2+b^2}+\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\int \frac {b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\int \frac {b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}d\cot (d+e x)}{2 \sqrt {a^2-2 a c+b^2+c^2}}}{(a-c)^2+b^2}+\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

\(\Big \downarrow \) 1363

\(\displaystyle -\frac {\frac {\frac {b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \left (b^2-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \int \frac {1}{\frac {b \left (b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )^2}{c \cot ^2(d+e x)+b \cot (d+e x)+a}+2 b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}}{\sqrt {a^2-2 a c+b^2+c^2}}-\frac {b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \left (b^2-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \int \frac {1}{\frac {b \left (b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )^2}{c \cot ^2(d+e x)+b \cot (d+e x)+a}+2 b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right )}d\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}}{\sqrt {a^2-2 a c+b^2+c^2}}}{(a-c)^2+b^2}+\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \left (b^2-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \text {arctanh}\left (\frac {-b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2}}-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \left (b^2-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )\right ) \text {arctanh}\left (\frac {-b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2-2 a c+b^2+c^2} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2}}}{(a-c)^2+b^2}+\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

input

Int[Cot[d + e*x]/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
 

output

-(((-((Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*(b^2 - (a - c)*(a - 
 c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[ 
a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])* 
Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqr 
t[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a 
+ b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + 
c^2]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]] 
)) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*(b^2 - (a - c)*(a - 
c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a 
^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*C 
ot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt 
[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + 
 b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c 
^2]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]) 
)/(b^2 + (a - c)^2) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x] 
))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e* 
x]^2]))/e)
 

3.1.14.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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

rule 1351

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^ 
(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((g*c)*((-b)*(c* 
d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2 
*f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x), x] + Simp[1/((b^2 - 4*a*c)*(b^2*d* 
f + (c*d - a*f)^2)*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*S 
imp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*((-b)*f))*(p + 1) + (b^2*(g*f) - b 
*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f* 
((g*c)*((-b)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + 
 q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1 
)))*x - c*f*(b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2* 
q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4 
*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[ 
p] && ILtQ[q, -1])
 

rule 1363

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f 
_.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h   Subst[Int[1/Simp[2*a^2*g*h*c + a 
*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ 
[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
 

rule 1369

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp 
[1/(2*q)   Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c 
*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q)   Int[ 
Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + 
 c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] 
&& NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

3.1.14.4 Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 1.24 (sec) , antiderivative size = 13066366, normalized size of antiderivative = 20576.95

\[\text {output too large to display}\]

input

int(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
 

output

result too large to display
 

3.1.14.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21090 vs. \(2 (580) = 1160\).

Time = 11.87 (sec) , antiderivative size = 21090, normalized size of antiderivative = 33.21 \[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

input

integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm="f 
ricas")
 

output

Too large to include
 

3.1.14.6 Sympy [F]

\[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\left (a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(3/2),x)
 

output

Integral(cot(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)**2)**(3/2), x)
 

3.1.14.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

input

integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm="m 
axima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

3.1.14.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(cot(e*x+d)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm="g 
iac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument 
Value
 

3.1.14.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{{\left (c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \]

input

int(cot(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2),x)
 

output

int(cot(d + e*x)/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2), x)