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

Optimal result

Integrand size = 31, antiderivative size = 749 \[ \int \frac {\tan (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{a^{3/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac {2 \left (b^2-2 a c+b c \cot (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.26 (sec) , antiderivative size = 934, normalized size of antiderivative = 1.25 \[ \int \frac {\tan (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\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 )}{-4 a+4 i b+4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\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 )}{-4 a-4 i b+4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \tan ^3(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b^3+a b (a-3 c)+a \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)\right )}{\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)}}+\frac {4 \left (b^2-4 a c\right ) \left (\frac {a^2}{\left (b^2-4 a c\right ) \left (\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}\right )}\right )^{3/2} \left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right ) \left (-\frac {a \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}{b^2-4 a c}\right )^{3/2}}{a^2 \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2} \sqrt {1-\frac {\left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right )^2}{\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}}}}-\frac {2 \left (b \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (-6 a^2 b^2 c+24 a^3 c^2\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (6 a^2 b c-12 a^3 c \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}} \] Input:

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

Output:

(Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*((2*((-4*Sqrt[a 
- I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan[( 
I*b + 2*c - ((-2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*T 
an[d + e*x] + a*Tan[d + e*x]^2])])/(-4*a + (4*I)*b + 4*c) - (4*Sqrt[a + I* 
b - c]*(-1/4*(b*(b^2 - 4*a*c)) - (I/4)*(a - c)*(b^2 - 4*a*c))*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])])/(-4*a - (4*I)*b + 4*c)))/((b^2 + (a - c)^ 
2)*(b^2 - 4*a*c)) - (2*Tan[d + e*x]^3*(-b^2 + 2*a*c - a*b*Tan[d + e*x]))/( 
c*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - (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]) + (4*(b^2 - 4*a* 
c)*(a^2/((b^2 - 4*a*c)*((a^2*b^2)/(b^2 - 4*a*c)^2 - (4*a^3*c)/(b^2 - 4*a*c 
)^2)))^(3/2)*(-((a*b)/(b^2 - 4*a*c)) - (2*a^2*Tan[d + e*x])/(b^2 - 4*a*c)) 
*(-((a*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2))/(b^2 - 4*a*c)))^(3/2))/(a^ 
2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)^(3/2)*Sqrt[1 - (-((a*b)/(b^2 - 4 
*a*c)) - (2*a^2*Tan[d + e*x])/(b^2 - 4*a*c))^2/((a^2*b^2)/(b^2 - 4*a*c)^2 
- (4*a^3*c)/(b^2 - 4*a*c)^2)]) - (2*(b*Tan[d + e*x]^2*Sqrt[c + b*Tan[d + e 
*x] + a*Tan[d + e*x]^2] + (((-6*a^2*b^2*c + 24*a^3*c^2)*ArcTanh[(b + 2*a*T 
an[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])])/(4* 
a^(5/2)) + ((6*a^2*b*c - 12*a^3*c*Tan[d + e*x])*Sqrt[c + b*Tan[d + e*x]...
 

Rubi [A] (verified)

Time = 3.60 (sec) , antiderivative size = 740, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3042, 4184, 7276, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4184

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{a^{3/2}}-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \text {arctanh}\left (\frac {-b \left (-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \text {arctanh}\left (\frac {-b \left (\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c\right ) \cot (d+e x)-(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt {-(a-c) \sqrt {a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac {2 \left (-2 a c+b^2+b c \cot (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \left (a \left (b^2-2 c (a-c)\right )+b c (a+c) \cot (d+e x)\right )}{\left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}}{e}\)

Input:

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

Output:

-((-(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c 
*Cot[d + e*x]^2])]/a^(3/2)) - (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]] 
*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]]*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]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) + (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]]*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])/(Sqr 
t[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]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) + (2*(b^2 
- 2*a*c + b*c*Cot[d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x] + c* 
Cot[d + e*x]^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)
 

Defintions of rubi rules used

rule 2009

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

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]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19942 vs. \(2 (684) = 1368\).

Time = 7.21 (sec) , antiderivative size = 39885, normalized size of antiderivative = 53.25 \[ \int \frac {\tan (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(tan(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
 

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-2)]

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

integrate(tan(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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\tan (d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}\, \tan \left (e x +d \right )}{\cot \left (e x +d \right )^{4} c^{2}+2 \cot \left (e x +d \right )^{3} b c +2 \cot \left (e x +d \right )^{2} a c +\cot \left (e x +d \right )^{2} b^{2}+2 \cot \left (e x +d \right ) a b +a^{2}}d x \] Input:

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

Output:

int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*tan(d + e*x))/(cot(d + e 
*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a*c + cot(d + e*x) 
**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)