3.1.10 \(\int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx\) [10]

3.1.10.1 Optimal result

Integrand size = 33, antiderivative size = 691 \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx=-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{3/2} e}+\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {(2 a+b \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{4 a e} \]

output

-1/8*(-4*a*c+b^2)*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-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^(1/2)/e+1/2*arctanh(1/2*(b^2+b*cot(e 
*x+d)*(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)))/(a^ 
2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(a^2+ 
b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/ 
2))*(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1 
/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)-1/2*arctan(1/2*(b^2+(a-c)* 
(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*cot(e*x+d)*(a^2-2*a*c+b^2+c^2)^(1/2))/(a 
^2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(a^2 
+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1 
/2))*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^( 
1/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)+1/4*(2*a+b*cot(e*x+d))*(a 
+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^2/a/e
 

3.1.10.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.48 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.53 \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx=-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x) \left (\left (8 a^2+b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \cot (d+e x)-2 \sqrt {a} \left (-2 i a \sqrt {a+i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \cot (d+e x)-2 i a \sqrt {a-i b-c} \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \cot (d+e x)+(2 a+b \cot (d+e x)) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )}{8 a^{3/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \]

input

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

output

-1/8*(Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2*((8*a^2 + 
 b^2 - 4*a*c)*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + 
 e*x] + a*Tan[d + e*x]^2])]*Cot[d + e*x] - 2*Sqrt[a]*((-2*I)*a*Sqrt[a + I* 
b - 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])]*Cot[d + e*x] - (2*I)*a*Sqr 
t[a - I*b - 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])]*Cot[d + e*x] + (2* 
a + b*Cot[d + e*x])*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])))/(a^(3/2 
)*e*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])
 

3.1.10.3 Rubi [F]

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4184

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

input

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

output

$Aborted
 

3.1.10.3.1 Defintions of rubi rules used

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 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

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]
 

3.1.10.4 Maple [F(-1)]

Timed out.

hanged

input

int((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x)
 

output

int((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x)
 

3.1.10.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2508 vs. \(2 (622) = 1244\).

Time = 0.98 (sec) , antiderivative size = 5018, normalized size of antiderivative = 7.26 \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^3(d+e x) \, dx=\text {Too large to display} \]

input

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x, algorithm= 
"fricas")
 

output

Too large to include
 

3.1.10.6 Sympy [F]

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

input

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(1/2)*tan(e*x+d)**3,x)
 

output

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

3.1.10.7 Maxima [F]

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

input

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x, algorithm= 
"maxima")
 

output

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

3.1.10.8 Giac [F]

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

input

integrate((a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x, algorithm= 
"giac")
 

output

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

3.1.10.9 Mupad [F(-1)]

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

input

int(tan(d + e*x)^3*(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2),x)
 

output

int(tan(d + e*x)^3*(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2), x)