\(\int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) [23]

Optimal result

Integrand size = 33, antiderivative size = 179 \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e} \] Output:

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

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.37 \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x) \left ((b-2 c) \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+2 \sqrt {c} \left (-\sqrt {a-b+c} \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}\right )\right )}{4 \sqrt {c} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \] Input:

Integrate[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
 

Output:

-1/4*(Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2*((b - 2 
*c)*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 
+ a*Tan[d + e*x]^4])] + 2*Sqrt[c]*(-(Sqrt[a - b + c]*ArcTanh[(b - 2*c + (2 
*a - b)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[c + b*Tan[d + e*x]^2 + a*T 
an[d + e*x]^4])]) + Cot[d + e*x]^2*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e 
*x]^4])))/(Sqrt[c]*e*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3042, 4184, 1576, 1162, 25, 1269, 1092, 219, 1154, 219}

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.

\(\displaystyle \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (d+e x) \sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}dx\)

\(\Big \downarrow \) 4184

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

\(\Big \downarrow \) 1576

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

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]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}-\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\)	\(217\)
default	\(\frac {-\frac {\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{2}-\frac {\left (b -2 c \right ) \ln \left (\frac {\frac {b}{2}-c +\left (\cot \left (e x +d \right )^{2}+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}\right )}{4 \sqrt {c}}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot \left (e x +d \right )^{2}+1\right )^{2}+\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2}}{e}\)	\(217\)

Input:

int(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (155) = 310\).

Time = 1.44 (sec) , antiderivative size = 1932, normalized size of antiderivative = 10.79 \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm= 
"fricas")
 

Output:

[1/8*(2*sqrt(a - b + c)*c*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*co 
s(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*cos(2*e*x + 2*d)^2 
 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*c 
os(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2 
*d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2 
*d)) - (b - 2*c)*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*e*x + 2* 
d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b 
*cos(2*e*x + 2*d) + b + 2*c)*sqrt(c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 
- 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e* 
x + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2* 
d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*c*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 
 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e 
*x + 2*d) + 1)))/(c*e), -1/4*((b - 2*c)*sqrt(-c)*arctan(-1/2*((b - 2*c)*co 
s(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + 
 c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e* 
x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*e*x + 2*d)^ 
2 + (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))) - sqrt(a - b + c)*c 
*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + 2*a^2 
- b^2 + 2*c^2 + 2*((a - b + c)*cos(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 
2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*...
 

Sympy [F]

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

integrate(cot(e*x+d)*(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)
 

Output:

Integral(sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)*cot(d + e*x), x)
 

Maxima [F]

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

integrate(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm= 
"maxima")
 

Output:

integrate(sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a)*cot(e*x + d), x)
 

Giac [F(-1)]

Timed out. \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\text {Timed out} \] Input:

integrate(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm= 
"giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx=\int \mathrm {cot}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \] Input:

int(cot(d + e*x)*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)
 

Output:

int(cot(d + e*x)*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)
 

Reduce [F]

\[ \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx =\text {Too large to display} \] Input:

int(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x)
 

Output:

( - sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*b + int((sqrt(cot(d + 
e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x))/(cot(d + e*x)**4*b*c + 2* 
cot(d + e*x)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)**2*b*c + a*b 
+ 2*a*c),x)*a*b**2*e + 4*int((sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + 
 a)*cot(d + e*x))/(cot(d + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 + cot(d + 
e*x)**2*b**2 + 2*cot(d + e*x)**2*b*c + a*b + 2*a*c),x)*a*b*c*e + 4*int((sq 
rt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x))/(cot(d + e*x)* 
*4*b*c + 2*cot(d + e*x)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)**2 
*b*c + a*b + 2*a*c),x)*a*c**2*e - int((sqrt(cot(d + e*x)**4*c + cot(d + e* 
x)**2*b + a)*cot(d + e*x))/(cot(d + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 + 
 cot(d + e*x)**2*b**2 + 2*cot(d + e*x)**2*b*c + a*b + 2*a*c),x)*b**3*e - 2 
*int((sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + e*x))/(cot(d 
 + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d + 
 e*x)**2*b*c + a*b + 2*a*c),x)*b**2*c*e - int((sqrt(cot(d + e*x)**4*c + co 
t(d + e*x)**2*b + a)*cot(d + e*x)**5)/(cot(d + e*x)**4*b*c + 2*cot(d + e*x 
)**4*c**2 + cot(d + e*x)**2*b**2 + 2*cot(d + e*x)**2*b*c + a*b + 2*a*c),x) 
*b**2*c*e + 4*int((sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*cot(d + 
 e*x)**5)/(cot(d + e*x)**4*b*c + 2*cot(d + e*x)**4*c**2 + cot(d + e*x)**2* 
b**2 + 2*cot(d + e*x)**2*b*c + a*b + 2*a*c),x)*c**3*e)/(e*(b + 2*c))