\(\int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\) [8]

Optimal result

Integrand size = 19, antiderivative size = 326 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=-\frac {2 b c \left (1+\frac {b^2-2 c (a+c)}{b \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{(a-b+c) (a+b+c) \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {2 b c \left (1-\frac {b^2-2 c (a+c)}{b \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{(a-b+c) (a+b+c) \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}-\frac {\sin (x)}{2 (a+b+c) (1-\cos (x))}+\frac {\sin (x)}{2 (a-b+c) (1+\cos (x))} \] Output:

-2*b*c*(1+(b^2-2*c*(a+c))/b/(-4*a*c+b^2)^(1/2))*arctan((b-2*c-(-4*a*c+b^2) 
^(1/2))^(1/2)*tan(1/2*x)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2))/(a-b+c)/(a+b+c) 
/(b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)-2*b*c*( 
1-(b^2-2*c*(a+c))/b/(-4*a*c+b^2)^(1/2))*arctan((b-2*c+(-4*a*c+b^2)^(1/2))^ 
(1/2)*tan(1/2*x)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2))/(a-b+c)/(a+b+c)/(b-2*c+ 
(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)-1/2*sin(x)/(a+b 
+c)/(1-cos(x))+1/2*sin(x)/(a-b+c)/(1+cos(x))
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {\sqrt {2} c \left (-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)-2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (a^2-b^2+2 a c+c^2\right ) \sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}-\frac {\sqrt {2} c \left (b^2-2 c (a+c)+b \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\left (-b+2 c+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (a^2-b^2+2 a c+c^2\right ) \sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}-\frac {\cot \left (\frac {x}{2}\right )}{2 (a+b+c)}+\frac {\tan \left (\frac {x}{2}\right )}{2 (a-b+c)} \] Input:

Integrate[Csc[x]^2/(a + b*Cos[x] + c*Cos[x]^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 3748, 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.

\(\displaystyle \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (x)^2 \left (a+b \cos (x)+c \cos (x)^2\right )}dx\)

\(\Big \downarrow \) 3748

\(\displaystyle \int \left (\frac {-\left (b^2 \left (1-\frac {c (a+c)}{b^2}\right )\right )-b c \cos (x)}{(a-b+c) (a+b+c) \left (a+b \cos (x)+c \cos ^2(x)\right )}-\frac {1}{2 (\cos (x)-1) (a+b+c)}+\frac {1}{2 (\cos (x)+1) (a-b+c)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b c \left (\frac {b^2-2 c (a+c)}{b \sqrt {b^2-4 a c}}+1\right ) \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {-\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{(a-b+c) (a+b+c) \sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {2 b c \left (1-\frac {b^2-2 c (a+c)}{b \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{(a-b+c) (a+b+c) \sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {\sin (x)}{2 (1-\cos (x)) (a+b+c)}+\frac {\sin (x)}{2 (\cos (x)+1) (a-b+c)}\)

Input:

Int[Csc[x]^2/(a + b*Cos[x] + c*Cos[x]^2),x]
 

Output:

(-2*b*c*(1 + (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[b - 2 
*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/((a 
- b + c)*(a + b + c)*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt 
[b^2 - 4*a*c]]) - (2*b*c*(1 - (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*A 
rcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 
 - 4*a*c]]])/((a - b + c)*(a + b + c)*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sq 
rt[b + 2*c + Sqrt[b^2 - 4*a*c]]) - Sin[x]/(2*(a + b + c)*(1 - Cos[x])) + S 
in[x]/(2*(a - b + c)*(1 + Cos[x]))
 

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 3748

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]^(n_.)*(b_.) + cos[(d_.) + (e_.)*(x_)]^ 
(n2_.)*(c_.))^(p_.)*sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandT 
rig[(1 - cos[d + e*x]^2)^(m/2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^(2*n) 
)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] & 
& NeQ[b^2 - 4*a*c, 0] && IntegersQ[n, p]
 

Maple [A] (verified)

Time = 7.57 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {\tan \left (\frac {x}{2}\right )}{2 a -2 b +2 c}-\frac {1}{2 \left (a +b +c \right ) \tan \left (\frac {x}{2}\right )}+\frac {\left (4 a -4 b +4 c \right ) \left (\frac {\left (\sqrt {-4 a c +b^{2}}\, a c -\sqrt {-4 a c +b^{2}}\, b^{2}+\sqrt {-4 a c +b^{2}}\, b c +\sqrt {-4 a c +b^{2}}\, c^{2}-3 c a b +2 a \,c^{2}+b^{3}-b^{2} c -b \,c^{2}+2 c^{3}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b -c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}+\frac {\left (\sqrt {-4 a c +b^{2}}\, a c -\sqrt {-4 a c +b^{2}}\, b^{2}+\sqrt {-4 a c +b^{2}}\, b c +\sqrt {-4 a c +b^{2}}\, c^{2}+3 c a b -2 a \,c^{2}-b^{3}+b^{2} c +b \,c^{2}-2 c^{3}\right ) \arctan \left (\frac {\left (a -b +c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \left (a +b +c \right ) \left (a -b +c \right )}\)	\(393\)
risch	\(\text {Expression too large to display}\)	\(6094\)

Input:

int(csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x,method=_RETURNVERBOSE)
 

Output:

1/2/(a-b+c)*tan(1/2*x)-1/2/(a+b+c)/tan(1/2*x)+1/2/(a+b+c)/(a-b+c)*(4*a-4*b 
+4*c)*(1/2*((-4*a*c+b^2)^(1/2)*a*c-(-4*a*c+b^2)^(1/2)*b^2+(-4*a*c+b^2)^(1/ 
2)*b*c+(-4*a*c+b^2)^(1/2)*c^2-3*c*a*b+2*a*c^2+b^3-b^2*c-b*c^2+2*c^3)/(-4*a 
*c+b^2)^(1/2)/(a-b+c)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a 
+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))+1/2*((-4*a*c+b^ 
2)^(1/2)*a*c-(-4*a*c+b^2)^(1/2)*b^2+(-4*a*c+b^2)^(1/2)*b*c+(-4*a*c+b^2)^(1 
/2)*c^2+3*c*a*b-2*a*c^2-b^3+b^2*c+b*c^2-2*c^3)/(-4*a*c+b^2)^(1/2)/(a-b+c)/ 
(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a 
*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16741 vs. \(2 (279) = 558\).

Time = 3.16 (sec) , antiderivative size = 16741, normalized size of antiderivative = 51.35 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display} \] Input:

integrate(csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\int \frac {\csc ^{2}{\left (x \right )}}{a + b \cos {\left (x \right )} + c \cos ^{2}{\left (x \right )}}\, dx \] Input:

integrate(csc(x)**2/(a+b*cos(x)+c*cos(x)**2),x)
 

Output:

Integral(csc(x)**2/(a + b*cos(x) + c*cos(x)**2), x)
 

Maxima [F]

\[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\int { \frac {\csc \left (x\right )^{2}}{c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a} \,d x } \] Input:

integrate(csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x, algorithm="maxima")
 

Output:

-(2*b*cos(2*x)*sin(x) + ((a^2 - b^2 + 2*a*c + c^2)*cos(2*x)^2 + (a^2 - b^2 
 + 2*a*c + c^2)*sin(2*x)^2 + a^2 - b^2 + 2*a*c + c^2 - 2*(a^2 - b^2 + 2*a* 
c + c^2)*cos(2*x))*integrate(2*(2*b^2*c*cos(3*x)^2 + 2*b^2*c*cos(x)^2 + 2* 
b^2*c*sin(3*x)^2 + 2*b^2*c*sin(x)^2 + b*c^2*cos(x) + 4*(2*a*b^2 - 3*a*c^2 
- c^3 - (2*a^2 - b^2)*c)*cos(2*x)^2 + 4*(2*a*b^2 - 3*a*c^2 - c^3 - (2*a^2 
- b^2)*c)*sin(2*x)^2 + 2*(2*b^3 - b*c^2)*sin(2*x)*sin(x) + (b*c^2*cos(3*x) 
 + b*c^2*cos(x) + 2*(b^2*c - a*c^2 - c^3)*cos(2*x))*cos(4*x) + (4*b^2*c*co 
s(x) + b*c^2 + 2*(2*b^3 - b*c^2)*cos(2*x))*cos(3*x) + 2*(b^2*c - a*c^2 - c 
^3 + (2*b^3 - b*c^2)*cos(x))*cos(2*x) + (b*c^2*sin(3*x) + b*c^2*sin(x) + 2 
*(b^2*c - a*c^2 - c^3)*sin(2*x))*sin(4*x) + 2*(2*b^2*c*sin(x) + (2*b^3 - b 
*c^2)*sin(2*x))*sin(3*x))/(2*a*c^3 + c^4 + (a^2 - b^2)*c^2 + (2*a*c^3 + c^ 
4 + (a^2 - b^2)*c^2)*cos(4*x)^2 + 4*(a^2*b^2 - b^4 + 2*a*b^2*c + b^2*c^2)* 
cos(3*x)^2 + 4*(4*a^4 - 4*a^2*b^2 + 6*a*c^3 + c^4 + (13*a^2 - b^2)*c^2 + 4 
*(3*a^3 - a*b^2)*c)*cos(2*x)^2 + 4*(a^2*b^2 - b^4 + 2*a*b^2*c + b^2*c^2)*c 
os(x)^2 + (2*a*c^3 + c^4 + (a^2 - b^2)*c^2)*sin(4*x)^2 + 4*(a^2*b^2 - b^4 
+ 2*a*b^2*c + b^2*c^2)*sin(3*x)^2 + 4*(4*a^4 - 4*a^2*b^2 + 6*a*c^3 + c^4 + 
 (13*a^2 - b^2)*c^2 + 4*(3*a^3 - a*b^2)*c)*sin(2*x)^2 + 8*(2*a^3*b - 2*a*b 
^3 + 4*a*b*c^2 + b*c^3 + (5*a^2*b - b^3)*c)*sin(2*x)*sin(x) + 4*(a^2*b^2 - 
 b^4 + 2*a*b^2*c + b^2*c^2)*sin(x)^2 + 2*(2*a*c^3 + c^4 + (a^2 - b^2)*c^2 
+ 2*(2*a*b*c^2 + b*c^3 + (a^2*b - b^3)*c)*cos(3*x) + 2*(4*a*c^3 + c^4 +...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21752 vs. \(2 (279) = 558\).

Time = 2.12 (sec) , antiderivative size = 21752, normalized size of antiderivative = 66.72 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display} \] Input:

integrate(csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x, algorithm="giac")
 

Output:

-((2*a^2*b^6 - 4*a*b^7 + 2*b^8 - 18*a^3*b^4*c + 34*a^2*b^5*c - 10*a*b^6*c 
- 6*b^7*c + 48*a^4*b^2*c^2 - 80*a^3*b^3*c^2 - 22*a^2*b^4*c^2 + 52*a*b^5*c^ 
2 + 4*b^6*c^2 - 32*a^5*c^3 + 32*a^4*b*c^3 + 144*a^3*b^2*c^3 - 128*a^2*b^3* 
c^3 - 38*a*b^4*c^3 + 2*b^5*c^3 - 96*a^4*c^4 + 64*a^3*b*c^4 + 112*a^2*b^2*c 
^4 - 16*a*b^3*c^4 - 2*b^4*c^4 - 96*a^3*c^5 + 32*a^2*b*c^5 + 16*a*b^2*c^5 - 
 32*a^2*c^6 + 3*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c) 
)*sqrt(b^2 - 4*a*c)*a^2*b^4 - 2*(b^2 - 4*a*c)*a^2*b^4 - 2*sqrt(a^2 - a*b + 
 b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b^5 + 4*(b 
^2 - 4*a*c)*a*b^5 - 5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - 
b + c))*sqrt(b^2 - 4*a*c)*b^6 - 2*(b^2 - 4*a*c)*b^6 - 15*sqrt(a^2 - a*b + 
b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^3*b^2*c + 1 
0*(b^2 - 4*a*c)*a^3*b^2*c + 7*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a* 
c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*b^3*c - 18*(b^2 - 4*a*c)*a^2*b^3*c + 
 41*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 
 4*a*c)*a*b^4*c + 2*(b^2 - 4*a*c)*a*b^4*c + 11*sqrt(a^2 - a*b + b*c - c^2 
- sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^5*c + 6*(b^2 - 4*a*c) 
*b^5*c + 12*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sq 
rt(b^2 - 4*a*c)*a^4*c^2 - 8*(b^2 - 4*a*c)*a^4*c^2 + 4*sqrt(a^2 - a*b + b*c 
 - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^3*b*c^2 + 8*(b 
^2 - 4*a*c)*a^3*b*c^2 - 101*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a...
 

Mupad [B] (verification not implemented)

Time = 8.49 (sec) , antiderivative size = 39229, normalized size of antiderivative = 120.33 \[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display} \] Input:

int(1/(sin(x)^2*(a + b*cos(x) + c*cos(x)^2)),x)
 

Output:

atan((((-(8*a*c^7 + b^8 + 24*a^2*c^6 + 24*a^3*c^5 + 8*a^4*c^4 + b^5*(-(4*a 
*c - b^2)^3)^(1/2) - 2*b^2*c^6 + 3*b^4*c^4 - 3*b^6*c^2 - 18*a*b^2*c^5 + 24 
*a*b^4*c^3 + 3*b*c^4*(-(4*a*c - b^2)^3)^(1/2) - 54*a^2*b^2*c^4 + 33*a^2*b^ 
4*c^2 - 38*a^3*b^2*c^3 - 3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c + 
 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b*c^3*(-(4*a*c - b^2)^3)^(1/2) 
 - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(3*a^2*b^8 - b^10 - 3*a^4*b^6 + 
a^6*b^4 + 16*a^2*c^8 + 96*a^3*c^7 + 240*a^4*c^6 + 320*a^5*c^5 + 240*a^6*c^ 
4 + 96*a^7*c^3 + 16*a^8*c^2 + b^4*c^6 - 3*b^6*c^4 + 3*b^8*c^2 - 8*a*b^2*c^ 
7 + 30*a*b^4*c^5 - 36*a*b^6*c^3 - 36*a^3*b^6*c + 30*a^5*b^4*c - 8*a^7*b^2* 
c - 96*a^2*b^2*c^6 + 159*a^2*b^4*c^4 - 82*a^2*b^6*c^2 - 312*a^3*b^2*c^5 + 
260*a^3*b^4*c^3 - 448*a^4*b^2*c^4 + 159*a^4*b^4*c^2 - 312*a^5*b^2*c^3 - 96 
*a^6*b^2*c^2 + 14*a*b^8*c)))^(1/2)*(128*a*c^13 - 64*a*b^13 - 32*b^13*c + 3 
2*b^14 - 96*a^2*b^12 + 256*a^3*b^11 + 64*a^4*b^10 - 384*a^5*b^9 + 64*a^6*b 
^8 + 256*a^7*b^7 - 96*a^8*b^6 - 64*a^9*b^5 + 32*a^10*b^4 + 1408*a^2*c^12 + 
 7040*a^3*c^11 + 21120*a^4*c^10 + 42240*a^5*c^9 + 59136*a^6*c^8 + 59136*a^ 
7*c^7 + 42240*a^8*c^6 + 21120*a^9*c^5 + 7040*a^10*c^4 + 1408*a^11*c^3 + 12 
8*a^12*c^2 - 32*b^2*c^12 + 96*b^3*c^11 + 64*b^4*c^10 - 416*b^5*c^9 + 96*b^ 
6*c^8 + 704*b^7*c^7 - 384*b^8*c^6 - 576*b^9*c^5 + 416*b^10*c^4 + 224*b^11* 
c^3 - 192*b^12*c^2 + tan(x/2)*(-(8*a*c^7 + b^8 + 24*a^2*c^6 + 24*a^3*c^5 + 
 8*a^4*c^4 + b^5*(-(4*a*c - b^2)^3)^(1/2) - 2*b^2*c^6 + 3*b^4*c^4 - 3*b...
 

Reduce [F]

\[ \int \frac {\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\int \frac {\csc \left (x \right )^{2}}{a +\cos \left (x \right ) b +c \cos \left (x \right )^{2}}d x \] Input:

int(csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x)
 

Output:

int(csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x)