### 3.8 $$\int \frac{\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx$$

Optimal. Leaf size=326 $-\frac{2 b c \left (\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{(a-b+c) (a+b+c) \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{2 b c \left (1-\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{(a-b+c) (a+b+c) \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sin (x)}{2 (1-\cos (x)) (a+b+c)}+\frac{\sin (x)}{2 (\cos (x)+1) (a-b+c)}$

[Out]

(-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])/Sq
rt[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 - Sq
rt[b^2 - 4*a*c]]) - (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]]) - Sin[x]/(2*(a + b + c)*(1 - Cos[x])) + Sin[x]/(2*(a - b + c)*(1 + Cos[x]
))

________________________________________________________________________________________

Rubi [A]  time = 3.33951, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.263, Rules used = {3267, 2648, 3293, 2659, 205} $-\frac{2 b c \left (\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{(a-b+c) (a+b+c) \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{2 b c \left (1-\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{(a-b+c) (a+b+c) \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sin (x)}{2 (1-\cos (x)) (a+b+c)}+\frac{\sin (x)}{2 (\cos (x)+1) (a-b+c)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-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])/Sq
rt[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 - Sq
rt[b^2 - 4*a*c]]) - (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]]) - Sin[x]/(2*(a + b + c)*(1 - Cos[x])) + Sin[x]/(2*(a - b + c)*(1 + Cos[x]
))

Rule 3267

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]^(n_.)*(b_.) + cos[(d_.) + (e_.)*(x_)]^(n2_.)*(c_.))^(p_.)*sin[(d_.) + (e_
.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrig[(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] && Integ
ersQ[n, p]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3293

Int[(cos[(d_.) + (e_.)*(x_)]*(B_.) + (A_))/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + cos[(d_.) + (e_.)*(x_)]^2*
(c_.)), x_Symbol] :> Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Cos[d + e*x
]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Cos[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B
}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\csc ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx &=\int \left (-\frac{1}{2 (a+b+c) (-1+\cos (x))}+\frac{1}{2 (a-b+c) (1+\cos (x))}+\frac{-b^2 \left (1-\frac{c (a+c)}{b^2}\right )-b c \cos (x)}{(a-b+c) (a+b+c) \left (a+b \cos (x)+c \cos ^2(x)\right )}\right ) \, dx\\ &=\frac{\int \frac{1}{1+\cos (x)} \, dx}{2 (a-b+c)}-\frac{\int \frac{1}{-1+\cos (x)} \, dx}{2 (a+b+c)}+\frac{\int \frac{-b^2 \left (1-\frac{c (a+c)}{b^2}\right )-b c \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx}{(a-b+c) (a+b+c)}\\ &=-\frac{\sin (x)}{2 (a+b+c) (1-\cos (x))}+\frac{\sin (x)}{2 (a-b+c) (1+\cos (x))}-\frac{\left (c \left (b+\frac{b^2-2 c (a+c)}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{b-\sqrt{b^2-4 a c}+2 c \cos (x)} \, dx}{(a-b+c) (a+b+c)}-\frac{\left (b c \left (1-\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{b+\sqrt{b^2-4 a c}+2 c \cos (x)} \, dx}{(a-b+c) (a+b+c)}\\ &=-\frac{\sin (x)}{2 (a+b+c) (1-\cos (x))}+\frac{\sin (x)}{2 (a-b+c) (1+\cos (x))}-\frac{\left (2 c \left (b+\frac{b^2-2 c (a+c)}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+2 c-\sqrt{b^2-4 a c}+\left (b-2 c-\sqrt{b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{(a-b+c) (a+b+c)}-\frac{\left (2 b c \left (1-\frac{b^2-2 c (a+c)}{b \sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+2 c+\sqrt{b^2-4 a c}+\left (b-2 c+\sqrt{b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{(a-b+c) (a+b+c)}\\ &=-\frac{2 c \left (b+\frac{b^2-2 c (a+c)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\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 ) \tan ^{-1}\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))}\\ \end{align*}

Mathematica [A]  time = 0.92149, size = 335, normalized size = 1.03 $\frac{\sqrt{2} c \left (b \sqrt{b^2-4 a c}+2 c (a+c)-b^2\right ) \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}+b-2 c\right )}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right )}{\sqrt{b^2-4 a c} \left (a^2+2 a c-b^2+c^2\right ) \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{\sqrt{2} c \left (b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right ) \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}-b+2 c\right )}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right )}{\sqrt{b^2-4 a c} \left (a^2+2 a c-b^2+c^2\right ) \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}+\frac{\tan \left (\frac{x}{2}\right )}{2 (a-b+c)}-\frac{\cot \left (\frac{x}{2}\right )}{2 (a+b+c)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(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))

________________________________________________________________________________________

Maple [B]  time = 0.061, size = 2816, normalized size = 8.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(a-b+c)*tan(1/2*x)-1/2/(a+b+c)/tan(1/2*x)+3/(a+b+c)/(a-b+c)^2*b/(-4*a*c+b^2)^(1/2)/(((-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))*c^3+2/(a+b+c)/(a-b+c)^2/
(-4*a*c+b^2)^(1/2)/(((-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))*b^3*c-3/(a+b+c)/(a-b+c)^2*b/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*ar
ctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*c^3-2/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)
/(((-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))*
a^2*c^2-1/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^2)^(1/2)/(((-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))*b^3-4/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^2)^(1/2)/(((-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))*c^3+2/(a+b+c)/(a-b
+c)^2/(-4*a*c+b^2)^(1/2)/(((-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))*a^2*c^2+1/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^2)^(1/2)/(((-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))*b^3+4/(a+b+c)/(a-b+c)^2*a/(-4*a*c+
b^2)^(1/2)/(((-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))*c^3-2/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)/(((-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))*b^3*c+6/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^2)^(1/2)/(((-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))*b*c^2+1/(a
+b+c)/(a-b+c)^2/(((-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))*c^3+1/(a+b+c)/(a-b+c)^2/(((-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))*c^3+1/(a+b+c)/(a-b+c)^2/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arcta
n((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*b^3+1/(a+b+c)/(a-b+c)^2/(((-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))*b^3-3/(a+b+c)/(a-b+c)^
2/(-4*a*c+b^2)^(1/2)/(((-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))*a^2*b*c+2/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^2)^(1/2)/(((-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))*b^2*c-6/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^
2)^(1/2)/(((-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))*b*c^2+3/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)/(((-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))*a^2*b*c-2/(a+b+c)/(a-b+c)^2*a/(-4*a*c+b^2)^(1/2)/(((-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))*b^2*c-2/
(a+b+c)/(a-b+c)^2/(((-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))*b^2*c-1/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arcta
nh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b^4-2/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)/((
(-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))*c^4
+2/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)/(((-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))*c^4+1/(a+b+c)/(a-b+c)^2/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arct
an((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*a^2*c-1/(a+b+c)/(a-b+c)^2*a/(((-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))*b^2+2/(a+b+c)/(a-b+
c)^2*a/(((-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))*c^2+1/(a+b+c)/(a-b+c)^2/(((-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))*a^2*c-1/(a+b+c)/(a-b+c)^2*a/(((-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))*b^2+2/(a+b+c)/(a-b+c)^2*a/(((-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))*c^2-2/(a+b+c)/(a-b+c)^2
/(((-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))*
b^2*c+1/(a+b+c)/(a-b+c)^2/(-4*a*c+b^2)^(1/2)/(((-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))*b^4

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (x \right )}}{a + b \cos{\left (x \right )} + c \cos ^{2}{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out