3.2 \(\int \frac {\sin ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx\)

Optimal. Leaf size=298 \[ \frac {\sqrt {2} b \left (-\frac {b^2}{\sqrt {b^2-4 a c}}+\frac {3 a c}{\sqrt {b^2-4 a c}}-\frac {a c}{b}+b\right ) \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right ) \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}\right )}{c^2 \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}+\frac {\sqrt {2} b \left (\frac {b^2}{\sqrt {b^2-4 a c}}-\frac {3 a c}{\sqrt {b^2-4 a c}}-\frac {a c}{b}+b\right ) \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}\right )}{c^2 \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {b x}{c^2}-\frac {\cos (x)}{c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 3.74, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3256, 2638, 3292, 2660, 618, 204} \[ \frac {\sqrt {2} b \left (-\frac {b^2}{\sqrt {b^2-4 a c}}+\frac {3 a c}{\sqrt {b^2-4 a c}}-\frac {a c}{b}+b\right ) \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right ) \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}\right )}{c^2 \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}+\frac {\sqrt {2} b \left (\frac {b^2}{\sqrt {b^2-4 a c}}-\frac {3 a c}{\sqrt {b^2-4 a c}}-\frac {a c}{b}+b\right ) \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}\right )}{c^2 \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}-\frac {b x}{c^2}-\frac {\cos (x)}{c} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]

[Out]

-((b*x)/c^2) + (Sqrt[2]*b*(b - (a*c)/b - b^2/Sqrt[b^2 - 4*a*c] + (3*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b -
Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c^2*Sqrt[b^2 - 2*c*(a
+ c) - b*Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*b*(b - (a*c)/b + b^2/Sqrt[b^2 - 4*a*c] - (3*a*c)/Sqrt[b^2 - 4*a*c])*A
rcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c^2*
Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - Cos[x]/c

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 3256

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

Rule 3292

Int[((A_) + (B_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)] + (c_.)*sin[(d_.) + (e_.)*(x
_)]^2), 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*Sin[d + e*x
]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B
}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {\sin ^3(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx &=\int \left (-\frac {b}{c^2}+\frac {\sin (x)}{c}+\frac {a b+b^2 \left (1-\frac {a c}{b^2}\right ) \sin (x)}{c^2 \left (a+b \sin (x)+c \sin ^2(x)\right )}\right ) \, dx\\ &=-\frac {b x}{c^2}+\frac {\int \frac {a b+b^2 \left (1-\frac {a c}{b^2}\right ) \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx}{c^2}+\frac {\int \sin (x) \, dx}{c}\\ &=-\frac {b x}{c^2}-\frac {\cos (x)}{c}+\frac {\left (b^2-a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {3 a b c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b+\sqrt {b^2-4 a c}+2 c \sin (x)} \, dx}{c^2}+\frac {\left (b^2-a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {3 a b c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b-\sqrt {b^2-4 a c}+2 c \sin (x)} \, dx}{c^2}\\ &=-\frac {b x}{c^2}-\frac {\cos (x)}{c}+\frac {\left (2 \left (b^2-a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {3 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}+4 c x+\left (b+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{c^2}+\frac {\left (2 \left (b^2-a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {3 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}+4 c x+\left (b-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{c^2}\\ &=-\frac {b x}{c^2}-\frac {\cos (x)}{c}-\frac {\left (4 \left (b^2-a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {3 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2\right )-x^2} \, dx,x,4 c+2 \left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )\right )}{c^2}-\frac {\left (4 \left (b^2-a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {3 a b c}{\sqrt {b^2-4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-8 \left (b^2-2 c (a+c)-b \sqrt {b^2-4 a c}\right )-x^2} \, dx,x,4 c+2 \left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )\right )}{c^2}\\ &=-\frac {b x}{c^2}+\frac {\sqrt {2} \left (b^2-a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {3 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{c^2 \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^2-a c+\frac {b^3}{\sqrt {b^2-4 a c}}-\frac {3 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{c^2 \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}-\frac {\cos (x)}{c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.96, size = 358, normalized size = 1.20 \[ \frac {\frac {\left (b^2 \sqrt {4 a c-b^2}-a c \sqrt {4 a c-b^2}-3 i a b c+i b^3\right ) \tan ^{-1}\left (\frac {2 c+\tan \left (\frac {x}{2}\right ) \left (b-i \sqrt {4 a c-b^2}\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a+c)+b^2}}\right )}{\sqrt {2 a c-\frac {b^2}{2}} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a+c)+b^2}}+\frac {\left (b^2 \sqrt {4 a c-b^2}-a c \sqrt {4 a c-b^2}+3 i a b c-i b^3\right ) \tan ^{-1}\left (\frac {2 c+\tan \left (\frac {x}{2}\right ) \left (b+i \sqrt {4 a c-b^2}\right )}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a+c)+b^2}}\right )}{\sqrt {2 a c-\frac {b^2}{2}} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a+c)+b^2}}-b x-c \cos (x)}{c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(a + b*Sin[x] + c*Sin[x]^2),x]

[Out]

(-(b*x) + ((I*b^3 - (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sqrt[-b^2 + 4*a*c])*ArcTan[(2*c + (b - I*Sqrt[-
b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*S
qrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]]) + (((-I)*b^3 + (3*I)*a*b*c + b^2*Sqrt[-b^2 + 4*a*c] - a*c*Sqr
t[-b^2 + 4*a*c])*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt
[-b^2 + 4*a*c]])])/(Sqrt[-1/2*b^2 + 2*a*c]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]]) - c*Cos[x])/c^2

________________________________________________________________________________________

fricas [B]  time = 2.27, size = 6531, normalized size = 21.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a+b*sin(x)+c*sin(x)^2),x, algorithm="fricas")

[Out]

-1/4*(sqrt(2)*c^2*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c - (4*a
*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 +
9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^
3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c
^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2*
(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4))*log(12*a^5*b*c^3 + 8*(a^6*b - 2*a^4*b^3)*c^2 + 2*(4*a^4*c^7 + (8
*a^5 - a^3*b^2)*c^6 + 2*(2*a^6 - 3*a^4*b^2)*c^5 - (a^5*b^2 - a^3*b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 +
9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a
^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*
c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*sin(x) - 4*(a^5*b^3 - a^3*b^5)*c
- sqrt(2)*((12*a^2*b*c^9 + 7*(4*a^3*b - a*b^3)*c^8 + (20*a^4*b - 27*a^2*b^3 + b^5)*c^7 + (4*a^5*b - 13*a^3*b^3
+ 9*a*b^5)*c^6 - (a^4*b^3 - 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 + 12*(a^5
*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^6 + 2*a*b^8)*c)/(4*
a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 - 3*a^3*b^
2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*cos(x) + (12*a^4*b*c^5 + (20*a^5*b - 31*a^3*b^3)*c^4 + (8
*a^6*b - 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 - 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 - 2*a^2*b^7 + b^9)*
c)*cos(x))*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c - (4*a*c^7 +
(8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^4*b
^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^6 +
2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10 + 4
*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3
- 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)) - 2*(a^5*b^4 - a^3*b^6 - 3*a^5*b^2*c^2 - 2*(a^6*b^2 - 2*a^4*b^4)*c)*sin
(x)) - sqrt(2)*c^2*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c + (4*
a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 +
9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a
^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*
c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2
*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4))*log(12*a^5*b*c^3 + 8*(a^6*b - 2*a^4*b^3)*c^2 - 2*(4*a^4*c^7 + (
8*a^5 - a^3*b^2)*c^6 + 2*(2*a^6 - 3*a^4*b^2)*c^5 - (a^5*b^2 - a^3*b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10
+ 9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*
a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)
*c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*sin(x) - 4*(a^5*b^3 - a^3*b^5)*c
- sqrt(2)*((12*a^2*b*c^9 + 7*(4*a^3*b - a*b^3)*c^8 + (20*a^4*b - 27*a^2*b^3 + b^5)*c^7 + (4*a^5*b - 13*a^3*b^
3 + 9*a*b^5)*c^6 - (a^4*b^3 - 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 + 12*(a^
5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^6 + 2*a*b^8)*c)/(4
*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 - 3*a^3*b
^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*cos(x) - (12*a^4*b*c^5 + (20*a^5*b - 31*a^3*b^3)*c^4 + (
8*a^6*b - 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 - 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 - 2*a^2*b^7 + b^9)
*c)*cos(x))*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c + (4*a*c^7 +
(8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^4*
b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^6
+ 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10 +
4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3
- 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)) - 2*(a^5*b^4 - a^3*b^6 - 3*a^5*b^2*c^2 - 2*(a^6*b^2 - 2*a^4*b^4)*c)*si
n(x)) + sqrt(2)*c^2*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c + (4
*a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10
+ 9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*
a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)
*c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 +
2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4))*log(-12*a^5*b*c^3 - 8*(a^6*b - 2*a^4*b^3)*c^2 + 2*(4*a^4*c^7 +
(8*a^5 - a^3*b^2)*c^6 + 2*(2*a^6 - 3*a^4*b^2)*c^5 - (a^5*b^2 - a^3*b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^1
0 + 9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 -
3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^
4)*c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*sin(x) + 4*(a^5*b^3 - a^3*b^5)
*c - sqrt(2)*((12*a^2*b*c^9 + 7*(4*a^3*b - a*b^3)*c^8 + (20*a^4*b - 27*a^2*b^3 + b^5)*c^7 + (4*a^5*b - 13*a^3*
b^3 + 9*a*b^5)*c^6 - (a^4*b^3 - 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 + 12*(
a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^6 + 2*a*b^8)*c)/
(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 - 3*a^3
*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*cos(x) - (12*a^4*b*c^5 + (20*a^5*b - 31*a^3*b^3)*c^4 +
(8*a^6*b - 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 - 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 - 2*a^2*b^7 + b^
9)*c)*cos(x))*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c + (4*a*c^7
+ (8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^
4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^
6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10
+ 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a
^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)) + 2*(a^5*b^4 - a^3*b^6 - 3*a^5*b^2*c^2 - 2*(a^6*b^2 - 2*a^4*b^4)*c)*
sin(x)) - sqrt(2)*c^2*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c -
(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^1
0 + 9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 -
3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^
4)*c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6
+ 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4))*log(-12*a^5*b*c^3 - 8*(a^6*b - 2*a^4*b^3)*c^2 - 2*(4*a^4*c^7
+ (8*a^5 - a^3*b^2)*c^6 + 2*(2*a^6 - 3*a^4*b^2)*c^5 - (a^5*b^2 - a^3*b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b
^10 + 9*a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4
- 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 +
b^4)*c^10 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*sin(x) + 4*(a^5*b^3 - a^3*b^
5)*c - sqrt(2)*((12*a^2*b*c^9 + 7*(4*a^3*b - a*b^3)*c^8 + (20*a^4*b - 27*a^2*b^3 + b^5)*c^7 + (4*a^5*b - 13*a^
3*b^3 + 9*a*b^5)*c^6 - (a^4*b^3 - 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 + 12
*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*b^6 + 2*a*b^8)*c
)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 - 3*a
^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8))*cos(x) + (12*a^4*b*c^5 + (20*a^5*b - 31*a^3*b^3)*c^4
+ (8*a^6*b - 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 - 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 - 2*a^2*b^7 +
b^9)*c)*cos(x))*sqrt((a^2*b^4 - b^6 + 2*a^3*c^3 + (2*a^4 - 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 - 3*a*b^4)*c - (4*a*c
^7 + (8*a^2 - b^2)*c^6 + 2*(2*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)*sqrt(-(a^4*b^6 - 2*a^2*b^8 + b^10 + 9*
a^4*b^2*c^4 + 12*(a^5*b^2 - 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 - 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 - 3*a^3*
b^6 + 2*a*b^8)*c)/(4*a*c^13 + (16*a^2 - b^2)*c^12 + 12*(2*a^3 - a*b^2)*c^11 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^1
0 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 + (8*a^2 - b^2)*c^6 + 2*(2
*a^3 - 3*a*b^2)*c^5 - (a^2*b^2 - b^4)*c^4)) + 2*(a^5*b^4 - a^3*b^6 - 3*a^5*b^2*c^2 - 2*(a^6*b^2 - 2*a^4*b^4)*c
)*sin(x)) + 4*b*x + 4*c*cos(x))/c^2

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a+b*sin(x)+c*sin(x)^2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.29, size = 890, normalized size = 2.99 \[ -\frac {8 a^{2} \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 c a +b^{2}}-b}{\sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) \sqrt {-4 c a +b^{2}}}{c \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}+\frac {4 a \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 c a +b^{2}}-b}{\sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}}{c^{2} \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}-\frac {16 a^{2} \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 c a +b^{2}}-b}{\sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) b}{c \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}+\frac {4 a \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 c a +b^{2}}-b}{\sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) b^{3}}{c^{2} \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}+2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}-\frac {8 a^{2} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 c a +b^{2}}}{\sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) \sqrt {-4 c a +b^{2}}}{c \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}+\frac {4 a \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 c a +b^{2}}}{\sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}}{c^{2} \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}+\frac {16 a^{2} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 c a +b^{2}}}{\sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) b}{c \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}-\frac {4 a \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 c a +b^{2}}}{\sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}\right ) b^{3}}{c^{2} \left (8 c a -2 b^{2}\right ) \sqrt {4 c a -2 b^{2}-2 b \sqrt {-4 c a +b^{2}}+4 a^{2}}}-\frac {2}{c \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}-\frac {2 b \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a+b*sin(x)+c*sin(x)^2),x)

[Out]

-8/c*a^2/(8*a*c-2*b^2)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b^2)^(
1/2)-b)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*(-4*a*c+b^2)^(1/2)+4/c^2*a/(8*a*c-2*b^2)/(4*c*a-2*b^
2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b^2)^(1/2)-b)/(4*c*a-2*b^2+2*b*(-4*a*c+b
^2)^(1/2)+4*a^2)^(1/2))*(-4*a*c+b^2)^(1/2)*b^2-16/c*a^2/(8*a*c-2*b^2)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^
2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b^2)^(1/2)-b)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*b+4/c
^2*a/(8*a*c-2*b^2)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((-2*a*tan(1/2*x)+(-4*a*c+b^2)^(1/2)
-b)/(4*c*a-2*b^2+2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*b^3-8/c*a^2/(8*a*c-2*b^2)/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^
(1/2)+4*a^2)^(1/2)*arctan((2*a*tan(1/2*x)+b+(-4*a*c+b^2)^(1/2))/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/
2))*(-4*a*c+b^2)^(1/2)+4/c^2*a/(8*a*c-2*b^2)/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((2*a*tan(
1/2*x)+b+(-4*a*c+b^2)^(1/2))/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*(-4*a*c+b^2)^(1/2)*b^2+16/c*a^2
/(8*a*c-2*b^2)/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2)*arctan((2*a*tan(1/2*x)+b+(-4*a*c+b^2)^(1/2))/(
4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*b-4/c^2*a/(8*a*c-2*b^2)/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4
*a^2)^(1/2)*arctan((2*a*tan(1/2*x)+b+(-4*a*c+b^2)^(1/2))/(4*c*a-2*b^2-2*b*(-4*a*c+b^2)^(1/2)+4*a^2)^(1/2))*b^3
-2/c/(tan(1/2*x)^2+1)-2/c^2*b*arctan(tan(1/2*x))

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a+b*sin(x)+c*sin(x)^2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [B]  time = 25.03, size = 21407, normalized size = 71.84 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a + c*sin(x)^2 + b*sin(x)),x)

[Out]

- 2/(c*(tan(x/2)^2 + 1)) - atan((((8192*(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3*b^5*c + 9*a^5*b^3*c + 20*a^4*b^3*c^2))
/c^4 + ((8192*(4*a*b^7*c^2 - 2*a^2*b^7*c + 2*a^4*b^5*c + 12*a^5*b*c^4 + 8*a^6*b*c^3 - 24*a^2*b^5*c^3 + 32*a^3*
b^3*c^4 + 10*a^3*b^5*c^2 - 10*a^4*b^3*c^3 - 10*a^5*b^3*c^2))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b
^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3
- 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2
*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7
+ 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^7*c^3 - 4*a*b^5*c^5 + 20*
a^4*b*c^6 + 9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))/c^4 + ((b^8 - a^2*
b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) +
33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3
*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b
^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(
3*a*b^5*c^6 + 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b^3*c^6 + 2*a^
4*b^3*c^5))/c^4 + ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a^3*b*c^9 + 12*a^4*b*c^8 - 17*a^2*b^3
*c^8 - 3*a^3*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 + 104*a^4*c^9 + 24*a^5*c^8 - 16*a*b^2*
c^10 + 17*a*b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4*c^6 - 14*a^4
*b^2*c^7))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3
*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a
*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 +
32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a
^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^6*c^5 - 40*a^
2*b^2*c^8 + 28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c^6))/c^4) - (8192*tan(x/2)*(16*a^4*c
^7 + 24*a^5*c^6 + 10*a^6*c^5 + 16*a*b^4*c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a^2*b^2*c^7 + 144*a^2*b^4*c^5 -
18*a^2*b^6*c^3 - 200*a^3*b^2*c^6 + 75*a^3*b^4*c^4 - 2*a^3*b^6*c^2 - 142*a^4*b^2*c^5 + 14*a^4*b^4*c^3 - 27*a^5*
b^2*c^4))/c^4) - (8192*tan(x/2)*(8*a^5*c^5 + 4*a^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^6*c + 40*a^2*b^4*c^4 - 28*a^2*b
^6*c^2 - 32*a^3*b^2*c^5 + 60*a^3*b^4*c^3 - 56*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)
*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2
)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1
/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 1
6*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(
1/2) + (8192*tan(x/2)*(8*a*b^8 - 8*a^3*b^6 + a^5*b^4 + a^7*c^2 - 48*a^2*b^6*c + 32*a^4*b^4*c - 2*a^6*b^2*c + 7
2*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 16*a^5*b^2*c^2))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c
- b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^
2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(
4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c
^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*1i + ((8192*(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3*b^5*c +
9*a^5*b^3*c + 20*a^4*b^3*c^2))/c^4 - ((8192*(4*a*b^7*c^2 - 2*a^2*b^7*c + 2*a^4*b^5*c + 12*a^5*b*c^4 + 8*a^6*b
*c^3 - 24*a^2*b^5*c^3 + 32*a^3*b^3*c^4 + 10*a^3*b^5*c^2 - 10*a^4*b^3*c^3 - 10*a^5*b^3*c^2))/c^4 + ((b^8 - a^2*
b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) +
33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3
*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b
^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*(((b^8 -
a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/
2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*
a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^
6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8
192*(3*a*b^5*c^6 + 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b^3*c^6 +
2*a^4*b^3*c^5))/c^4 - ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a^3*b*c^9 + 12*a^4*b*c^8 - 17*a^
2*b^3*c^8 - 3*a^3*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 + 104*a^4*c^9 + 24*a^5*c^8 - 16*a
*b^2*c^10 + 17*a*b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4*c^6 - 1
4*a^4*b^2*c^7))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^
2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(
-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*
c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4
- 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^6*c^5 -
40*a^2*b^2*c^8 + 28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c^6))/c^4) - (8192*(3*a*b^7*c^3
- 4*a*b^5*c^5 + 20*a^4*b*c^6 + 9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))
/c^4 + (8192*tan(x/2)*(16*a^4*c^7 + 24*a^5*c^6 + 10*a^6*c^5 + 16*a*b^4*c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a
^2*b^2*c^7 + 144*a^2*b^4*c^5 - 18*a^2*b^6*c^3 - 200*a^3*b^2*c^6 + 75*a^3*b^4*c^4 - 2*a^3*b^6*c^2 - 142*a^4*b^2
*c^5 + 14*a^4*b^4*c^3 - 27*a^5*b^2*c^4))/c^4) - (8192*tan(x/2)*(8*a^5*c^5 + 4*a^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^
6*c + 40*a^2*b^4*c^4 - 28*a^2*b^6*c^2 - 32*a^3*b^2*c^5 + 60*a^3*b^4*c^3 - 56*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16
*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3
*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*
a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a
^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(8*a*b^8 - 8*a^3*b^6 + a^5*b^4 + a^7*c^2 - 48*a^2*b^6*c +
32*a^4*b^4*c - 2*a^6*b^2*c + 72*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 16*a^5*b^2*c^2))/c^4)*((b^8 - a^2*b^6 + 8*a^4*
c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c
^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c
- b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6
*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*1i)/(((8192*(4*a^2*b
^7 - 3*a^4*b^5 - 20*a^3*b^5*c + 9*a^5*b^3*c + 20*a^4*b^3*c^2))/c^4 - ((8192*(4*a*b^7*c^2 - 2*a^2*b^7*c + 2*a^4
*b^5*c + 12*a^5*b*c^4 + 8*a^6*b*c^3 - 24*a^2*b^5*c^3 + 32*a^3*b^3*c^4 + 10*a^3*b^5*c^2 - 10*a^4*b^3*c^3 - 10*a
^5*b^3*c^2))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*
b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(
4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^
8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 -
8*a^3*b^2*c^5)))^(1/2)*(((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c -
a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^
2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a
^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c
^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^5*c^6 + 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 -
2*a^2*b^5*c^5 + 9*a^3*b^3*c^6 + 2*a^4*b^3*c^5))/c^4 - ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a
^3*b*c^9 + 12*a^4*b*c^8 - 17*a^2*b^3*c^8 - 3*a^3*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 +
104*a^4*c^9 + 24*a^5*c^8 - 16*a*b^2*c^10 + 17*a*b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*
a^3*b^2*c^8 + 2*a^3*b^4*c^6 - 14*a^4*b^2*c^7))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b
^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c
^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a
*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5
- 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7
+ 8*a*b^4*c^7 - 4*a*b^6*c^5 - 40*a^2*b^2*c^8 + 28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c
^6))/c^4) - (8192*(3*a*b^7*c^3 - 4*a*b^5*c^5 + 20*a^4*b*c^6 + 9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 -
3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))/c^4 + (8192*tan(x/2)*(16*a^4*c^7 + 24*a^5*c^6 + 10*a^6*c^5 + 16*a*b^4*c^6 - 24
*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a^2*b^2*c^7 + 144*a^2*b^4*c^5 - 18*a^2*b^6*c^3 - 200*a^3*b^2*c^6 + 75*a^3*b^4*c^
4 - 2*a^3*b^6*c^2 - 142*a^4*b^2*c^5 + 14*a^4*b^4*c^3 - 27*a^5*b^2*c^4))/c^4) - (8192*tan(x/2)*(8*a^5*c^5 + 4*a
^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^6*c + 40*a^2*b^4*c^4 - 28*a^2*b^6*c^2 - 32*a^3*b^2*c^5 + 60*a^3*b^4*c^3 - 56*a^
4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(
-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 1
8*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3
*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10
*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(8*a*b^8 - 8*a^3*b^6 + a^5
*b^4 + a^7*c^2 - 48*a^2*b^6*c + 32*a^4*b^4*c - 2*a^6*b^2*c + 72*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 16*a^5*b^2*c^2)
)/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c
- b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)
^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c
^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^
5)))^(1/2) - ((8192*(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3*b^5*c + 9*a^5*b^3*c + 20*a^4*b^3*c^2))/c^4 + ((8192*(4*a*b
^7*c^2 - 2*a^2*b^7*c + 2*a^4*b^5*c + 12*a^5*b*c^4 + 8*a^6*b*c^3 - 24*a^2*b^5*c^3 + 32*a^3*b^3*c^4 + 10*a^3*b^5
*c^2 - 10*a^4*b^3*c^3 - 10*a^5*b^3*c^2))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3
)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 -
10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c -
b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*
a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^7*c^3 - 4*a*b^5*c^5 + 20*a^4*b*c^6 + 9*a^5*b*
c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*
a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*
a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3
)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8
*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^5*c^6 + 16*a^3
*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b^3*c^6 + 2*a^4*b^3*c^5))/c^4 + ((
8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a^3*b*c^9 + 12*a^4*b*c^8 - 17*a^2*b^3*c^8 - 3*a^3*b^3*c^7
))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 + 104*a^4*c^9 + 24*a^5*c^8 - 16*a*b^2*c^10 + 17*a*b^4*c^8
- 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4*c^6 - 14*a^4*b^2*c^7))/c^4)*((b^
8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^
(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) -
4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4
*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)
+ (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^6*c^5 - 40*a^2*b^2*c^8 + 28*a^2*b
^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c^6))/c^4) - (8192*tan(x/2)*(16*a^4*c^7 + 24*a^5*c^6 + 10
*a^6*c^5 + 16*a*b^4*c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a^2*b^2*c^7 + 144*a^2*b^4*c^5 - 18*a^2*b^6*c^3 - 200
*a^3*b^2*c^6 + 75*a^3*b^4*c^4 - 2*a^3*b^6*c^2 - 142*a^4*b^2*c^5 + 14*a^4*b^4*c^3 - 27*a^5*b^2*c^4))/c^4) - (81
92*tan(x/2)*(8*a^5*c^5 + 4*a^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^6*c + 40*a^2*b^4*c^4 - 28*a^2*b^6*c^2 - 32*a^3*b^2*
c^5 + 60*a^3*b^4*c^3 - 56*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)*((b^8 - a^2*b^6 + 8
*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*
b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4
*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6
- b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2
)*(8*a*b^8 - 8*a^3*b^6 + a^5*b^4 + a^7*c^2 - 48*a^2*b^6*c + 32*a^4*b^4*c - 2*a^6*b^2*c + 72*a^3*b^4*c^2 - 16*a
^4*b^2*c^3 - 16*a^5*b^2*c^2))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*
a^3*b^4*c - a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c +
3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2
))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6
+ a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (16384*(a^7*b - 4*a^5*b^3))/c^4 + (16384*tan(x/2)*(4*a^6*b^2 - 8*a^4*
b^4 + 8*a^5*b^2*c))/c^4))*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 + b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c
- a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c + 3*a^2*b*
c^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) + 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16
*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4
*c^4 - 8*a^3*b^2*c^5)))^(1/2)*2i - atan((((8192*(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3*b^5*c + 9*a^5*b^3*c + 20*a^4*b
^3*c^2))/c^4 + ((8192*(4*a*b^7*c^2 - 2*a^2*b^7*c + 2*a^4*b^5*c + 12*a^5*b*c^4 + 8*a^6*b*c^3 - 24*a^2*b^5*c^3 +
32*a^3*b^3*c^4 + 10*a^3*b^5*c^2 - 10*a^4*b^3*c^3 - 10*a^5*b^3*c^2))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5
*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3
*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(
1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*
b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^7*c^3 - 4*a*b^5*c
^5 + 20*a^4*b*c^6 + 9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))/c^4 + ((b^
8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^
(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) +
4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4
*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*
((8192*(3*a*b^5*c^6 + 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b^3*c^
6 + 2*a^4*b^3*c^5))/c^4 + ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a^3*b*c^9 + 12*a^4*b*c^8 - 17
*a^2*b^3*c^8 - 3*a^3*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 + 104*a^4*c^9 + 24*a^5*c^8 - 1
6*a*b^2*c^10 + 17*a*b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4*c^6
- 14*a^4*b^2*c^7))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c +
a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^
2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a
^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c
^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^6*c^5
- 40*a^2*b^2*c^8 + 28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c^6))/c^4) - (8192*tan(x/2)*(
16*a^4*c^7 + 24*a^5*c^6 + 10*a^6*c^5 + 16*a*b^4*c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a^2*b^2*c^7 + 144*a^2*b^
4*c^5 - 18*a^2*b^6*c^3 - 200*a^3*b^2*c^6 + 75*a^3*b^4*c^4 - 2*a^3*b^6*c^2 - 142*a^4*b^2*c^5 + 14*a^4*b^4*c^3 -
27*a^5*b^2*c^4))/c^4) - (8192*tan(x/2)*(8*a^5*c^5 + 4*a^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^6*c + 40*a^2*b^4*c^4 -
28*a^2*b^6*c^2 - 32*a^3*b^2*c^5 + 60*a^3*b^4*c^3 - 56*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*
c))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a
*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^
2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3
*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*
c^5)))^(1/2) + (8192*tan(x/2)*(8*a*b^8 - 8*a^3*b^6 + a^5*b^4 + a^7*c^2 - 48*a^2*b^6*c + 32*a^4*b^4*c - 2*a^6*b
^2*c + 72*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 16*a^5*b^2*c^2))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(
-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 1
8*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3
*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10
*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*1i + ((8192*(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3
*b^5*c + 9*a^5*b^3*c + 20*a^4*b^3*c^2))/c^4 - ((8192*(4*a*b^7*c^2 - 2*a^2*b^7*c + 2*a^4*b^5*c + 12*a^5*b*c^4 +
8*a^6*b*c^3 - 24*a^2*b^5*c^3 + 32*a^3*b^3*c^4 + 10*a^3*b^5*c^2 - 10*a^4*b^3*c^3 - 10*a^5*b^3*c^2))/c^4 + ((b^
8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^
(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) +
4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4
*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*
(((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2
)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1
/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 1
6*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(
1/2)*((8192*(3*a*b^5*c^6 + 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b
^3*c^6 + 2*a^4*b^3*c^5))/c^4 - ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a^3*b*c^9 + 12*a^4*b*c^8
- 17*a^2*b^3*c^8 - 3*a^3*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 + 104*a^4*c^9 + 24*a^5*c^
8 - 16*a*b^2*c^10 + 17*a*b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4
*c^6 - 14*a^4*b^2*c^7))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^
4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2
*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*
(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*
b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^
6*c^5 - 40*a^2*b^2*c^8 + 28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c^6))/c^4) - (8192*(3*a*
b^7*c^3 - 4*a*b^5*c^5 + 20*a^4*b*c^6 + 9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b
^3*c^4))/c^4 + (8192*tan(x/2)*(16*a^4*c^7 + 24*a^5*c^6 + 10*a^6*c^5 + 16*a*b^4*c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^
2 - 64*a^2*b^2*c^7 + 144*a^2*b^4*c^5 - 18*a^2*b^6*c^3 - 200*a^3*b^2*c^6 + 75*a^3*b^4*c^4 - 2*a^3*b^6*c^2 - 142
*a^4*b^2*c^5 + 14*a^4*b^4*c^3 - 27*a^5*b^2*c^4))/c^4) - (8192*tan(x/2)*(8*a^5*c^5 + 4*a^6*c^4 - 8*a*b^6*c^3 -
4*a^3*b^6*c + 40*a^2*b^4*c^4 - 28*a^2*b^6*c^2 - 32*a^3*b^2*c^5 + 60*a^3*b^4*c^3 - 56*a^4*b^2*c^4 + 20*a^4*b^4*
c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2)
+ 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^
6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)
^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2
*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(8*a*b^8 - 8*a^3*b^6 + a^5*b^4 + a^7*c^2 - 48*a^2
*b^6*c + 32*a^4*b^4*c - 2*a^6*b^2*c + 72*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 16*a^5*b^2*c^2))/c^4)*((b^8 - a^2*b^6
+ 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a
^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(
-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c
^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*1i)/(((8192*
(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3*b^5*c + 9*a^5*b^3*c + 20*a^4*b^3*c^2))/c^4 - ((8192*(4*a*b^7*c^2 - 2*a^2*b^7*c
+ 2*a^4*b^5*c + 12*a^5*b*c^4 + 8*a^6*b*c^3 - 24*a^2*b^5*c^3 + 32*a^3*b^3*c^4 + 10*a^3*b^5*c^2 - 10*a^4*b^3*c^
3 - 10*a^5*b^3*c^2))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*
c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b
*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(1
6*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^
4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*(((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3
*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*
a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/
(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a
^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^5*c^6 + 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^
3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b^3*c^6 + 2*a^4*b^3*c^5))/c^4 - ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^1
0 + 20*a^3*b*c^9 + 12*a^4*b*c^8 - 17*a^2*b^3*c^8 - 3*a^3*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3
*c^10 + 104*a^4*c^9 + 24*a^5*c^8 - 16*a*b^2*c^10 + 17*a*b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c
^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4*c^6 - 14*a^4*b^2*c^7))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4
*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a
^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*
c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*
b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16
*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^6*c^5 - 40*a^2*b^2*c^8 + 28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a
^4*b^2*c^6))/c^4) - (8192*(3*a*b^7*c^3 - 4*a*b^5*c^5 + 20*a^4*b*c^6 + 9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^
5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))/c^4 + (8192*tan(x/2)*(16*a^4*c^7 + 24*a^5*c^6 + 10*a^6*c^5 + 16*a*b^4*
c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a^2*b^2*c^7 + 144*a^2*b^4*c^5 - 18*a^2*b^6*c^3 - 200*a^3*b^2*c^6 + 75*a^
3*b^4*c^4 - 2*a^3*b^6*c^2 - 142*a^4*b^2*c^5 + 14*a^4*b^4*c^3 - 27*a^5*b^2*c^4))/c^4) - (8192*tan(x/2)*(8*a^5*c
^5 + 4*a^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^6*c + 40*a^2*b^4*c^4 - 28*a^2*b^6*c^2 - 32*a^3*b^2*c^5 + 60*a^3*b^4*c^3
- 56*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3
- b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2
*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2)
- 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*
c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192*tan(x/2)*(8*a*b^8 - 8*a^3*b
^6 + a^5*b^4 + a^7*c^2 - 48*a^2*b^6*c + 32*a^4*b^4*c - 2*a^6*b^2*c + 72*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 16*a^5*
b^2*c^2))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*
(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*
c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 +
32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^
3*b^2*c^5)))^(1/2) - ((8192*(4*a^2*b^7 - 3*a^4*b^5 - 20*a^3*b^5*c + 9*a^5*b^3*c + 20*a^4*b^3*c^2))/c^4 + ((819
2*(4*a*b^7*c^2 - 2*a^2*b^7*c + 2*a^4*b^5*c + 12*a^5*b*c^4 + 8*a^6*b*c^3 - 24*a^2*b^5*c^3 + 32*a^3*b^3*c^4 + 10
*a^3*b^5*c^2 - 10*a^4*b^3*c^3 - 10*a^5*b^3*c^2))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c
- b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^
2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(
4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c
^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^7*c^3 - 4*a*b^5*c^5 + 20*a^4*b*c^6 +
9*a^5*b*c^5 + 16*a^2*b^3*c^6 - 13*a^2*b^5*c^4 - 3*a^3*b^5*c^3 + 9*a^4*b^3*c^4))/c^4 + ((b^8 - a^2*b^6 + 8*a^4*
c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c
^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c
- b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6
*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*((8192*(3*a*b^5*c^6
+ 16*a^3*b*c^8 - 4*a^4*b*c^7 - 8*a^5*b*c^6 - 16*a^2*b^3*c^7 - 2*a^2*b^5*c^5 + 9*a^3*b^3*c^6 + 2*a^4*b^3*c^5))/
c^4 + ((8192*(3*a*b^5*c^7 - 4*a*b^3*c^9 + 16*a^2*b*c^10 + 20*a^3*b*c^9 + 12*a^4*b*c^8 - 17*a^2*b^3*c^8 - 3*a^3
*b^3*c^7))/c^4 + (8192*tan(x/2)*(64*a^2*c^11 + 144*a^3*c^10 + 104*a^4*c^9 + 24*a^5*c^8 - 16*a*b^2*c^10 + 17*a*
b^4*c^8 - 2*a*b^6*c^6 - 104*a^2*b^2*c^9 + 18*a^2*b^4*c^7 - 66*a^3*b^2*c^8 + 2*a^3*b^4*c^6 - 14*a^4*b^2*c^7))/c
^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c -
b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)
^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7
+ 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5))
)^(1/2) + (8192*tan(x/2)*(32*a^3*c^9 + 48*a^4*c^8 + 16*a^5*c^7 + 8*a*b^4*c^7 - 4*a*b^6*c^5 - 40*a^2*b^2*c^8 +
28*a^2*b^4*c^6 - 60*a^3*b^2*c^7 + 4*a^3*b^4*c^5 - 20*a^4*b^2*c^6))/c^4) - (8192*tan(x/2)*(16*a^4*c^7 + 24*a^5*
c^6 + 10*a^6*c^5 + 16*a*b^4*c^6 - 24*a*b^6*c^4 + 2*a*b^8*c^2 - 64*a^2*b^2*c^7 + 144*a^2*b^4*c^5 - 18*a^2*b^6*c
^3 - 200*a^3*b^2*c^6 + 75*a^3*b^4*c^4 - 2*a^3*b^6*c^2 - 142*a^4*b^2*c^5 + 14*a^4*b^4*c^3 - 27*a^5*b^2*c^4))/c^
4) - (8192*tan(x/2)*(8*a^5*c^5 + 4*a^6*c^4 - 8*a*b^6*c^3 - 4*a^3*b^6*c + 40*a^2*b^4*c^4 - 28*a^2*b^6*c^2 - 32*
a^3*b^2*c^5 + 60*a^3*b^4*c^3 - 56*a^4*b^2*c^4 + 20*a^4*b^4*c^2 - 16*a^5*b^2*c^3 + 4*a*b^8*c))/c^4)*((b^8 - a^2
*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) +
33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^
3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 +
b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (8192
*tan(x/2)*(8*a*b^8 - 8*a^3*b^6 + a^5*b^4 + a^7*c^2 - 48*a^2*b^6*c + 32*a^4*b^4*c - 2*a^6*b^2*c + 72*a^3*b^4*c^
2 - 16*a^4*b^2*c^3 - 16*a^5*b^2*c^2))/c^4)*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1
/2) + 8*a^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a
*b^6*c - 3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)
^3)^(1/2))/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*
b^2*c^6 + a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2) + (16384*(a^7*b - 4*a^5*b^3))/c^4 + (16384*tan(x/2)*(4*a^6*b^2
- 8*a^4*b^4 + 8*a^5*b^2*c))/c^4))*((b^8 - a^2*b^6 + 8*a^4*c^4 + 8*a^5*c^3 - b^5*(-(4*a*c - b^2)^3)^(1/2) + 8*a
^3*b^4*c + a^2*b^3*(-(4*a*c - b^2)^3)^(1/2) + 33*a^2*b^4*c^2 - 38*a^3*b^2*c^3 - 18*a^4*b^2*c^2 - 10*a*b^6*c -
3*a^2*b*c^2*(-(4*a*c - b^2)^3)^(1/2) + 4*a*b^3*c*(-(4*a*c - b^2)^3)^(1/2) - 2*a^3*b*c*(-(4*a*c - b^2)^3)^(1/2)
)/(2*(16*a^2*c^8 + 32*a^3*c^7 + 16*a^4*c^6 + b^4*c^6 - b^6*c^4 - 8*a*b^2*c^7 + 10*a*b^4*c^5 - 32*a^2*b^2*c^6 +
a^2*b^4*c^4 - 8*a^3*b^2*c^5)))^(1/2)*2i - (2*b*atan((16384*a*b^9*tan(x/2))/(16384*a*b^9 + 16384*a^3*b^7 - 327
68*a^5*b^5 - 131072*a^2*b^7*c - 98304*a^4*b^5*c + 131072*a^6*b^3*c + 16384*a^7*b*c^2 + 262144*a^3*b^5*c^2 + 13
1072*a^5*b^3*c^2) + (16384*a^7*b*tan(x/2))/(16384*a^7*b + 262144*a^3*b^5 + 131072*a^5*b^3 + (16384*a*b^9)/c^2
- (131072*a^2*b^7)/c - (98304*a^4*b^5)/c + (131072*a^6*b^3)/c + (16384*a^3*b^7)/c^2 - (32768*a^5*b^5)/c^2) - (
131072*a^2*b^7*tan(x/2))/(131072*a^6*b^3 - 98304*a^4*b^5 - 131072*a^2*b^7 + 262144*a^3*b^5*c + 131072*a^5*b^3*
c + (16384*a*b^9)/c + (16384*a^3*b^7)/c - (32768*a^5*b^5)/c + 16384*a^7*b*c) - (98304*a^4*b^5*tan(x/2))/(13107
2*a^6*b^3 - 98304*a^4*b^5 - 131072*a^2*b^7 + 262144*a^3*b^5*c + 131072*a^5*b^3*c + (16384*a*b^9)/c + (16384*a^
3*b^7)/c - (32768*a^5*b^5)/c + 16384*a^7*b*c) + (131072*a^6*b^3*tan(x/2))/(131072*a^6*b^3 - 98304*a^4*b^5 - 13
1072*a^2*b^7 + 262144*a^3*b^5*c + 131072*a^5*b^3*c + (16384*a*b^9)/c + (16384*a^3*b^7)/c - (32768*a^5*b^5)/c +
16384*a^7*b*c) + (16384*a^3*b^7*tan(x/2))/(16384*a*b^9 + 16384*a^3*b^7 - 32768*a^5*b^5 - 131072*a^2*b^7*c - 9
8304*a^4*b^5*c + 131072*a^6*b^3*c + 16384*a^7*b*c^2 + 262144*a^3*b^5*c^2 + 131072*a^5*b^3*c^2) - (32768*a^5*b^
5*tan(x/2))/(16384*a*b^9 + 16384*a^3*b^7 - 32768*a^5*b^5 - 131072*a^2*b^7*c - 98304*a^4*b^5*c + 131072*a^6*b^3
*c + 16384*a^7*b*c^2 + 262144*a^3*b^5*c^2 + 131072*a^5*b^3*c^2) + (262144*a^3*b^5*tan(x/2))/(16384*a^7*b + 262
144*a^3*b^5 + 131072*a^5*b^3 + (16384*a*b^9)/c^2 - (131072*a^2*b^7)/c - (98304*a^4*b^5)/c + (131072*a^6*b^3)/c
+ (16384*a^3*b^7)/c^2 - (32768*a^5*b^5)/c^2) + (131072*a^5*b^3*tan(x/2))/(16384*a^7*b + 262144*a^3*b^5 + 1310
72*a^5*b^3 + (16384*a*b^9)/c^2 - (131072*a^2*b^7)/c - (98304*a^4*b^5)/c + (131072*a^6*b^3)/c + (16384*a^3*b^7)
/c^2 - (32768*a^5*b^5)/c^2)))/c^2

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/(a+b*sin(x)+c*sin(x)**2),x)

[Out]

Timed out

________________________________________________________________________________________