3.1.6 \(\int \frac {\sin ^4(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\) [6]

Optimal. Leaf size=388 \[ \frac {x}{2 c}+\frac {\left (b^2-c (a+2 c)\right ) x}{c^3}+\frac {2 \left (b^2 \left (b^2-2 c (a+c)\right )-b \sqrt {b^2-4 a c} \left (b^2-2 c (a+c)\right )-2 c \left (a b^2-c (a+c)^2\right )\right ) \text {ArcTan}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{c^3 \sqrt {b^2-4 a c} \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {2 \left (b^4+2 c^2 (a+c)^2-2 b^2 c (2 a+c)+b^3 \sqrt {b^2-4 a c}-2 b c (a+c) \sqrt {b^2-4 a c}\right ) \text {ArcTan}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{c^3 \sqrt {b^2-4 a c} \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}-\frac {b \sin (x)}{c^2}+\frac {\cos (x) \sin (x)}{2 c} \]

[Out]

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

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Rubi [A]
time = 9.71, antiderivative size = 386, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3348, 2717, 2715, 8, 3374, 2738, 211} \begin {gather*} \frac {2 \left (-2 b^2 c (a+c)-b \sqrt {b^2-4 a c} \left (b^2-2 c (a+c)\right )-2 c \left (a b^2-c (a+c)^2\right )+b^4\right ) \text {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 )}{c^3 \sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {2 \left (-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}+2 c^2 (a+c)^2+b^4\right ) \text {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 )}{c^3 \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}+\frac {x \left (b^2-c (a+2 c)\right )}{c^3}-\frac {b \sin (x)}{c^2}+\frac {x}{2 c}+\frac {\sin (x) \cos (x)}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[x]^4/(a + b*Cos[x] + c*Cos[x]^2),x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 211

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2738

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 3348

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 3374

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]

Rubi steps

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

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Mathematica [A]
time = 0.92, size = 374, normalized size = 0.96 \begin {gather*} \frac {4 b^2 x-2 c (2 a+3 c) x+\frac {4 \sqrt {2} \left (b^4+2 c^2 (a+c)^2-2 b^2 c (2 a+c)+b^3 \sqrt {b^2-4 a c}-2 b c (a+c) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\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} \sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}-\frac {4 \sqrt {2} \left (-b^4-2 c^2 (a+c)^2+2 b^2 c (2 a+c)+b^3 \sqrt {b^2-4 a c}-2 b c (a+c) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\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} \sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}-4 b c \sin (x)+c^2 \sin (2 x)}{4 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^4/(a + b*Cos[x] + c*Cos[x]^2),x]

[Out]

(4*b^2*x - 2*c*(2*a + 3*c)*x + (4*Sqrt[2]*(b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] -
2*b*c*(a + c)*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]*Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (4*Sqrt[2]*(-b^
4 - 2*c^2*(a + c)^2 + 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*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]
*Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - 4*b*c*Sin[x] + c^2*Sin[2*x])/(4*c^3)

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Maple [A]
time = 2.92, size = 406, normalized size = 1.05

method	result	size
default	\(-\frac {2 \left (\frac {\left (b c +\frac {1}{2} c^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (b c -\frac {1}{2} c^{2}\right ) \tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}+\frac {\left (2 a c -2 b^{2}+3 c^{2}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{2}\right )}{c^{3}}+\frac {2 \left (a -b +c \right ) \left (\frac {\left (\sqrt {-4 a c +b^{2}}\, a c -\sqrt {-4 a c +b^{2}}\, b^{2}-\sqrt {-4 a c +b^{2}}\, b c +\sqrt {-4 a c +b^{2}}\, c^{2}-3 c a b -2 a \,c^{2}+b^{3}+b^{2} c -c^{2} b -2 c^{3}\right ) \arctan \left (\frac {\left (a -b +c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}+\frac {\left (\sqrt {-4 a c +b^{2}}\, a c -\sqrt {-4 a c +b^{2}}\, b^{2}-\sqrt {-4 a c +b^{2}}\, b c +\sqrt {-4 a c +b^{2}}\, c^{2}+3 c a b +2 a \,c^{2}-b^{3}-b^{2} c +c^{2} b +2 c^{3}\right ) \arctanh \left (\frac {\left (-a +b -c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{c^{3}}\)	\(406\)
risch	\(\text {Expression too large to display}\)	\(5253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a+b*cos(x)+c*cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

-2/c^3*(((b*c+1/2*c^2)*tan(1/2*x)^3+(b*c-1/2*c^2)*tan(1/2*x))/(1+tan(1/2*x)^2)^2+1/2*(2*a*c-2*b^2+3*c^2)*arcta
n(tan(1/2*x)))+2/c^3*(a-b+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-c^2*b-2*c^3)/(-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))+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+c^2*b+2*c^3)/(-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)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+b*cos(x)+c*cos(x)^2),x, algorithm="maxima")

[Out]

1/4*(4*c^3*integrate(-2*(2*(b^4 - 2*a*b^2*c - 2*b^2*c^2)*cos(3*x)^2 + 4*(2*a^2*b^2 - 5*a^2*c^2 - 4*a*c^3 - c^4
- (2*a^3 - a*b^2)*c)*cos(2*x)^2 + 2*(b^4 - 2*a*b^2*c - 2*b^2*c^2)*cos(x)^2 + 2*(b^4 - 2*a*b^2*c - 2*b^2*c^2)*
sin(3*x)^2 + 4*(2*a^2*b^2 - 5*a^2*c^2 - 4*a*c^3 - c^4 - (2*a^3 - a*b^2)*c)*sin(2*x)^2 + 2*(4*a*b^3 - 10*a*b*c^
2 - 4*b*c^3 - (6*a^2*b - b^3)*c)*sin(2*x)*sin(x) + 2*(b^4 - 2*a*b^2*c - 2*b^2*c^2)*sin(x)^2 + ((b^3*c - 2*a*b*
c^2 - 2*b*c^3)*cos(3*x) + 2*(a*b^2*c - a^2*c^2 - 2*a*c^3 - c^4)*cos(2*x) + (b^3*c - 2*a*b*c^2 - 2*b*c^3)*cos(x
))*cos(4*x) + (b^3*c - 2*a*b*c^2 - 2*b*c^3 + 2*(4*a*b^3 - 10*a*b*c^2 - 4*b*c^3 - (6*a^2*b - b^3)*c)*cos(2*x) +
4*(b^4 - 2*a*b^2*c - 2*b^2*c^2)*cos(x))*cos(3*x) + 2*(a*b^2*c - a^2*c^2 - 2*a*c^3 - c^4 + (4*a*b^3 - 10*a*b*c
^2 - 4*b*c^3 - (6*a^2*b - b^3)*c)*cos(x))*cos(2*x) + (b^3*c - 2*a*b*c^2 - 2*b*c^3)*cos(x) + ((b^3*c - 2*a*b*c^
2 - 2*b*c^3)*sin(3*x) + 2*(a*b^2*c - a^2*c^2 - 2*a*c^3 - c^4)*sin(2*x) + (b^3*c - 2*a*b*c^2 - 2*b*c^3)*sin(x))
*sin(4*x) + 2*((4*a*b^3 - 10*a*b*c^2 - 4*b*c^3 - (6*a^2*b - b^3)*c)*sin(2*x) + 2*(b^4 - 2*a*b^2*c - 2*b^2*c^2)
*sin(x))*sin(3*x))/(c^5*cos(4*x)^2 + 4*b^2*c^3*cos(3*x)^2 + 4*b^2*c^3*cos(x)^2 + c^5*sin(4*x)^2 + 4*b^2*c^3*si
n(3*x)^2 + 4*b^2*c^3*sin(x)^2 + 4*b*c^4*cos(x) + c^5 + 4*(4*a^2*c^3 + 4*a*c^4 + c^5)*cos(2*x)^2 + 4*(4*a^2*c^3
+ 4*a*c^4 + c^5)*sin(2*x)^2 + 8*(2*a*b*c^3 + b*c^4)*sin(2*x)*sin(x) + 2*(2*b*c^4*cos(3*x) + 2*b*c^4*cos(x) +
c^5 + 2*(2*a*c^4 + c^5)*cos(2*x))*cos(4*x) + 4*(2*b^2*c^3*cos(x) + b*c^4 + 2*(2*a*b*c^3 + b*c^4)*cos(2*x))*cos
(3*x) + 4*(2*a*c^4 + c^5 + 2*(2*a*b*c^3 + b*c^4)*cos(x))*cos(2*x) + 4*(b*c^4*sin(3*x) + b*c^4*sin(x) + (2*a*c^
4 + c^5)*sin(2*x))*sin(4*x) + 8*(b^2*c^3*sin(x) + (2*a*b*c^3 + b*c^4)*sin(2*x))*sin(3*x)), x) + c^2*sin(2*x) -
4*b*c*sin(x) + 2*(2*b^2 - 2*a*c - 3*c^2)*x)/c^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5045 vs. \(2 (323) = 646\).
time = 2.42, size = 5045, normalized size = 13.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+b*cos(x)+c*cos(x)^2),x, algorithm="fricas")

[Out]

1/4*(sqrt(2)*c^3*sqrt(-(b^6 - 6*a*b^4*c - 6*a*c^5 - 2*c^6 - 3*(2*a^2 - b^2)*c^4 - 2*(a^3 - 6*a*b^2)*c^3 + 3*(3
*a^2*b^2 - b^4)*c^2 + (b^2*c^6 - 4*a*c^7)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 -
b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^
3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13)))/(b^2*c^6 - 4*a*c^7))*log(24*a*b*c^6 + 6*b*c^7 + 12*(3*
a^2*b - b^3)*c^5 + 8*(3*a^3*b - 4*a*b^3)*c^4 + 2*(3*a^4*b - 14*a^2*b^3 + 4*b^5)*c^3 - 4*(2*a^3*b^3 - 3*a*b^5)*
c^2 - (4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6)*sqrt((b^10 - 8*a*b^8*c + 3
6*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4
+ 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13))*cos(x) + 2*(a
^2*b^5 - b^7)*c + 1/2*sqrt(2)*((b^4*c^7 - 6*a*b^2*c^8 + 8*a*c^10 + 2*(4*a^2 - b^2)*c^9)*sqrt((b^10 - 8*a*b^8*c
+ 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*
b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13))*sin(x) -
(b^8*c - 8*a*b^6*c^2 - 12*a*b^2*c^6 - 3*(8*a^2*b^2 - b^4)*c^5 - 6*(2*a^3*b^2 - 3*a*b^4)*c^4 + (19*a^2*b^4 - 3*
b^6)*c^3)*sin(x))*sqrt(-(b^6 - 6*a*b^4*c - 6*a*c^5 - 2*c^6 - 3*(2*a^2 - b^2)*c^4 - 2*(a^3 - 6*a*b^2)*c^3 + 3*(
3*a^2*b^2 - b^4)*c^2 + (b^2*c^6 - 4*a*c^7)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 -
b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c
^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13)))/(b^2*c^6 - 4*a*c^7)) + (a^2*b^6 - b^8 + 12*a*b^2*c^5
+ 3*b^2*c^6 + 6*(3*a^2*b^2 - b^4)*c^4 + 4*(3*a^3*b^2 - 4*a*b^4)*c^3 + (3*a^4*b^2 - 14*a^2*b^4 + 4*b^6)*c^2 - 2
*(2*a^3*b^4 - 3*a*b^6)*c)*cos(x)) - sqrt(2)*c^3*sqrt(-(b^6 - 6*a*b^4*c - 6*a*c^5 - 2*c^6 - 3*(2*a^2 - b^2)*c^4
- 2*(a^3 - 6*a*b^2)*c^3 + 3*(3*a^2*b^2 - b^4)*c^2 + (b^2*c^6 - 4*a*c^7)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7
+ 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^
4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13)))/(b^2*c^6 - 4*a*c^7))*lo
g(24*a*b*c^6 + 6*b*c^7 + 12*(3*a^2*b - b^3)*c^5 + 8*(3*a^3*b - 4*a*b^3)*c^4 + 2*(3*a^4*b - 14*a^2*b^3 + 4*b^5)
*c^3 - 4*(2*a^3*b^3 - 3*a*b^5)*c^2 - (4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3 - 3*a*b^2)*c^7 - (a^2*b^2 - b^4)*
c^6)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c
^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*
c^12 - 4*a*c^13))*cos(x) + 2*(a^2*b^5 - b^7)*c - 1/2*sqrt(2)*((b^4*c^7 - 6*a*b^2*c^8 + 8*a*c^10 + 2*(4*a^2 - b
^2)*c^9)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^
4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(
b^2*c^12 - 4*a*c^13))*sin(x) - (b^8*c - 8*a*b^6*c^2 - 12*a*b^2*c^6 - 3*(8*a^2*b^2 - b^4)*c^5 - 6*(2*a^3*b^2 -
3*a*b^4)*c^4 + (19*a^2*b^4 - 3*b^6)*c^3)*sin(x))*sqrt(-(b^6 - 6*a*b^4*c - 6*a*c^5 - 2*c^6 - 3*(2*a^2 - b^2)*c^
4 - 2*(a^3 - 6*a*b^2)*c^3 + 3*(3*a^2*b^2 - b^4)*c^2 + (b^2*c^6 - 4*a*c^7)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^
7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c
^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13)))/(b^2*c^6 - 4*a*c^7)) +
(a^2*b^6 - b^8 + 12*a*b^2*c^5 + 3*b^2*c^6 + 6*(3*a^2*b^2 - b^4)*c^4 + 4*(3*a^3*b^2 - 4*a*b^4)*c^3 + (3*a^4*b^
2 - 14*a^2*b^4 + 4*b^6)*c^2 - 2*(2*a^3*b^4 - 3*a*b^6)*c)*cos(x)) + sqrt(2)*c^3*sqrt(-(b^6 - 6*a*b^4*c - 6*a*c^
5 - 2*c^6 - 3*(2*a^2 - b^2)*c^4 - 2*(a^3 - 6*a*b^2)*c^3 + 3*(3*a^2*b^2 - b^4)*c^2 - (b^2*c^6 - 4*a*c^7)*sqrt((
b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a
^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*
c^13)))/(b^2*c^6 - 4*a*c^7))*log(-24*a*b*c^6 - 6*b*c^7 - 12*(3*a^2*b - b^3)*c^5 - 8*(3*a^3*b - 4*a*b^3)*c^4 -
2*(3*a^4*b - 14*a^2*b^3 + 4*b^5)*c^3 + 4*(2*a^3*b^3 - 3*a*b^5)*c^2 - (4*a*c^9 + (8*a^2 - b^2)*c^8 + 2*(2*a^3 -
3*a*b^2)*c^7 - (a^2*b^2 - b^4)*c^6)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b^4)*
c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3 + 2
*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*c^13))*cos(x) - 2*(a^2*b^5 - b^7)*c + 1/2*sqrt(2)*((b^4*c^7 - 6*a*b
^2*c^8 + 8*a*c^10 + 2*(4*a^2 - b^2)*c^9)*sqrt((b^10 - 8*a*b^8*c + 36*a*b^2*c^7 + 9*b^2*c^8 + 18*(3*a^2*b^2 - b
^4)*c^6 + 12*(3*a^3*b^2 - 5*a*b^4)*c^5 + 3*(3*a^4*b^2 - 22*a^2*b^4 + 5*b^6)*c^4 - 12*(2*a^3*b^4 - 3*a*b^6)*c^3
+ 2*(11*a^2*b^6 - 3*b^8)*c^2)/(b^2*c^12 - 4*a*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**4/(a+b*cos(x)+c*cos(x)**2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11373 vs. \(2 (323) = 646\).
time = 2.95, size = 11373, normalized size = 29.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^4/(a+b*cos(x)+c*cos(x)^2),x, algorithm="giac")

[Out]

((2*a^2*b^6 - 4*a*b^7 + 2*b^8 - 18*a^3*b^4*c + 38*a^2*b^5*c - 18*a*b^6*c - 2*b^7*c + 48*a^4*b^2*c^2 - 112*a^3*
b^3*c^2 + 42*a^2*b^4*c^2 + 28*a*b^5*c^2 - 4*b^6*c^2 - 32*a^5*c^3 + 96*a^4*b*c^3 + 16*a^3*b^2*c^3 - 128*a^2*b^3
*c^3 + 26*a*b^4*c^3 + 6*b^5*c^3 - 96*a^4*c^4 + 192*a^3*b*c^4 - 16*a^2*b^2*c^4 - 48*a*b^3*c^4 - 2*b^4*c^4 - 96*
a^3*c^5 + 96*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 + 10*(b^2 - 4*a*c)*a^3*b^2*c + 13*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 - 22*(b^2 - 4*a*c)*a^2*b^3*c + 37*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 + 10*(b^2 - 4*a*c)*a*b^4*c + 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 + 2*(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))*sqrt(b^2 - 4*a*c)*a^4*c^2 - 8*(b^2 - 4*a*c)*a^4*c^2 - 20*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 + 24*(b^2 - 4*a*c)*a^3*b*c^2 -
85*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^2*c^2 - 2*(b^2 - 4*a*c)
*a^2*b^2*c^2 + 6*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^3*c^2 - 20*
(b^2 - 4*a*c)*a*b^3*c^2 + 6*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^4*
c^2 + 4*(b^2 - 4*a*c)*b^4*c^2 + 68*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*c^3 - 24*(b^2 - 4*a*c)*a^3*c^3 - 40*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*c^3 + 48*(b^2 - 4*a*c)*a^2*b*c^3 - 33*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^2*c^3 - 10*(b^2 - 4*a*c)*a*b^2*c^3 - 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^3*c^3 - 6*(b^2 - 4*a*c)*b^3*c^3 + 36*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*c^4 - 24*(b^2 - 4*a*c)*a^2*c^4 + 44*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*c^4 + 24*(b^2 - 4*a*c)*a*b*c^4 + 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^2*c^4 + 2*(b^2 - 4*a*c)*b^2*c^4 - 20*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*c^5 - 8*(b^2 - 4*a*c)*a*c^5)*c^2*abs(a - b +
c) + (4*a^2*b^6*c - 4*b^8*c - 36*a^3*b^4*c^2 - 4*a^2*b^5*c^2 + 44*a*b^6*c^2 + 4*b^7*c^2 + 96*a^4*b^2*c^3 + 32
*a^3*b^3*c^3 - 172*a^2*b^4*c^3 - 40*a*b^5*c^3 + 8*b^6*c^3 - 64*a^5*c^4 - 64*a^4*b*c^4 + 288*a^3*b^2*c^4 + 128*
a^2*b^3*c^4 - 76*a*b^4*c^4 - 4*b^5*c^4 - 192*a^4*c^5 - 128*a^3*b*c^5 + 224*a^2*b^2*c^5 + 32*a*b^3*c^5 - 4*b^4*
c^5 - 192*a^3*c^6 - 64*a^2*b*c^6 + 32*a*b^2*c^6 - 64*a^2*c^7 - 3*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c
)*(a - b + c))*a^3*b^4*c - sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b^5*c + 7*sqrt(a^2
- a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^6*c + 5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*
(a - b + c))*b^7*c + 15*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*b^2*c^2 + 8*sqrt(a^2 -
a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^3*c^2 - 51*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a
*c)*(a - b + c))*a^2*b^4*c^2 - 50*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^5*c^2 - 6*sq
rt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^6*c^2 - 12*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 -
4*a*c)*(a - b + c))*a^5*c^3 - 16*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*b*c^3 + 112*
sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^2*c^3 + 156*sqrt(a^2 - a*b + b*c - c^2 - sqr
t(b^2 - 4*a*c)*(a - b + c))*a^2*b^3*c^3 + 27*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^4
*c^3 - 5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^5*c^3 - 80*sqrt(a^2 - a*b + b*c - c^2 -
sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*c^4 - 144*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*
b*c^4 + 14*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b^2*c^4 + 48*sqrt(a^2 - a*b + b*c -
c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^3*c^4 + 7*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c)
)*b^4*c^4 - 104*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*c^5 - 112*sqrt(a^2 - a*b + b*c
- c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b*c^5 - 24*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b +
c))*a*b^2*c^5 - 4*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^3*c^5 - 16*sqrt(a^2 - a*b + b
*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*c...

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Mupad [B]
time = 13.77, size = 2500, normalized size = 6.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^4/(a + b*cos(x) + c*cos(x)^2),x)

[Out]

atan(((((((2048*(48*a*c^15 + 272*a^2*c^14 + 576*a^3*c^13 + 576*a^4*c^12 + 272*a^5*c^11 + 48*a^6*c^10 - 12*b^2*
c^14 + 20*b^3*c^13 + 18*b^4*c^12 - 46*b^5*c^11 + 6*b^6*c^10 + 26*b^7*c^9 - 12*b^8*c^8 - 140*a*b^2*c^13 + 288*a
*b^3*c^12 + 30*a*b^4*c^11 - 240*a*b^5*c^10 + 74*a*b^6*c^9 + 20*a*b^7*c^8 - 416*a^2*b*c^13 - 736*a^3*b*c^12 - 5
44*a^4*b*c^11 - 144*a^5*b*c^10 - 360*a^2*b^2*c^12 + 728*a^2*b^3*c^11 - 50*a^2*b^4*c^10 - 182*a^2*b^5*c^9 + 4*a
^2*b^6*c^8 - 360*a^3*b^2*c^11 + 544*a^3*b^3*c^10 + 10*a^3*b^4*c^9 - 20*a^3*b^5*c^8 - 172*a^4*b^2*c^10 + 116*a^
4*b^3*c^9 + 8*a^4*b^4*c^8 - 44*a^5*b^2*c^9 - 80*a*b*c^14))/c^8 - (2048*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^6*c^2 - 18*a*b^2*c^5 + 2
4*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*(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7)))^(1/2)*(32*a*c^16 - 64*a^2*
c^15 - 128*a^3*c^14 + 64*a^4*c^13 + 96*a^5*c^12 - 8*b^2*c^15 + 24*b^3*c^14 - 32*b^4*c^13 + 32*b^5*c^12 - 24*b^
6*c^11 + 8*b^7*c^10 + 144*a*b^2*c^14 - 200*a*b^3*c^13 + 184*a*b^4*c^12 - 56*a*b^5*c^11 - 8*a*b^6*c^10 + 288*a^
2*b*c^14 + 352*a^3*b*c^13 - 32*a^4*b*c^12 - 320*a^2*b^2*c^13 + 8*a^2*b^3*c^12 + 96*a^2*b^4*c^11 - 8*a^2*b^5*c^
10 - 272*a^3*b^2*c^12 + 40*a^3*b^3*c^11 + 8*a^3*b^4*c^10 - 56*a^4*b^2*c^11 - 96*a*b*c^15))/c^8)*(-(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*(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7)))^(1/2) + (2
048*tan(x/2)*(24*b*c^14 - 96*a*c^14 - 8*c^15 + 152*a^2*c^13 + 952*a^3*c^12 + 1096*a^4*c^11 + 304*a^5*c^10 - 15
2*a^6*c^9 - 72*a^7*c^8 + 2*b^2*c^13 - 38*b^3*c^12 - 7*b^4*c^11 + 39*b^5*c^10 - 15*b^6*c^9 + 35*b^7*c^8 - 44*b^
8*c^7 - 4*b^9*c^6 + 24*b^10*c^5 - 8*b^11*c^4 + 68*a*b^2*c^12 + 42*a*b^3*c^11 - 159*a*b^4*c^10 - 400*a*b^5*c^9
+ 537*a*b^6*c^8 + 68*a*b^7*c^7 - 276*a*b^8*c^6 + 72*a*b^9*c^5 + 8*a*b^10*c^4 - 944*a^2*b*c^12 - 2520*a^3*b*c^1
1 - 1824*a^4*b*c^10 - 272*a^5*b*c^9 + 88*a^6*b*c^8 + 584*a^2*b^2*c^11 + 1742*a^2*b^3*c^10 - 1645*a^2*b^4*c^9 -
795*a^2*b^5*c^8 + 1132*a^2*b^6*c^7 - 112*a^2*b^7*c^6 - 112*a^2*b^8*c^5 + 8*a^2*b^9*c^4 + 476*a^3*b^2*c^10 + 2
766*a^3*b^3*c^9 - 1705*a^3*b^4*c^8 - 396*a^3*b^5*c^7 + 456*a^3*b^6*c^6 - 56*a^3*b^7*c^5 - 8*a^3*b^8*c^4 + 230*
a^4*b^2*c^9 + 880*a^4*b^3*c^8 - 656*a^4*b^4*c^7 + 140*a^4*b^5*c^6 + 72*a^4*b^6*c^5 + 464*a^5*b^2*c^8 - 192*a^5
*b^3*c^7 - 220*a^5*b^4*c^6 + 256*a^6*b^2*c^7 + 136*a*b*c^13))/c^8)*(-(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*(16*a^2*c^8 + b^4*c^6 - 8*a*b^2*c^7)))^(1/2) + (2048*(236*a*c^13 - 32*b*c^13 +
12*c^14 + 1084*a^2*c^12 + 2328*a^3*c^11 + 2784*a^4*c^10 + 1948*a^5*c^9 + 780*a^6*c^8 + 160*a^7*c^7 + 12*a^8*c
^6 - 39*b^2*c^12 + 121*b^3*c^11 + 61*b^4*c^10 - 220*b^5*c^9 - 36*b^6*c^8 + 232*b^7*c^7 - 28*b^8*c^6 - 127*b^9*
c^5 + 42*b^10*c^4 + 26*b^11*c^3 - 12*b^12*c^2 - 635*a*b^2*c^11 + 1300*a*b^3*c^10 + 608*a*b^4*c^9 - 1792*a*b^5*
c^8 - 60*a*b^6*c^7 + 1218*a*b^7*c^6 - 249*a*b^8*c^5 - 340*a*b^9*c^4 + 98*a*b^10*c^3 + 20*a*b^11*c^2 - 1616*a^2
*b*c^11 - 3160*a^3*b*c^10 - 3440*a^4*b*c^9 - 2132*a^5*b*c^8 - 704*a^6*b*c^7 - 96*a^7*b*c^6 - 2242*a^2*b^2*c^10
+ 4146*a^2*b^3*c^9 + 1420*a^2*b^4*c^8 - 4158*a^2*b^5*c^7 + 77*a^2*b^6*c^6 + 1735*a^2*b^7*c^5 - 234*a^2*b^8*c^
4 - 222*a^2*b^9*c^3 + 4*a^2*b^10*c^2 - 3714*a^3*b^2*c^9 + 6252*a^3*b^3*c^8 + 1730*a^3*b^4*c^7 - 4300*a^3*b^5*c
^6 - 79*a^3*b^6*c^5 + 968*a^3*b^7*c^4 + 2*a^3*b^8*c^3 - 20*a^3*b^9*c^2 - 3523*a^4*b^2*c^8 + 5025*a^4*b^3*c^7 +
1339*a^4*b^4*c^6 - 2082*a^4*b^5*c^5 - 192*a^4*b^6*c^4 + 156*a^4*b^7*c^3 + 8*a^4*b^8*c^2 - 2031*a^5*b^2*c^7 +
2104*a^5*b^3*c^6 + 634*a^5*b^4*c^5 - 388*a^5*b^5*c^4 - 60*a^5*b^6*c^3 - 676*a^6*b^2*c^6 + 364*a^6*b^3*c^5 + 13
6*a^6*b^4*c^4 - 100*a^7*b^2*c^5 - 404*a*b*c^12))/c^8)*(-(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*(16*a^2*c^8 + b^4*c^6 - 8*a*...

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