3.1.15 \(\int \frac {\cos ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\) [15]

Optimal. Leaf size=255 \[ \frac {x}{c}-\frac {2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{c \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{c \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

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+(2*a*c-b^2)/(-4*
a*c+b^2)^(1/2))/c/(b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)-2*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+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))/c/(b-2*c+(-4*
a*c+b^2)^(1/2))^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.90, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3338, 3374, 2738, 211} \begin {gather*} -\frac {2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\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 \sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {2 \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\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 \sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}+\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 3338

Int[cos[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + cos[(d_.) + (e_.)*(x_)]^(n_.)*(b_.) + cos[(d_.) + (e_.)*(x_)]^(n2_.
)*(c_.))^(p_), x_Symbol] :> Int[ExpandTrig[cos[d + e*x]^m*(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] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, 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 {\cos ^2(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx &=\int \left (\frac {1}{c}+\frac {-a-b \cos (x)}{c \left (a+b \cos (x)+c \cos ^2(x)\right )}\right ) \, dx\\ &=\frac {x}{c}+\frac {\int \frac {-a-b \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx}{c}\\ &=\frac {x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b-\sqrt {b^2-4 a c}+2 c \cos (x)} \, dx}{c}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b+\sqrt {b^2-4 a c}+2 c \cos (x)} \, dx}{c}\\ &=\frac {x}{c}-\frac {\left (2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\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}-\frac {\left (2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\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}\\ &=\frac {x}{c}-\frac {2 \left (b-\frac {b^2-2 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 )}{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+\frac {b^2-2 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 )}{c \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 264, normalized size = 1.04 \begin {gather*} \frac {x+\frac {\sqrt {2} \left (b^2-2 a c+b \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 {\sqrt {2} \left (-b^2+2 a c+b \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}}}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.13, size = 261, normalized size = 1.02

method	result	size
default	\(\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{c}+\frac {2 \left (a -b +c \right ) \left (\frac {\left (-a \sqrt {-4 a c +b^{2}}+b \sqrt {-4 a c +b^{2}}+a b +2 a c -b^{2}\right ) \arctan \left (\frac {\left (a -b +c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}+\frac {\left (-a \sqrt {-4 a c +b^{2}}+b \sqrt {-4 a c +b^{2}}-a b -2 a c +b^{2}\right ) \arctanh \left (\frac {\left (-a +b -c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{c}\)	\(261\)
risch	\(\frac {x}{c}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{4} c^{4}-8 a^{3} b^{2} c^{3}+32 a^{3} c^{5}+a^{2} b^{4} c^{2}-32 a^{2} b^{2} c^{4}+16 a^{2} c^{6}+10 a \,b^{4} c^{3}-8 a \,b^{2} c^{5}-c^{2} b^{6}+c^{4} b^{4}\right ) \textit {\_Z}^{4}+\left (8 a^{4} c^{2}-6 a^{3} b^{2} c +8 a^{3} c^{3}+a^{2} b^{4}-18 b^{2} a^{2} c^{2}+8 a \,b^{4} c -b^{6}\right ) \textit {\_Z}^{2}+a^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (-\frac {2 i c^{2} a^{4} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {18 i c^{3} a^{3} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {3 i c^{2} a^{2} b^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {22 i c^{4} a^{2} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {8 i c^{3} a \,b^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {6 i c^{5} b^{2} a}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {8 i c^{3} a^{5}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {24 i c^{4} a^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {24 i c^{5} a^{3}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {8 i c^{6} a^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {i c^{2} b^{6}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {i c^{4} b^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}\right ) \textit {\_R}^{3}+\left (\frac {4 c^{2} a^{5}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {8 c^{3} a^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {4 c^{4} a^{3}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {c \,a^{4} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {6 c^{2} a^{3} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {c \,a^{2} b^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {c^{3} b^{2} a^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}\right ) \textit {\_R}^{2}+\left (-\frac {8 i c \,a^{3} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {9 i c^{2} b^{2} a^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {6 i c \,b^{4} a}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {i b^{6}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {4 i c^{2} a^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {2 i c^{3} a^{3}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {2 i a^{2} b^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {2 i c \,a^{5}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {i a^{4} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}\right ) \textit {\_R} +\frac {a^{5}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}+\frac {c \,a^{4}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}-\frac {a^{3} b^{2}}{a^{4} b +2 a^{3} b c -a^{2} b^{3}}\right )\right )\)	\(1185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-(c*integrate(2*(2*b^2*cos(3*x)^2 + 2*b^2*cos(x)^2 + 2*b^2*sin(3*x)^2 + 2*b^2*sin(x)^2 + 4*(2*a^2 + a*c)*cos(2
*x)^2 + b*c*cos(x) + 4*(2*a^2 + a*c)*sin(2*x)^2 + 2*(4*a*b + b*c)*sin(2*x)*sin(x) + (b*c*cos(3*x) + 2*a*c*cos(
2*x) + b*c*cos(x))*cos(4*x) + (4*b^2*cos(x) + b*c + 2*(4*a*b + b*c)*cos(2*x))*cos(3*x) + 2*(a*c + (4*a*b + b*c
)*cos(x))*cos(2*x) + (b*c*sin(3*x) + 2*a*c*sin(2*x) + b*c*sin(x))*sin(4*x) + 2*(2*b^2*sin(x) + (4*a*b + b*c)*s
in(2*x))*sin(3*x))/(c^3*cos(4*x)^2 + 4*b^2*c*cos(3*x)^2 + 4*b^2*c*cos(x)^2 + c^3*sin(4*x)^2 + 4*b^2*c*sin(3*x)
^2 + 4*b^2*c*sin(x)^2 + 4*b*c^2*cos(x) + c^3 + 4*(4*a^2*c + 4*a*c^2 + c^3)*cos(2*x)^2 + 4*(4*a^2*c + 4*a*c^2 +
c^3)*sin(2*x)^2 + 8*(2*a*b*c + b*c^2)*sin(2*x)*sin(x) + 2*(2*b*c^2*cos(3*x) + 2*b*c^2*cos(x) + c^3 + 2*(2*a*c
^2 + c^3)*cos(2*x))*cos(4*x) + 4*(2*b^2*c*cos(x) + b*c^2 + 2*(2*a*b*c + b*c^2)*cos(2*x))*cos(3*x) + 4*(2*a*c^2
+ c^3 + 2*(2*a*b*c + b*c^2)*cos(x))*cos(2*x) + 4*(b*c^2*sin(3*x) + b*c^2*sin(x) + (2*a*c^2 + c^3)*sin(2*x))*s
in(4*x) + 8*(b^2*c*sin(x) + (2*a*b*c + b*c^2)*sin(2*x))*sin(3*x)), x) - x)/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4983 vs. \(2 (215) = 430\).
time = 1.21, size = 4983, normalized size = 19.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2)*c*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c - (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^
3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)
*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3
*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^
3 - (a^2*b^2 - b^4)*c^2))*log(4*a^3*b*c^2 - (4*a^3*c^5 + (8*a^4 - a^2*b^2)*c^4 + 2*(2*a^5 - 3*a^3*b^2)*c^3 - (
a^4*b^2 - a^2*b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (
16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*
c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4))*cos(x) + 2*(a^4*b - a^2*b^3)*c + 1/2*sqrt(2)*((8*a^2*c^7 + 6*(4*a^3 -
a*b^2)*c^6 + (24*a^4 - 22*a^2*b^2 + b^4)*c^5 + 2*(4*a^5 - 9*a^3*b^2 + 4*a*b^4)*c^4 - (2*a^4*b^2 - 3*a^2*b^4 +
b^6)*c^3)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*
c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2
- 2*a^2*b^4 + b^6)*c^4))*sin(x) + (8*a^2*b^2*c^3 + 2*(2*a^3*b^2 - 3*a*b^4)*c^2 - (a^2*b^4 - b^6)*c)*sin(x))*s
qrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c - (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3
- (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (1
6*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c
^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^
4)*c^2)) + (a^4*b^2 - a^2*b^4 + 2*a^3*b^2*c)*cos(x)) - sqrt(2)*c*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*
a*b^2)*c - (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^
2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 +
2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a
*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2))*log(4*a^3*b*c^2 - (4*a^3*c^5 + (8*a
^4 - a^2*b^2)*c^4 + 2*(2*a^5 - 3*a^3*b^2)*c^3 - (a^4*b^2 - a^2*b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*
a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^
2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4))*cos(x) + 2*(a^4*b - a
^2*b^3)*c - 1/2*sqrt(2)*((8*a^2*c^7 + 6*(4*a^3 - a*b^2)*c^6 + (24*a^4 - 22*a^2*b^2 + b^4)*c^5 + 2*(4*a^5 - 9*a
^3*b^2 + 4*a*b^4)*c^4 - (2*a^4*b^2 - 3*a^2*b^4 + b^6)*c^3)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 +
4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c
^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4))*sin(x) + (8*a^2*b^2*c^3 + 2*(2*a^3*
b^2 - 3*a*b^4)*c^2 - (a^2*b^4 - b^6)*c)*sin(x))*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c - (4*a*c
^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a
^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2
*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b
^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2 - b^4)*c^2)) + (a^4*b^2 - a^2*b^4 + 2*a^3*b^2*c)*cos(x)) + sqrt(2
)*c*sqrt((a^2*b^2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c + (4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)
*c^3 - (a^2*b^2 - b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9
+ (16*a^2 - b^2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b
^4)*c^5 - (a^4*b^2 - 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 + (8*a^2 - b^2)*c^4 + 2*(2*a^3 - 3*a*b^2)*c^3 - (a^2*b^2
- b^4)*c^2))*log(-4*a^3*b*c^2 - (4*a^3*c^5 + (8*a^4 - a^2*b^2)*c^4 + 2*(2*a^5 - 3*a^3*b^2)*c^3 - (a^4*b^2 - a
^2*b^4)*c^2)*sqrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^
2)*c^8 + 12*(2*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*
b^2 - 2*a^2*b^4 + b^6)*c^4))*cos(x) - 2*(a^4*b - a^2*b^3)*c + 1/2*sqrt(2)*((8*a^2*c^7 + 6*(4*a^3 - a*b^2)*c^6
+ (24*a^4 - 22*a^2*b^2 + b^4)*c^5 + 2*(4*a^5 - 9*a^3*b^2 + 4*a*b^4)*c^4 - (2*a^4*b^2 - 3*a^2*b^4 + b^6)*c^3)*s
qrt(-(a^4*b^2 - 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 + 4*(a^3*b^2 - a*b^4)*c)/(4*a*c^9 + (16*a^2 - b^2)*c^8 + 12*(2
*a^3 - a*b^2)*c^7 + 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 - 2*a^2*b^
4 + b^6)*c^4))*sin(x) - (8*a^2*b^2*c^3 + 2*(2*a^3*b^2 - 3*a*b^4)*c^2 - (a^2*b^4 - b^6)*c)*sin(x))*sqrt((a^2*b^
2 - b^4 - 2*a^2*c^2 - 2*(a^3 - 2*a*b^2)*c + (4*...

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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(cos(x)**2/(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. 9030 vs. \(2 (215) = 430\).
time = 2.52, size = 9030, normalized size = 35.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c - ((2*a^3*b^4 - 6*a^2*b^5 + 6*a*b^6 - 2*b^7 - 16*a^4*b^2*c + 48*a^3*b^3*c - 44*a^2*b^4*c + 8*a*b^5*c + 4*b
^6*c + 32*a^5*c^2 - 96*a^4*b*c^2 + 64*a^3*b^2*c^2 + 32*a^2*b^3*c^2 - 30*a*b^4*c^2 - 2*b^5*c^2 + 64*a^4*c^3 - 1
28*a^3*b*c^3 + 48*a^2*b^2*c^3 + 16*a*b^3*c^3 + 32*a^3*c^4 - 32*a^2*b*c^4 + 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^3*b^2 - 2*(b^2 - 4*a*c)*a^3*b^2 - 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)*a^2*b^3 + 6*(b^2 - 4*a*c)*a^2*b^3 - 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*b^4 - 6*(b^2 - 4*a*c)*a*b^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^5 + 2*(b^2 - 4*a*c)*b^5 - 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 + 8*(b^2 - 4*a*c)*a^4*c + 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 - 24*(b^2 - 4*a*c)*a^3*b*c + 26*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 + 20*(b^2 - 4*a*c)*a^2*b^2*c -
28*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 - 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 - 4*(b^2 - 4*a*c)*b^4*c - 56*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^2 + 16*(b^2 - 4*a*c)*a^3*c^2 + 32*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^2 - 32*(b^2 - 4*a*c)*a^2*b*c^2
+ 19*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^2 + 14*(b^2 - 4*a*
c)*a*b^2*c^2 + 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^3*c^2 + 2*(b^
2 - 4*a*c)*b^3*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^2*c^3
+ 8*(b^2 - 4*a*c)*a^2*c^3 - 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
*b*c^3 - 8*(b^2 - 4*a*c)*a*b*c^3)*c^2*abs(a - b + c) - (4*a^3*b^4*c - 4*a^2*b^5*c - 4*a*b^6*c + 4*b^7*c - 32*a
^4*b^2*c^2 + 32*a^3*b^3*c^2 + 40*a^2*b^4*c^2 - 32*a*b^5*c^2 - 8*b^6*c^2 + 64*a^5*c^3 - 64*a^4*b*c^3 - 128*a^3*
b^2*c^3 + 64*a^2*b^3*c^3 + 68*a*b^4*c^3 + 4*b^5*c^3 + 128*a^4*c^4 - 160*a^2*b^2*c^4 - 32*a*b^3*c^4 + 64*a^3*c^
5 + 64*a^2*b*c^5 - 3*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*b^2*c + 2*sqrt(a^2 - a*b
+ b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^3*c + 8*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a -
b + c))*a^2*b^4*c - 2*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^5*c - 5*sqrt(a^2 - a*b
+ b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^6*c + 12*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b
+ c))*a^5*c^2 - 8*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*b*c^2 - 49*sqrt(a^2 - a*b +
b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^2*c^2 + sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b
+ c))*a^2*b^3*c^2 + 41*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^4*c^2 + 11*sqrt(a^2 -
a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^5*c^2 + 68*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(
a - b + c))*a^4*c^3 + 28*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b*c^3 - 93*sqrt(a^2 -
a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b^2*c^3 - 64*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a
*c)*(a - b + c))*a*b^3*c^3 - 11*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^4*c^3 + 36*sqrt(
a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*c^4 + 80*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*
a*c)*(a - b + c))*a^2*b*c^4 + 49*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^2*c^4 + 5*sqr
t(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^3*c^4 - 20*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 -
4*a*c)*(a - b + c))*a^2*c^5 - 20*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b*c^5 - 4*(b^2
- 4*a*c)*a^3*b^2*c + 4*(b^2 - 4*a*c)*a^2*b^3*c + 4*(b^2 - 4*a*c)*a*b^4*c - 4*(b^2 - 4*a*c)*b^5*c + 16*(b^2 - 4
*a*c)*a^4*c^2 - 16*(b^2 - 4*a*c)*a^3*b*c^2 - 24*(b^2 - 4*a*c)*a^2*b^2*c^2 + 16*(b^2 - 4*a*c)*a*b^3*c^2 + 8*(b^
2 - 4*a*c)*b^4*c^2 + 32*(b^2 - 4*a*c)*a^3*c^3 - 36*(b^2 - 4*a*c)*a*b^2*c^3 - 4*(b^2 - 4*a*c)*b^3*c^3 + 16*(b^2
- 4*a*c)*a^2*c^4 + 16*(b^2 - 4*a*c)*a*b*c^4)*abs(a - b + c)*abs(c) + (2*a^4*b^3*c^2 - 6*a^3*b^4*c^2 + 6*a^2*b
^5*c^2 - 2*a*b^6*c^2 - 8*a^5*b*c^3 + 28*a^4*b^2*c^3 - 30*a^3*b^3*c^3 + 10*a^2*b^4*c^3 - 2*a*b^5*c^3 + 2*b^6*c^
3 - 16*a^5*c^4 + 24*a^4*b*c^4 - 4*a^3*b^2*c^4 + 6*a^2*b^3*c^4 - 6*a*b^4*c^4 - 4*b^5*c^4 - 16*a^4*c^5 + 8*a^3*b
*c^5 - 12*a^2*b^2*c^5 + 22*a*b^3*c^5 + 2*b^4*c^5 + 16*a^3*c^6 - 24*a^2*b*c^6 - 12*a*b^2*c^6 + 16*a^2*c^7 + 3*s
qrt(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*b*c^2 - 2*(b^2 - 4*a*c)*a^4*b
*c^2 - 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)*a^3*b^2*c^2 + 6*(b^2 -
4*a*c)*a^3*b^2*c^2 - 3*sqrt(a^2 - a*b + b*c - c...

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Mupad [B]
time = 14.56, size = 2500, normalized size = 9.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atan((540672*a^4*tan(x/2))/(131072*a*b^3 - 393216*a^3*b + 655360*a^3*c + 540672*a^4 + 16384*b^4 - 360448*a^
2*b^2 + 262144*a^2*c^2 + (147456*a^5)/c - (16384*b^5)/c + (49152*a*b^4)/c - (147456*a^4*b)/c - (32768*a*b^5)/c
^2 + (229376*a^2*b^3)/c - (262144*a^3*b^2)/c + (32768*a^2*b^4)/c^2 + (32768*a^3*b^3)/c^2 - (32768*a^4*b^2)/c^2
- 131072*a*b^2*c - 262144*a^2*b*c) + (16384*b^4*tan(x/2))/(131072*a*b^3 - 393216*a^3*b + 655360*a^3*c + 54067
2*a^4 + 16384*b^4 - 360448*a^2*b^2 + 262144*a^2*c^2 + (147456*a^5)/c - (16384*b^5)/c + (49152*a*b^4)/c - (1474
56*a^4*b)/c - (32768*a*b^5)/c^2 + (229376*a^2*b^3)/c - (262144*a^3*b^2)/c + (32768*a^2*b^4)/c^2 + (32768*a^3*b
^3)/c^2 - (32768*a^4*b^2)/c^2 - 131072*a*b^2*c - 262144*a^2*b*c) + (147456*a^5*tan(x/2))/(49152*a*b^4 - 147456
*a^4*b + 540672*a^4*c + 16384*b^4*c + 147456*a^5 - 16384*b^5 + 229376*a^2*b^3 - 262144*a^3*b^2 + 262144*a^2*c^
3 + 655360*a^3*c^2 - 131072*a*b^2*c^2 - 262144*a^2*b*c^2 - 360448*a^2*b^2*c - (32768*a*b^5)/c + (32768*a^2*b^4
)/c + (32768*a^3*b^3)/c - (32768*a^4*b^2)/c + 131072*a*b^3*c - 393216*a^3*b*c) - (16384*b^5*tan(x/2))/(49152*a
*b^4 - 147456*a^4*b + 540672*a^4*c + 16384*b^4*c + 147456*a^5 - 16384*b^5 + 229376*a^2*b^3 - 262144*a^3*b^2 +
262144*a^2*c^3 + 655360*a^3*c^2 - 131072*a*b^2*c^2 - 262144*a^2*b*c^2 - 360448*a^2*b^2*c - (32768*a*b^5)/c + (
32768*a^2*b^4)/c + (32768*a^3*b^3)/c - (32768*a^4*b^2)/c + 131072*a*b^3*c - 393216*a^3*b*c) - (360448*a^2*b^2*
tan(x/2))/(131072*a*b^3 - 393216*a^3*b + 655360*a^3*c + 540672*a^4 + 16384*b^4 - 360448*a^2*b^2 + 262144*a^2*c
^2 + (147456*a^5)/c - (16384*b^5)/c + (49152*a*b^4)/c - (147456*a^4*b)/c - (32768*a*b^5)/c^2 + (229376*a^2*b^3
)/c - (262144*a^3*b^2)/c + (32768*a^2*b^4)/c^2 + (32768*a^3*b^3)/c^2 - (32768*a^4*b^2)/c^2 - 131072*a*b^2*c -
262144*a^2*b*c) + (262144*a^2*c^2*tan(x/2))/(131072*a*b^3 - 393216*a^3*b + 655360*a^3*c + 540672*a^4 + 16384*b
^4 - 360448*a^2*b^2 + 262144*a^2*c^2 + (147456*a^5)/c - (16384*b^5)/c + (49152*a*b^4)/c - (147456*a^4*b)/c - (
32768*a*b^5)/c^2 + (229376*a^2*b^3)/c - (262144*a^3*b^2)/c + (32768*a^2*b^4)/c^2 + (32768*a^3*b^3)/c^2 - (3276
8*a^4*b^2)/c^2 - 131072*a*b^2*c - 262144*a^2*b*c) + (49152*a*b^4*tan(x/2))/(49152*a*b^4 - 147456*a^4*b + 54067
2*a^4*c + 16384*b^4*c + 147456*a^5 - 16384*b^5 + 229376*a^2*b^3 - 262144*a^3*b^2 + 262144*a^2*c^3 + 655360*a^3
*c^2 - 131072*a*b^2*c^2 - 262144*a^2*b*c^2 - 360448*a^2*b^2*c - (32768*a*b^5)/c + (32768*a^2*b^4)/c + (32768*a
^3*b^3)/c - (32768*a^4*b^2)/c + 131072*a*b^3*c - 393216*a^3*b*c) - (147456*a^4*b*tan(x/2))/(49152*a*b^4 - 1474
56*a^4*b + 540672*a^4*c + 16384*b^4*c + 147456*a^5 - 16384*b^5 + 229376*a^2*b^3 - 262144*a^3*b^2 + 262144*a^2*
c^3 + 655360*a^3*c^2 - 131072*a*b^2*c^2 - 262144*a^2*b*c^2 - 360448*a^2*b^2*c - (32768*a*b^5)/c + (32768*a^2*b
^4)/c + (32768*a^3*b^3)/c - (32768*a^4*b^2)/c + 131072*a*b^3*c - 393216*a^3*b*c) - (32768*a*b^5*tan(x/2))/(147
456*a^5*c - 32768*a*b^5 - 16384*b^5*c + 32768*a^2*b^4 + 32768*a^3*b^3 - 32768*a^4*b^2 + 262144*a^2*c^4 + 65536
0*a^3*c^3 + 540672*a^4*c^2 + 16384*b^4*c^2 - 131072*a*b^2*c^3 + 131072*a*b^3*c^2 - 262144*a^2*b*c^3 + 229376*a
^2*b^3*c - 393216*a^3*b*c^2 - 262144*a^3*b^2*c - 360448*a^2*b^2*c^2 + 49152*a*b^4*c - 147456*a^4*b*c) + (22937
6*a^2*b^3*tan(x/2))/(49152*a*b^4 - 147456*a^4*b + 540672*a^4*c + 16384*b^4*c + 147456*a^5 - 16384*b^5 + 229376
*a^2*b^3 - 262144*a^3*b^2 + 262144*a^2*c^3 + 655360*a^3*c^2 - 131072*a*b^2*c^2 - 262144*a^2*b*c^2 - 360448*a^2
*b^2*c - (32768*a*b^5)/c + (32768*a^2*b^4)/c + (32768*a^3*b^3)/c - (32768*a^4*b^2)/c + 131072*a*b^3*c - 393216
*a^3*b*c) - (262144*a^3*b^2*tan(x/2))/(49152*a*b^4 - 147456*a^4*b + 540672*a^4*c + 16384*b^4*c + 147456*a^5 -
16384*b^5 + 229376*a^2*b^3 - 262144*a^3*b^2 + 262144*a^2*c^3 + 655360*a^3*c^2 - 131072*a*b^2*c^2 - 262144*a^2*
b*c^2 - 360448*a^2*b^2*c - (32768*a*b^5)/c + (32768*a^2*b^4)/c + (32768*a^3*b^3)/c - (32768*a^4*b^2)/c + 13107
2*a*b^3*c - 393216*a^3*b*c) + (131072*a*b^3*tan(x/2))/(131072*a*b^3 - 393216*a^3*b + 655360*a^3*c + 540672*a^4
+ 16384*b^4 - 360448*a^2*b^2 + 262144*a^2*c^2 + (147456*a^5)/c - (16384*b^5)/c + (49152*a*b^4)/c - (147456*a^
4*b)/c - (32768*a*b^5)/c^2 + (229376*a^2*b^3)/c - (262144*a^3*b^2)/c + (32768*a^2*b^4)/c^2 + (32768*a^3*b^3)/c
^2 - (32768*a^4*b^2)/c^2 - 131072*a*b^2*c - 262144*a^2*b*c) - (393216*a^3*b*tan(x/2))/(131072*a*b^3 - 393216*a
^3*b + 655360*a^3*c + 540672*a^4 + 16384*b^4 - 360448*a^2*b^2 + 262144*a^2*c^2 + (147456*a^5)/c - (16384*b^5)/
c + (49152*a*b^4)/c - (147456*a^4*b)/c - (32768*a*b^5)/c^2 + (229376*a^2*b^3)/c - (262144*a^3*b^2)/c + (32768*
a^2*b^4)/c^2 + (32768*a^3*b^3)/c^2 - (32768*a^4*b^2)/c^2 - 131072*a*b^2*c - 262144*a^2*b*c) + (655360*a^3*c*ta
n(x/2))/(131072*a*b^3 - 393216*a^3*b + 655360*a^3*c + 540672*a^4 + 16384*b^4 - 360448*a^2*b^2 + 262144*a^2*c^2
+ (147456*a^5)/c - (16384*b^5)/c + (49152*a*b^4)/c - (147456*a^4*b)/c - (32768*a*b^5)/c^2 + (229376*a^2*b^3)/
c - (262144*a^3*b^2)/c + (32768*a^2*b^4)/c^2 + (32768*a^3*b^3)/c^2 - (32768*a^4*b^2)/c^2 - 131072*a*b^2*c - 26
2144*a^2*b*c) + (32768*a^2*b^4*tan(x/2))/(14745...

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