3.1.20 \(\int \frac {\sec ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\) [20]

Optimal. Leaf size=334 \[ -\frac {2 c \left (b^3-3 a b c+\sqrt {b^2-4 a c} \left (b^2-a c\right )\right ) \text {ArcTan}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {2 c \left (b^3-3 a b c-\sqrt {b^2-4 a c} \left (b^2-a c\right )\right ) \text {ArcTan}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}+\frac {\tanh ^{-1}(\sin (x))}{2 a}+\frac {\left (b^2-a c\right ) \tanh ^{-1}(\sin (x))}{a^3}-\frac {b \tan (x)}{a^2}+\frac {\sec (x) \tan (x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 3.25, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3338, 3374, 2738, 211, 3855, 3852, 8, 3853} \begin {gather*} -\frac {2 c \left (\sqrt {b^2-4 a c} \left (b^2-a c\right )-3 a b c+b^3\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 )}{a^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 c \left (-\sqrt {b^2-4 a c} \left (b^2-a c\right )-3 a b c+b^3\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 )}{a^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 {\left (b^2-a c\right ) \tanh ^{-1}(\sin (x))}{a^3}-\frac {b \tan (x)}{a^2}+\frac {\tanh ^{-1}(\sin (x))}{2 a}+\frac {\tan (x) \sec (x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

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

Rule 3855

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

Rubi steps

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

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Mathematica [A]
time = 3.17, size = 446, normalized size = 1.34 \begin {gather*} -\frac {\frac {4 \sqrt {2} c \left (b^3-3 a b c-b^2 \sqrt {b^2-4 a 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} c \left (b^3-3 a b c+b^2 \sqrt {b^2-4 a 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}}}+2 \left (a^2+2 b^2-2 a c\right ) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-2 \left (a^2+2 b^2-2 a c\right ) \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {4 a b \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}+\frac {a^2}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {4 a b \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}+\frac {a^2}{-1+\sin (x)}}{4 a^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*((4*Sqrt[2]*c*(b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*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]*c*(b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*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]]) + 2*(a^2 + 2*b^2 - 2*a*c)*Log[Cos[x/2] - Sin[x/2]
] - 2*(a^2 + 2*b^2 - 2*a*c)*Log[Cos[x/2] + Sin[x/2]] + (4*a*b*Sin[x/2])/(Cos[x/2] - Sin[x/2]) + a^2/(Cos[x/2]
+ Sin[x/2])^2 + (4*a*b*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + a^2/(-1 + Sin[x]))/a^3

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Maple [A]
time = 4.21, size = 481, normalized size = 1.44

method	result	size
default	\(\frac {1}{2 a \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a -2 b}{2 a^{2} \left (\tan \left (\frac {x}{2}\right )-1\right )}+\frac {\left (-a^{2}+2 a c -2 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{2 a^{3}}+\frac {2 \left (a -b +c \right ) \left (\frac {\left (2 c a b \sqrt {-4 a c +b^{2}}-a \,c^{2} \sqrt {-4 a c +b^{2}}-b^{3} \sqrt {-4 a c +b^{2}}+b^{2} c \sqrt {-4 a c +b^{2}}-2 a^{2} c^{2}+4 a \,b^{2} c -3 c^{2} a b -b^{4}+b^{3} c \right ) \arctan \left (\frac {\left (a -b +c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}+\frac {\left (2 c a b \sqrt {-4 a c +b^{2}}-a \,c^{2} \sqrt {-4 a c +b^{2}}-b^{3} \sqrt {-4 a c +b^{2}}+b^{2} c \sqrt {-4 a c +b^{2}}+2 a^{2} c^{2}-4 a \,b^{2} c +3 c^{2} a b +b^{4}-b^{3} c \right ) \arctanh \left (\frac {\left (-a +b -c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{a^{3}}-\frac {1}{2 a \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-a -2 b}{2 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {\left (a^{2}-2 a c +2 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{2 a^{3}}\)	\(481\)
risch	\(\text {Expression too large to display}\)	\(4131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^3/(a+b*cos(x)+c*cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

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

[Out]

-1/4*(8*a^2*cos(3*x)*sin(2*x) + 8*a^2*cos(2*x)*sin(x) + 4*a^2*sin(x) - 4*(a^2*sin(3*x) + 2*a*b*sin(2*x) - a^2*
sin(x))*cos(4*x) - 4*(a^3*cos(4*x)^2 + 4*a^3*cos(2*x)^2 + a^3*sin(4*x)^2 + 4*a^3*sin(4*x)*sin(2*x) + 4*a^3*sin
(2*x)^2 + 4*a^3*cos(2*x) + a^3 + 2*(2*a^3*cos(2*x) + a^3)*cos(4*x))*integrate(-2*(2*(b^3*c - a*b*c^2)*cos(3*x)
^2 + 4*(2*a*b^3 - 2*a*b*c^2 - (4*a^2*b - b^3)*c)*cos(2*x)^2 + 2*(b^3*c - a*b*c^2)*cos(x)^2 + 2*(b^3*c - a*b*c^
2)*sin(3*x)^2 + 4*(2*a*b^3 - 2*a*b*c^2 - (4*a^2*b - b^3)*c)*sin(2*x)^2 + 2*(2*b^4 - 2*a*b^2*c - a*c^3 - (2*a^2
- b^2)*c^2)*sin(2*x)*sin(x) + 2*(b^3*c - a*b*c^2)*sin(x)^2 + ((b^2*c^2 - a*c^3)*cos(3*x) + 2*(b^3*c - 2*a*b*c
^2)*cos(2*x) + (b^2*c^2 - a*c^3)*cos(x))*cos(4*x) + (b^2*c^2 - a*c^3 + 2*(2*b^4 - 2*a*b^2*c - a*c^3 - (2*a^2 -
b^2)*c^2)*cos(2*x) + 4*(b^3*c - a*b*c^2)*cos(x))*cos(3*x) + 2*(b^3*c - 2*a*b*c^2 + (2*b^4 - 2*a*b^2*c - a*c^3
- (2*a^2 - b^2)*c^2)*cos(x))*cos(2*x) + (b^2*c^2 - a*c^3)*cos(x) + ((b^2*c^2 - a*c^3)*sin(3*x) + 2*(b^3*c - 2
*a*b*c^2)*sin(2*x) + (b^2*c^2 - a*c^3)*sin(x))*sin(4*x) + 2*((2*b^4 - 2*a*b^2*c - a*c^3 - (2*a^2 - b^2)*c^2)*s
in(2*x) + 2*(b^3*c - a*b*c^2)*sin(x))*sin(3*x))/(a^3*c^2*cos(4*x)^2 + 4*a^3*b^2*cos(3*x)^2 + 4*a^3*b^2*cos(x)^
2 + a^3*c^2*sin(4*x)^2 + 4*a^3*b^2*sin(3*x)^2 + 4*a^3*b^2*sin(x)^2 + 4*a^3*b*c*cos(x) + a^3*c^2 + 4*(4*a^5 + 4
*a^4*c + a^3*c^2)*cos(2*x)^2 + 4*(4*a^5 + 4*a^4*c + a^3*c^2)*sin(2*x)^2 + 8*(2*a^4*b + a^3*b*c)*sin(2*x)*sin(x
) + 2*(2*a^3*b*c*cos(3*x) + 2*a^3*b*c*cos(x) + a^3*c^2 + 2*(2*a^4*c + a^3*c^2)*cos(2*x))*cos(4*x) + 4*(2*a^3*b
^2*cos(x) + a^3*b*c + 2*(2*a^4*b + a^3*b*c)*cos(2*x))*cos(3*x) + 4*(2*a^4*c + a^3*c^2 + 2*(2*a^4*b + a^3*b*c)*
cos(x))*cos(2*x) + 4*(a^3*b*c*sin(3*x) + a^3*b*c*sin(x) + (2*a^4*c + a^3*c^2)*sin(2*x))*sin(4*x) + 8*(a^3*b^2*
sin(x) + (2*a^4*b + a^3*b*c)*sin(2*x))*sin(3*x)), x) - ((a^2 + 2*b^2 - 2*a*c)*cos(4*x)^2 + 4*(a^2 + 2*b^2 - 2*
a*c)*cos(2*x)^2 + (a^2 + 2*b^2 - 2*a*c)*sin(4*x)^2 + 4*(a^2 + 2*b^2 - 2*a*c)*sin(4*x)*sin(2*x) + 4*(a^2 + 2*b^
2 - 2*a*c)*sin(2*x)^2 + a^2 + 2*b^2 - 2*a*c + 2*(a^2 + 2*b^2 - 2*a*c + 2*(a^2 + 2*b^2 - 2*a*c)*cos(2*x))*cos(4
*x) + 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x))*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + ((a^2 + 2*b^2 - 2*a*c)*cos(4
*x)^2 + 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x)^2 + (a^2 + 2*b^2 - 2*a*c)*sin(4*x)^2 + 4*(a^2 + 2*b^2 - 2*a*c)*sin(4*
x)*sin(2*x) + 4*(a^2 + 2*b^2 - 2*a*c)*sin(2*x)^2 + a^2 + 2*b^2 - 2*a*c + 2*(a^2 + 2*b^2 - 2*a*c + 2*(a^2 + 2*b
^2 - 2*a*c)*cos(2*x))*cos(4*x) + 4*(a^2 + 2*b^2 - 2*a*c)*cos(2*x))*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) + 4
*(a^2*cos(3*x) + 2*a*b*cos(2*x) - a^2*cos(x) + 2*a*b)*sin(4*x) - 4*(2*a^2*cos(2*x) + a^2)*sin(3*x) - 8*(a^2*co
s(x) - a*b)*sin(2*x))/(a^3*cos(4*x)^2 + 4*a^3*cos(2*x)^2 + a^3*sin(4*x)^2 + 4*a^3*sin(4*x)*sin(2*x) + 4*a^3*si
n(2*x)^2 + 4*a^3*cos(2*x) + a^3 + 2*(2*a^3*cos(2*x) + a^3)*cos(4*x))

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Fricas [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(sec(x)^3/(a+b*cos(x)+c*cos(x)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{3}{\left (x \right )}}{a + b \cos {\left (x \right )} + c \cos ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(x)**3/(a + b*cos(x) + c*cos(x)**2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 12613 vs. \(2 (286) = 572\).
time = 2.78, size = 12613, normalized size = 37.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*a^2*b^7 - 4*a*b^8 + 2*b^9 - 20*a^3*b^5*c + 38*a^2*b^6*c - 12*a*b^7*c - 6*b^8*c + 64*a^4*b^3*c^2 - 110*a^3*
b^4*c^2 - 12*a^2*b^5*c^2 + 54*a*b^6*c^2 + 6*b^7*c^2 - 64*a^5*b*c^3 + 80*a^4*b^2*c^3 + 160*a^3*b^3*c^3 - 140*a^
2*b^4*c^3 - 56*a*b^5*c^3 - 2*b^6*c^3 + 32*a^5*c^4 - 192*a^4*b*c^4 + 64*a^3*b^2*c^4 + 160*a^2*b^3*c^4 + 18*a*b^
4*c^4 + 64*a^4*c^5 - 128*a^3*b*c^5 - 48*a^2*b^2*c^5 + 32*a^3*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^5 - 2*(b^2 - 4*a*c)*a^2*b^5 - 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^6 + 4*(b^2 - 4*a*c)*a*b^6 - 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^7 - 2*(b^2 - 4*a*c)*b^7 - 18*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^3*c + 12*(b^2 - 4*a*c)*a^3*b^3*c + 9*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*c - 22*(b^2 - 4*a*c)*a^2*b^4*c + 46*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*c + 4*(b^2 - 4*a*c)*a*b^5*c + 11*sqrt
(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^6*c + 6*(b^2 - 4*a*c)*b^6*c + 24*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 - 16*(b^2 - 4*a*c)*a^4*
b*c^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^3*b^2*c^2 + 22*(b^2 -
4*a*c)*a^3*b^2*c^2 - 134*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^2 + 28*(b^2 - 4*a*c)*a^2*b^3*c^2 - 75*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2
- 4*a*c)*a*b^4*c^2 - 30*(b^2 - 4*a*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
))*sqrt(b^2 - 4*a*c)*b^5*c^2 - 6*(b^2 - 4*a*c)*b^5*c^2 - 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^3 + 8*(b^2 - 4*a*c)*a^4*c^3 + 120*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^3 - 48*(b^2 - 4*a*c)*a^3*b*c^3 + 138*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^3 + 20*(b^2 - 4*a*c)*a^2*b^2*c^3 + 60*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^3 + 32*(b^2 - 4*a*c)*a*b^3*c^3 + 5*sq
rt(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^3 + 2*(b^2 - 4*a*c)*b^4*c^3
- 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^4 + 16*(b^2 - 4*a*c)*
a^3*c^4 - 64*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^4 - 32*(b^2
- 4*a*c)*a^2*b*c^4 - 25*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
^4 - 10*(b^2 - 4*a*c)*a*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^2*c^5 + 8*(b^2 - 4*a*c)*a^2*c^5)*a^2*abs(a - b + c) - (4*a^3*b^7 - 8*a^2*b^8 + 4*a*b^9 - 40*a^4*b^5*c +
84*a^3*b^6*c - 40*a^2*b^7*c - 4*a*b^8*c + 128*a^5*b^3*c^2 - 292*a^4*b^4*c^2 + 120*a^3*b^5*c^2 + 52*a^2*b^6*c^
2 - 4*a*b^7*c^2 - 128*a^6*b*c^3 + 352*a^5*b^2*c^3 - 64*a^4*b^3*c^3 - 232*a^3*b^4*c^3 + 32*a^2*b^5*c^3 + 4*a*b^
6*c^3 - 64*a^6*c^4 - 128*a^5*b*c^4 + 384*a^4*b^2*c^4 - 64*a^3*b^3*c^4 - 36*a^2*b^4*c^4 - 128*a^5*c^5 + 96*a^3*
b^2*c^5 - 64*a^4*c^6 + 3*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*b^5 - 5*sqrt(a^2 - a*
b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^6 - 3*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a -
b + c))*a^2*b^7 + 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^8 - 18*sqrt(a^2 - a*b + b
*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^5*b^3*c + 33*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b
+ c))*a^4*b^4*c + 30*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^5*c - 43*sqrt(a^2 - a*
b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b^6*c - 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a
- b + c))*a*b^7*c + 24*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^6*b*c^2 - 55*sqrt(a^2 -
a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^5*b^2*c^2 - 101*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a
*c)*(a - b + c))*a^4*b^3*c^2 + 132*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^4*c^2 + 4
0*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b^5*c^2 + 12*sqrt(a^2 - a*b + b*c - c^2 + sq
rt(b^2 - 4*a*c)*(a - b + c))*a^6*c^3 + 116*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^5*b*c
^3 - 177*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^4*b^2*c^3 - 72*sqrt(a^2 - a*b + b*c - c
^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^3*b^3*c^3 - 2*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c)
)*a^2*b^4*c^3 + 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^5*c^3 + 68*sqrt(a^2 - a*b +
b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^5*c^4 + 32*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b
+ c))*a^4*b*c^4 - sqrt(a^2 - a*b + b*c - c^2 + ...

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Mupad [B]
time = 14.82, size = 2500, normalized size = 7.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^3*(a + b*cos(x) + c*cos(x)^2)),x)

[Out]

((tan(x/2)^3*(a + 2*b))/a^2 + (tan(x/2)*(a - 2*b))/a^2)/(tan(x/2)^4 - 2*tan(x/2)^2 + 1) - atan(((((((2048*(26*
a^9*b^7 - 12*a^8*b^8 - 18*a^10*b^6 + 6*a^11*b^5 - 2*a^12*b^4 + 48*a^10*c^6 + 176*a^11*c^5 + 176*a^12*c^4 + 16*
a^13*c^3 - 32*a^14*c^2 + 20*a^8*b^7*c + 74*a^9*b^6*c - 144*a^10*b*c^5 - 192*a^10*b^5*c - 352*a^11*b*c^4 + 122*
a^11*b^4*c - 144*a^12*b*c^3 - 40*a^12*b^3*c + 64*a^13*b*c^2 + 16*a^13*b^2*c + 8*a^8*b^4*c^4 - 20*a^8*b^5*c^3 +
4*a^8*b^6*c^2 - 44*a^9*b^2*c^5 + 116*a^9*b^3*c^4 + 10*a^9*b^4*c^3 - 182*a^9*b^5*c^2 - 148*a^10*b^2*c^4 + 496*
a^10*b^3*c^3 - 50*a^10*b^4*c^2 - 260*a^11*b^2*c^3 + 388*a^11*b^3*c^2 - 204*a^12*b^2*c^2))/a^8 - (2048*tan(x/2)
*((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4 - 52*
a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^8*c
- 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^3*(-(4*a*c - b^2)^3
)^(1/2) - 10*a^2*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^8*b^4 - a^6*b^6
+ 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^7*b^2*c^3 - 32*a^8*b^
2*c^2)))^(1/2)*(32*a^16*c + 8*a^10*b^7 - 24*a^11*b^6 + 32*a^12*b^5 - 32*a^13*b^4 + 24*a^14*b^3 - 8*a^15*b^2 +
96*a^12*c^5 + 64*a^13*c^4 - 128*a^14*c^3 - 64*a^15*c^2 - 8*a^10*b^6*c - 56*a^11*b^5*c - 32*a^12*b*c^4 + 184*a^
12*b^4*c + 352*a^13*b*c^3 - 200*a^13*b^3*c + 288*a^14*b*c^2 + 144*a^14*b^2*c + 8*a^10*b^4*c^3 - 8*a^10*b^5*c^2
- 56*a^11*b^2*c^4 + 40*a^11*b^3*c^3 + 96*a^11*b^4*c^2 - 272*a^12*b^2*c^3 + 8*a^12*b^3*c^2 - 320*a^13*b^2*c^2
- 96*a^15*b*c))/a^8)*((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 +
33*a^2*b^4*c^4 - 52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3
)^(1/2) + 12*a*b^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c
^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2
*(a^8*b^4 - a^6*b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^7
*b^2*c^3 - 32*a^8*b^2*c^2)))^(1/2) + (2048*tan(x/2)*(8*a^14*c + 8*a^4*b^11 - 24*a^5*b^10 + 36*a^6*b^9 - 52*a^7
*b^8 + 61*a^8*b^7 - 49*a^9*b^6 + 33*a^10*b^5 - 17*a^11*b^4 + 6*a^12*b^3 - 2*a^13*b^2 + 72*a^8*c^7 - 136*a^9*c^
6 - 192*a^10*c^5 + 168*a^11*c^4 + 80*a^12*c^3 - 64*a^13*c^2 - 8*a^4*b^10*c - 72*a^5*b^9*c + 244*a^6*b^8*c - 30
8*a^7*b^7*c - 88*a^8*b*c^6 + 375*a^8*b^6*c + 496*a^9*b*c^5 - 416*a^9*b^5*c - 16*a^10*b*c^4 + 295*a^10*b^4*c -
328*a^11*b*c^3 - 178*a^11*b^3*c + 184*a^12*b*c^2 + 84*a^12*b^2*c + 8*a^4*b^8*c^3 - 8*a^4*b^9*c^2 - 72*a^5*b^6*
c^4 + 56*a^5*b^7*c^3 + 112*a^5*b^8*c^2 + 220*a^6*b^4*c^5 - 140*a^6*b^5*c^4 - 424*a^6*b^6*c^3 + 80*a^6*b^7*c^2
- 256*a^7*b^2*c^6 + 192*a^7*b^3*c^5 + 416*a^7*b^4*c^4 + 572*a^7*b^5*c^3 - 732*a^7*b^6*c^2 + 64*a^8*b^2*c^5 - 1
152*a^8*b^3*c^4 + 521*a^8*b^4*c^3 + 779*a^8*b^5*c^2 + 234*a^9*b^2*c^4 - 494*a^9*b^3*c^3 - 723*a^9*b^4*c^2 + 18
0*a^10*b^2*c^3 + 770*a^10*b^3*c^2 - 416*a^11*b^2*c^2 - 24*a^13*b*c))/a^8)*((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7
*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4 - 52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3
*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/
2) + 3*a^2*b*c^4*(-(4*a*c - b^2)^3)^(1/2) + 4*a^3*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a^2*b^3*c^2*(-(4*a*c - b
^2)^3)^(1/2) + 6*a*b^5*c*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^8*b^4 - a^6*b^6 + 16*a^8*c^4 + 32*a^9*c^3 + 16*a^10*c
^2 + 10*a^7*b^4*c - 8*a^9*b^2*c + a^6*b^4*c^2 - 8*a^7*b^2*c^3 - 32*a^8*b^2*c^2)))^(1/2) - (2048*(26*a^3*b^11 -
12*a^2*b^12 - 30*a^4*b^10 + 29*a^5*b^9 - 20*a^6*b^8 + 10*a^7*b^7 - 4*a^8*b^6 + a^9*b^5 + 12*a^6*c^8 + 88*a^7*
c^7 + 72*a^8*c^6 - 44*a^9*c^5 - 28*a^10*c^4 + 12*a^11*c^3 + 20*a^2*b^11*c + 98*a^3*b^10*c - 228*a^4*b^9*c + 25
1*a^5*b^8*c - 96*a^6*b*c^7 - 238*a^6*b^7*c - 200*a^7*b*c^6 + 154*a^7*b^6*c + 100*a^8*b*c^5 - 72*a^8*b^5*c + 11
2*a^9*b*c^4 + 27*a^9*b^4*c - 68*a^10*b*c^3 - 6*a^10*b^3*c + 8*a^11*b*c^2 + 8*a^2*b^8*c^4 - 20*a^2*b^9*c^3 + 4*
a^2*b^10*c^2 - 60*a^3*b^6*c^5 + 156*a^3*b^7*c^4 + 2*a^3*b^8*c^3 - 222*a^3*b^9*c^2 + 136*a^4*b^4*c^6 - 388*a^4*
b^5*c^5 - 152*a^4*b^6*c^4 + 856*a^4*b^7*c^3 - 202*a^4*b^8*c^2 - 100*a^5*b^2*c^7 + 364*a^5*b^3*c^6 + 394*a^5*b^
4*c^5 - 1362*a^5*b^5*c^4 - 115*a^5*b^6*c^3 + 635*a^5*b^7*c^2 - 340*a^6*b^2*c^6 + 904*a^6*b^3*c^5 + 583*a^6*b^4
*c^4 - 564*a^6*b^5*c^3 - 655*a^6*b^6*c^2 - 399*a^7*b^2*c^5 + 9*a^7*b^3*c^4 + 536*a^7*b^4*c^3 + 612*a^7*b^5*c^2
- 37*a^8*b^2*c^4 - 524*a^8*b^3*c^3 - 354*a^8*b^4*c^2 + 239*a^9*b^2*c^3 + 145*a^9*b^3*c^2 - 47*a^10*b^2*c^2))/
a^8)*((8*a^4*c^6 - b^10 + 8*a^5*c^5 - b^7*(-(4*a*c - b^2)^3)^(1/2) + b^8*c^2 - 10*a*b^6*c^3 + 33*a^2*b^4*c^4 -
52*a^2*b^6*c^2 - 38*a^3*b^2*c^5 + 96*a^3*b^4*c^3 - 66*a^4*b^2*c^4 + b^5*c^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b
^8*c - 4*a*b^3*c^3*(-(4*a*c - b^2)^3)^(1/2) + 3...

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