3.6 \(\int \frac{\sin ^4(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\)

Optimal. Leaf size=388 \[ \frac{x \left (b^2-c (a+2 c)\right )}{c^3}+\frac{2 \left (b^2 \left (b^2-2 c (a+c)\right )-b \sqrt{b^2-4 a c} \left (b^2-2 c (a+c)\right )-2 c \left (a b^2-c (a+c)^2\right )\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{c^3 \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{2 \left (b^3 \sqrt{b^2-4 a c}-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{c^3 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b \sin (x)}{c^2}+\frac{x}{2 c}+\frac{\sin (x) \cos (x)}{2 c} \]

[Out]

x/(2*c) + ((b^2 - c*(a + 2*c))*x)/c^3 + (2*(b^2*(b^2 - 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)

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Rubi [A]  time = 11.0127, 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 = {3267, 2637, 2635, 8, 3293, 2659, 205} \[ \frac{x \left (b^2-c (a+2 c)\right )}{c^3}+\frac{2 \left (-2 b^2 c (a+c)-b \sqrt{b^2-4 a c} \left (b^2-2 c (a+c)\right )-2 c \left (a b^2-c (a+c)^2\right )+b^4\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{c^3 \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{2 \left (b^3 \sqrt{b^2-4 a c}-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right ) \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{c^3 \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{b \sin (x)}{c^2}+\frac{x}{2 c}+\frac{\sin (x) \cos (x)}{2 c} \]

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 3267

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]^(n_.)*(b_.) + cos[(d_.) + (e_.)*(x_)]^(n2_.)*(c_.))^(p_.)*sin[(d_.) + (e_
.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrig[(1 - cos[d + e*x]^2)^(m/2)*(a + b*cos[d + e*x]^n + c*cos[d + e*x]^
(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && IntegerQ[m/2] && NeQ[b^2 - 4*a*c, 0] && Integ
ersQ[n, p]

Rule 2637

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

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 3293

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

Rule 2659

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

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.843071, size = 374, normalized size = 0.96 \[ \frac{\frac{4 \sqrt{2} \left (b^3 \sqrt{b^2-4 a c}-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right ) \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}+b-2 c\right )}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{4 \sqrt{2} \left (b^3 \sqrt{b^2-4 a c}+2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}-2 c^2 (a+c)^2-b^4\right ) \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}-b+2 c\right )}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-2 c x (2 a+3 c)+4 b^2 x-4 b c \sin (x)+c^2 \sin (2 x)}{4 c^3} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.086, size = 2608, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}

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]

-3/c*arctan(tan(1/2*x))+1/(((-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/(((-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))+2/c*a/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)
*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b-3/c^2/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-
b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*a^2*b+4/c^2*a/(-4*a*c+b^2)^(1/
2)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)
)*b^2-2/c*a/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^
2)^(1/2)+a-c)*(a-b+c))^(1/2))*b+3/c^2/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+
b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*a^2*b-4/c^2*a/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2
)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b^2+1/c^3*a/(-4*a*
c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b
+c))^(1/2))*b^3-1/c^3*a/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x
)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b^3+1/c^2/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*
tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*a^2+2/c*a/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan
((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))+1/c^2/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*a
rctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*a^2+2/c*a/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+
c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))-2/c*b/(((-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))-2/c*b/(((-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))-2*c/(-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))+2*c/(-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))+1/c^3/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c
+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*b^3+1/c^3/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x
)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b^3-4*a/(-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))+4*a/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1
/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))+b/(-4*a*c+b^2)^(
1/2)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/
2))-b/(-4*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))-2/c^2/(tan(1/2*x)^2+1)^2*tan(1/2*x)^3*b-2/c^2/(tan(1/2*x)^2+1)^2*tan(1/2*x)*b-1/c/(t
an(1/2*x)^2+1)^2*tan(1/2*x)^3+1/c/(tan(1/2*x)^2+1)^2*tan(1/2*x)-2/c^2*arctan(tan(1/2*x))*a+2/c^3*arctan(tan(1/
2*x))*b^2-2/c/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*
c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b^2-1/c^3*a/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*
x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*b^2-1/c^3*a/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+
b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b^2-1/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))*b^4+1/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))*b^4-2/c^2*a/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)
*(a-b+c))^(1/2))*b-2/c/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(
((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*a^2-2/c^2*a/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)
*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*b+2/c/(-4*a*c+b^2)^(1/2)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+
c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2))*a^2+2/c/(-4*a*c+b^2)^(1/2)/(((
-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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]  time = 13.3403, size = 10364, normalized size = 26.71 \begin{align*} \text{result too large to display} \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out