3.21 \(\int \frac{1}{(\cos (x)+\cos (3 x))^5} \, dx\)

Optimal. Leaf size=108 \[ -\frac{437 \sin (x)}{512 \left (1-2 \sin ^2(x)\right )}+\frac{203 \sin (x)}{768 \left (1-2 \sin ^2(x)\right )^2}-\frac{17 \sin (x)}{192 \left (1-2 \sin ^2(x)\right )^3}+\frac{\sin (x)}{32 \left (1-2 \sin ^2(x)\right )^4}-\frac{523}{256} \tanh ^{-1}(\sin (x))+\frac{1483 \tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{512 \sqrt{2}}-\frac{1}{128} \tan (x) \sec ^3(x)-\frac{43}{256} \tan (x) \sec (x) \]

[Out]

(-523*ArcTanh[Sin[x]])/256 + (1483*ArcTanh[Sqrt[2]*Sin[x]])/(512*Sqrt[2]) + Sin[x]/(32*(1 - 2*Sin[x]^2)^4) - (
17*Sin[x])/(192*(1 - 2*Sin[x]^2)^3) + (203*Sin[x])/(768*(1 - 2*Sin[x]^2)^2) - (437*Sin[x])/(512*(1 - 2*Sin[x]^
2)) - (43*Sec[x]*Tan[x])/256 - (Sec[x]^3*Tan[x])/128

________________________________________________________________________________________

Rubi [B]  time = 1.11659, antiderivative size = 786, normalized size of antiderivative = 7.28, number of steps used = 45, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {12, 2073, 207, 638, 614, 618, 206} \[ \frac{451 \left (\tan \left (\frac{x}{2}\right )+1\right )}{512 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )}-\frac{15 \tan \left (\frac{x}{2}\right )+89}{64 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )}-\frac{451 \left (1-\tan \left (\frac{x}{2}\right )\right )}{512 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )^2}-\frac{65 \left (\tan \left (\frac{x}{2}\right )+1\right )}{384 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )^2}+\frac{65 \left (1-\tan \left (\frac{x}{2}\right )\right )}{384 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )^2}+\frac{43 \tan \left (\frac{x}{2}\right )+1}{32 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )^2}+\frac{119 \left (\tan \left (\frac{x}{2}\right )+1\right )}{48 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )^3}-\frac{11 \left (3 \tan \left (\frac{x}{2}\right )+1\right )}{12 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )^3}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )^3}-\frac{119 \left (1-\tan \left (\frac{x}{2}\right )\right )}{48 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )^3}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (-\tan ^2\left (\frac{x}{2}\right )-2 \tan \left (\frac{x}{2}\right )+1\right )^4}+\frac{17 \tan \left (\frac{x}{2}\right )+7}{4 \left (-\tan ^2\left (\frac{x}{2}\right )+2 \tan \left (\frac{x}{2}\right )+1\right )^4}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}-\frac{45}{256 \left (\tan \left (\frac{x}{2}\right )+1\right )}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{47}{256 \left (\tan \left (\frac{x}{2}\right )+1\right )^2}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{1}{64 \left (\tan \left (\frac{x}{2}\right )+1\right )^3}-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{128 \left (\tan \left (\frac{x}{2}\right )+1\right )^4}-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{1483 \log \left (-\sqrt{2} \sin (x)-\sin (x)+\sqrt{2} \cos (x)+\cos (x)+\sqrt{2}+2\right )}{2048 \sqrt{2}}-\frac{1483 \log \left (-\sqrt{2} \sin (x)+\sin (x)-\sqrt{2} \cos (x)+\cos (x)-\sqrt{2}+2\right )}{2048 \sqrt{2}}+\frac{1483 \log \left (\sqrt{2} \sin (x)-\sin (x)-\sqrt{2} \cos (x)+\cos (x)-\sqrt{2}+2\right )}{2048 \sqrt{2}}+\frac{1483 \log \left (\sqrt{2} \sin (x)+\sin (x)+\sqrt{2} \cos (x)+\cos (x)+\sqrt{2}+2\right )}{2048 \sqrt{2}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(Cos[x] + Cos[3*x])^(-5),x]

[Out]

(-523*ArcTanh[Sin[x]])/256 - (1483*Log[2 + Sqrt[2] + Cos[x] + Sqrt[2]*Cos[x] - Sin[x] - Sqrt[2]*Sin[x]])/(2048
*Sqrt[2]) - (1483*Log[2 - Sqrt[2] + Cos[x] - Sqrt[2]*Cos[x] + Sin[x] - Sqrt[2]*Sin[x]])/(2048*Sqrt[2]) + (1483
*Log[2 - Sqrt[2] + Cos[x] - Sqrt[2]*Cos[x] - Sin[x] + Sqrt[2]*Sin[x]])/(2048*Sqrt[2]) + (1483*Log[2 + Sqrt[2]
+ Cos[x] + Sqrt[2]*Cos[x] + Sin[x] + Sqrt[2]*Sin[x]])/(2048*Sqrt[2]) - 1/(128*(1 - Tan[x/2])^4) + 1/(64*(1 - T
an[x/2])^3) - 47/(256*(1 - Tan[x/2])^2) + 45/(256*(1 - Tan[x/2])) + 1/(128*(1 + Tan[x/2])^4) - 1/(64*(1 + Tan[
x/2])^3) + 47/(256*(1 + Tan[x/2])^2) - 45/(256*(1 + Tan[x/2])) - (7 - 17*Tan[x/2])/(4*(1 - 2*Tan[x/2] - Tan[x/
2]^2)^4) + (119*(1 + Tan[x/2]))/(48*(1 - 2*Tan[x/2] - Tan[x/2]^2)^3) - (11*(1 + 3*Tan[x/2]))/(12*(1 - 2*Tan[x/
2] - Tan[x/2]^2)^3) - (1 - 43*Tan[x/2])/(32*(1 - 2*Tan[x/2] - Tan[x/2]^2)^2) - (65*(1 + Tan[x/2]))/(384*(1 - 2
*Tan[x/2] - Tan[x/2]^2)^2) + (451*(1 + Tan[x/2]))/(512*(1 - 2*Tan[x/2] - Tan[x/2]^2)) - (89 + 15*Tan[x/2])/(64
*(1 - 2*Tan[x/2] - Tan[x/2]^2)) + (7 + 17*Tan[x/2])/(4*(1 + 2*Tan[x/2] - Tan[x/2]^2)^4) + (11*(1 - 3*Tan[x/2])
)/(12*(1 + 2*Tan[x/2] - Tan[x/2]^2)^3) - (119*(1 - Tan[x/2]))/(48*(1 + 2*Tan[x/2] - Tan[x/2]^2)^3) + (65*(1 -
Tan[x/2]))/(384*(1 + 2*Tan[x/2] - Tan[x/2]^2)^2) + (1 + 43*Tan[x/2])/(32*(1 + 2*Tan[x/2] - Tan[x/2]^2)^2) + (8
9 - 15*Tan[x/2])/(64*(1 + 2*Tan[x/2] - Tan[x/2]^2)) - (451*(1 - Tan[x/2]))/(512*(1 + 2*Tan[x/2] - Tan[x/2]^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rule 207

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

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{(\cos (x)+\cos (3 x))^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^{14}}{32 \left (1-7 x^2+7 x^4-x^6\right )^5} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{16} \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^{14}}{\left (1-7 x^2+7 x^4-x^6\right )^5} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{16} \operatorname{Subst}\left (\int \left (\frac{1}{2 (-1+x)^5}+\frac{3}{4 (-1+x)^4}+\frac{47}{8 (-1+x)^3}+\frac{45}{16 (-1+x)^2}-\frac{1}{2 (1+x)^5}+\frac{3}{4 (1+x)^4}-\frac{47}{8 (1+x)^3}+\frac{45}{16 (1+x)^2}+\frac{523}{8 \left (-1+x^2\right )}-\frac{64 (5+12 x)}{\left (-1-2 x+x^2\right )^5}-\frac{176 (2+x)}{\left (-1-2 x+x^2\right )^4}-\frac{4 (21+22 x)}{\left (-1-2 x+x^2\right )^3}+\frac{-52+37 x}{\left (-1-2 x+x^2\right )^2}-\frac{36}{-1-2 x+x^2}+\frac{64 (-5+12 x)}{\left (-1+2 x+x^2\right )^5}+\frac{176 (-2+x)}{\left (-1+2 x+x^2\right )^4}+\frac{4 (-21+22 x)}{\left (-1+2 x+x^2\right )^3}+\frac{-52-37 x}{\left (-1+2 x+x^2\right )^2}-\frac{36}{-1+2 x+x^2}\right ) \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{-52+37 x}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{1}{16} \operatorname{Subst}\left (\int \frac{-52-37 x}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{21+22 x}{\left (-1-2 x+x^2\right )^3} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-21+22 x}{\left (-1+2 x+x^2\right )^3} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{9}{4} \operatorname{Subst}\left (\int \frac{1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{9}{4} \operatorname{Subst}\left (\int \frac{1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-4 \operatorname{Subst}\left (\int \frac{5+12 x}{\left (-1-2 x+x^2\right )^5} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+4 \operatorname{Subst}\left (\int \frac{-5+12 x}{\left (-1+2 x+x^2\right )^5} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{523}{128} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-11 \operatorname{Subst}\left (\int \frac{2+x}{\left (-1-2 x+x^2\right )^4} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+11 \operatorname{Subst}\left (\int \frac{-2+x}{\left (-1+2 x+x^2\right )^4} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}-\frac{11 \left (1+3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{89+15 \tan \left (\frac{x}{2}\right )}{64 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{7+17 \tan \left (\frac{x}{2}\right )}{4 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}+\frac{1+43 \tan \left (\frac{x}{2}\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{15}{64} \operatorname{Subst}\left (\int \frac{1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{15}{64} \operatorname{Subst}\left (\int \frac{1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{129}{32} \operatorname{Subst}\left (\int \frac{1}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{129}{32} \operatorname{Subst}\left (\int \frac{1}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac{x}{2}\right )\right )+\frac{9}{2} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 \tan \left (\frac{x}{2}\right )\right )+\frac{55}{4} \operatorname{Subst}\left (\int \frac{1}{\left (-1-2 x+x^2\right )^3} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{55}{4} \operatorname{Subst}\left (\int \frac{1}{\left (-1+2 x+x^2\right )^3} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{119}{4} \operatorname{Subst}\left (\int \frac{1}{\left (-1-2 x+x^2\right )^4} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{119}{4} \operatorname{Subst}\left (\int \frac{1}{\left (-1+2 x+x^2\right )^4} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{9 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)-\sin (x)-\sqrt{2} \sin (x)\right )}{16 \sqrt{2}}-\frac{9 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)+\sin (x)-\sqrt{2} \sin (x)\right )}{16 \sqrt{2}}+\frac{9 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)-\sin (x)+\sqrt{2} \sin (x)\right )}{16 \sqrt{2}}+\frac{9 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)+\sin (x)+\sqrt{2} \sin (x)\right )}{16 \sqrt{2}}-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{119 \left (1+\tan \left (\frac{x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{11 \left (1+3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{55 \left (1+\tan \left (\frac{x}{2}\right )\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{129 \left (1+\tan \left (\frac{x}{2}\right )\right )}{128 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{89+15 \tan \left (\frac{x}{2}\right )}{64 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{7+17 \tan \left (\frac{x}{2}\right )}{4 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{119 \left (1-\tan \left (\frac{x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}+\frac{55 \left (1-\tan \left (\frac{x}{2}\right )\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{1+43 \tan \left (\frac{x}{2}\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{129 \left (1-\tan \left (\frac{x}{2}\right )\right )}{128 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{15}{32} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac{x}{2}\right )\right )-\frac{15}{32} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 \tan \left (\frac{x}{2}\right )\right )-\frac{129}{128} \operatorname{Subst}\left (\int \frac{1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{129}{128} \operatorname{Subst}\left (\int \frac{1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{165}{32} \operatorname{Subst}\left (\int \frac{1}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{165}{32} \operatorname{Subst}\left (\int \frac{1}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{595}{48} \operatorname{Subst}\left (\int \frac{1}{\left (-1-2 x+x^2\right )^3} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{595}{48} \operatorname{Subst}\left (\int \frac{1}{\left (-1+2 x+x^2\right )^3} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{129 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)-\sin (x)-\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}-\frac{129 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)+\sin (x)-\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}+\frac{129 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)-\sin (x)+\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}+\frac{129 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)+\sin (x)+\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{119 \left (1+\tan \left (\frac{x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{11 \left (1+3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{65 \left (1+\tan \left (\frac{x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{9 \left (1+\tan \left (\frac{x}{2}\right )\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{89+15 \tan \left (\frac{x}{2}\right )}{64 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{7+17 \tan \left (\frac{x}{2}\right )}{4 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{119 \left (1-\tan \left (\frac{x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}+\frac{65 \left (1-\tan \left (\frac{x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{1+43 \tan \left (\frac{x}{2}\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{9 \left (1-\tan \left (\frac{x}{2}\right )\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{165}{128} \operatorname{Subst}\left (\int \frac{1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{165}{128} \operatorname{Subst}\left (\int \frac{1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{129}{64} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac{x}{2}\right )\right )+\frac{129}{64} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 \tan \left (\frac{x}{2}\right )\right )+\frac{595}{128} \operatorname{Subst}\left (\int \frac{1}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )+\frac{595}{128} \operatorname{Subst}\left (\int \frac{1}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{387 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)-\sin (x)-\sqrt{2} \sin (x)\right )}{512 \sqrt{2}}-\frac{387 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)+\sin (x)-\sqrt{2} \sin (x)\right )}{512 \sqrt{2}}+\frac{387 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)-\sin (x)+\sqrt{2} \sin (x)\right )}{512 \sqrt{2}}+\frac{387 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)+\sin (x)+\sqrt{2} \sin (x)\right )}{512 \sqrt{2}}-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{119 \left (1+\tan \left (\frac{x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{11 \left (1+3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{65 \left (1+\tan \left (\frac{x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{451 \left (1+\tan \left (\frac{x}{2}\right )\right )}{512 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{89+15 \tan \left (\frac{x}{2}\right )}{64 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{7+17 \tan \left (\frac{x}{2}\right )}{4 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{119 \left (1-\tan \left (\frac{x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}+\frac{65 \left (1-\tan \left (\frac{x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{1+43 \tan \left (\frac{x}{2}\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{451 \left (1-\tan \left (\frac{x}{2}\right )\right )}{512 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{595}{512} \operatorname{Subst}\left (\int \frac{1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{595}{512} \operatorname{Subst}\left (\int \frac{1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{165}{64} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac{x}{2}\right )\right )-\frac{165}{64} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 \tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{111 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)-\sin (x)-\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}-\frac{111 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)+\sin (x)-\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}+\frac{111 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)-\sin (x)+\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}+\frac{111 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)+\sin (x)+\sqrt{2} \sin (x)\right )}{256 \sqrt{2}}-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{119 \left (1+\tan \left (\frac{x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{11 \left (1+3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{65 \left (1+\tan \left (\frac{x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{451 \left (1+\tan \left (\frac{x}{2}\right )\right )}{512 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{89+15 \tan \left (\frac{x}{2}\right )}{64 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{7+17 \tan \left (\frac{x}{2}\right )}{4 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{119 \left (1-\tan \left (\frac{x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}+\frac{65 \left (1-\tan \left (\frac{x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{1+43 \tan \left (\frac{x}{2}\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{451 \left (1-\tan \left (\frac{x}{2}\right )\right )}{512 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{595}{256} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac{x}{2}\right )\right )+\frac{595}{256} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 \tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{523}{256} \tanh ^{-1}(\sin (x))-\frac{1483 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)-\sin (x)-\sqrt{2} \sin (x)\right )}{2048 \sqrt{2}}-\frac{1483 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)+\sin (x)-\sqrt{2} \sin (x)\right )}{2048 \sqrt{2}}+\frac{1483 \log \left (2-\sqrt{2}+\cos (x)-\sqrt{2} \cos (x)-\sin (x)+\sqrt{2} \sin (x)\right )}{2048 \sqrt{2}}+\frac{1483 \log \left (2+\sqrt{2}+\cos (x)+\sqrt{2} \cos (x)+\sin (x)+\sqrt{2} \sin (x)\right )}{2048 \sqrt{2}}-\frac{1}{128 \left (1-\tan \left (\frac{x}{2}\right )\right )^4}+\frac{1}{64 \left (1-\tan \left (\frac{x}{2}\right )\right )^3}-\frac{47}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )^2}+\frac{45}{256 \left (1-\tan \left (\frac{x}{2}\right )\right )}+\frac{1}{128 \left (1+\tan \left (\frac{x}{2}\right )\right )^4}-\frac{1}{64 \left (1+\tan \left (\frac{x}{2}\right )\right )^3}+\frac{47}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )^2}-\frac{45}{256 \left (1+\tan \left (\frac{x}{2}\right )\right )}-\frac{7-17 \tan \left (\frac{x}{2}\right )}{4 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{119 \left (1+\tan \left (\frac{x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{11 \left (1+3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{1-43 \tan \left (\frac{x}{2}\right )}{32 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}-\frac{65 \left (1+\tan \left (\frac{x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{451 \left (1+\tan \left (\frac{x}{2}\right )\right )}{512 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{89+15 \tan \left (\frac{x}{2}\right )}{64 \left (1-2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}+\frac{7+17 \tan \left (\frac{x}{2}\right )}{4 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^4}+\frac{11 \left (1-3 \tan \left (\frac{x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}-\frac{119 \left (1-\tan \left (\frac{x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^3}+\frac{65 \left (1-\tan \left (\frac{x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{1+43 \tan \left (\frac{x}{2}\right )}{32 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )^2}+\frac{89-15 \tan \left (\frac{x}{2}\right )}{64 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}-\frac{451 \left (1-\tan \left (\frac{x}{2}\right )\right )}{512 \left (1+2 \tan \left (\frac{x}{2}\right )-\tan ^2\left (\frac{x}{2}\right )\right )}\\ \end{align*}

Mathematica [C]  time = 6.29602, size = 478, normalized size = 4.43 \[ \frac{1483 \log \left (2 \sin (x)+\sqrt{2}\right )}{1024 \sqrt{2}}+\frac{83 \sin (x)}{512 (\cos (x)-\sin (x))^2}+\frac{\sin (x)}{128 (\cos (x)-\sin (x))^4}-\frac{437}{1024 (\cos (x)-\sin (x))}+\frac{437}{1024 (\sin (x)+\cos (x))}-\frac{43}{512 \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^2}+\frac{43}{512 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2}+\frac{83 \sin (x)}{512 (\sin (x)+\cos (x))^2}-\frac{17}{768 (\cos (x)-\sin (x))^3}+\frac{17}{768 (\sin (x)+\cos (x))^3}-\frac{1}{512 \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^4}+\frac{1}{512 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4}+\frac{\sin (x)}{128 (\sin (x)+\cos (x))^4}+\frac{523}{256} \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{523}{256} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\frac{1483 \log \left (-\sqrt{2} \sin (x)-\sqrt{2} \cos (x)+2\right )}{2048 \sqrt{2}}+\frac{\left (\frac{1483}{4096}-\frac{1483 i}{4096}\right ) \left (\sqrt{2}+(-1-i)\right ) \log \left (-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+2\right )}{\sqrt{2}+(-1+i)}-\frac{1483 i \tan ^{-1}\left (\frac{-\sqrt{2} \sin \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}{-\sin \left (\frac{x}{2}\right )+\sqrt{2} \cos \left (\frac{x}{2}\right )-\cos \left (\frac{x}{2}\right )}\right )}{1024 \sqrt{2}}+\frac{\left (\frac{1483}{2048}+\frac{1483 i}{2048}\right ) \left (\sqrt{2}+(-1-i)\right ) \tan ^{-1}\left (\frac{-\sqrt{2} \sin \left (\frac{x}{2}\right )+\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}{-\sin \left (\frac{x}{2}\right )+\sqrt{2} \cos \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}\right )}{\sqrt{2}+(-1+i)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cos[x] + Cos[3*x])^(-5),x]

[Out]

(((-1483*I)/1024)*ArcTan[(Cos[x/2] - Sin[x/2] - Sqrt[2]*Sin[x/2])/(-Cos[x/2] + Sqrt[2]*Cos[x/2] - Sin[x/2])])/
Sqrt[2] + ((1483/2048 + (1483*I)/2048)*((-1 - I) + Sqrt[2])*ArcTan[(Cos[x/2] + Sin[x/2] - Sqrt[2]*Sin[x/2])/(C
os[x/2] + Sqrt[2]*Cos[x/2] - Sin[x/2])])/((-1 + I) + Sqrt[2]) + (523*Log[Cos[x/2] - Sin[x/2]])/256 - (523*Log[
Cos[x/2] + Sin[x/2]])/256 + (1483*Log[Sqrt[2] + 2*Sin[x]])/(1024*Sqrt[2]) - (1483*Log[2 - Sqrt[2]*Cos[x] - Sqr
t[2]*Sin[x]])/(2048*Sqrt[2]) + ((1483/4096 - (1483*I)/4096)*((-1 - I) + Sqrt[2])*Log[2 + Sqrt[2]*Cos[x] - Sqrt
[2]*Sin[x]])/((-1 + I) + Sqrt[2]) - 1/(512*(Cos[x/2] - Sin[x/2])^4) - 43/(512*(Cos[x/2] - Sin[x/2])^2) + 1/(51
2*(Cos[x/2] + Sin[x/2])^4) + 43/(512*(Cos[x/2] + Sin[x/2])^2) - 17/(768*(Cos[x] - Sin[x])^3) - 437/(1024*(Cos[
x] - Sin[x])) + Sin[x]/(128*(Cos[x] - Sin[x])^4) + (83*Sin[x])/(512*(Cos[x] - Sin[x])^2) + Sin[x]/(128*(Cos[x]
+ Sin[x])^4) + 17/(768*(Cos[x] + Sin[x])^3) + (83*Sin[x])/(512*(Cos[x] + Sin[x])^2) + 437/(1024*(Cos[x] + Sin
[x]))

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Maple [A]  time = 0.062, size = 95, normalized size = 0.9 \begin{align*} -4\,{\frac{1}{ \left ( 2\, \left ( \sin \left ( x \right ) \right ) ^{2}-1 \right ) ^{4}} \left ( -{\frac{437\, \left ( \sin \left ( x \right ) \right ) ^{7}}{256}}+{\frac{3527\, \left ( \sin \left ( x \right ) \right ) ^{5}}{1536}}-{\frac{3257\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3072}}+{\frac{331\,\sin \left ( x \right ) }{2048}} \right ) }+{\frac{1483\,{\it Artanh} \left ( \sin \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{1024}}-{\frac{1}{512\, \left ( -1+\sin \left ( x \right ) \right ) ^{2}}}+{\frac{43}{-512+512\,\sin \left ( x \right ) }}+{\frac{523\,\ln \left ( -1+\sin \left ( x \right ) \right ) }{512}}+{\frac{1}{512\, \left ( 1+\sin \left ( x \right ) \right ) ^{2}}}+{\frac{43}{512+512\,\sin \left ( x \right ) }}-{\frac{523\,\ln \left ( 1+\sin \left ( x \right ) \right ) }{512}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)+cos(3*x))^5,x)

[Out]

-4*(-437/256*sin(x)^7+3527/1536*sin(x)^5-3257/3072*sin(x)^3+331/2048*sin(x))/(2*sin(x)^2-1)^4+1483/1024*arctan
h(sin(x)*2^(1/2))*2^(1/2)-1/512/(-1+sin(x))^2+43/512/(-1+sin(x))+523/512*ln(-1+sin(x))+1/512/(1+sin(x))^2+43/5
12/(1+sin(x))-523/512*ln(1+sin(x))

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Maxima [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(1/(cos(x)+cos(3*x))^5,x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 2.92037, size = 718, normalized size = 6.65 \begin{align*} \frac{4449 \,{\left (16 \, \sqrt{2} \cos \left (x\right )^{12} - 32 \, \sqrt{2} \cos \left (x\right )^{10} + 24 \, \sqrt{2} \cos \left (x\right )^{8} - 8 \, \sqrt{2} \cos \left (x\right )^{6} + \sqrt{2} \cos \left (x\right )^{4}\right )} \log \left (-\frac{2 \, \cos \left (x\right )^{2} - 2 \, \sqrt{2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - 6276 \,{\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )} \log \left (\sin \left (x\right ) + 1\right ) + 6276 \,{\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 4 \,{\left (14616 \, \cos \left (x\right )^{10} - 25420 \, \cos \left (x\right )^{8} + 15570 \, \cos \left (x\right )^{6} - 3677 \, \cos \left (x\right )^{4} + 162 \, \cos \left (x\right )^{2} + 12\right )} \sin \left (x\right )}{6144 \,{\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)+cos(3*x))^5,x, algorithm="fricas")

[Out]

1/6144*(4449*(16*sqrt(2)*cos(x)^12 - 32*sqrt(2)*cos(x)^10 + 24*sqrt(2)*cos(x)^8 - 8*sqrt(2)*cos(x)^6 + sqrt(2)
*cos(x)^4)*log(-(2*cos(x)^2 - 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1)) - 6276*(16*cos(x)^12 - 32*cos(x)^10 + 24
*cos(x)^8 - 8*cos(x)^6 + cos(x)^4)*log(sin(x) + 1) + 6276*(16*cos(x)^12 - 32*cos(x)^10 + 24*cos(x)^8 - 8*cos(x
)^6 + cos(x)^4)*log(-sin(x) + 1) - 4*(14616*cos(x)^10 - 25420*cos(x)^8 + 15570*cos(x)^6 - 3677*cos(x)^4 + 162*
cos(x)^2 + 12)*sin(x))/(16*cos(x)^12 - 32*cos(x)^10 + 24*cos(x)^8 - 8*cos(x)^6 + cos(x)^4)

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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(1/(cos(x)+cos(3*x))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.11205, size = 140, normalized size = 1.3 \begin{align*} -\frac{1483}{2048} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}\right ) + \frac{43 \, \sin \left (x\right )^{3} - 45 \, \sin \left (x\right )}{256 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac{10488 \, \sin \left (x\right )^{7} - 14108 \, \sin \left (x\right )^{5} + 6514 \, \sin \left (x\right )^{3} - 993 \, \sin \left (x\right )}{1536 \,{\left (2 \, \sin \left (x\right )^{2} - 1\right )}^{4}} - \frac{523}{512} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{523}{512} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)+cos(3*x))^5,x, algorithm="giac")

[Out]

-1483/2048*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x))) + 1/256*(43*sin(x)^3 - 45*sin(x))
/(sin(x)^2 - 1)^2 + 1/1536*(10488*sin(x)^7 - 14108*sin(x)^5 + 6514*sin(x)^3 - 993*sin(x))/(2*sin(x)^2 - 1)^4 -
523/512*log(sin(x) + 1) + 523/512*log(-sin(x) + 1)