3.1.21 \(\int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx\) [21]

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

[Out]

-523/256*arctanh(sin(x))+1/32*sin(x)/(1-2*sin(x)^2)^4-17/192*sin(x)/(1-2*sin(x)^2)^3+203/768*sin(x)/(1-2*sin(x
)^2)^2-437/512*sin(x)/(1-2*sin(x)^2)+1483/1024*arctanh(sin(x)*2^(1/2))*2^(1/2)-43/256*sec(x)*tan(x)-1/128*sec(
x)^3*tan(x)

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(786\) vs. \(2(108)=216\).
time = 0.79, 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, 2098, 213, 652, 628, 632, 212} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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 212

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

Rule 213

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

Rule 628

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 632

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 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 2098

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]

Rubi steps

\begin {align*} \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx &=2 \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {9}{4} \text {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-4 \text {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 \text {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} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-11 \text {Subst}\left (\int \frac {2+x}{\left (-1-2 x+x^2\right )^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+11 \text {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} \text {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {15}{64} \text {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {129}{32} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {9}{2} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {55}{4} \text {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} \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )-\frac {15}{32} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )-\frac {129}{128} \text {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {129}{128} \text {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {165}{32} \text {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} \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {165}{128} \text {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {129}{64} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {129}{64} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {595}{128} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {595}{512} \text {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {165}{64} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )-\frac {165}{64} \text {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} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {595}{256} \text {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] Result contains complex when optimal does not.
time = 6.23, size = 478, normalized size = 4.43 \begin {gather*} -\frac {1483 i \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )-\sqrt {2} \sin \left (\frac {x}{2}\right )}{-\cos \left (\frac {x}{2}\right )+\sqrt {2} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )}{1024 \sqrt {2}}+\frac {\left (\frac {1483}{2048}+\frac {1483 i}{2048}\right ) \left ((-1-i)+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )-\sqrt {2} \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sqrt {2} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )}{(-1+i)+\sqrt {2}}+\frac {523}{256} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\frac {523}{256} \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {1483 \log \left (\sqrt {2}+2 \sin (x)\right )}{1024 \sqrt {2}}-\frac {1483 \log \left (2-\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )}{2048 \sqrt {2}}+\frac {\left (\frac {1483}{4096}-\frac {1483 i}{4096}\right ) \left ((-1-i)+\sqrt {2}\right ) \log \left (2+\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )}{(-1+i)+\sqrt {2}}-\frac {1}{512 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}-\frac {43}{512 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {1}{512 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4}+\frac {43}{512 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}-\frac {17}{768 (\cos (x)-\sin (x))^3}-\frac {437}{1024 (\cos (x)-\sin (x))}+\frac {\sin (x)}{128 (\cos (x)-\sin (x))^4}+\frac {83 \sin (x)}{512 (\cos (x)-\sin (x))^2}+\frac {\sin (x)}{128 (\cos (x)+\sin (x))^4}+\frac {17}{768 (\cos (x)+\sin (x))^3}+\frac {83 \sin (x)}{512 (\cos (x)+\sin (x))^2}+\frac {437}{1024 (\cos (x)+\sin (x))} \end {gather*}

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

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(Cos[x] + Cos[3*x])^5,x]')

[Out]

Timed out

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Maple [A]
time = 0.13, size = 95, normalized size = 0.88

method	result	size
default	\(-\frac {4 \left (-\frac {437 \left (\sin ^{7}\left (x \right )\right )}{256}+\frac {3527 \left (\sin ^{5}\left (x \right )\right )}{1536}-\frac {3257 \left (\sin ^{3}\left (x \right )\right )}{3072}+\frac {331 \sin \left (x \right )}{2048}\right )}{\left (2 \left (\sin ^{2}\left (x \right )\right )-1\right )^{4}}+\frac {1483 \arctanh \left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{1024}-\frac {1}{512 \left (\sin \left (x \right )-1\right )^{2}}+\frac {43}{512 \left (\sin \left (x \right )-1\right )}+\frac {523 \ln \left (\sin \left (x \right )-1\right )}{512}+\frac {1}{512 \left (\sin \left (x \right )+1\right )^{2}}+\frac {43}{512 \left (\sin \left (x \right )+1\right )}-\frac {523 \ln \left (\sin \left (x \right )+1\right )}{512}\)	\(95\)
risch	\(\frac {i \left (1827 \,{\mathrm e}^{23 i x}+3733 \,{\mathrm e}^{21 i x}+6115 \,{\mathrm e}^{19 i x}+9109 \,{\mathrm e}^{17 i x}+5746 \,{\mathrm e}^{15 i x}+2382 \,{\mathrm e}^{13 i x}-2382 \,{\mathrm e}^{11 i x}-5746 \,{\mathrm e}^{9 i x}-9109 \,{\mathrm e}^{7 i x}-6115 \,{\mathrm e}^{5 i x}-3733 \,{\mathrm e}^{3 i x}-1827 \,{\mathrm e}^{i x}\right )}{1536 \left ({\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}+1\right )^{4}}+\frac {523 \ln \left ({\mathrm e}^{i x}-i\right )}{256}-\frac {523 \ln \left ({\mathrm e}^{i x}+i\right )}{256}+\frac {1483 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{2048}-\frac {1483 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{2048}\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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/(sin(x)-1)^2+43/512/(sin(x)-1)+523/512*ln(sin(x)-1)+1/512/(sin(x)+1)^2+43/512/
(sin(x)+1)-523/512*ln(sin(x)+1)

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

[Out]

Timed out

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (88) = 176\).
time = 0.39, size = 219, normalized size = 2.03 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\cos {\left (x \right )} + \cos {\left (3 x \right )}\right )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cos(x) + cos(3*x))**(-5), x)

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Giac [A]
time = 0.00, size = 128, normalized size = 1.19 \begin {gather*} 2 \left (-\frac {-43 \sin ^{3}x+45 \sin x}{512 \left (\sin ^{2}x-1\right )^{2}}+\frac {10488 \sin ^{7}x-14108 \sin ^{5}x+6514 \sin ^{3}x-993 \sin x}{3072 \left (2 \sin ^{2}x-1\right )^{4}}+\frac {523}{1024} \ln \left (-\sin x+1\right )-\frac {523}{1024} \ln \left (\sin x+1\right )-\frac {1483 \ln \left (\frac {\left |4 \sin x-2 \sqrt {2}\right |}{\left |4 \sin x+2 \sqrt {2}\right |}\right )}{2\cdot 1024 \sqrt {2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Mupad [B]
time = 1.25, size = 307, normalized size = 2.84 \begin {gather*} -\frac {11492\,\sin \left (3\,x\right )+18218\,\sin \left (5\,x\right )+12230\,\sin \left (7\,x\right )+7466\,\sin \left (9\,x\right )+3654\,\sin \left (11\,x\right )+276144\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+4764\,\sin \left (x\right )+502080\,\cos \left (2\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+389112\,\cos \left (4\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+251040\,\cos \left (6\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+125520\,\cos \left (8\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+50208\,\cos \left (10\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+12552\,\cos \left (12\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )-97878\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )-177960\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (2\,x\right )-137919\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (4\,x\right )-88980\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (6\,x\right )-44490\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (8\,x\right )-17796\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (10\,x\right )-4449\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (12\,x\right )}{122880\,\cos \left (2\,x\right )+95232\,\cos \left (4\,x\right )+61440\,\cos \left (6\,x\right )+30720\,\cos \left (8\,x\right )+12288\,\cos \left (10\,x\right )+3072\,\cos \left (12\,x\right )+67584} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(3*x) + cos(x))^5,x)

[Out]

-(11492*sin(3*x) + 18218*sin(5*x) + 12230*sin(7*x) + 7466*sin(9*x) + 3654*sin(11*x) + 276144*atanh(sin(x/2)/co
s(x/2)) + 4764*sin(x) + 502080*cos(2*x)*atanh(sin(x/2)/cos(x/2)) + 389112*cos(4*x)*atanh(sin(x/2)/cos(x/2)) +
251040*cos(6*x)*atanh(sin(x/2)/cos(x/2)) + 125520*cos(8*x)*atanh(sin(x/2)/cos(x/2)) + 50208*cos(10*x)*atanh(si
n(x/2)/cos(x/2)) + 12552*cos(12*x)*atanh(sin(x/2)/cos(x/2)) - 97878*2^(1/2)*atanh(2^(1/2)*sin(x)) - 177960*2^(
1/2)*atanh(2^(1/2)*sin(x))*cos(2*x) - 137919*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(4*x) - 88980*2^(1/2)*atanh(2^(1
/2)*sin(x))*cos(6*x) - 44490*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(8*x) - 17796*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(
10*x) - 4449*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(12*x))/(122880*cos(2*x) + 95232*cos(4*x) + 61440*cos(6*x) + 307
20*cos(8*x) + 12288*cos(10*x) + 3072*cos(12*x) + 67584)

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