\(\int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx\) [48]

Optimal result

Integrand size = 9, antiderivative size = 227 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\frac {33243}{128} \text {arctanh}(2 \cos (x) \sin (x))+\frac {109312 \log \left (\cos (x)-\sqrt {3} \sin (x)\right )}{243 \sqrt {3}}-\frac {109312 \log \left (\cos (x)+\sqrt {3} \sin (x)\right )}{243 \sqrt {3}}+\frac {715 \tan (x)}{139968}+\frac {11 \tan ^3(x)}{69984}+\frac {\tan ^5(x)}{233280}+\frac {32 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (1-4 \tan ^2(x)+3 \tan ^4(x)\right )^5}-\frac {2 \tan (x) \left (11066113+8997573 \tan ^2(x)\right )}{2657205 \left (1-4 \tan ^2(x)+3 \tan ^4(x)\right )^4}+\frac {\tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{590490 \left (1-4 \tan ^2(x)+3 \tan ^4(x)\right )^3}-\frac {\tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{4723920 \left (1-4 \tan ^2(x)+3 \tan ^4(x)\right )^2}+\frac {7 \tan (x) \left (82529173-118536693 \tan ^2(x)\right )}{1259712 \left (1-4 \tan ^2(x)+3 \tan ^4(x)\right )} \] Output:

33243/128*arctanh(2*cos(x)*sin(x))+109312/729*ln(cos(x)-sin(x)*3^(1/2))*3^ 
(1/2)-109312/729*ln(cos(x)+sin(x)*3^(1/2))*3^(1/2)+715/139968*tan(x)+11/69 
984*tan(x)^3+1/233280*tan(x)^5+32/2657205*tan(x)*(547825-1610707*tan(x)^2) 
/(1-4*tan(x)^2+3*tan(x)^4)^5-2/2657205*tan(x)*(11066113+8997573*tan(x)^2)/ 
(1-4*tan(x)^2+3*tan(x)^4)^4+1/590490*tan(x)*(13572421-26063973*tan(x)^2)/( 
1-4*tan(x)^2+3*tan(x)^4)^3-1/4723920*tan(x)*(470396279-677695389*tan(x)^2) 
/(1-4*tan(x)^2+3*tan(x)^4)^2+7*tan(x)*(82529173-118536693*tan(x)^2)/(12597 
12-5038848*tan(x)^2+3779136*tan(x)^4)
 

Mathematica [A] (verified)

Time = 5.61 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\frac {-839516160 \sqrt {3} \text {arctanh}\left (\sqrt {3} \tan (x)\right )-727024410 \log (\cos (x)-\sin (x))[phone] \log (\cos (x)+\sin (x))+\frac {102789}{(\cos (x)-\sin (x))^4}+\frac {12381336}{(\cos (x)-\sin (x))^2}+\frac {729}{2} (404327 \cos (x)+266847 \cos (3 x)+269093 \cos (5 x)+65353 \cos (7 x)+66452 \cos (9 x)) \sec ^5(2 x) \sin (x)-\frac {102789}{(\cos (x)+\sin (x))^4}-\frac {12381336}{(\cos (x)+\sin (x))^2}-\frac {3538944 \sin (2 x)}{(1-2 \cos (2 x))^4}-\frac {76890112 \sin (2 x)}{(1-2 \cos (2 x))^2}+\frac {442368 \sin (2 x)}{(-1+2 \cos (2 x))^5}+\frac {17842176 \sin (2 x)}{(-1+2 \cos (2 x))^3}+\frac {373850112 \sin (2 x)}{-1+2 \cos (2 x)}+13872 \tan (x)+416 \sec ^2(x) \tan (x)+12 \sec ^4(x) \tan (x)}{2799360} \] Input:

Integrate[(Cos[x] + Cos[5*x])^(-6),x]
 

Output:

(-839516160*Sqrt[3]*ArcTanh[Sqrt[3]*Tan[x]] - 727024410*Log[Cos[x] - Sin[x 
]] + 727024410*Log[Cos[x] + Sin[x]] + 102789/(Cos[x] - Sin[x])^4 + 1238133 
6/(Cos[x] - Sin[x])^2 + (729*(404327*Cos[x] + 266847*Cos[3*x] + 269093*Cos 
[5*x] + 65353*Cos[7*x] + 66452*Cos[9*x])*Sec[2*x]^5*Sin[x])/2 - 102789/(Co 
s[x] + Sin[x])^4 - 12381336/(Cos[x] + Sin[x])^2 - (3538944*Sin[2*x])/(1 - 
2*Cos[2*x])^4 - (76890112*Sin[2*x])/(1 - 2*Cos[2*x])^2 + (442368*Sin[2*x]) 
/(-1 + 2*Cos[2*x])^5 + (17842176*Sin[2*x])/(-1 + 2*Cos[2*x])^3 + (37385011 
2*Sin[2*x])/(-1 + 2*Cos[2*x]) + 13872*Tan[x] + 416*Sec[x]^2*Tan[x] + 12*Se 
c[x]^4*Tan[x])/2799360
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.667, Rules used = {3042, 4823, 27, 1517, 27, 2206, 27, 2206, 27, 2206, 27, 2206, 27, 2205, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\cos (x)+\cos (5 x))^6}dx\)

\(\Big \downarrow \) 4823

\(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^{14}}{64 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^6}d\tan (x)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \int \frac {\left (\tan ^2(x)+1\right )^{14}}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^6}d\tan (x)\)

\(\Big \downarrow \) 1517

\(\displaystyle \frac {1}{64} \left (\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}-\frac {1}{40} \int \frac {8 \left (-885735 \tan ^{24}(x)-13581270 \tan ^{22}(x)-98415000 \tan ^{20}(x)-449100450 \tan ^{18}(x)-1452616335 \tan ^{16}(x)-3560363100 \tan ^{14}(x)-6922807560 \tan ^{12}(x)-11083464900 \tan ^{10}(x)-15130212885 \tan ^8(x)-18252370350 \tan ^6(x)-20179710240 \tan ^4(x)[phone] \tan ^2(x)[phone]\right )}{531441 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}d\tan (x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}-\frac {\int \frac {-885735 \tan ^{24}(x)-13581270 \tan ^{22}(x)-98415000 \tan ^{20}(x)-449100450 \tan ^{18}(x)-1452616335 \tan ^{16}(x)-3560363100 \tan ^{14}(x)-6922807560 \tan ^{12}(x)-11083464900 \tan ^{10}(x)-15130212885 \tan ^8(x)-18252370350 \tan ^6(x)-20179710240 \tan ^4(x)[phone] \tan ^2(x)[phone]}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}d\tan (x)}{2657205}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {1}{64} \left (\frac {\frac {1}{32} \int \frac {864 \left (10935 \tan ^{20}(x)+182250 \tan ^{18}(x)+1454355 \tan ^{16}(x)+7422840 \tan ^{14}(x)[phone] \tan ^{12}(x)[phone] \tan ^{10}(x)[phone] \tan ^8(x)[phone] \tan ^6(x)[phone] \tan ^4(x)[phone] \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}d\tan (x)-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (\frac {27 \int \frac {10935 \tan ^{20}(x)+182250 \tan ^{18}(x)+1454355 \tan ^{16}(x)+7422840 \tan ^{14}(x)[phone] \tan ^{12}(x)[phone] \tan ^{10}(x)[phone] \tan ^8(x)[phone] \tan ^6(x)[phone] \tan ^4(x)[phone] \tan ^2(x)[phone]}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}d\tan (x)-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}-\frac {1}{24} \int \frac {8 \left (-10935 \tan ^{16}(x)-196830 \tan ^{14}(x)-1713150 \tan ^{12}(x)-9641430 \tan ^{10}(x)-39630060 \tan ^8(x)-127568250 \tan ^6(x)-337155090 \tan ^4(x)[phone] \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}d\tan (x)\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}-\frac {1}{3} \int \frac {-10935 \tan ^{16}(x)-196830 \tan ^{14}(x)-1713150 \tan ^{12}(x)-9641430 \tan ^{10}(x)-39630060 \tan ^8(x)-127568250 \tan ^6(x)-337155090 \tan ^4(x)[phone] \tan ^2(x)[phone]}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}d\tan (x)\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {1}{3} \left (\frac {1}{16} \int \frac {240 \left (243 \tan ^{12}(x)+4698 \tan ^{10}(x)+44253 \tan ^8(x)+271692 \tan ^6(x)+1228173 \tan ^4(x)[phone] \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}d\tan (x)-\frac {4 \tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}\right )+\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {1}{3} \left (15 \int \frac {243 \tan ^{12}(x)+4698 \tan ^{10}(x)+44253 \tan ^8(x)+271692 \tan ^6(x)+1228173 \tan ^4(x)[phone] \tan ^2(x)[phone]}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}d\tan (x)-\frac {4 \tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}\right )+\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {1}{3} \left (15 \left (\frac {7 \tan (x) \left (82529173-118536693 \tan ^2(x)\right )}{3 \tan ^4(x)-4 \tan ^2(x)+1}-\frac {1}{8} \int \frac {72 \left (-9 \tan ^8(x)-186 \tan ^6(x)-1884 \tan ^4(x)[phone] \tan ^2(x)[phone]\right )}{3 \tan ^4(x)-4 \tan ^2(x)+1}d\tan (x)\right )-\frac {4 \tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}\right )+\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {1}{3} \left (15 \left (\frac {7 \tan (x) \left (82529173-118536693 \tan ^2(x)\right )}{3 \tan ^4(x)-4 \tan ^2(x)+1}-9 \int \frac {-9 \tan ^8(x)-186 \tan ^6(x)-1884 \tan ^4(x)[phone] \tan ^2(x)[phone]}{3 \tan ^4(x)-4 \tan ^2(x)+1}d\tan (x)\right )-\frac {4 \tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}\right )+\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 2205

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {1}{3} \left (15 \left (\frac {7 \tan (x) \left (82529173-118536693 \tan ^2(x)\right )}{3 \tan ^4(x)-4 \tan ^2(x)+1}-9 \int \left (-3 \tan ^4(x)-66 \tan ^2(x)+\frac {63 \left (1463173 \tan ^2(x)+844841\right )}{3 \tan ^4(x)-4 \tan ^2(x)+1}-715\right )d\tan (x)\right )-\frac {4 \tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}\right )+\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{64} \left (\frac {27 \left (\frac {1}{3} \left (15 \left (\frac {7 \tan (x) \left (82529173-118536693 \tan ^2(x)\right )}{3 \tan ^4(x)-4 \tan ^2(x)+1}-9 \left (-72702441 \text {arctanh}(\tan (x))[phone] \sqrt {3} \text {arctanh}\left (\sqrt {3} \tan (x)\right )-\frac {3}{5} \tan ^5(x)-22 \tan ^3(x)-715 \tan (x)\right )\right )-\frac {4 \tan (x) \left (470396279-677695389 \tan ^2(x)\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^2}\right )+\frac {32 \tan (x) \left (13572421-26063973 \tan ^2(x)\right )}{3 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^3}\right )-\frac {128 \tan (x) \left (8997573 \tan ^2(x)[phone]\right )}{\left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^4}}{2657205}+\frac {2048 \tan (x) \left (547825-1610707 \tan ^2(x)\right )}{2657205 \left (3 \tan ^4(x)-4 \tan ^2(x)+1\right )^5}\right )\)

Input:

Int[(Cos[x] + Cos[5*x])^(-6),x]
 

Output:

((2048*Tan[x]*(547825 - 1610707*Tan[x]^2))/(2657205*(1 - 4*Tan[x]^2 + 3*Ta 
n[x]^4)^5) + ((-128*Tan[x]*(11066113 + 8997573*Tan[x]^2))/(1 - 4*Tan[x]^2 
+ 3*Tan[x]^4)^4 + 27*((32*Tan[x]*(13572421 - 26063973*Tan[x]^2))/(3*(1 - 4 
*Tan[x]^2 + 3*Tan[x]^4)^3) + ((-4*Tan[x]*(470396279 - 677695389*Tan[x]^2)) 
/(1 - 4*Tan[x]^2 + 3*Tan[x]^4)^2 + 15*((7*Tan[x]*(82529173 - 118536693*Tan 
[x]^2))/(1 - 4*Tan[x]^2 + 3*Tan[x]^4) - 9*(-72702441*ArcTanh[Tan[x]] + 419 
75808*Sqrt[3]*ArcTanh[Sqrt[3]*Tan[x]] - 715*Tan[x] - 22*Tan[x]^3 - (3*Tan[ 
x]^5)/5)))/3))/2657205)/64
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1517

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + 
c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + 
c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* 
a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* 
(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p 
 + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] 
 + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* 
x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ 
[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2205

Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte 
grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 
2] && Expon[Px, x^2] > 1
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4823

Int[(cos[(m_.)*((c_.) + (d_.)*(x_))]*(a_.) + cos[(n_.)*((c_.) + (d_.)*(x_)) 
]*(b_.))^(p_), x_Symbol] :> Simp[1/d   Subst[Int[Simplify[TrigExpand[a*Cos[ 
m*ArcTan[x]] + b*Cos[n*ArcTan[x]]]]^p/(1 + x^2), x], x, Tan[c + d*x]], x] / 
; FreeQ[{a, b, c, d}, x] && ILtQ[p/2, 0] && IntegerQ[(m - 1)/2] && IntegerQ 
[(n - 1)/2]
 

Maple [A] (verified)

Time = 165.00 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {\tan \left (x \right )^{5}}{233280}+\frac {11 \tan \left (x \right )^{3}}{69984}+\frac {715 \tan \left (x \right )}{139968}+\frac {-\frac {1882496 \tan \left (x \right )^{9}}{81}+28672 \tan \left (x \right )^{7}-\frac {16312576 \tan \left (x \right )^{5}}{1215}+\frac {6157312 \tan \left (x \right )^{3}}{2187}-\frac {487040 \tan \left (x \right )}{2187}}{\left (3 \tan \left (x \right )^{2}-1\right )^{5}}-\frac {218624 \sqrt {3}\, \operatorname {arctanh}\left (\tan \left (x \right ) \sqrt {3}\right )}{729}-\frac {1}{80 \left (\tan \left (x \right )+1\right )^{5}}-\frac {7}{64 \left (\tan \left (x \right )+1\right )^{4}}-\frac {173}{192 \left (\tan \left (x \right )+1\right )^{3}}-\frac {109}{16 \left (\tan \left (x \right )+1\right )^{2}}-\frac {7931}{128 \left (\tan \left (x \right )+1\right )}+\frac {33243 \ln \left (\tan \left (x \right )+1\right )}{128}-\frac {1}{80 \left (\tan \left (x \right )-1\right )^{5}}+\frac {7}{64 \left (\tan \left (x \right )-1\right )^{4}}-\frac {173}{192 \left (\tan \left (x \right )-1\right )^{3}}+\frac {109}{16 \left (\tan \left (x \right )-1\right )^{2}}-\frac {7931}{128 \left (\tan \left (x \right )-1\right )}-\frac {33243 \ln \left (\tan \left (x \right )-1\right )}{128}\)	\(165\)
risch	\(-\frac {i \left (-21150720-5410405 \,{\mathrm e}^{2 i x}-88263680 \,{\mathrm e}^{4 i x}-702189680 \,{\mathrm e}^{14 i x}-359783050 \,{\mathrm e}^{12 i x}-905873280 \,{\mathrm e}^{16 i x}-1305280104 \,{\mathrm e}^{20 i x}-443179392 \,{\mathrm e}^{10 i x}-159854585 \,{\mathrm e}^{8 i x}-119342210 \,{\mathrm e}^{6 i x}-742867375 \,{\mathrm e}^{18 i x}-1206807525 \,{\mathrm e}^{26 i x}-612290560 \,{\mathrm e}^{28 i x}-1014018075 \,{\mathrm e}^{24 i x}-974013440 \,{\mathrm e}^{22 i x}-830942616 \,{\mathrm e}^{30 i x}-257416320 \,{\mathrm e}^{34 i x}-63231350 \,{\mathrm e}^{38 i x}+5410405 \,{\mathrm e}^{48 i x}-526175825 \,{\mathrm e}^{32 i x}-17489920 \,{\mathrm e}^{46 i x}-355346320 \,{\mathrm e}^{36 i x}-51652615 \,{\mathrm e}^{42 i x}[phone] \,{\mathrm e}^{44 i x}-85588608 \,{\mathrm e}^{40 i x}\right )}{77760 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+1\right )^{5}}-\frac {33243 \ln \left ({\mathrm e}^{2 i x}-i\right )}{128}+\frac {109312 \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{729}-\frac {109312 \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{729}+\frac {33243 \ln \left ({\mathrm e}^{2 i x}+i\right )}{128}\)	\(254\)

Input:

int(1/(cos(x)+cos(5*x))^6,x,method=_RETURNVERBOSE)
 

Output:

1/233280*tan(x)^5+11/69984*tan(x)^3+715/139968*tan(x)+128/9*(-14707/9*tan( 
x)^9+2016*tan(x)^7-127442/135*tan(x)^5+48104/243*tan(x)^3-3805/243*tan(x)) 
/(3*tan(x)^2-1)^5-218624/729*3^(1/2)*arctanh(tan(x)*3^(1/2))-1/80/(tan(x)+ 
1)^5-7/64/(tan(x)+1)^4-173/192/(tan(x)+1)^3-109/16/(tan(x)+1)^2-7931/128/( 
tan(x)+1)+33243/128*ln(tan(x)+1)-1/80/(tan(x)-1)^5+7/64/(tan(x)-1)^4-173/1 
92/(tan(x)-1)^3+109/16/(tan(x)-1)^2-7931/128/(tan(x)-1)-33243/128*ln(tan(x 
)-1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (197) = 394\).

Time = 0.33 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.01 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx =\text {Too large to display} \] Input:

integrate(1/(cos(x)+cos(5*x))^6,x, algorithm="fricas")
 

Output:

1/933120*(121170735*(32768*cos(x)^25 - 204800*cos(x)^23 + 573440*cos(x)^21 
 - 947200*cos(x)^19 + 1022080*cos(x)^17 - 752800*cos(x)^15 + 383280*cos(x) 
^13 - 133200*cos(x)^11 + 30240*cos(x)^9 - 4050*cos(x)^7 + 243*cos(x)^5)*lo 
g(2*cos(x)*sin(x) + 1) - 121170735*(32768*cos(x)^25 - 204800*cos(x)^23 + 5 
73440*cos(x)^21 - 947200*cos(x)^19 + 1022080*cos(x)^17 - 752800*cos(x)^15 
+ 383280*cos(x)^13 - 133200*cos(x)^11 + 30240*cos(x)^9 - 4050*cos(x)^7 + 2 
43*cos(x)^5)*log(-2*cos(x)*sin(x) + 1) + 69959680*(32768*sqrt(3)*cos(x)^25 
 - 204800*sqrt(3)*cos(x)^23 + 573440*sqrt(3)*cos(x)^21 - 947200*sqrt(3)*co 
s(x)^19 + 1022080*sqrt(3)*cos(x)^17 - 752800*sqrt(3)*cos(x)^15 + 383280*sq 
rt(3)*cos(x)^13 - 133200*sqrt(3)*cos(x)^11 + 30240*sqrt(3)*cos(x)^9 - 4050 
*sqrt(3)*cos(x)^7 + 243*sqrt(3)*cos(x)^5)*log(-(8*cos(x)^4 - 4*(2*sqrt(3)* 
cos(x)^3 - 3*sqrt(3)*cos(x))*sin(x) - 9)/(16*cos(x)^4 - 24*cos(x)^2 + 9)) 
+ 12*(346533396480*cos(x)^24 - 1948244992000*cos(x)^22 + 4842857543680*cos 
(x)^20 - 6985286225920*cos(x)^18 + 6442664756480*cos(x)^16 - 3940300651616 
*cos(x)^14 + 1598007058712*cos(x)^12 - 414416003556*cos(x)^10 + 6236466975 
0*cos(x)^8 - 4150039815*cos(x)^6 + 56916*cos(x)^4 + 1458*cos(x)^2 + 81)*si 
n(x))/(32768*cos(x)^25 - 204800*cos(x)^23 + 573440*cos(x)^21 - 947200*cos( 
x)^19 + 1022080*cos(x)^17 - 752800*cos(x)^15 + 383280*cos(x)^13 - 133200*c 
os(x)^11 + 30240*cos(x)^9 - 4050*cos(x)^7 + 243*cos(x)^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\text {Timed out} \] Input:

integrate(1/(cos(x)+cos(5*x))**6,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(cos(x)+cos(5*x))^6,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\frac {1}{233280} \, \tan \left (x\right )^{5} + \frac {11}{69984} \, \tan \left (x\right )^{3} + \frac {109312}{729} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 6 \, \tan \left (x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 6 \, \tan \left (x\right ) \right |}}\right ) - \frac {37339058295 \, \tan \left (x\right )^{19} - 225138333735 \, \tan \left (x\right )^{17} + 584006928624 \, \tan \left (x\right )^{15} - 852406796600 \, \tan \left (x\right )^{13} + 769144293830 \, \tan \left (x\right )^{11} - 444064396750 \, \tan \left (x\right )^{9} + 164043621960 \, \tan \left (x\right )^{7} - 37463326112 \, \tan \left (x\right )^{5} + 4814040715 \, \tan \left (x\right )^{3} - 266132275 \, \tan \left (x\right )}{699840 \, {\left (3 \, \tan \left (x\right )^{4} - 4 \, \tan \left (x\right )^{2} + 1\right )}^{5}} + \frac {33243}{128} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) - \frac {33243}{128} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) + \frac {715}{139968} \, \tan \left (x\right ) \] Input:

integrate(1/(cos(x)+cos(5*x))^6,x, algorithm="giac")
 

Output:

1/233280*tan(x)^5 + 11/69984*tan(x)^3 + 109312/729*sqrt(3)*log(abs(-2*sqrt 
(3) + 6*tan(x))/abs(2*sqrt(3) + 6*tan(x))) - 1/699840*(37339058295*tan(x)^ 
19 - 225138333735*tan(x)^17 + 584006928624*tan(x)^15 - 852406796600*tan(x) 
^13 + 769144293830*tan(x)^11 - 444064396750*tan(x)^9 + 164043621960*tan(x) 
^7 - 37463326112*tan(x)^5 + 4814040715*tan(x)^3 - 266132275*tan(x))/(3*tan 
(x)^4 - 4*tan(x)^2 + 1)^5 + 33243/128*log(abs(tan(x) + 1)) - 33243/128*log 
(abs(tan(x) - 1)) + 715/139968*tan(x)
 

Mupad [B] (verification not implemented)

Time = 23.76 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\frac {33243\,\mathrm {atanh}\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )}{64}-\frac {218624\,\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sin \left (x\right )}{\cos \left (x\right )}\right )}{729}+\frac {1039600189440\,\sin \left (x\right )\,{\cos \left (x\right )}^{24}-5844734976000\,\sin \left (x\right )\,{\cos \left (x\right )}^{22}[phone]\,\sin \left (x\right )\,{\cos \left (x\right )}^{20}-20955858677760\,\sin \left (x\right )\,{\cos \left (x\right )}^{18}[phone]\,\sin \left (x\right )\,{\cos \left (x\right )}^{16}-11820901954848\,\sin \left (x\right )\,{\cos \left (x\right )}^{14}[phone]\,\sin \left (x\right )\,{\cos \left (x\right )}^{12}-1243248010668\,\sin \left (x\right )\,{\cos \left (x\right )}^{10}[phone]\,\sin \left (x\right )\,{\cos \left (x\right )}^8-12450119445\,\sin \left (x\right )\,{\cos \left (x\right )}^6+170748\,\sin \left (x\right )\,{\cos \left (x\right )}^4+4374\,\sin \left (x\right )\,{\cos \left (x\right )}^2+243\,\sin \left (x\right )}{7644119040\,{\cos \left (x\right )}^{25}-47775744000\,{\cos \left (x\right )}^{23}[phone]\,{\cos \left (x\right )}^{21}-220962816000\,{\cos \left (x\right )}^{19}[phone]\,{\cos \left (x\right )}^{17}-175613184000\,{\cos \left (x\right )}^{15}[phone]\,{\cos \left (x\right )}^{13}-31072896000\,{\cos \left (x\right )}^{11}[phone]\,{\cos \left (x\right )}^9-944784000\,{\cos \left (x\right )}^7+56687040\,{\cos \left (x\right )}^5} \] Input:

int(1/(cos(5*x) + cos(x))^6,x)
                                                                                    
                                                                                    
 

Output:

(33243*atanh(sin(x)/cos(x)))/64 - (218624*3^(1/2)*atanh((3^(1/2)*sin(x))/c 
os(x)))/729 + (243*sin(x) + 4374*cos(x)^2*sin(x) + 170748*cos(x)^4*sin(x) 
- 12450119445*cos(x)^6*sin(x) + 187094009250*cos(x)^8*sin(x) - 12432480106 
68*cos(x)^10*sin(x) + 4794021176136*cos(x)^12*sin(x) - 11820901954848*cos( 
x)^14*sin(x) + 19327994269440*cos(x)^16*sin(x) - 20955858677760*cos(x)^18* 
sin(x) + 14528572631040*cos(x)^20*sin(x) - 5844734976000*cos(x)^22*sin(x) 
+ 1039600189440*cos(x)^24*sin(x))/(56687040*cos(x)^5 - 944784000*cos(x)^7 
+ 7054387200*cos(x)^9 - 31072896000*cos(x)^11 + 89411558400*cos(x)^13 - 17 
5613184000*cos(x)^15 + 238430822400*cos(x)^17 - 220962816000*cos(x)^19 + 1 
33772083200*cos(x)^21 - 47775744000*cos(x)^23 + 7644119040*cos(x)^25)
 

Reduce [F]

\[ \int \frac {1}{(\cos (x)+\cos (5 x))^6} \, dx=\int \frac {1}{\cos \left (5 x \right )^{6}+6 \cos \left (5 x \right )^{5} \cos \left (x \right )+15 \cos \left (5 x \right )^{4} \cos \left (x \right )^{2}+20 \cos \left (5 x \right )^{3} \cos \left (x \right )^{3}+15 \cos \left (5 x \right )^{2} \cos \left (x \right )^{4}+6 \cos \left (5 x \right ) \cos \left (x \right )^{5}+\cos \left (x \right )^{6}}d x \] Input:

int(1/(cos(x)+cos(5*x))^6,x)
 

Output:

int(1/(cos(5*x)**6 + 6*cos(5*x)**5*cos(x) + 15*cos(5*x)**4*cos(x)**2 + 20* 
cos(5*x)**3*cos(x)**3 + 15*cos(5*x)**2*cos(x)**4 + 6*cos(5*x)*cos(x)**5 + 
cos(x)**6),x)