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

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Rubi [B]  time = 2.33316, 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 \[ \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}} \]

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] - S
in[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] + C
os[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 - Tan[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) + (89 - 15*Tan[x/2])/(64*(1 + 2*Tan[x/2] - T
an[x/2]^2)) - (451*(1 - Tan[x/2]))/(512*(1 + 2*Tan[x/2] - Tan[x/2]^2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(cos(x)+cos(3*x))**5,x)

[Out]

Timed out

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Mathematica [C]  time = 6.58471, size = 478, normalized size = 4.43 \[ \frac{1483 \log \left (2 \sin (x)+\sqrt{2}\right )}{1024 \sqrt{2}}+\frac{83 \sin (x)}{512 (\cos (x)-\sin (x))^2}+\frac{\sin (x)}{128 (\cos (x)-\sin (x))^4}-\frac{437}{1024 (\cos (x)-\sin (x))}+\frac{437}{1024 (\sin (x)+\cos (x))}-\frac{43}{512 \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^2}+\frac{43}{512 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2}+\frac{83 \sin (x)}{512 (\sin (x)+\cos (x))^2}-\frac{17}{768 (\cos (x)-\sin (x))^3}+\frac{17}{768 (\sin (x)+\cos (x))^3}-\frac{1}{512 \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^4}+\frac{1}{512 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4}+\frac{\sin (x)}{128 (\sin (x)+\cos (x))^4}+\frac{523}{256} \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{523}{256} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\frac{1483 \log \left (-\sqrt{2} \sin (x)-\sqrt{2} \cos (x)+2\right )}{2048 \sqrt{2}}+\frac{\left (\frac{1483}{4096}-\frac{1483 i}{4096}\right ) \left (\sqrt{2}+(-1-i)\right ) \log \left (-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+2\right )}{\sqrt{2}+(-1+i)}-\frac{1483 i \tan ^{-1}\left (\frac{-\sqrt{2} \sin \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}{-\sin \left (\frac{x}{2}\right )+\sqrt{2} \cos \left (\frac{x}{2}\right )-\cos \left (\frac{x}{2}\right )}\right )}{1024 \sqrt{2}}+\frac{\left (\frac{1483}{2048}+\frac{1483 i}{2048}\right ) \left (\sqrt{2}+(-1-i)\right ) \tan ^{-1}\left (\frac{-\sqrt{2} \sin \left (\frac{x}{2}\right )+\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}{-\sin \left (\frac{x}{2}\right )+\sqrt{2} \cos \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}\right )}{\sqrt{2}+(-1+i)} \]

Antiderivative was successfully verified.

[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] + S
qrt[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])/(Cos[x/2] + Sqrt[2]*Co
s[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] - Sqrt[2]*Sin[x]])/(2048*Sqrt[2]) + ((1483/40
96 - (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/(512*(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*(C
os[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 = 0.096, size = 95, normalized size = 0.9 \[{\frac{1}{512\, \left ( 1+\sin \left ( x \right ) \right ) ^{2}}}+{\frac{43}{512+512\,\sin \left ( x \right ) }}-{\frac{523\,\ln \left ( 1+\sin \left ( x \right ) \right ) }{512}}-{\frac{1}{512\, \left ( -1+\sin \left ( x \right ) \right ) ^{2}}}+{\frac{43}{-512+512\,\sin \left ( x \right ) }}+{\frac{523\,\ln \left ( -1+\sin \left ( x \right ) \right ) }{512}}-4\,{\frac{1}{ \left ( 2\, \left ( \sin \left ( x \right ) \right ) ^{2}-1 \right ) ^{4}} \left ( -{\frac{437\, \left ( \sin \left ( x \right ) \right ) ^{7}}{256}}+{\frac{3527\, \left ( \sin \left ( x \right ) \right ) ^{5}}{1536}}-{\frac{3257\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3072}}+{\frac{331\,\sin \left ( x \right ) }{2048}} \right ) }+{\frac{1483\,{\it Artanh} \left ( \sin \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{1024}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(cos(x)+cos(3*x))^5,x)

[Out]

1/512/(1+sin(x))^2+43/512/(1+sin(x))-523/512*ln(1+sin(x))-1/512/(-1+sin(x))^2+43
/512/(-1+sin(x))+523/512*ln(-1+sin(x))-4*(-437/256*sin(x)^7+3527/1536*sin(x)^5-3
257/3072*sin(x)^3+331/2048*sin(x))/(2*sin(x)^2-1)^4+1483/1024*arctanh(sin(x)*2^(
1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(3*x) + cos(x))^(-5),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.299649, size = 370, normalized size = 3.43 \[ \frac{4449 \,{\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )} \log \left (-\frac{2 \, \sqrt{2} \cos \left (x\right )^{2} - 3 \, \sqrt{2} - 4 \, \sin \left (x\right )}{2 \, \cos \left (x\right )^{2} - 1}\right ) - 3138 \,{\left (16 \, \sqrt{2} \cos \left (x\right )^{12} - 32 \, \sqrt{2} \cos \left (x\right )^{10} + 24 \, \sqrt{2} \cos \left (x\right )^{8} - 8 \, \sqrt{2} \cos \left (x\right )^{6} + \sqrt{2} \cos \left (x\right )^{4}\right )} \log \left (\sin \left (x\right ) + 1\right ) + 3138 \,{\left (16 \, \sqrt{2} \cos \left (x\right )^{12} - 32 \, \sqrt{2} \cos \left (x\right )^{10} + 24 \, \sqrt{2} \cos \left (x\right )^{8} - 8 \, \sqrt{2} \cos \left (x\right )^{6} + \sqrt{2} \cos \left (x\right )^{4}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \,{\left (14616 \, \sqrt{2} \cos \left (x\right )^{10} - 25420 \, \sqrt{2} \cos \left (x\right )^{8} + 15570 \, \sqrt{2} \cos \left (x\right )^{6} - 3677 \, \sqrt{2} \cos \left (x\right )^{4} + 162 \, \sqrt{2} \cos \left (x\right )^{2} + 12 \, \sqrt{2}\right )} \sin \left (x\right )}{3072 \,{\left (16 \, \sqrt{2} \cos \left (x\right )^{12} - 32 \, \sqrt{2} \cos \left (x\right )^{10} + 24 \, \sqrt{2} \cos \left (x\right )^{8} - 8 \, \sqrt{2} \cos \left (x\right )^{6} + \sqrt{2} \cos \left (x\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(3*x) + cos(x))^(-5),x, algorithm="fricas")

[Out]

1/3072*(4449*(16*cos(x)^12 - 32*cos(x)^10 + 24*cos(x)^8 - 8*cos(x)^6 + cos(x)^4)
*log(-(2*sqrt(2)*cos(x)^2 - 3*sqrt(2) - 4*sin(x))/(2*cos(x)^2 - 1)) - 3138*(16*s
qrt(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(sin(x) + 1) + 3138*(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(-s
in(x) + 1) - 2*(14616*sqrt(2)*cos(x)^10 - 25420*sqrt(2)*cos(x)^8 + 15570*sqrt(2)
*cos(x)^6 - 3677*sqrt(2)*cos(x)^4 + 162*sqrt(2)*cos(x)^2 + 12*sqrt(2))*sin(x))/(
16*sqrt(2)*cos(x)^12 - 32*sqrt(2)*cos(x)^10 + 24*sqrt(2)*cos(x)^8 - 8*sqrt(2)*co
s(x)^6 + sqrt(2)*cos(x)^4)

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

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/XCAS [A]  time = 0.213761, size = 140, normalized size = 1.3 \[ -\frac{1483}{2048} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}\right ) + \frac{43 \, \sin \left (x\right )^{3} - 45 \, \sin \left (x\right )}{256 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac{10488 \, \sin \left (x\right )^{7} - 14108 \, \sin \left (x\right )^{5} + 6514 \, \sin \left (x\right )^{3} - 993 \, \sin \left (x\right )}{1536 \,{\left (2 \, \sin \left (x\right )^{2} - 1\right )}^{4}} - \frac{523}{512} \,{\rm ln}\left (\sin \left (x\right ) + 1\right ) + \frac{523}{512} \,{\rm ln}\left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(3*x) + cos(x))^(-5),x, algorithm="giac")

[Out]

-1483/2048*sqrt(2)*ln(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*ln(sin(x) +
1) + 523/512*ln(-sin(x) + 1)