Integrand size = 11, antiderivative size = 729 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx =\text {Too large to display} \] Output:
1/512/(1-sin(x))^2-1/512/(1+sin(x))^2-1/24*sin(x)*(75-121*sin(x)^2)/(1-8*s in(x)^2+8*sin(x)^4)^2+1/512*sin(x)*(57-286*sin(x)^2)/(1-8*sin(x)^2+8*sin(x )^4)^2+1/16*sin(x)*(1-6*sin(x)^2)/(1-8*sin(x)^2+8*sin(x)^4)^4-15*sin(x)*(1 3-21*sin(x)^2)/(16-128*sin(x)^2+128*sin(x)^4)-5*sin(x)*(295-441*sin(x)^2)/ (1024-8192*sin(x)^2+8192*sin(x)^4)+1/6*sin(x)*(5-14*sin(x)^2)/(1-8*sin(x)^ 2+8*sin(x)^4)^3+3/8*sin(x)*(9-22*sin(x)^2)/(1-8*sin(x)^2+8*sin(x)^4)^2-1/3 2*sin(x)*(23-35*sin(x)^2)/(1-8*sin(x)^2+8*sin(x)^4)^3+7523/256*arctanh(sin (x))-1/512*(775268+166546*2^(1/2))^(1/2)*arctanh(2*sin(x)/(2-2^(1/2))^(1/2 ))+1/512*(775268-166546*2^(1/2))^(1/2)*arctanh(2*sin(x)/(2+2^(1/2))^(1/2)) -15/16384*(84116+43202*2^(1/2))^(1/2)*arctanh(2*sin(x)/(2+2^(1/2))^(1/2))+ 15/16384*(84116-43202*2^(1/2))^(1/2)*arctanh(2*sin(x)/(2-2^(1/2))^(1/2))-5 /8*(1780+1042*2^(1/2))^(1/2)*arctanh(2*sin(x)/(2+2^(1/2))^(1/2))+5/8*(1780 -1042*2^(1/2))^(1/2)*arctanh(2*sin(x)/(2-2^(1/2))^(1/2))-3/8*(772+146*2^(1 /2))^(1/2)*arctanh(2*sin(x)/(2-2^(1/2))^(1/2))+3/8*(772-146*2^(1/2))^(1/2) *arctanh(2*sin(x)/(2+2^(1/2))^(1/2))-45/256*(340+82*2^(1/2))^(1/2)*arctanh (2*sin(x)/(2+2^(1/2))^(1/2))+45/256*(340-82*2^(1/2))^(1/2)*arctanh(2*sin(x )/(2-2^(1/2))^(1/2))+sin(x)*(13-30*sin(x)^2)/(1-8*sin(x)^2+8*sin(x)^4)+sin (x)*(359-882*sin(x)^2)/(192-1536*sin(x)^2+1536*sin(x)^4)+163/(512-512*sin( x))-163/(512+512*sin(x))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 6.08 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=-\frac {7523}{256} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {7523}{256} \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-\frac {\text {RootSum}\left [1+\text {$\#$1}^8\&,\frac {444922 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-222461 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-184578 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+92289 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-184578 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4+92289 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+444922 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^6-222461 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{\text {$\#$1}^7}\&\right ]}{65536}+\frac {1}{512 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}+\frac {163}{512 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {1}{512 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4}-\frac {163}{512 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {3 \cos (x)-7 \sin (x)}{128 (\cos (2 x)-\sin (2 x))^4}+\frac {-115 \cos (x)+234 \sin (x)}{768 (\cos (2 x)-\sin (2 x))^3}+\frac {4289 \cos (x)-8381 \sin (x)}{6144 (\cos (2 x)-\sin (2 x))^2}+\frac {-13375 \cos (x)+26434 \sin (x)}{4096 (\cos (2 x)-\sin (2 x))}+\frac {-3 \cos (x)-7 \sin (x)}{128 (\cos (2 x)+\sin (2 x))^4}+\frac {115 \cos (x)+234 \sin (x)}{768 (\cos (2 x)+\sin (2 x))^3}+\frac {-4289 \cos (x)-8381 \sin (x)}{6144 (\cos (2 x)+\sin (2 x))^2}+\frac {13375 \cos (x)+26434 \sin (x)}{4096 (\cos (2 x)+\sin (2 x))} \] Input:
Integrate[(Cos[3*x] + Cos[5*x])^(-5),x]
Output:
(-7523*Log[Cos[x/2] - Sin[x/2]])/256 + (7523*Log[Cos[x/2] + Sin[x/2]])/256 - RootSum[1 + #1^8 & , (444922*ArcTan[Sin[x]/(Cos[x] - #1)] - (222461*I)* Log[1 - 2*Cos[x]*#1 + #1^2] - 184578*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 + ( 92289*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 - 184578*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 + (92289*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 + 444922*ArcTan[Si n[x]/(Cos[x] - #1)]*#1^6 - (222461*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/#1 ^7 & ]/65536 + 1/(512*(Cos[x/2] - Sin[x/2])^4) + 163/(512*(Cos[x/2] - Sin[ x/2])^2) - 1/(512*(Cos[x/2] + Sin[x/2])^4) - 163/(512*(Cos[x/2] + Sin[x/2] )^2) + (3*Cos[x] - 7*Sin[x])/(128*(Cos[2*x] - Sin[2*x])^4) + (-115*Cos[x] + 234*Sin[x])/(768*(Cos[2*x] - Sin[2*x])^3) + (4289*Cos[x] - 8381*Sin[x])/ (6144*(Cos[2*x] - Sin[2*x])^2) + (-13375*Cos[x] + 26434*Sin[x])/(4096*(Cos [2*x] - Sin[2*x])) + (-3*Cos[x] - 7*Sin[x])/(128*(Cos[2*x] + Sin[2*x])^4) + (115*Cos[x] + 234*Sin[x])/(768*(Cos[2*x] + Sin[2*x])^3) + (-4289*Cos[x] - 8381*Sin[x])/(6144*(Cos[2*x] + Sin[2*x])^2) + (13375*Cos[x] + 26434*Sin[ x])/(4096*(Cos[2*x] + Sin[2*x]))
Time = 1.04 (sec) , antiderivative size = 710, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 4825, 27, 1567, 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.
| \(\displaystyle \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (3 x)+\cos (5 x))^5}dx\) |
| \(\Big \downarrow \) 4825 |
| \(\displaystyle \int \frac {1}{32 \left (1-\sin ^2(x)\right )^3 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle \frac {1}{32} \int \left (\frac {64 \left (7 \sin ^2(x)-1\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^5}-\frac {7523}{8 \left (\sin ^2(x)-1\right )}+\frac {320 \left (23 \sin ^2(x)-1\right )}{8 \sin ^4(x)-8 \sin ^2(x)+1}+\frac {163}{16 (\sin (x)-1)^2}+\frac {163}{16 (\sin (x)+1)^2}+\frac {256 \left (19 \sin ^2(x)-1\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{8 (\sin (x)+1)^3}+\frac {192 \left (15 \sin ^2(x)-1\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^3}+\frac {128 \left (11 \sin ^2(x)-1\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^4}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{32} \left (\frac {7523}{8} \text {arctanh}(\sin (x))-\frac {1}{8} \sqrt {\frac {1}{2} \left (387634+83273 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {2}}}\right )-12 \sqrt {2 \left (386+73 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {2}}}\right )+\frac {45}{4} \sqrt {\frac {1}{2} \left (170-41 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {2}}}\right )+20 \sqrt {2 \left (890-521 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {2}}}\right )+\frac {15}{256} \sqrt {\frac {1}{2} \left (42058-21601 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {15}{256} \sqrt {\frac {1}{2} \left (42058+21601 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {2}}}\right )-20 \sqrt {2 \left (890+521 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {2}}}\right )-\frac {45}{4} \sqrt {\frac {1}{2} \left (170+41 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {2}}}\right )+12 \sqrt {2 \left (386-73 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (387634-83273 \sqrt {2}\right )} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {2}}}\right )+\frac {\sin (x) \left (359-882 \sin ^2(x)\right )}{6 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )}-\frac {5 \sin (x) \left (295-441 \sin ^2(x)\right )}{32 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )}+\frac {32 \sin (x) \left (13-30 \sin ^2(x)\right )}{8 \sin ^4(x)-8 \sin ^2(x)+1}-\frac {30 \sin (x) \left (13-21 \sin ^2(x)\right )}{8 \sin ^4(x)-8 \sin ^2(x)+1}+\frac {\sin (x) \left (57-286 \sin ^2(x)\right )}{16 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^2}-\frac {4 \sin (x) \left (75-121 \sin ^2(x)\right )}{3 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^2}+\frac {12 \sin (x) \left (9-22 \sin ^2(x)\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^2}-\frac {\sin (x) \left (23-35 \sin ^2(x)\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^3}+\frac {16 \sin (x) \left (5-14 \sin ^2(x)\right )}{3 \left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^3}+\frac {2 \sin (x) \left (1-6 \sin ^2(x)\right )}{\left (8 \sin ^4(x)-8 \sin ^2(x)+1\right )^4}+\frac {163}{16 (1-\sin (x))}-\frac {163}{16 (\sin (x)+1)}+\frac {1}{16 (1-\sin (x))^2}-\frac {1}{16 (\sin (x)+1)^2}\right )\) |
Input:
Int[(Cos[3*x] + Cos[5*x])^(-5),x]
Output:
((7523*ArcTanh[Sin[x]])/8 + (15*Sqrt[(42058 - 21601*Sqrt[2])/2]*ArcTanh[(2 *Sin[x])/Sqrt[2 - Sqrt[2]]])/256 + 20*Sqrt[2*(890 - 521*Sqrt[2])]*ArcTanh[ (2*Sin[x])/Sqrt[2 - Sqrt[2]]] + (45*Sqrt[(170 - 41*Sqrt[2])/2]*ArcTanh[(2* Sin[x])/Sqrt[2 - Sqrt[2]]])/4 - 12*Sqrt[2*(386 + 73*Sqrt[2])]*ArcTanh[(2*S in[x])/Sqrt[2 - Sqrt[2]]] - (Sqrt[(387634 + 83273*Sqrt[2])/2]*ArcTanh[(2*S in[x])/Sqrt[2 - Sqrt[2]]])/8 + (Sqrt[(387634 - 83273*Sqrt[2])/2]*ArcTanh[( 2*Sin[x])/Sqrt[2 + Sqrt[2]]])/8 + 12*Sqrt[2*(386 - 73*Sqrt[2])]*ArcTanh[(2 *Sin[x])/Sqrt[2 + Sqrt[2]]] - (45*Sqrt[(170 + 41*Sqrt[2])/2]*ArcTanh[(2*Si n[x])/Sqrt[2 + Sqrt[2]]])/4 - 20*Sqrt[2*(890 + 521*Sqrt[2])]*ArcTanh[(2*Si n[x])/Sqrt[2 + Sqrt[2]]] - (15*Sqrt[(42058 + 21601*Sqrt[2])/2]*ArcTanh[(2* Sin[x])/Sqrt[2 + Sqrt[2]]])/256 + 1/(16*(1 - Sin[x])^2) + 163/(16*(1 - Sin [x])) - 1/(16*(1 + Sin[x])^2) - 163/(16*(1 + Sin[x])) + (2*Sin[x]*(1 - 6*S in[x]^2))/(1 - 8*Sin[x]^2 + 8*Sin[x]^4)^4 - (Sin[x]*(23 - 35*Sin[x]^2))/(1 - 8*Sin[x]^2 + 8*Sin[x]^4)^3 + (16*Sin[x]*(5 - 14*Sin[x]^2))/(3*(1 - 8*Si n[x]^2 + 8*Sin[x]^4)^3) + (Sin[x]*(57 - 286*Sin[x]^2))/(16*(1 - 8*Sin[x]^2 + 8*Sin[x]^4)^2) - (4*Sin[x]*(75 - 121*Sin[x]^2))/(3*(1 - 8*Sin[x]^2 + 8* Sin[x]^4)^2) + (12*Sin[x]*(9 - 22*Sin[x]^2))/(1 - 8*Sin[x]^2 + 8*Sin[x]^4) ^2 + (Sin[x]*(359 - 882*Sin[x]^2))/(6*(1 - 8*Sin[x]^2 + 8*Sin[x]^4)) - (5* Sin[x]*(295 - 441*Sin[x]^2))/(32*(1 - 8*Sin[x]^2 + 8*Sin[x]^4)) + (32*Sin[ x]*(13 - 30*Sin[x]^2))/(1 - 8*Sin[x]^2 + 8*Sin[x]^4) - (30*Sin[x]*(13 -...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
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*ArcSin[x]] + b*Cos[n*ArcSin[x]]]]^p/Sqrt[1 - x^2], x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[(m - 1)/2] & & IntegerQ[(n - 1)/2]
Time = 70.04 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.25
| method | result | size |
| default | \(\frac {1}{512 \left (\sin \left (x \right )-1\right )^{2}}-\frac {163}{512 \left (\sin \left (x \right )-1\right )}-\frac {7523 \ln \left (\sin \left (x \right )-1\right )}{512}+\frac {\frac {1823 \sin \left (x \right )}{1024}-\frac {167845 \sin \left (x \right )^{3}}{3072}+\frac {64735 \sin \left (x \right )^{5}}{96}-\frac {1622347 \sin \left (x \right )^{7}}{384}+\frac {675737 \sin \left (x \right )^{9}}{48}-\frac {1160911 \sin \left (x \right )^{11}}{48}+\frac {60673 \sin \left (x \right )^{13}}{3}-\frac {13059 \sin \left (x \right )^{15}}{2}}{\left (1-8 \sin \left (x \right )^{2}+8 \sin \left (x \right )^{4}\right )^{4}}-\frac {\left (314750+222461 \sqrt {2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {2 \sin \left (x \right )}{\sqrt {2+\sqrt {2}}}\right )}{16384 \sqrt {2+\sqrt {2}}}-\frac {\sqrt {2}\, \left (-314750+222461 \sqrt {2}\right ) \operatorname {arctanh}\left (\frac {2 \sin \left (x \right )}{\sqrt {2-\sqrt {2}}}\right )}{16384 \sqrt {2-\sqrt {2}}}-\frac {1}{512 \left (1+\sin \left (x \right )\right )^{2}}-\frac {163}{512 \left (1+\sin \left (x \right )\right )}+\frac {7523 \ln \left (1+\sin \left (x \right )\right )}{512}\) | \(179\) |
| risch | \(-\frac {i \left (54825 \,{\mathrm e}^{39 i x}+70853 \,{\mathrm e}^{37 i x}-65023 \,{\mathrm e}^{35 i x}-70635 \,{\mathrm e}^{33 i x}+239574 \,{\mathrm e}^{31 i x}+269918 \,{\mathrm e}^{29 i x}-256978 \,{\mathrm e}^{27 i x}-275034 \,{\mathrm e}^{25 i x}+395292 \,{\mathrm e}^{23 i x}+384876 \,{\mathrm e}^{21 i x}-384876 \,{\mathrm e}^{19 i x}-395292 \,{\mathrm e}^{17 i x}+275034 \,{\mathrm e}^{15 i x}+256978 \,{\mathrm e}^{13 i x}-269918 \,{\mathrm e}^{11 i x}-239574 \,{\mathrm e}^{9 i x}+70635 \,{\mathrm e}^{7 i x}+65023 \,{\mathrm e}^{5 i x}-70853 \,{\mathrm e}^{3 i x}-54825 \,{\mathrm e}^{i x}\right )}{12288 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{8 i x}+{\mathrm e}^{2 i x}+1\right )^{4}}-\frac {7523 \ln \left ({\mathrm e}^{i x}-i\right )}{256}+\frac {7523 \ln \left ({\mathrm e}^{i x}+i\right )}{256}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (144115188075855872 \textit {\_Z}^{4}-31141817895882850304 \textit {\_Z}^{2}+2014638897403441\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {811782628926554112}{1841015462328776441} i \textit {\_R}^{3}+\frac {175186457419243945984}{1841015462328776441} i \textit {\_R} \right ) {\mathrm e}^{i x}-1\right )\right )\) | \(230\) |
Input:
int(1/(cos(3*x)+cos(5*x))^5,x,method=_RETURNVERBOSE)
Output:
1/512/(sin(x)-1)^2-163/512/(sin(x)-1)-7523/512*ln(sin(x)-1)+8192*(1823/838 8608*sin(x)-167845/25165824*sin(x)^3+64735/786432*sin(x)^5-1622347/3145728 *sin(x)^7+675737/393216*sin(x)^9-1160911/393216*sin(x)^11+60673/24576*sin( x)^13-13059/16384*sin(x)^15)/(1-8*sin(x)^2+8*sin(x)^4)^4-1/16384*(314750+2 22461*2^(1/2))*2^(1/2)/(2+2^(1/2))^(1/2)*arctanh(2*sin(x)/(2+2^(1/2))^(1/2 ))-1/16384*2^(1/2)*(-314750+222461*2^(1/2))/(2-2^(1/2))^(1/2)*arctanh(2*si n(x)/(2-2^(1/2))^(1/2))-1/512/(1+sin(x))^2-163/512/(1+sin(x))+7523/512*ln( 1+sin(x))
Time = 0.22 (sec) , antiderivative size = 588, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(3*x)+cos(5*x))^5,x, algorithm="fricas")
Output:
-1/49152*(3*(4096*cos(x)^20 - 16384*cos(x)^18 + 26624*cos(x)^16 - 22528*co s(x)^14 + 10624*cos(x)^12 - 2816*cos(x)^10 + 416*cos(x)^8 - 32*cos(x)^6 + cos(x)^4)*sqrt(41016521729/2*sqrt(2) + 29003078021)*log(sqrt(41016521729/2 *sqrt(2) + 29003078021)*(92289*sqrt(2) - 130172) + 89769458*sin(x)) - 3*(4 096*cos(x)^20 - 16384*cos(x)^18 + 26624*cos(x)^16 - 22528*cos(x)^14 + 1062 4*cos(x)^12 - 2816*cos(x)^10 + 416*cos(x)^8 - 32*cos(x)^6 + cos(x)^4)*sqrt (41016521729/2*sqrt(2) + 29003078021)*log(sqrt(41016521729/2*sqrt(2) + 290 03078021)*(92289*sqrt(2) - 130172) - 89769458*sin(x)) - 3*(4096*cos(x)^20 - 16384*cos(x)^18 + 26624*cos(x)^16 - 22528*cos(x)^14 + 10624*cos(x)^12 - 2816*cos(x)^10 + 416*cos(x)^8 - 32*cos(x)^6 + cos(x)^4)*sqrt(-41016521729/ 2*sqrt(2) + 29003078021)*log((92289*sqrt(2) + 130172)*sqrt(-41016521729/2* sqrt(2) + 29003078021) + 89769458*sin(x)) + 3*(4096*cos(x)^20 - 16384*cos( x)^18 + 26624*cos(x)^16 - 22528*cos(x)^14 + 10624*cos(x)^12 - 2816*cos(x)^ 10 + 416*cos(x)^8 - 32*cos(x)^6 + cos(x)^4)*sqrt(-41016521729/2*sqrt(2) + 29003078021)*log((92289*sqrt(2) + 130172)*sqrt(-41016521729/2*sqrt(2) + 29 003078021) - 89769458*sin(x)) - 722208*(4096*cos(x)^20 - 16384*cos(x)^18 + 26624*cos(x)^16 - 22528*cos(x)^14 + 10624*cos(x)^12 - 2816*cos(x)^10 + 41 6*cos(x)^8 - 32*cos(x)^6 + cos(x)^4)*log(sin(x) + 1) + 722208*(4096*cos(x) ^20 - 16384*cos(x)^18 + 26624*cos(x)^16 - 22528*cos(x)^14 + 10624*cos(x)^1 2 - 2816*cos(x)^10 + 416*cos(x)^8 - 32*cos(x)^6 + cos(x)^4)*log(-sin(x)...
\[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=\int \frac {1}{\left (\cos {\left (3 x \right )} + \cos {\left (5 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(cos(3*x)+cos(5*x))**5,x)
Output:
Integral((cos(3*x) + cos(5*x))**(-5), x)
\[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=\int { \frac {1}{{\left (\cos \left (5 \, x\right ) + \cos \left (3 \, x\right )\right )}^{5}} \,d x } \] Input:
integrate(1/(cos(3*x)+cos(5*x))^5,x, algorithm="maxima")
Output:
1/12288*((54825*sin(39*x) + 70853*sin(37*x) - 65023*sin(35*x) - 70635*sin( 33*x) + 239574*sin(31*x) + 269918*sin(29*x) - 256978*sin(27*x) - 275034*si n(25*x) + 395292*sin(23*x) + 384876*sin(21*x) - 384876*sin(19*x) - 395292* sin(17*x) + 275034*sin(15*x) + 256978*sin(13*x) - 269918*sin(11*x) - 23957 4*sin(9*x) + 70635*sin(7*x) + 65023*sin(5*x) - 70853*sin(3*x) - 54825*sin( x))*cos(40*x) - 54825*(4*sin(38*x) + 6*sin(36*x) + 4*sin(34*x) + 5*sin(32* x) + 16*sin(30*x) + 24*sin(28*x) + 16*sin(26*x) + 10*sin(24*x) + 24*sin(22 *x) + 36*sin(20*x) + 24*sin(18*x) + 10*sin(16*x) + 16*sin(14*x) + 24*sin(1 2*x) + 16*sin(10*x) + 5*sin(8*x) + 4*sin(6*x) + 6*sin(4*x) + 4*sin(2*x))*c os(39*x) + 4*(70853*sin(37*x) - 65023*sin(35*x) - 70635*sin(33*x) + 239574 *sin(31*x) + 269918*sin(29*x) - 256978*sin(27*x) - 275034*sin(25*x) + 3952 92*sin(23*x) + 384876*sin(21*x) - 384876*sin(19*x) - 395292*sin(17*x) + 27 5034*sin(15*x) + 256978*sin(13*x) - 269918*sin(11*x) - 239574*sin(9*x) + 7 0635*sin(7*x) + 65023*sin(5*x) - 70853*sin(3*x) - 54825*sin(x))*cos(38*x) - 70853*(6*sin(36*x) + 4*sin(34*x) + 5*sin(32*x) + 16*sin(30*x) + 24*sin(2 8*x) + 16*sin(26*x) + 10*sin(24*x) + 24*sin(22*x) + 36*sin(20*x) + 24*sin( 18*x) + 10*sin(16*x) + 16*sin(14*x) + 24*sin(12*x) + 16*sin(10*x) + 5*sin( 8*x) + 4*sin(6*x) + 6*sin(4*x) + 4*sin(2*x))*cos(37*x) - 6*(65023*sin(35*x ) + 70635*sin(33*x) - 239574*sin(31*x) - 269918*sin(29*x) + 256978*sin(27* x) + 275034*sin(25*x) - 395292*sin(23*x) - 384876*sin(21*x) + 384876*si...
Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.28 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=-\frac {1}{32768} \, \sqrt {82033043458 \, \sqrt {2} + 116012312084} \log \left ({\left | \frac {1}{2} \, \sqrt {\sqrt {2} + 2} + \sin \left (x\right ) \right |}\right ) + \frac {1}{32768} \, \sqrt {82033043458 \, \sqrt {2} + 116012312084} \log \left ({\left | -\frac {1}{2} \, \sqrt {\sqrt {2} + 2} + \sin \left (x\right ) \right |}\right ) + \frac {1}{32768} \, \sqrt {-82033043458 \, \sqrt {2} + 116012312084} \log \left ({\left | \sqrt {-\frac {1}{4} \, \sqrt {2} + \frac {1}{2}} + \sin \left (x\right ) \right |}\right ) - \frac {1}{32768} \, \sqrt {-82033043458 \, \sqrt {2} + 116012312084} \log \left ({\left | -\sqrt {-\frac {1}{4} \, \sqrt {2} + \frac {1}{2}} + \sin \left (x\right ) \right |}\right ) - \frac {163 \, \sin \left (x\right )^{3} - 165 \, \sin \left (x\right )}{256 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{2}} - \frac {20058624 \, \sin \left (x\right )^{15} - 62129152 \, \sin \left (x\right )^{13} + 74298304 \, \sin \left (x\right )^{11} - 43247168 \, \sin \left (x\right )^{9} + 12978776 \, \sin \left (x\right )^{7} - 2071520 \, \sin \left (x\right )^{5} + 167845 \, \sin \left (x\right )^{3} - 5469 \, \sin \left (x\right )}{3072 \, {\left (8 \, \sin \left (x\right )^{4} - 8 \, \sin \left (x\right )^{2} + 1\right )}^{4}} + \frac {7523}{512} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {7523}{512} \, \log \left (-\sin \left (x\right ) + 1\right ) \] Input:
integrate(1/(cos(3*x)+cos(5*x))^5,x, algorithm="giac")
Output:
-1/32768*sqrt(82033043458*sqrt(2) + 116012312084)*log(abs(1/2*sqrt(sqrt(2) + 2) + sin(x))) + 1/32768*sqrt(82033043458*sqrt(2) + 116012312084)*log(ab s(-1/2*sqrt(sqrt(2) + 2) + sin(x))) + 1/32768*sqrt(-82033043458*sqrt(2) + 116012312084)*log(abs(sqrt(-1/4*sqrt(2) + 1/2) + sin(x))) - 1/32768*sqrt(- 82033043458*sqrt(2) + 116012312084)*log(abs(-sqrt(-1/4*sqrt(2) + 1/2) + si n(x))) - 1/256*(163*sin(x)^3 - 165*sin(x))/(sin(x)^2 - 1)^2 - 1/3072*(2005 8624*sin(x)^15 - 62129152*sin(x)^13 + 74298304*sin(x)^11 - 43247168*sin(x) ^9 + 12978776*sin(x)^7 - 2071520*sin(x)^5 + 167845*sin(x)^3 - 5469*sin(x)) /(8*sin(x)^4 - 8*sin(x)^2 + 1)^4 + 7523/512*log(sin(x) + 1) - 7523/512*log (-sin(x) + 1)
Time = 21.81 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=\text {Too large to display} \] Input:
int(1/(cos(3*x) + cos(5*x))^5,x)
Output:
(7523*atanh(tan(x/2)))/128 + (atanh((4398325912867474380053013309804055793 0563*tan(x/2)*(116012312084 - 82033043458*2^(1/2))^(1/2))/(576460752303423 488*((6920297007783883353938948953298666114673976983*2^(1/2))/576460752303 423488 + (6920297007783883353938948953298666114673976983*2^(1/2)*tan(x/2)^ 2)/576460752303423488 - (19573555768127569458466594778505314811645845293*t an(x/2)^2)/1152921504606846976 - 19573555768127569458466594778505314811645 845293/1152921504606846976)) - (31100860870860865060704879553239441632581* 2^(1/2)*tan(x/2)*(116012312084 - 82033043458*2^(1/2))^(1/2))/(576460752303 423488*((6920297007783883353938948953298666114673976983*2^(1/2))/576460752 303423488 + (6920297007783883353938948953298666114673976983*2^(1/2)*tan(x/ 2)^2)/576460752303423488 - (1957355576812756945846659477850531481164584529 3*tan(x/2)^2)/1152921504606846976 - 19573555768127569458466594778505314811 645845293/1152921504606846976)))*(116012312084 - 82033043458*2^(1/2))^(1/2 ))/16384 - (atanh((43983259128674743800530133098040557930563*tan(x/2)*(820 33043458*2^(1/2) + 116012312084)^(1/2))/(576460752303423488*((692029700778 3883353938948953298666114673976983*2^(1/2))/576460752303423488 + (69202970 07783883353938948953298666114673976983*2^(1/2)*tan(x/2)^2)/576460752303423 488 + (19573555768127569458466594778505314811645845293*tan(x/2)^2)/1152921 504606846976 + 19573555768127569458466594778505314811645845293/11529215046 06846976)) + (31100860870860865060704879553239441632581*2^(1/2)*tan(x/2...
\[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^5} \, dx=\int \frac {1}{\cos \left (5 x \right )^{5}+5 \cos \left (5 x \right )^{4} \cos \left (3 x \right )+10 \cos \left (5 x \right )^{3} \cos \left (3 x \right )^{2}+10 \cos \left (5 x \right )^{2} \cos \left (3 x \right )^{3}+5 \cos \left (5 x \right ) \cos \left (3 x \right )^{4}+\cos \left (3 x \right )^{5}}d x \] Input:
int(1/(cos(3*x)+cos(5*x))^5,x)
Output:
int(1/(cos(5*x)**5 + 5*cos(5*x)**4*cos(3*x) + 10*cos(5*x)**3*cos(3*x)**2 + 10*cos(5*x)**2*cos(3*x)**3 + 5*cos(5*x)*cos(3*x)**4 + cos(3*x)**5),x)