Integrand size = 9, antiderivative size = 221 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=\frac {3889 \text {arctanh}(\sin (x))}{186624}+\frac {332929 \text {arctanh}(2 \sin (x))}{2916}-\frac {82683 \text {arctanh}\left (\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}+\frac {1}{108 (1-2 \sin (x))^4}-\frac {19}{162 (1-2 \sin (x))^3}+\frac {749}{648 (1-2 \sin (x))^2}-\frac {71551}{5832 (1-2 \sin (x))}+\frac {1}{124416 (1-\sin (x))^2}+\frac {209}{373248 (1-\sin (x))}-\frac {11643}{512} \sec (2 x) \sin (x)-\frac {681}{256} \sec ^2(2 x) \sin (x)-\frac {21}{64} \sec ^3(2 x) \sin (x)-\frac {1}{32} \sec ^4(2 x) \sin (x)-\frac {1}{124416 (1+\sin (x))^2}-\frac {209}{373248 (1+\sin (x))}-\frac {1}{108 (1+2 \sin (x))^4}+\frac {19}{162 (1+2 \sin (x))^3}-\frac {749}{648 (1+2 \sin (x))^2}+\frac {71551}{5832 (1+2 \sin (x))} \] Output:
3889/186624*arctanh(sin(x))+332929/2916*arctanh(2*sin(x))-82683/1024*arcta nh(sin(x)*2^(1/2))*2^(1/2)+1/108/(1-2*sin(x))^4-19/162/(1-2*sin(x))^3+749/ 648/(1-2*sin(x))^2-71551/(5832-11664*sin(x))+1/124416/(1-sin(x))^2+209/(37 3248-373248*sin(x))-11643/512*sec(2*x)*sin(x)-681/256*sec(2*x)^2*sin(x)-21 /64*sec(2*x)^3*sin(x)-1/32*sec(2*x)^4*sin(x)-1/124416/(1+sin(x))^2-209/(37 3248+373248*sin(x))-1/108/(1+2*sin(x))^4+19/162/(1+2*sin(x))^3-749/648/(1+ 2*sin(x))^2+71551/(5832+11664*sin(x))
Time = 6.09 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=-\frac {3889 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{186624}+\frac {3889 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{186624}-\frac {332929 \log (1-2 \sin (x))}{5832}+\frac {82683 \log \left (\sqrt {2}-2 \sin (x)\right )}{1024 \sqrt {2}}+\frac {332929 \log (1+2 \sin (x))}{5832}-\frac {82683 \log \left (\sqrt {2}+2 \sin (x)\right )}{1024 \sqrt {2}}+\frac {1}{124416 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}+\frac {209}{373248 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {1}{124416 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4}-\frac {209}{373248 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}-\frac {21}{256 (\cos (x)-\sin (x))^3}-\frac {11643}{1024 (\cos (x)-\sin (x))}-\frac {\sin (x)}{128 (\cos (x)-\sin (x))^4}-\frac {643 \sin (x)}{512 (\cos (x)-\sin (x))^2}-\frac {\sin (x)}{128 (\cos (x)+\sin (x))^4}+\frac {21}{256 (\cos (x)+\sin (x))^3}-\frac {643 \sin (x)}{512 (\cos (x)+\sin (x))^2}+\frac {11643}{1024 (\cos (x)+\sin (x))}+\frac {1}{108 (-1+2 \sin (x))^4}+\frac {19}{162 (-1+2 \sin (x))^3}+\frac {749}{648 (-1+2 \sin (x))^2}+\frac {71551}{5832 (-1+2 \sin (x))}-\frac {1}{108 (1+2 \sin (x))^4}+\frac {19}{162 (1+2 \sin (x))^3}-\frac {749}{648 (1+2 \sin (x))^2}+\frac {71551}{5832 (1+2 \sin (x))} \] Input:
Integrate[(Cos[x] + Cos[5*x])^(-5),x]
Output:
(-3889*Log[Cos[x/2] - Sin[x/2]])/186624 + (3889*Log[Cos[x/2] + Sin[x/2]])/ 186624 - (332929*Log[1 - 2*Sin[x]])/5832 + (82683*Log[Sqrt[2] - 2*Sin[x]]) /(1024*Sqrt[2]) + (332929*Log[1 + 2*Sin[x]])/5832 - (82683*Log[Sqrt[2] + 2 *Sin[x]])/(1024*Sqrt[2]) + 1/(124416*(Cos[x/2] - Sin[x/2])^4) + 209/(37324 8*(Cos[x/2] - Sin[x/2])^2) - 1/(124416*(Cos[x/2] + Sin[x/2])^4) - 209/(373 248*(Cos[x/2] + Sin[x/2])^2) - 21/(256*(Cos[x] - Sin[x])^3) - 11643/(1024* (Cos[x] - Sin[x])) - Sin[x]/(128*(Cos[x] - Sin[x])^4) - (643*Sin[x])/(512* (Cos[x] - Sin[x])^2) - Sin[x]/(128*(Cos[x] + Sin[x])^4) + 21/(256*(Cos[x] + Sin[x])^3) - (643*Sin[x])/(512*(Cos[x] + Sin[x])^2) + 11643/(1024*(Cos[x ] + Sin[x])) + 1/(108*(-1 + 2*Sin[x])^4) + 19/(162*(-1 + 2*Sin[x])^3) + 74 9/(648*(-1 + 2*Sin[x])^2) + 71551/(5832*(-1 + 2*Sin[x])) - 1/(108*(1 + 2*S in[x])^4) + 19/(162*(1 + 2*Sin[x])^3) - 749/(648*(1 + 2*Sin[x])^2) + 71551 /(5832*(1 + 2*Sin[x]))
Time = 0.50 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, 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.
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (5 x))^5}dx\) |
| \(\Big \downarrow \) 4825 |
| \(\displaystyle \int \frac {1}{32 \left (1-\sin ^2(x)\right )^3 \left (8 \sin ^4(x)-6 \sin ^2(x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (8 \sin ^4(x)-6 \sin ^2(x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle \frac {1}{32} \int \left (\frac {4440}{2 \sin ^2(x)-1}-\frac {5326864}{729 \left (4 \sin ^2(x)-1\right )}+\frac {209}{11664 (\sin (x)-1)^2}+\frac {209}{11664 (\sin (x)+1)^2}-\frac {572408}{729 (2 \sin (x)-1)^2}-\frac {572408}{729 (2 \sin (x)+1)^2}-\frac {1200}{\left (2 \sin ^2(x)-1\right )^2}-\frac {1}{1944 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {11984}{81 (2 \sin (x)-1)^3}+\frac {11984}{81 (2 \sin (x)+1)^3}+\frac {288}{\left (2 \sin ^2(x)-1\right )^3}-\frac {608}{27 (2 \sin (x)-1)^4}-\frac {608}{27 (2 \sin (x)+1)^4}-\frac {56}{\left (2 \sin ^2(x)-1\right )^4}-\frac {64}{27 (2 \sin (x)-1)^5}+\frac {64}{27 (2 \sin (x)+1)^5}+\frac {8}{\left (2 \sin ^2(x)-1\right )^5}-\frac {3889}{5832 \left (\sin ^2(x)-1\right )}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{32} \left (\frac {3889 \text {arctanh}(\sin (x))}{5832}+\frac {2663432}{729} \text {arctanh}(2 \sin (x))-2574 \sqrt {2} \text {arctanh}\left (\sqrt {2} \sin (x)\right )-\frac {315 \text {arctanh}\left (\sqrt {2} \sin (x)\right )}{16 \sqrt {2}}-\frac {11643 \sin (x)}{16 \left (1-2 \sin ^2(x)\right )}-\frac {681 \sin (x)}{8 \left (1-2 \sin ^2(x)\right )^2}-\frac {21 \sin (x)}{2 \left (1-2 \sin ^2(x)\right )^3}-\frac {\sin (x)}{\left (1-2 \sin ^2(x)\right )^4}-\frac {286204}{729 (1-2 \sin (x))}+\frac {209}{11664 (1-\sin (x))}-\frac {209}{11664 (\sin (x)+1)}+\frac {286204}{729 (2 \sin (x)+1)}+\frac {2996}{81 (1-2 \sin (x))^2}+\frac {1}{3888 (1-\sin (x))^2}-\frac {1}{3888 (\sin (x)+1)^2}-\frac {2996}{81 (2 \sin (x)+1)^2}-\frac {304}{81 (1-2 \sin (x))^3}+\frac {304}{81 (2 \sin (x)+1)^3}+\frac {8}{27 (1-2 \sin (x))^4}-\frac {8}{27 (2 \sin (x)+1)^4}\right )\) |
Input:
Int[(Cos[x] + Cos[5*x])^(-5),x]
Output:
((3889*ArcTanh[Sin[x]])/5832 + (2663432*ArcTanh[2*Sin[x]])/729 - (315*ArcT anh[Sqrt[2]*Sin[x]])/(16*Sqrt[2]) - 2574*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[x]] + 8/(27*(1 - 2*Sin[x])^4) - 304/(81*(1 - 2*Sin[x])^3) + 2996/(81*(1 - 2*Sin [x])^2) - 286204/(729*(1 - 2*Sin[x])) + 1/(3888*(1 - Sin[x])^2) + 209/(116 64*(1 - Sin[x])) - 1/(3888*(1 + Sin[x])^2) - 209/(11664*(1 + Sin[x])) - 8/ (27*(1 + 2*Sin[x])^4) + 304/(81*(1 + 2*Sin[x])^3) - 2996/(81*(1 + 2*Sin[x] )^2) + 286204/(729*(1 + 2*Sin[x])) - Sin[x]/(1 - 2*Sin[x]^2)^4 - (21*Sin[x ])/(2*(1 - 2*Sin[x]^2)^3) - (681*Sin[x])/(8*(1 - 2*Sin[x]^2)^2) - (11643*S in[x])/(16*(1 - 2*Sin[x]^2)))/32
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 = 50.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {\frac {11643 \sin \left (x \right )^{7}}{64}-\frac {36291 \sin \left (x \right )^{5}}{128}+\frac {37821 \sin \left (x \right )^{3}}{256}-\frac {13189 \sin \left (x \right )}{512}}{\left (2 \sin \left (x \right )^{2}-1\right )^{4}}-\frac {82683 \,\operatorname {arctanh}\left (\sqrt {2}\, \sin \left (x \right )\right ) \sqrt {2}}{1024}-\frac {1}{108 \left (2 \sin \left (x \right )+1\right )^{4}}+\frac {19}{162 \left (2 \sin \left (x \right )+1\right )^{3}}-\frac {749}{648 \left (2 \sin \left (x \right )+1\right )^{2}}+\frac {71551}{5832 \left (2 \sin \left (x \right )+1\right )}+\frac {332929 \ln \left (2 \sin \left (x \right )+1\right )}{5832}-\frac {1}{124416 \left (1+\sin \left (x \right )\right )^{2}}-\frac {209}{373248 \left (1+\sin \left (x \right )\right )}+\frac {3889 \ln \left (1+\sin \left (x \right )\right )}{373248}+\frac {1}{124416 \left (\sin \left (x \right )-1\right )^{2}}-\frac {209}{373248 \left (\sin \left (x \right )-1\right )}-\frac {3889 \ln \left (\sin \left (x \right )-1\right )}{373248}+\frac {1}{108 \left (2 \sin \left (x \right )-1\right )^{4}}+\frac {19}{162 \left (2 \sin \left (x \right )-1\right )^{3}}+\frac {749}{648 \left (2 \sin \left (x \right )-1\right )^{2}}+\frac {71551}{5832 \left (2 \sin \left (x \right )-1\right )}-\frac {332929 \ln \left (2 \sin \left (x \right )-1\right )}{5832}\) | \(193\) |
| risch | \(\frac {i \left (5881813 \,{\mathrm e}^{39 i x}-2770929 \,{\mathrm e}^{37 i x}+16666827 \,{\mathrm e}^{35 i x}+11603277 \,{\mathrm e}^{33 i x}+2153987 \,{\mathrm e}^{31 i x}+49799073 \,{\mathrm e}^{29 i x}-11124845 \,{\mathrm e}^{27 i x}+29440353 \,{\mathrm e}^{25 i x}+33090774 \,{\mathrm e}^{23 i x}-41444690 \,{\mathrm e}^{21 i x}+41444690 \,{\mathrm e}^{19 i x}-33090774 \,{\mathrm e}^{17 i x}-29440353 \,{\mathrm e}^{15 i x}+11124845 \,{\mathrm e}^{13 i x}-49799073 \,{\mathrm e}^{11 i x}-2153987 \,{\mathrm e}^{9 i x}-11603277 \,{\mathrm e}^{7 i x}-16666827 \,{\mathrm e}^{5 i x}+2770929 \,{\mathrm e}^{3 i x}-5881813 \,{\mathrm e}^{i x}\right )}{124416 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+1\right )^{4}}+\frac {3889 \ln \left ({\mathrm e}^{i x}+i\right )}{186624}-\frac {3889 \ln \left ({\mathrm e}^{i x}-i\right )}{186624}+\frac {332929 \ln \left (i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{5832}+\frac {82683 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{2048}-\frac {82683 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{2048}-\frac {332929 \ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{5832}\) | \(271\) |
Input:
int(1/(cos(x)+cos(5*x))^5,x,method=_RETURNVERBOSE)
Output:
4*(11643/256*sin(x)^7-36291/512*sin(x)^5+37821/1024*sin(x)^3-13189/2048*si n(x))/(2*sin(x)^2-1)^4-82683/1024*arctanh(2^(1/2)*sin(x))*2^(1/2)-1/108/(2 *sin(x)+1)^4+19/162/(2*sin(x)+1)^3-749/648/(2*sin(x)+1)^2+71551/5832/(2*si n(x)+1)+332929/5832*ln(2*sin(x)+1)-1/124416/(1+sin(x))^2-209/373248/(1+sin (x))+3889/373248*ln(1+sin(x))+1/124416/(sin(x)-1)^2-209/373248/(sin(x)-1)- 3889/373248*ln(sin(x)-1)+1/108/(2*sin(x)-1)^4+19/162/(2*sin(x)-1)^3+749/64 8/(2*sin(x)-1)^2+71551/5832/(2*sin(x)-1)-332929/5832*ln(2*sin(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (175) = 350\).
Time = 0.22 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx =\text {Too large to display} \] Input:
integrate(1/(cos(x)+cos(5*x))^5,x, algorithm="fricas")
Output:
1/1492992*(60275907*(4096*sqrt(2)*cos(x)^20 - 20480*sqrt(2)*cos(x)^18 + 44 544*sqrt(2)*cos(x)^16 - 55040*sqrt(2)*cos(x)^14 + 42256*sqrt(2)*cos(x)^12 - 20640*sqrt(2)*cos(x)^10 + 6264*sqrt(2)*cos(x)^8 - 1080*sqrt(2)*cos(x)^6 + 81*sqrt(2)*cos(x)^4)*log(-(2*cos(x)^2 + 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^ 2 - 1)) + 85229824*(4096*cos(x)^20 - 20480*cos(x)^18 + 44544*cos(x)^16 - 5 5040*cos(x)^14 + 42256*cos(x)^12 - 20640*cos(x)^10 + 6264*cos(x)^8 - 1080* cos(x)^6 + 81*cos(x)^4)*log(2*sin(x) + 1) + 15556*(4096*cos(x)^20 - 20480* cos(x)^18 + 44544*cos(x)^16 - 55040*cos(x)^14 + 42256*cos(x)^12 - 20640*co s(x)^10 + 6264*cos(x)^8 - 1080*cos(x)^6 + 81*cos(x)^4)*log(sin(x) + 1) - 1 5556*(4096*cos(x)^20 - 20480*cos(x)^18 + 44544*cos(x)^16 - 55040*cos(x)^14 + 42256*cos(x)^12 - 20640*cos(x)^10 + 6264*cos(x)^8 - 1080*cos(x)^6 + 81* cos(x)^4)*log(-sin(x) + 1) - 85229824*(4096*cos(x)^20 - 20480*cos(x)^18 + 44544*cos(x)^16 - 55040*cos(x)^14 + 42256*cos(x)^12 - 20640*cos(x)^10 + 62 64*cos(x)^8 - 1080*cos(x)^6 + 81*cos(x)^4)*log(-2*sin(x) + 1) - 12*(120459 53024*cos(x)^18 - 52614016000*cos(x)^16 + 97798185216*cos(x)^14 - 10027023 7696*cos(x)^12 + 61237672232*cos(x)^10 - 22277937972*cos(x)^8 + 4470458046 *cos(x)^6 - 381752883*cos(x)^4 - 6966*cos(x)^2 - 324)*sin(x))/(4096*cos(x) ^20 - 20480*cos(x)^18 + 44544*cos(x)^16 - 55040*cos(x)^14 + 42256*cos(x)^1 2 - 20640*cos(x)^10 + 6264*cos(x)^8 - 1080*cos(x)^6 + 81*cos(x)^4)
\[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=\int \frac {1}{\left (\cos {\left (x \right )} + \cos {\left (5 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(cos(x)+cos(5*x))**5,x)
Output:
Integral((cos(x) + cos(5*x))**(-5), x)
Exception generated. \[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(x)+cos(5*x))^5,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=\frac {82683}{2048} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac {209 \, \sin \left (x\right )^{3} - 215 \, \sin \left (x\right )}{186624 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac {36139571200 \, \sin \left (x\right )^{15} - 95126438912 \, \sin \left (x\right )^{13} + 105240567552 \, \sin \left (x\right )^{11} - 63358060800 \, \sin \left (x\right )^{9} + 22400373144 \, \sin \left (x\right )^{7} - 4650907308 \, \sin \left (x\right )^{5} + 525480506 \, \sin \left (x\right )^{3} - 24950461 \, \sin \left (x\right )}{373248 \, {\left (8 \, \sin \left (x\right )^{4} - 6 \, \sin \left (x\right )^{2} + 1\right )}^{4}} + \frac {3889}{373248} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {3889}{373248} \, \log \left (-\sin \left (x\right ) + 1\right ) + \frac {332929}{5832} \, \log \left ({\left | 2 \, \sin \left (x\right ) + 1 \right |}\right ) - \frac {332929}{5832} \, \log \left ({\left | 2 \, \sin \left (x\right ) - 1 \right |}\right ) \] Input:
integrate(1/(cos(x)+cos(5*x))^5,x, algorithm="giac")
Output:
82683/2048*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x) )) - 1/186624*(209*sin(x)^3 - 215*sin(x))/(sin(x)^2 - 1)^2 + 1/373248*(361 39571200*sin(x)^15 - 95126438912*sin(x)^13 + 105240567552*sin(x)^11 - 6335 8060800*sin(x)^9 + 22400373144*sin(x)^7 - 4650907308*sin(x)^5 + 525480506* sin(x)^3 - 24950461*sin(x))/(8*sin(x)^4 - 6*sin(x)^2 + 1)^4 + 3889/373248* log(sin(x) + 1) - 3889/373248*log(-sin(x) + 1) + 332929/5832*log(abs(2*sin (x) + 1)) - 332929/5832*log(abs(2*sin(x) - 1))
Time = 22.58 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + cos(x))^5,x)
Output:
(332929*atanh((5293501072611728083288241268544485325492309354181*tan(x/2)) /(129075891691010291516178432*((529350107261172808328824126854448532549230 9354181*tan(x/2)^2)/516303566764041166064713728 + 529350107261172808328824 1268544485325492309354181/516303566764041166064713728))))/2916 + (3889*ata nh(tan(x/2)))/93312 - ((8316677*tan(x/2))/62208 - (7520297*tan(x/2)^3)/768 + (2102536375*tan(x/2)^5)/6912 - (108549008321*tan(x/2)^7)/20736 + (85152 5536249*tan(x/2)^9)/15552 - (1865668038367*tan(x/2)^11)/5184 + (2309100463 9391*tan(x/2)^13)/15552 - (19244414424625*tan(x/2)^15)/5184 + (52955516006 129*tan(x/2)^17)/10368 - (79937493546559*tan(x/2)^19)/31104 - (79937493546 559*tan(x/2)^21)/31104 + (52955516006129*tan(x/2)^23)/10368 - (19244414424 625*tan(x/2)^25)/5184 + (23091004639391*tan(x/2)^27)/15552 - (186566803836 7*tan(x/2)^29)/5184 + (851525536249*tan(x/2)^31)/15552 - (108549008321*tan (x/2)^33)/20736 + (2102536375*tan(x/2)^35)/6912 - (7520297*tan(x/2)^37)/76 8 + (8316677*tan(x/2)^39)/62208)/(3070*tan(x/2)^4 - 84*tan(x/2)^2 - 64180* tan(x/2)^6 + 849645*tan(x/2)^8 - 7459216*tan(x/2)^10 + 44289640*tan(x/2)^1 2 - 178563024*tan(x/2)^14 + 486234130*tan(x/2)^16 - 887655320*tan(x/2)^18 + 1084730676*tan(x/2)^20 - 887655320*tan(x/2)^22 + 486234130*tan(x/2)^24 - 178563024*tan(x/2)^26 + 44289640*tan(x/2)^28 - 7459216*tan(x/2)^30 + 8496 45*tan(x/2)^32 - 64180*tan(x/2)^34 + 3070*tan(x/2)^36 - 84*tan(x/2)^38 + t an(x/2)^40 + 1) - (82683*2^(1/2)*atanh((3870728759430982009595161229590...
\[ \int \frac {1}{(\cos (x)+\cos (5 x))^5} \, dx=\int \frac {1}{\cos \left (5 x \right )^{5}+5 \cos \left (5 x \right )^{4} \cos \left (x \right )+10 \cos \left (5 x \right )^{3} \cos \left (x \right )^{2}+10 \cos \left (5 x \right )^{2} \cos \left (x \right )^{3}+5 \cos \left (5 x \right ) \cos \left (x \right )^{4}+\cos \left (x \right )^{5}}d x \] Input:
int(1/(cos(x)+cos(5*x))^5,x)
Output:
int(1/(cos(5*x)**5 + 5*cos(5*x)**4*cos(x) + 10*cos(5*x)**3*cos(x)**2 + 10* cos(5*x)**2*cos(x)**3 + 5*cos(5*x)*cos(x)**4 + cos(x)**5),x)