Integrand size = 9, antiderivative size = 114 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=-\frac {217}{128} \text {arctanh}(2 \cos (x) \sin (x))+\frac {97 \tan (x)}{64}+\frac {7 \tan ^3(x)}{96}+\frac {\tan ^5(x)}{320}+\frac {2 \tan (x)}{5 \left (1-\tan ^2(x)\right )^5}-\frac {31 \tan (x)}{20 \left (1-\tan ^2(x)\right )^4}+\frac {343 \tan (x)}{120 \left (1-\tan ^2(x)\right )^3}-\frac {329 \tan (x)}{96 \left (1-\tan ^2(x)\right )^2}+\frac {231 \tan (x)}{64 \left (1-\tan ^2(x)\right )} \] Output:
-217/128*arctanh(2*cos(x)*sin(x))+97/64*tan(x)+7/96*tan(x)^3+1/320*tan(x)^ 5+2/5*tan(x)/(1-tan(x)^2)^5-31/20*tan(x)/(1-tan(x)^2)^4+343/120*tan(x)/(1- tan(x)^2)^3-329/96*tan(x)/(1-tan(x)^2)^2+231*tan(x)/(64-64*tan(x)^2)
Time = 0.68 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=\frac {6510 \log (\cos (x)-\sin (x))-6510 \log (\cos (x)+\sin (x))-\frac {39}{(\cos (x)-\sin (x))^4}-\frac {596}{(\cos (x)-\sin (x))^2}+\frac {1}{2} (7847 \cos (x)+4767 \cos (3 x)+5093 \cos (5 x)+1033 \cos (7 x)+1172 \cos (9 x)) \sec ^5(2 x) \sin (x)+\frac {39}{(\cos (x)+\sin (x))^4}+\frac {596}{(\cos (x)+\sin (x))^2}+4 \left (1388+64 \sec ^2(x)+3 \sec ^4(x)\right ) \tan (x)}{3840} \] Input:
Integrate[(Cos[x] + Cos[3*x])^(-6),x]
Output:
(6510*Log[Cos[x] - Sin[x]] - 6510*Log[Cos[x] + Sin[x]] - 39/(Cos[x] - Sin[ x])^4 - 596/(Cos[x] - Sin[x])^2 + ((7847*Cos[x] + 4767*Cos[3*x] + 5093*Cos [5*x] + 1033*Cos[7*x] + 1172*Cos[9*x])*Sec[2*x]^5*Sin[x])/2 + 39/(Cos[x] + Sin[x])^4 + 596/(Cos[x] + Sin[x])^2 + 4*(1388 + 64*Sec[x]^2 + 3*Sec[x]^4) *Tan[x])/3840
Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4823, 27, 300, 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.
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (3 x))^6}dx\) |
| \(\Big \downarrow \) 4823 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^8}{64 \left (1-\tan ^2(x)\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \int \frac {\left (\tan ^2(x)+1\right )^8}{\left (1-\tan ^2(x)\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {1}{64} \int \left (\tan ^4(x)+14 \tan ^2(x)-\frac {32 \left (-14 \tan ^{10}(x)+35 \tan ^8(x)-56 \tan ^6(x)+42 \tan ^4(x)-18 \tan ^2(x)+3\right )}{\left (1-\tan ^2(x)\right )^6}+97\right )d\tan (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{64} \left (-217 \text {arctanh}(\tan (x))+\frac {\tan ^5(x)}{5}+\frac {14 \tan ^3(x)}{3}+\frac {231 \tan (x)}{1-\tan ^2(x)}-\frac {658 \tan (x)}{3 \left (1-\tan ^2(x)\right )^2}+\frac {2744 \tan (x)}{15 \left (1-\tan ^2(x)\right )^3}-\frac {496 \tan (x)}{5 \left (1-\tan ^2(x)\right )^4}+\frac {128 \tan (x)}{5 \left (1-\tan ^2(x)\right )^5}+97 \tan (x)\right )\) |
Input:
Int[(Cos[x] + Cos[3*x])^(-6),x]
Output:
(-217*ArcTanh[Tan[x]] + 97*Tan[x] + (14*Tan[x]^3)/3 + Tan[x]^5/5 + (128*Ta n[x])/(5*(1 - Tan[x]^2)^5) - (496*Tan[x])/(5*(1 - Tan[x]^2)^4) + (2744*Tan [x])/(15*(1 - Tan[x]^2)^3) - (658*Tan[x])/(3*(1 - Tan[x]^2)^2) + (231*Tan[ x])/(1 - Tan[x]^2))/64
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
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]
Time = 165.96 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {\tan \left (x \right )^{5}}{320}+\frac {7 \tan \left (x \right )^{3}}{96}+\frac {97 \tan \left (x \right )}{64}-\frac {1}{80 \left (\tan \left (x \right )-1\right )^{5}}-\frac {5}{64 \left (\tan \left (x \right )-1\right )^{4}}-\frac {53}{192 \left (\tan \left (x \right )-1\right )^{3}}-\frac {23}{32 \left (\tan \left (x \right )-1\right )^{2}}-\frac {231}{128 \left (\tan \left (x \right )-1\right )}+\frac {217 \ln \left (\tan \left (x \right )-1\right )}{128}-\frac {1}{80 \left (\tan \left (x \right )+1\right )^{5}}+\frac {5}{64 \left (\tan \left (x \right )+1\right )^{4}}-\frac {53}{192 \left (\tan \left (x \right )+1\right )^{3}}+\frac {23}{32 \left (\tan \left (x \right )+1\right )^{2}}-\frac {231}{128 \left (\tan \left (x \right )+1\right )}-\frac {217 \ln \left (\tan \left (x \right )+1\right )}{128}\) | \(112\) |
| risch | \(\frac {i \left (3255 \,{\mathrm e}^{28 i x}+16275 \,{\mathrm e}^{26 i x}+47740 \,{\mathrm e}^{24 i x}+108500 \,{\mathrm e}^{22 i x}+195951 \,{\mathrm e}^{20 i x}+294035 \,{\mathrm e}^{18 i x}+378200 \,{\mathrm e}^{16 i x}+415400 \,{\mathrm e}^{14 i x}+397165 \,{\mathrm e}^{12 i x}+321169 \,{\mathrm e}^{10 i x}+224300 \,{\mathrm e}^{8 i x}+131460 \,{\mathrm e}^{6 i x}+60525 \,{\mathrm e}^{4 i x}+22345 \,{\mathrm e}^{2 i x}+5120\right )}{960 \left ({\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}+1\right )^{5}}+\frac {217 \ln \left ({\mathrm e}^{2 i x}-i\right )}{128}-\frac {217 \ln \left ({\mathrm e}^{2 i x}+i\right )}{128}\) | \(146\) |
Input:
int(1/(cos(x)+cos(3*x))^6,x,method=_RETURNVERBOSE)
Output:
1/320*tan(x)^5+7/96*tan(x)^3+97/64*tan(x)-1/80/(tan(x)-1)^5-5/64/(tan(x)-1 )^4-53/192/(tan(x)-1)^3-23/32/(tan(x)-1)^2-231/128/(tan(x)-1)+217/128*ln(t an(x)-1)-1/80/(tan(x)+1)^5+5/64/(tan(x)+1)^4-53/192/(tan(x)+1)^3+23/32/(ta n(x)+1)^2-231/128/(tan(x)+1)-217/128*ln(tan(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (86) = 172\).
Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=-\frac {3255 \, {\left (32 \, \cos \left (x\right )^{15} - 80 \, \cos \left (x\right )^{13} + 80 \, \cos \left (x\right )^{11} - 40 \, \cos \left (x\right )^{9} + 10 \, \cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )} \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - 3255 \, {\left (32 \, \cos \left (x\right )^{15} - 80 \, \cos \left (x\right )^{13} + 80 \, \cos \left (x\right )^{11} - 40 \, \cos \left (x\right )^{9} + 10 \, \cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )} \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - 4 \, {\left (81920 \, \cos \left (x\right )^{14} - 189880 \, \cos \left (x\right )^{12} + 172180 \, \cos \left (x\right )^{10} - 75070 \, \cos \left (x\right )^{8} + 15025 \, \cos \left (x\right )^{6} - 868 \, \cos \left (x\right )^{4} - 34 \, \cos \left (x\right )^{2} - 3\right )} \sin \left (x\right )}{3840 \, {\left (32 \, \cos \left (x\right )^{15} - 80 \, \cos \left (x\right )^{13} + 80 \, \cos \left (x\right )^{11} - 40 \, \cos \left (x\right )^{9} + 10 \, \cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )}} \] Input:
integrate(1/(cos(x)+cos(3*x))^6,x, algorithm="fricas")
Output:
-1/3840*(3255*(32*cos(x)^15 - 80*cos(x)^13 + 80*cos(x)^11 - 40*cos(x)^9 + 10*cos(x)^7 - cos(x)^5)*log(2*cos(x)*sin(x) + 1) - 3255*(32*cos(x)^15 - 80 *cos(x)^13 + 80*cos(x)^11 - 40*cos(x)^9 + 10*cos(x)^7 - cos(x)^5)*log(-2*c os(x)*sin(x) + 1) - 4*(81920*cos(x)^14 - 189880*cos(x)^12 + 172180*cos(x)^ 10 - 75070*cos(x)^8 + 15025*cos(x)^6 - 868*cos(x)^4 - 34*cos(x)^2 - 3)*sin (x))/(32*cos(x)^15 - 80*cos(x)^13 + 80*cos(x)^11 - 40*cos(x)^9 + 10*cos(x) ^7 - cos(x)^5)
Timed out. \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(cos(x)+cos(3*x))**6,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 10127 vs. \(2 (86) = 172\).
Time = 0.87 (sec) , antiderivative size = 10127, normalized size of antiderivative = 88.83 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(x)+cos(3*x))^6,x, algorithm="maxima")
Output:
-1/3840*(4*(3255*sin(28*x) + 16275*sin(26*x) + 47740*sin(24*x) + 108500*si n(22*x) + 195951*sin(20*x) + 294035*sin(18*x) + 378200*sin(16*x) + 415400* sin(14*x) + 397165*sin(12*x) + 321169*sin(10*x) + 224300*sin(8*x) + 131460 *sin(6*x) + 60525*sin(4*x) + 22345*sin(2*x))*cos(30*x) + 20*(6510*sin(26*x ) + 24955*sin(24*x) + 66185*sin(22*x) + 130200*sin(20*x) + 206150*sin(18*x ) + 277295*sin(16*x) + 314495*sin(14*x) + 309280*sin(12*x) + 255418*sin(10 *x) + 181985*sin(8*x) + 108675*sin(6*x) + 50760*sin(4*x) + 19090*sin(2*x)) *cos(28*x) + 60*(9765*sin(24*x) + 37975*sin(22*x) + 86366*sin(20*x) + 1475 60*sin(18*x) + 210025*sin(16*x) + 247225*sin(14*x) + 250690*sin(12*x) + 21 1584*sin(10*x) + 153775*sin(8*x) + 93485*sin(6*x) + 44250*sin(4*x) + 16920 *sin(2*x))*cos(26*x) + 140*(19840*sin(22*x) + 58187*sin(20*x) + 109895*sin (18*x) + 166780*sin(16*x) + 203980*sin(14*x) + 213025*sin(12*x) + 183405*s in(10*x) + 135640*sin(8*x) + 83720*sin(6*x) + 40065*sin(4*x) + 15525*sin(2 *x))*cos(24*x) + 20*(355663*sin(20*x) + 892955*sin(18*x) + 1553100*sin(16* x) + 2036700*sin(14*x) + 2233645*sin(12*x) + 1983497*sin(10*x) + 1505400*s in(8*x) + 949480*sin(6*x) + 461325*sin(4*x) + 181985*sin(2*x))*cos(22*x) + 4*(3244150*sin(18*x) + 7825795*sin(16*x) + 11582995*sin(14*x) + 13660280* sin(12*x) + 12647018*sin(10*x) + 9917485*sin(8*x) + 6419175*sin(6*x) + 317 3760*sin(4*x) + 1277090*sin(2*x))*cos(20*x) + 20*(1096315*sin(16*x) + 2100 715*sin(14*x) + 2784510*sin(12*x) + 2732056*sin(10*x) + 2233645*sin(8*x...
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=\frac {1}{320} \, \tan \left (x\right )^{5} + \frac {7}{96} \, \tan \left (x\right )^{3} - \frac {3465 \, \tan \left (x\right )^{9} - 10570 \, \tan \left (x\right )^{7} + 13664 \, \tan \left (x\right )^{5} - 7990 \, \tan \left (x\right )^{3} + 1815 \, \tan \left (x\right )}{960 \, {\left (\tan \left (x\right )^{2} - 1\right )}^{5}} - \frac {217}{128} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) + \frac {217}{128} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) + \frac {97}{64} \, \tan \left (x\right ) \] Input:
integrate(1/(cos(x)+cos(3*x))^6,x, algorithm="giac")
Output:
1/320*tan(x)^5 + 7/96*tan(x)^3 - 1/960*(3465*tan(x)^9 - 10570*tan(x)^7 + 1 3664*tan(x)^5 - 7990*tan(x)^3 + 1815*tan(x))/(tan(x)^2 - 1)^5 - 217/128*lo g(abs(tan(x) + 1)) + 217/128*log(abs(tan(x) - 1)) + 97/64*tan(x)
Time = 23.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=\frac {-81920\,\sin \left (x\right )\,{\cos \left (x\right )}^{14}+189880\,\sin \left (x\right )\,{\cos \left (x\right )}^{12}-172180\,\sin \left (x\right )\,{\cos \left (x\right )}^{10}+75070\,\sin \left (x\right )\,{\cos \left (x\right )}^8-15025\,\sin \left (x\right )\,{\cos \left (x\right )}^6+868\,\sin \left (x\right )\,{\cos \left (x\right )}^4+34\,\sin \left (x\right )\,{\cos \left (x\right )}^2+3\,\sin \left (x\right )}{-30720\,{\cos \left (x\right )}^{15}+76800\,{\cos \left (x\right )}^{13}-76800\,{\cos \left (x\right )}^{11}+38400\,{\cos \left (x\right )}^9-9600\,{\cos \left (x\right )}^7+960\,{\cos \left (x\right )}^5}-\frac {217\,\mathrm {atanh}\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )}{64} \] Input:
int(1/(cos(3*x) + cos(x))^6,x)
Output:
(3*sin(x) + 34*cos(x)^2*sin(x) + 868*cos(x)^4*sin(x) - 15025*cos(x)^6*sin( x) + 75070*cos(x)^8*sin(x) - 172180*cos(x)^10*sin(x) + 189880*cos(x)^12*si n(x) - 81920*cos(x)^14*sin(x))/(960*cos(x)^5 - 9600*cos(x)^7 + 38400*cos(x )^9 - 76800*cos(x)^11 + 76800*cos(x)^13 - 30720*cos(x)^15) - (217*atanh(si n(x)/cos(x)))/64
\[ \int \frac {1}{(\cos (x)+\cos (3 x))^6} \, dx=\int \frac {1}{\cos \left (3 x \right )^{6}+6 \cos \left (3 x \right )^{5} \cos \left (x \right )+15 \cos \left (3 x \right )^{4} \cos \left (x \right )^{2}+20 \cos \left (3 x \right )^{3} \cos \left (x \right )^{3}+15 \cos \left (3 x \right )^{2} \cos \left (x \right )^{4}+6 \cos \left (3 x \right ) \cos \left (x \right )^{5}+\cos \left (x \right )^{6}}d x \] Input:
int(1/(cos(x)+cos(3*x))^6,x)
Output:
int(1/(cos(3*x)**6 + 6*cos(3*x)**5*cos(x) + 15*cos(3*x)**4*cos(x)**2 + 20* cos(3*x)**3*cos(x)**3 + 15*cos(3*x)**2*cos(x)**4 + 6*cos(3*x)*cos(x)**5 + cos(x)**6),x)