Integrand size = 9, antiderivative size = 108 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=-\frac {523}{256} \text {arctanh}(\sin (x))+\frac {1483 \text {arctanh}\left (\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}+\frac {\sin (x)}{32 \left (1-2 \sin ^2(x)\right )^4}-\frac {17 \sin (x)}{192 \left (1-2 \sin ^2(x)\right )^3}+\frac {203 \sin (x)}{768 \left (1-2 \sin ^2(x)\right )^2}-\frac {437 \sin (x)}{512 \left (1-2 \sin ^2(x)\right )}-\frac {43}{256} \sec (x) \tan (x)-\frac {1}{128} \sec ^3(x) \tan (x) \]
-523/256*arctanh(sin(x))+1/32*sin(x)/(1-2*sin(x)^2)^4-17/192*sin(x)/(1-2*s in(x)^2)^3+203/768*sin(x)/(1-2*sin(x)^2)^2-437/512*sin(x)/(1-2*sin(x)^2)+1 483/1024*arctanh(sin(x)*2^(1/2))*2^(1/2)-43/256*sec(x)*tan(x)-1/128*sec(x) ^3*tan(x)
Time = 3.80 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=\frac {12552 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-12552 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-4449 \sqrt {2} \log \left (\sqrt {2}-2 \sin (x)\right )+4449 \sqrt {2} \log \left (\sqrt {2}+2 \sin (x)\right )-\frac {12}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}-\frac {516}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {12}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4}+\frac {516}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}-\frac {136}{(\cos (x)-\sin (x))^3}-\frac {2622}{\cos (x)-\sin (x)}+\frac {136}{(\cos (x)+\sin (x))^3}+\frac {2622}{\cos (x)+\sin (x)}+6 \sec ^4(2 x) (190 \sin (x)+79 (-\sin (3 x)+\sin (5 x)))}{6144} \]
(12552*Log[Cos[x/2] - Sin[x/2]] - 12552*Log[Cos[x/2] + Sin[x/2]] - 4449*Sq rt[2]*Log[Sqrt[2] - 2*Sin[x]] + 4449*Sqrt[2]*Log[Sqrt[2] + 2*Sin[x]] - 12/ (Cos[x/2] - Sin[x/2])^4 - 516/(Cos[x/2] - Sin[x/2])^2 + 12/(Cos[x/2] + Sin [x/2])^4 + 516/(Cos[x/2] + Sin[x/2])^2 - 136/(Cos[x] - Sin[x])^3 - 2622/(C os[x] - Sin[x]) + 136/(Cos[x] + Sin[x])^3 + 2622/(Cos[x] + Sin[x]) + 6*Sec [2*x]^4*(190*Sin[x] + 79*(-Sin[3*x] + Sin[5*x])))/6144
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.70, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.556, Rules used = {3042, 4825, 27, 316, 27, 402, 402, 402, 402, 27, 402, 27, 397, 219}
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.
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (3 x))^5}dx\) |
| \(\Big \downarrow \) 4825 |
| \(\displaystyle \int \frac {1}{32 \left (1-2 \sin ^2(x)\right )^5 \left (1-\sin ^2(x)\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \frac {1}{\left (1-2 \sin ^2(x)\right )^5 \left (1-\sin ^2(x)\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{8} \int \frac {2 \left (3-11 \sin ^2(x)\right )}{\left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^3}d\sin (x)+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \int \frac {3-11 \sin ^2(x)}{\left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^3}d\sin (x)+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \int \frac {45 \sin ^2(x)+23}{\left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^3}d\sin (x)-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {1-637 \sin ^2(x)}{\left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^3}d\sin (x)+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {3175 \sin ^2(x)+637}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^3}d\sin (x)-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\frac {1}{4} \int -\frac {24 \left (953 \sin ^2(x)+265\right )}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}d\sin (x)-\frac {953 \sin (x)}{\left (1-\sin ^2(x)\right )^2}\right )-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (6 \int \frac {953 \sin ^2(x)+265}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}d\sin (x)-\frac {953 \sin (x)}{\left (1-\sin ^2(x)\right )^2}\right )-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (6 \left (-\frac {1}{2} \int -\frac {4 \left (609 \sin ^2(x)+437\right )}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}d\sin (x)-\frac {609 \sin (x)}{1-\sin ^2(x)}\right )-\frac {953 \sin (x)}{\left (1-\sin ^2(x)\right )^2}\right )-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (6 \left (2 \int \frac {609 \sin ^2(x)+437}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}d\sin (x)-\frac {609 \sin (x)}{1-\sin ^2(x)}\right )-\frac {953 \sin (x)}{\left (1-\sin ^2(x)\right )^2}\right )-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (6 \left (2 \left (1483 \int \frac {1}{1-2 \sin ^2(x)}d\sin (x)-1046 \int \frac {1}{1-\sin ^2(x)}d\sin (x)\right )-\frac {609 \sin (x)}{1-\sin ^2(x)}\right )-\frac {953 \sin (x)}{\left (1-\sin ^2(x)\right )^2}\right )-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (6 \left (2 \left (\frac {1483 \text {arctanh}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}}-1046 \text {arctanh}(\sin (x))\right )-\frac {609 \sin (x)}{1-\sin ^2(x)}\right )-\frac {953 \sin (x)}{\left (1-\sin ^2(x)\right )^2}\right )-\frac {635 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}\right )+\frac {91 \sin (x)}{4 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}\right )-\frac {5 \sin (x)}{6 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}\right )+\frac {\sin (x)}{4 \left (1-2 \sin ^2(x)\right )^4 \left (1-\sin ^2(x)\right )^2}\right )\) |
(Sin[x]/(4*(1 - 2*Sin[x]^2)^4*(1 - Sin[x]^2)^2) + ((-5*Sin[x])/(6*(1 - 2*S in[x]^2)^3*(1 - Sin[x]^2)^2) + ((91*Sin[x])/(4*(1 - 2*Sin[x]^2)^2*(1 - Sin [x]^2)^2) + ((-635*Sin[x])/(2*(1 - 2*Sin[x]^2)*(1 - Sin[x]^2)^2) + ((-953* Sin[x])/(1 - Sin[x]^2)^2 + 6*(2*(-1046*ArcTanh[Sin[x]] + (1483*ArcTanh[Sqr t[2]*Sin[x]])/Sqrt[2]) - (609*Sin[x])/(1 - Sin[x]^2)))/2)/4)/6)/4)/32
3.1.21.3.1 Defintions of rubi rules used
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)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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 = 119.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(-\frac {4 \left (-\frac {437 \left (\sin ^{7}\left (x \right )\right )}{256}+\frac {3527 \left (\sin ^{5}\left (x \right )\right )}{1536}-\frac {3257 \left (\sin ^{3}\left (x \right )\right )}{3072}+\frac {331 \sin \left (x \right )}{2048}\right )}{{\left (2 \left (\sin ^{2}\left (x \right )\right )-1\right )}^{4}}+\frac {1483 \,\operatorname {arctanh}\left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{1024}-\frac {1}{512 \left (\sin \left (x \right )-1\right )^{2}}+\frac {43}{512 \left (\sin \left (x \right )-1\right )}+\frac {523 \ln \left (\sin \left (x \right )-1\right )}{512}+\frac {1}{512 \left (\sin \left (x \right )+1\right )^{2}}+\frac {43}{512 \left (\sin \left (x \right )+1\right )}-\frac {523 \ln \left (\sin \left (x \right )+1\right )}{512}\) | \(95\) |
| risch | \(\frac {i \left (1827 \,{\mathrm e}^{23 i x}+3733 \,{\mathrm e}^{21 i x}+6115 \,{\mathrm e}^{19 i x}+9109 \,{\mathrm e}^{17 i x}+5746 \,{\mathrm e}^{15 i x}+2382 \,{\mathrm e}^{13 i x}-2382 \,{\mathrm e}^{11 i x}-5746 \,{\mathrm e}^{9 i x}-9109 \,{\mathrm e}^{7 i x}-6115 \,{\mathrm e}^{5 i x}-3733 \,{\mathrm e}^{3 i x}-1827 \,{\mathrm e}^{i x}\right )}{1536 \left ({\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}+1\right )^{4}}+\frac {523 \ln \left ({\mathrm e}^{i x}-i\right )}{256}-\frac {523 \ln \left (i+{\mathrm e}^{i x}\right )}{256}+\frac {1483 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{2048}-\frac {1483 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{2048}\) | \(179\) |
-4*(-437/256*sin(x)^7+3527/1536*sin(x)^5-3257/3072*sin(x)^3+331/2048*sin(x ))/(2*sin(x)^2-1)^4+1483/1024*arctanh(sin(x)*2^(1/2))*2^(1/2)-1/512/(sin(x )-1)^2+43/512/(sin(x)-1)+523/512*ln(sin(x)-1)+1/512/(sin(x)+1)^2+43/512/(s in(x)+1)-523/512*ln(sin(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (88) = 176\).
Time = 0.31 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.03 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=\frac {4449 \, {\left (16 \, \sqrt {2} \cos \left (x\right )^{12} - 32 \, \sqrt {2} \cos \left (x\right )^{10} + 24 \, \sqrt {2} \cos \left (x\right )^{8} - 8 \, \sqrt {2} \cos \left (x\right )^{6} + \sqrt {2} \cos \left (x\right )^{4}\right )} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - 6276 \, {\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )} \log \left (\sin \left (x\right ) + 1\right ) + 6276 \, {\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 4 \, {\left (14616 \, \cos \left (x\right )^{10} - 25420 \, \cos \left (x\right )^{8} + 15570 \, \cos \left (x\right )^{6} - 3677 \, \cos \left (x\right )^{4} + 162 \, \cos \left (x\right )^{2} + 12\right )} \sin \left (x\right )}{6144 \, {\left (16 \, \cos \left (x\right )^{12} - 32 \, \cos \left (x\right )^{10} + 24 \, \cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + \cos \left (x\right )^{4}\right )}} \]
1/6144*(4449*(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(-(2*cos(x)^2 - 2*sqrt(2 )*sin(x) - 3)/(2*cos(x)^2 - 1)) - 6276*(16*cos(x)^12 - 32*cos(x)^10 + 24*c os(x)^8 - 8*cos(x)^6 + cos(x)^4)*log(sin(x) + 1) + 6276*(16*cos(x)^12 - 32 *cos(x)^10 + 24*cos(x)^8 - 8*cos(x)^6 + cos(x)^4)*log(-sin(x) + 1) - 4*(14 616*cos(x)^10 - 25420*cos(x)^8 + 15570*cos(x)^6 - 3677*cos(x)^4 + 162*cos( x)^2 + 12)*sin(x))/(16*cos(x)^12 - 32*cos(x)^10 + 24*cos(x)^8 - 8*cos(x)^6 + cos(x)^4)
\[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=\int \frac {1}{\left (\cos {\left (x \right )} + \cos {\left (3 x \right )}\right )^{5}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 12209 vs. \(2 (88) = 176\).
Time = 0.85 (sec) , antiderivative size = 12209, normalized size of antiderivative = 113.05 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=\text {Too large to display} \]
-1/12288*(8*(1827*sin(23*x) + 3733*sin(21*x) + 6115*sin(19*x) + 9109*sin(1 7*x) + 5746*sin(15*x) + 2382*sin(13*x) - 2382*sin(11*x) - 5746*sin(9*x) - 9109*sin(7*x) - 6115*sin(5*x) - 3733*sin(3*x) - 1827*sin(x))*cos(24*x) - 1 4616*(4*sin(22*x) + 10*sin(20*x) + 20*sin(18*x) + 31*sin(16*x) + 40*sin(14 *x) + 44*sin(12*x) + 40*sin(10*x) + 31*sin(8*x) + 20*sin(6*x) + 10*sin(4*x ) + 4*sin(2*x))*cos(23*x) + 32*(3733*sin(21*x) + 6115*sin(19*x) + 9109*sin (17*x) + 5746*sin(15*x) + 2382*sin(13*x) - 2382*sin(11*x) - 5746*sin(9*x) - 9109*sin(7*x) - 6115*sin(5*x) - 3733*sin(3*x) - 1827*sin(x))*cos(22*x) - 29864*(10*sin(20*x) + 20*sin(18*x) + 31*sin(16*x) + 40*sin(14*x) + 44*sin (12*x) + 40*sin(10*x) + 31*sin(8*x) + 20*sin(6*x) + 10*sin(4*x) + 4*sin(2* x))*cos(21*x) + 80*(6115*sin(19*x) + 9109*sin(17*x) + 5746*sin(15*x) + 238 2*sin(13*x) - 2382*sin(11*x) - 5746*sin(9*x) - 9109*sin(7*x) - 6115*sin(5* x) - 3733*sin(3*x) - 1827*sin(x))*cos(20*x) - 48920*(20*sin(18*x) + 31*sin (16*x) + 40*sin(14*x) + 44*sin(12*x) + 40*sin(10*x) + 31*sin(8*x) + 20*sin (6*x) + 10*sin(4*x) + 4*sin(2*x))*cos(19*x) + 160*(9109*sin(17*x) + 5746*s in(15*x) + 2382*sin(13*x) - 2382*sin(11*x) - 5746*sin(9*x) - 9109*sin(7*x) - 6115*sin(5*x) - 3733*sin(3*x) - 1827*sin(x))*cos(18*x) - 72872*(31*sin( 16*x) + 40*sin(14*x) + 44*sin(12*x) + 40*sin(10*x) + 31*sin(8*x) + 20*sin( 6*x) + 10*sin(4*x) + 4*sin(2*x))*cos(17*x) + 248*(5746*sin(15*x) + 2382*si n(13*x) - 2382*sin(11*x) - 5746*sin(9*x) - 9109*sin(7*x) - 6115*sin(5*x...
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=-\frac {1483}{2048} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) + \frac {43 \, \sin \left (x\right )^{3} - 45 \, \sin \left (x\right )}{256 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac {10488 \, \sin \left (x\right )^{7} - 14108 \, \sin \left (x\right )^{5} + 6514 \, \sin \left (x\right )^{3} - 993 \, \sin \left (x\right )}{1536 \, {\left (2 \, \sin \left (x\right )^{2} - 1\right )}^{4}} - \frac {523}{512} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {523}{512} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
-1483/2048*sqrt(2)*log(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 - 5 23/512*log(sin(x) + 1) + 523/512*log(-sin(x) + 1)
Time = 1.25 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.84 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx=-\frac {11492\,\sin \left (3\,x\right )+18218\,\sin \left (5\,x\right )+12230\,\sin \left (7\,x\right )+7466\,\sin \left (9\,x\right )+3654\,\sin \left (11\,x\right )+276144\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+4764\,\sin \left (x\right )+502080\,\cos \left (2\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+389112\,\cos \left (4\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+251040\,\cos \left (6\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+125520\,\cos \left (8\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+50208\,\cos \left (10\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+12552\,\cos \left (12\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )-97878\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )-177960\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (2\,x\right )-137919\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (4\,x\right )-88980\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (6\,x\right )-44490\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (8\,x\right )-17796\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (10\,x\right )-4449\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (12\,x\right )}{122880\,\cos \left (2\,x\right )+95232\,\cos \left (4\,x\right )+61440\,\cos \left (6\,x\right )+30720\,\cos \left (8\,x\right )+12288\,\cos \left (10\,x\right )+3072\,\cos \left (12\,x\right )+67584} \]
-(11492*sin(3*x) + 18218*sin(5*x) + 12230*sin(7*x) + 7466*sin(9*x) + 3654* sin(11*x) + 276144*atanh(sin(x/2)/cos(x/2)) + 4764*sin(x) + 502080*cos(2*x )*atanh(sin(x/2)/cos(x/2)) + 389112*cos(4*x)*atanh(sin(x/2)/cos(x/2)) + 25 1040*cos(6*x)*atanh(sin(x/2)/cos(x/2)) + 125520*cos(8*x)*atanh(sin(x/2)/co s(x/2)) + 50208*cos(10*x)*atanh(sin(x/2)/cos(x/2)) + 12552*cos(12*x)*atanh (sin(x/2)/cos(x/2)) - 97878*2^(1/2)*atanh(2^(1/2)*sin(x)) - 177960*2^(1/2) *atanh(2^(1/2)*sin(x))*cos(2*x) - 137919*2^(1/2)*atanh(2^(1/2)*sin(x))*cos (4*x) - 88980*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(6*x) - 44490*2^(1/2)*atanh (2^(1/2)*sin(x))*cos(8*x) - 17796*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(10*x) - 4449*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(12*x))/(122880*cos(2*x) + 95232*c os(4*x) + 61440*cos(6*x) + 30720*cos(8*x) + 12288*cos(10*x) + 3072*cos(12* x) + 67584)