Integrand size = 11, antiderivative size = 331 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\frac {7 \sqrt {\frac {1}{2} \left (194686841-137664384 \sqrt {2}\right )} \log \left (\sqrt {3-2 \sqrt {2}} \cos (x)-\sin (x)\right )}{2048}+\frac {7 \sqrt {\frac {1}{2} \left (194686841+137664384 \sqrt {2}\right )} \log \left (\sqrt {3+2 \sqrt {2}} \cos (x)-\sin (x)\right )}{2048}-\frac {7 \sqrt {\frac {1}{2} \left (194686841-137664384 \sqrt {2}\right )} \log \left (\sqrt {3-2 \sqrt {2}} \cos (x)+\sin (x)\right )}{2048}-\frac {7 \sqrt {\frac {1}{2} \left (194686841+137664384 \sqrt {2}\right )} \log \left (\sqrt {3+2 \sqrt {2}} \cos (x)+\sin (x)\right )}{2048}+\frac {1345 \tan (x)}{64}+\frac {25 \tan ^3(x)}{96}+\frac {\tan ^5(x)}{320}+\frac {1024 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )^5}-\frac {16 \tan (x) \left (161861+349117 \tan ^2(x)\right )}{5 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )^4}-\frac {\tan (x) \left (2334795+1718923 \tan ^2(x)\right )}{15 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )^3}-\frac {\tan (x) \left (2320917+901229 \tan ^2(x)\right )}{160 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )^2}-\frac {7 \tan (x) \left (44613+12853 \tan ^2(x)\right )}{512 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
7/2048*(9867/2*2^(1/2)-6976)*ln((2^(1/2)-1)*cos(x)-sin(x))+7/2048*(9867/2* 2^(1/2)+6976)*ln((1+2^(1/2))*cos(x)-sin(x))-7/2048*(9867/2*2^(1/2)-6976)*l n((2^(1/2)-1)*cos(x)+sin(x))-7/2048*(9867/2*2^(1/2)+6976)*ln((1+2^(1/2))*c os(x)+sin(x))+1345/64*tan(x)+25/96*tan(x)^3+1/320*tan(x)^5+1024/5*tan(x)*( 3363-19601*tan(x)^2)/(1-6*tan(x)^2+tan(x)^4)^5-16/5*tan(x)*(161861+349117* tan(x)^2)/(1-6*tan(x)^2+tan(x)^4)^4-1/15*tan(x)*(2334795+1718923*tan(x)^2) /(1-6*tan(x)^2+tan(x)^4)^3-1/160*tan(x)*(2320917+901229*tan(x)^2)/(1-6*tan (x)^2+tan(x)^4)^2-7*tan(x)*(44613+12853*tan(x)^2)/(512-3072*tan(x)^2+512*t an(x)^4)
Time = 1.60 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\frac {2929920 \text {arctanh}\left (\sqrt {2}-\tan (x)\right )-2929920 \text {arctanh}\left (\sqrt {2}+\tan (x)\right )+1036035 \sqrt {2} \log \left (\sqrt {2}-2 \sin (2 x)\right )-1036035 \sqrt {2} \log \left (\sqrt {2}+2 \sin (2 x)\right )+\frac {240 (40-51 \sin (2 x))}{(\cos (2 x)-\sin (2 x))^4}+\frac {20 (9224-11223 \sin (2 x))}{(\cos (2 x)-\sin (2 x))^2}+\frac {192 (-7+10 \sin (2 x))}{(\cos (2 x)-\sin (2 x))^5}+\frac {192 (7+10 \sin (2 x))}{(\cos (2 x)+\sin (2 x))^5}-\frac {240 (40+51 \sin (2 x))}{(\cos (2 x)+\sin (2 x))^4}+\frac {40 (-1133+1400 \sin (2 x))}{(\cos (2 x)-\sin (2 x))^3}+\frac {40 (1133+1400 \sin (2 x))}{(\cos (2 x)+\sin (2 x))^3}-\frac {20 (9224+11223 \sin (2 x))}{(\cos (2 x)+\sin (2 x))^2}+\frac {10 (-82791+102656 \sin (2 x))}{\cos (2 x)-\sin (2 x)}+\frac {10 (82791+102656 \sin (2 x))}{\cos (2 x)+\sin (2 x)}+1275392 \tan (x)+15616 \sec ^2(x) \tan (x)+192 \sec ^4(x) \tan (x)}{61440} \] Input:
Integrate[(Cos[3*x] + Cos[5*x])^(-6),x]
Output:
(2929920*ArcTanh[Sqrt[2] - Tan[x]] - 2929920*ArcTanh[Sqrt[2] + Tan[x]] + 1 036035*Sqrt[2]*Log[Sqrt[2] - 2*Sin[2*x]] - 1036035*Sqrt[2]*Log[Sqrt[2] + 2 *Sin[2*x]] + (240*(40 - 51*Sin[2*x]))/(Cos[2*x] - Sin[2*x])^4 + (20*(9224 - 11223*Sin[2*x]))/(Cos[2*x] - Sin[2*x])^2 + (192*(-7 + 10*Sin[2*x]))/(Cos [2*x] - Sin[2*x])^5 + (192*(7 + 10*Sin[2*x]))/(Cos[2*x] + Sin[2*x])^5 - (2 40*(40 + 51*Sin[2*x]))/(Cos[2*x] + Sin[2*x])^4 + (40*(-1133 + 1400*Sin[2*x ]))/(Cos[2*x] - Sin[2*x])^3 + (40*(1133 + 1400*Sin[2*x]))/(Cos[2*x] + Sin[ 2*x])^3 - (20*(9224 + 11223*Sin[2*x]))/(Cos[2*x] + Sin[2*x])^2 + (10*(-827 91 + 102656*Sin[2*x]))/(Cos[2*x] - Sin[2*x]) + (10*(82791 + 102656*Sin[2*x ]))/(Cos[2*x] + Sin[2*x]) + 1275392*Tan[x] + 15616*Sec[x]^2*Tan[x] + 192*S ec[x]^4*Tan[x])/61440
Time = 1.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.76, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.364, Rules used = {3042, 4823, 27, 1517, 27, 2206, 27, 2206, 27, 2206, 27, 2206, 27, 2205, 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.
| \(\displaystyle \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (3 x)+\cos (5 x))^6}dx\) |
| \(\Big \downarrow \) 4823 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^{14}}{64 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \int \frac {\left (\tan ^2(x)+1\right )^{14}}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {1}{64} \left (\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}-\frac {1}{320} \int \frac {64 \left (-5 \tan ^{24}(x)-100 \tan ^{22}(x)-1050 \tan ^{20}(x)-8020 \tan ^{18}(x)-52075 \tan ^{16}(x)-314440 \tan ^{14}(x)-1849580 \tan ^{12}(x)-10800200 \tan ^{10}(x)-62966635 \tan ^8(x)-367009620 \tan ^6(x)-2139096090 \tan ^4(x)+9370140572 \tan ^2(x)+220397563\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}d\tan (x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}-\frac {1}{5} \int \frac {-5 \tan ^{24}(x)-100 \tan ^{22}(x)-1050 \tan ^{20}(x)-8020 \tan ^{18}(x)-52075 \tan ^{16}(x)-314440 \tan ^{14}(x)-1849580 \tan ^{12}(x)-10800200 \tan ^{10}(x)-62966635 \tan ^8(x)-367009620 \tan ^6(x)-2139096090 \tan ^4(x)+9370140572 \tan ^2(x)+220397563}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}d\tan (x)\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (\frac {1}{256} \int -\frac {256 \left (-5 \tan ^{20}(x)-130 \tan ^{18}(x)-1825 \tan ^{16}(x)-18840 \tan ^{14}(x)-163290 \tan ^{12}(x)-1275340 \tan ^{10}(x)-9338330 \tan ^8(x)-65554840 \tan ^6(x)-446957345 \tan ^4(x)+1664246654 \tan ^2(x)+54651899\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}d\tan (x)-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-\int \frac {-5 \tan ^{20}(x)-130 \tan ^{18}(x)-1825 \tan ^{16}(x)-18840 \tan ^{14}(x)-163290 \tan ^{12}(x)-1275340 \tan ^{10}(x)-9338330 \tan ^8(x)-65554840 \tan ^6(x)-446957345 \tan ^4(x)+1664246654 \tan ^2(x)+54651899}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}d\tan (x)-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (\frac {1}{192} \int -\frac {192 \left (-5 \tan ^{16}(x)-160 \tan ^{14}(x)-2780 \tan ^{12}(x)-35360 \tan ^{10}(x)-372670 \tan ^8(x)-3476000 \tan ^6(x)-29821660 \tan ^4(x)+89024416 \tan ^2(x)+4842939\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}d\tan (x)-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-\int \frac {-5 \tan ^{16}(x)-160 \tan ^{14}(x)-2780 \tan ^{12}(x)-35360 \tan ^{10}(x)-372670 \tan ^8(x)-3476000 \tan ^6(x)-29821660 \tan ^4(x)+89024416 \tan ^2(x)+4842939}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}d\tan (x)-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (\frac {1}{128} \int -\frac {640 \left (-\tan ^{12}(x)-38 \tan ^{10}(x)-783 \tan ^8(x)-11732 \tan ^6(x)-144143 \tan ^4(x)+254132 \tan ^2(x)+40221\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}d\tan (x)-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {2 \tan (x) \left (901229 \tan ^2(x)+2320917\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-5 \int \frac {-\tan ^{12}(x)-38 \tan ^{10}(x)-783 \tan ^8(x)-11732 \tan ^6(x)-144143 \tan ^4(x)+254132 \tan ^2(x)+40221}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}d\tan (x)-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {2 \tan (x) \left (901229 \tan ^2(x)+2320917\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-5 \left (\frac {7 \tan (x) \left (12853 \tan ^2(x)+44613\right )}{8 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )}-\frac {1}{64} \int -\frac {8 \left (-8 \tan ^8(x)-352 \tan ^6(x)-8368 \tan ^4(x)-53741 \tan ^2(x)+9477\right )}{\tan ^4(x)-6 \tan ^2(x)+1}d\tan (x)\right )-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {2 \tan (x) \left (901229 \tan ^2(x)+2320917\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-5 \left (\frac {1}{8} \int \frac {-8 \tan ^8(x)-352 \tan ^6(x)-8368 \tan ^4(x)-53741 \tan ^2(x)+9477}{\tan ^4(x)-6 \tan ^2(x)+1}d\tan (x)+\frac {7 \tan (x) \left (12853 \tan ^2(x)+44613\right )}{8 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )}\right )-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {2 \tan (x) \left (901229 \tan ^2(x)+2320917\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-5 \left (\frac {1}{8} \int \left (-8 \tan ^4(x)-400 \tan ^2(x)+\frac {7 \left (2891-16843 \tan ^2(x)\right )}{\tan ^4(x)-6 \tan ^2(x)+1}-10760\right )d\tan (x)+\frac {7 \tan (x) \left (12853 \tan ^2(x)+44613\right )}{8 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )}\right )-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {2 \tan (x) \left (901229 \tan ^2(x)+2320917\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{5} \left (-5 \left (\frac {1}{8} \left (\frac {7}{2} \sqrt {\frac {1}{2} \left (194686841-137664384 \sqrt {2}\right )} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3-2 \sqrt {2}}}\right )+\frac {7}{2} \sqrt {\frac {1}{2} \left (194686841+137664384 \sqrt {2}\right )} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3+2 \sqrt {2}}}\right )-\frac {8}{5} \tan ^5(x)-\frac {400 \tan ^3(x)}{3}-10760 \tan (x)\right )+\frac {7 \tan (x) \left (12853 \tan ^2(x)+44613\right )}{8 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )}\right )-\frac {1024 \tan (x) \left (349117 \tan ^2(x)+161861\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}-\frac {2 \tan (x) \left (901229 \tan ^2(x)+2320917\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {64 \tan (x) \left (1718923 \tan ^2(x)+2334795\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )+\frac {65536 \tan (x) \left (3363-19601 \tan ^2(x)\right )}{5 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^5}\right )\) |
Input:
Int[(Cos[3*x] + Cos[5*x])^(-6),x]
Output:
((65536*Tan[x]*(3363 - 19601*Tan[x]^2))/(5*(1 - 6*Tan[x]^2 + Tan[x]^4)^5) + ((-1024*Tan[x]*(161861 + 349117*Tan[x]^2))/(1 - 6*Tan[x]^2 + Tan[x]^4)^4 - (64*Tan[x]*(2334795 + 1718923*Tan[x]^2))/(3*(1 - 6*Tan[x]^2 + Tan[x]^4) ^3) - (2*Tan[x]*(2320917 + 901229*Tan[x]^2))/(1 - 6*Tan[x]^2 + Tan[x]^4)^2 - 5*((7*Tan[x]*(44613 + 12853*Tan[x]^2))/(8*(1 - 6*Tan[x]^2 + Tan[x]^4)) + ((7*Sqrt[(194686841 - 137664384*Sqrt[2])/2]*ArcTanh[Tan[x]/Sqrt[3 - 2*Sq rt[2]]])/2 + (7*Sqrt[(194686841 + 137664384*Sqrt[2])/2]*ArcTanh[Tan[x]/Sqr t[3 + 2*Sqrt[2]]])/2 - 10760*Tan[x] - (400*Tan[x]^3)/3 - (8*Tan[x]^5)/5)/8 ))/5)/64
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] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && 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*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 = 254.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(\frac {\tan \left (x \right )^{5}}{320}+\frac {25 \tan \left (x \right )^{3}}{96}+\frac {1345 \tan \left (x \right )}{64}-\frac {\frac {89971 \tan \left (x \right )^{9}}{64}+\frac {622347 \tan \left (x \right )^{8}}{64}+\frac {76717 \tan \left (x \right )^{7}}{4}-\frac {23969 \tan \left (x \right )^{6}}{8}-\frac {5258653 \tan \left (x \right )^{5}}{160}+\frac {564905 \tan \left (x \right )^{4}}{96}+\frac {573689 \tan \left (x \right )^{3}}{24}-\frac {209671 \tan \left (x \right )^{2}}{12}+\frac {903029 \tan \left (x \right )}{192}-\frac {437287}{960}}{16 \left (\tan \left (x \right )^{2}+2 \tan \left (x \right )-1\right )^{5}}-\frac {763 \ln \left (\tan \left (x \right )^{2}+2 \tan \left (x \right )-1\right )}{32}-\frac {69069 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (x \right )+2\right ) \sqrt {2}}{4}\right )}{2048}+\frac {-\frac {89971 \tan \left (x \right )^{9}}{64}+\frac {622347 \tan \left (x \right )^{8}}{64}-\frac {76717 \tan \left (x \right )^{7}}{4}-\frac {23969 \tan \left (x \right )^{6}}{8}+\frac {5258653 \tan \left (x \right )^{5}}{160}+\frac {564905 \tan \left (x \right )^{4}}{96}-\frac {573689 \tan \left (x \right )^{3}}{24}-\frac {209671 \tan \left (x \right )^{2}}{12}-\frac {903029 \tan \left (x \right )}{192}-\frac {437287}{960}}{16 \left (\tan \left (x \right )^{2}-2 \tan \left (x \right )-1\right )^{5}}+\frac {763 \ln \left (\tan \left (x \right )^{2}-2 \tan \left (x \right )-1\right )}{32}-\frac {69069 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (x \right )-2\right ) \sqrt {2}}{4}\right )}{2048}\) | \(214\) |
| risch | \(\frac {i \left (1150976+4718845 \,{\mathrm e}^{2 i x}+7062065 \,{\mathrm e}^{4 i x}+35211445 \,{\mathrm e}^{36 i x}+4447695 \,{\mathrm e}^{46 i x}+22337355 \,{\mathrm e}^{14 i x}+35275115 \,{\mathrm e}^{12 i x}+16497935 \,{\mathrm e}^{16 i x}+70481978 \,{\mathrm e}^{20 i x}+23643583 \,{\mathrm e}^{10 i x}+6747485 \,{\mathrm e}^{8 i x}+4466465 \,{\mathrm e}^{6 i x}+47394610 \,{\mathrm e}^{18 i x}+47506110 \,{\mathrm e}^{26 i x}+70424790 \,{\mathrm e}^{28 i x}+21552450 \,{\mathrm e}^{24 i x}+44672810 \,{\mathrm e}^{22 i x}+44615622 \,{\mathrm e}^{30 i x}+23786225 \,{\mathrm e}^{34 i x}+22273685 \,{\mathrm e}^{38 i x}+1036035 \,{\mathrm e}^{48 i x}+15909070 \,{\mathrm e}^{32 i x}+4762275 \,{\mathrm e}^{42 i x}+7043295 \,{\mathrm e}^{44 i x}+6281793 \,{\mathrm e}^{40 i x}\right )}{15360 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{8 i x}+{\mathrm e}^{2 i x}+1\right )^{5}}+\frac {763 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right )}{32}+\frac {69069 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{4096}+\frac {763 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right )}{32}-\frac {69069 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{4096}-\frac {763 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right )}{32}+\frac {69069 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{4096}-\frac {763 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right )}{32}-\frac {69069 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{4096}\) | \(366\) |
Input:
int(1/(cos(3*x)+cos(5*x))^6,x,method=_RETURNVERBOSE)
Output:
1/320*tan(x)^5+25/96*tan(x)^3+1345/64*tan(x)-1/16*(89971/64*tan(x)^9+62234 7/64*tan(x)^8+76717/4*tan(x)^7-23969/8*tan(x)^6-5258653/160*tan(x)^5+56490 5/96*tan(x)^4+573689/24*tan(x)^3-209671/12*tan(x)^2+903029/192*tan(x)-4372 87/960)/(tan(x)^2+2*tan(x)-1)^5-763/32*ln(tan(x)^2+2*tan(x)-1)-69069/2048* 2^(1/2)*arctanh(1/4*(2*tan(x)+2)*2^(1/2))+1/16*(-89971/64*tan(x)^9+622347/ 64*tan(x)^8-76717/4*tan(x)^7-23969/8*tan(x)^6+5258653/160*tan(x)^5+564905/ 96*tan(x)^4-573689/24*tan(x)^3-209671/12*tan(x)^2-903029/192*tan(x)-437287 /960)/(tan(x)^2-2*tan(x)-1)^5+763/32*ln(tan(x)^2-2*tan(x)-1)-69069/2048*2^ (1/2)*arctanh(1/4*(2*tan(x)-2)*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (235) = 470\).
Time = 0.34 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(3*x)+cos(5*x))^6,x, algorithm="fricas")
Output:
1/122880*(1464960*(32768*cos(x)^25 - 163840*cos(x)^23 + 348160*cos(x)^21 - 409600*cos(x)^19 + 291840*cos(x)^17 - 130048*cos(x)^15 + 36480*cos(x)^13 - 6400*cos(x)^11 + 680*cos(x)^9 - 40*cos(x)^7 + cos(x)^5)*log(4*(2*cos(x)^ 3 - cos(x))*sin(x) + 1) - 1464960*(32768*cos(x)^25 - 163840*cos(x)^23 + 34 8160*cos(x)^21 - 409600*cos(x)^19 + 291840*cos(x)^17 - 130048*cos(x)^15 + 36480*cos(x)^13 - 6400*cos(x)^11 + 680*cos(x)^9 - 40*cos(x)^7 + cos(x)^5)* log(-4*(2*cos(x)^3 - cos(x))*sin(x) + 1) + 1036035*(32768*sqrt(2)*cos(x)^2 5 - 163840*sqrt(2)*cos(x)^23 + 348160*sqrt(2)*cos(x)^21 - 409600*sqrt(2)*c os(x)^19 + 291840*sqrt(2)*cos(x)^17 - 130048*sqrt(2)*cos(x)^15 + 36480*sqr t(2)*cos(x)^13 - 6400*sqrt(2)*cos(x)^11 + 680*sqrt(2)*cos(x)^9 - 40*sqrt(2 )*cos(x)^7 + sqrt(2)*cos(x)^5)*log((4*sqrt(2)*cos(x)^2 - 4*(2*cos(x)^3 + ( sqrt(2) - 1)*cos(x))*sin(x) - 2*sqrt(2) + 3)/(4*(2*cos(x)^3 - cos(x))*sin( x) + 1)) + 1036035*(32768*sqrt(2)*cos(x)^25 - 163840*sqrt(2)*cos(x)^23 + 3 48160*sqrt(2)*cos(x)^21 - 409600*sqrt(2)*cos(x)^19 + 291840*sqrt(2)*cos(x) ^17 - 130048*sqrt(2)*cos(x)^15 + 36480*sqrt(2)*cos(x)^13 - 6400*sqrt(2)*co s(x)^11 + 680*sqrt(2)*cos(x)^9 - 40*sqrt(2)*cos(x)^7 + sqrt(2)*cos(x)^5)*l og(-(4*sqrt(2)*cos(x)^2 - 4*(2*cos(x)^3 - (sqrt(2) + 1)*cos(x))*sin(x) - 2 *sqrt(2) - 3)/(4*(2*cos(x)^3 - cos(x))*sin(x) - 1)) + 16*(9428795392*cos(x )^24 - 46673178624*cos(x)^22 + 97869217792*cos(x)^20 - 113064197120*cos(x) ^18 + 78516024320*cos(x)^16 - 33708463104*cos(x)^14 + 8952107008*cos(x)...
Timed out. \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(cos(3*x)+cos(5*x))**6,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 44647 vs. \(2 (235) = 470\).
Time = 7.60 (sec) , antiderivative size = 44647, normalized size of antiderivative = 134.89 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(3*x)+cos(5*x))^6,x, algorithm="maxima")
Output:
-1/122880*(8*(1036035*sin(48*x) + 4447695*sin(46*x) + 7043295*sin(44*x) + 4762275*sin(42*x) + 6281793*sin(40*x) + 22273685*sin(38*x) + 35211445*sin( 36*x) + 23786225*sin(34*x) + 15909070*sin(32*x) + 44615622*sin(30*x) + 704 24790*sin(28*x) + 47506110*sin(26*x) + 21552450*sin(24*x) + 44672810*sin(2 2*x) + 70481978*sin(20*x) + 47394610*sin(18*x) + 16497935*sin(16*x) + 2233 7355*sin(14*x) + 35275115*sin(12*x) + 23643583*sin(10*x) + 6747485*sin(8*x ) + 4466465*sin(6*x) + 7062065*sin(4*x) + 4718845*sin(2*x))*cos(50*x) + 40 *(2375625*sin(46*x) + 4971225*sin(44*x) + 2690205*sin(42*x) + 894411*sin(4 0*x) + 11913335*sin(38*x) + 24851095*sin(36*x) + 16533980*sin(34*x) + 4512 685*sin(32*x) + 23894922*sin(30*x) + 49704090*sin(28*x) + 35073690*sin(26* x) + 9120030*sin(24*x) + 23952110*sin(22*x) + 49761278*sin(20*x) + 3599822 5*sin(18*x) + 9245690*sin(16*x) + 11977005*sin(14*x) + 24914765*sin(12*x) + 18256201*sin(10*x) + 4675415*sin(8*x) + 2394395*sin(6*x) + 4989995*sin(4 *x) + 3682810*sin(2*x))*cos(48*x) + 40*(5191200*sin(44*x) + 629160*sin(42* x) - 10564428*sin(40*x) + 70420*sin(38*x) + 25945940*sin(36*x) + 16438585* sin(34*x) - 17106505*sin(32*x) + 277344*sin(30*x) + 51895680*sin(28*x) + 4 1639880*sin(26*x) - 10267440*sin(24*x) + 391720*sin(22*x) + 52010056*sin(2 0*x) + 45864575*sin(18*x) + 1862005*sin(16*x) + 197760*sin(14*x) + 2607328 0*sin(12*x) + 24159152*sin(10*x) + 4599580*sin(8*x) + 37540*sin(6*x) + 522 8740*sin(4*x) + 4989995*sin(2*x))*cos(46*x) - 40*(4562040*sin(42*x) + 2...
Time = 0.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\frac {1}{320} \, \tan \left (x\right )^{5} + \frac {25}{96} \, \tan \left (x\right )^{3} + \frac {69069}{4096} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \tan \left (x\right ) + 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \tan \left (x\right ) + 2 \right |}}\right ) + \frac {69069}{4096} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \tan \left (x\right ) - 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \tan \left (x\right ) - 2 \right |}}\right ) - \frac {1349565 \, \tan \left (x\right )^{19} - 27705195 \, \tan \left (x\right )^{17} + 227738532 \, \tan \left (x\right )^{15} - 899890380 \, \tan \left (x\right )^{13} + 1632144470 \, \tan \left (x\right )^{11} - 1041572010 \, \tan \left (x\right )^{9} + 318559540 \, \tan \left (x\right )^{7} - 51437148 \, \tan \left (x\right )^{5} + 4250645 \, \tan \left (x\right )^{3} - 142275 \, \tan \left (x\right )}{7680 \, {\left (\tan \left (x\right )^{4} - 6 \, \tan \left (x\right )^{2} + 1\right )}^{5}} - \frac {763}{32} \, \log \left ({\left | \tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) - 1 \right |}\right ) + \frac {763}{32} \, \log \left ({\left | \tan \left (x\right )^{2} - 2 \, \tan \left (x\right ) - 1 \right |}\right ) + \frac {1345}{64} \, \tan \left (x\right ) \] Input:
integrate(1/(cos(3*x)+cos(5*x))^6,x, algorithm="giac")
Output:
1/320*tan(x)^5 + 25/96*tan(x)^3 + 69069/4096*sqrt(2)*log(abs(-2*sqrt(2) + 2*tan(x) + 2)/abs(2*sqrt(2) + 2*tan(x) + 2)) + 69069/4096*sqrt(2)*log(abs( -2*sqrt(2) + 2*tan(x) - 2)/abs(2*sqrt(2) + 2*tan(x) - 2)) - 1/7680*(134956 5*tan(x)^19 - 27705195*tan(x)^17 + 227738532*tan(x)^15 - 899890380*tan(x)^ 13 + 1632144470*tan(x)^11 - 1041572010*tan(x)^9 + 318559540*tan(x)^7 - 514 37148*tan(x)^5 + 4250645*tan(x)^3 - 142275*tan(x))/(tan(x)^4 - 6*tan(x)^2 + 1)^5 - 763/32*log(abs(tan(x)^2 + 2*tan(x) - 1)) + 763/32*log(abs(tan(x)^ 2 - 2*tan(x) - 1)) + 1345/64*tan(x)
Time = 23.69 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx =\text {Too large to display} \] Input:
int(1/(cos(3*x) + cos(5*x))^6,x)
Output:
(atan((sin(x)*9353073240i - 2^(1/2)*sin(x)*6613621513i)/(22580316266*cos(x ) - 15966694753*2^(1/2)*cos(x)))*763i)/16 + (atan((sin(x)*9353073240i + 2^ (1/2)*sin(x)*6613621513i)/(22580316266*cos(x) + 15966694753*2^(1/2)*cos(x) ))*763i)/16 - (2^(1/2)*atan((sin(x)*9353073240i - 2^(1/2)*sin(x)*661362151 3i)/(22580316266*cos(x) - 15966694753*2^(1/2)*cos(x)))*69069i)/2048 + (2^( 1/2)*atan((sin(x)*9353073240i + 2^(1/2)*sin(x)*6613621513i)/(22580316266*c os(x) + 15966694753*2^(1/2)*cos(x)))*69069i)/2048 + (96*sin(x) + 3968*cos( x)^2*sin(x) + 390656*cos(x)^4*sin(x) - 26211060*cos(x)^6*sin(x) + 54656928 0*cos(x)^8*sin(x) - 5800760896*cos(x)^10*sin(x) + 35808428032*cos(x)^12*si n(x) - 134833852416*cos(x)^14*sin(x) + 314064097280*cos(x)^16*sin(x) - 452 256788480*cos(x)^18*sin(x) + 391476871168*cos(x)^20*sin(x) - 186692714496* cos(x)^22*sin(x) + 37715181568*cos(x)^24*sin(x))/(30720*cos(x)^5 - 1228800 *cos(x)^7 + 20889600*cos(x)^9 - 196608000*cos(x)^11 + 1120665600*cos(x)^13 - 3995074560*cos(x)^15 + 8965324800*cos(x)^17 - 12582912000*cos(x)^19 + 1 0695475200*cos(x)^21 - 5033164800*cos(x)^23 + 1006632960*cos(x)^25)
\[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^6} \, dx=\int \frac {1}{\cos \left (5 x \right )^{6}+6 \cos \left (5 x \right )^{5} \cos \left (3 x \right )+15 \cos \left (5 x \right )^{4} \cos \left (3 x \right )^{2}+20 \cos \left (5 x \right )^{3} \cos \left (3 x \right )^{3}+15 \cos \left (5 x \right )^{2} \cos \left (3 x \right )^{4}+6 \cos \left (5 x \right ) \cos \left (3 x \right )^{5}+\cos \left (3 x \right )^{6}}d x \] Input:
int(1/(cos(3*x)+cos(5*x))^6,x)
Output:
int(1/(cos(5*x)**6 + 6*cos(5*x)**5*cos(3*x) + 15*cos(5*x)**4*cos(3*x)**2 + 20*cos(5*x)**3*cos(3*x)**3 + 15*cos(5*x)**2*cos(3*x)**4 + 6*cos(5*x)*cos( 3*x)**5 + cos(3*x)**6),x)