Integrand size = 11, antiderivative size = 190 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=-\frac {637 \text {arctanh}(2 \cos (x) \sin (x))}{8192}-\frac {3955 \cot (x)}{1048576}-\frac {2131 \cot ^3(x)}{3145728}-\frac {173 \cot ^5(x)}{2621440}+\frac {158879 \tan (x)}{1048576}+\frac {128051 \tan ^3(x)}{3145728}+\frac {78617 \tan ^5(x)}{5242880}+\frac {4751 \tan ^7(x)}{1048576}+\frac {8501 \tan ^9(x)}{9437184}+\frac {983 \tan ^{11}(x)}{11534336}+\frac {\csc ^5(x) \sec ^{21}(x)}{1310720 \left (1-\tan ^2(x)\right )^5}-\frac {3 \csc ^5(x) \sec ^{19}(x)}{5242880 \left (1-\tan ^2(x)\right )^4}+\frac {61 \csc ^5(x) \sec ^{17}(x)}{15728640 \left (1-\tan ^2(x)\right )^3}-\frac {97 \csc ^5(x) \sec ^{15}(x)}{10485760 \left (1-\tan ^2(x)\right )^2}+\frac {443 \csc ^5(x) \sec ^{13}(x)}{6291456 \left (1-\tan ^2(x)\right )} \] Output:
-637/8192*arctanh(2*cos(x)*sin(x))-3955/1048576*cot(x)-2131/3145728*cot(x) ^3-173/2621440*cot(x)^5+158879/1048576*tan(x)+128051/3145728*tan(x)^3+7861 7/5242880*tan(x)^5+4751/1048576*tan(x)^7+8501/9437184*tan(x)^9+983/1153433 6*tan(x)^11+1/1310720*csc(x)^5*sec(x)^21/(1-tan(x)^2)^5-3/5242880*csc(x)^5 *sec(x)^19/(1-tan(x)^2)^4+61/15728640*csc(x)^5*sec(x)^17/(1-tan(x)^2)^3-97 /10485760*csc(x)^5*sec(x)^15/(1-tan(x)^2)^2+443*csc(x)^5*sec(x)^13/(629145 6-6291456*tan(x)^2)
Time = 1.85 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=\frac {\cos (2 x) \left (330631741440 \cos ^{12}(x) \cos ^5(2 x) (\log (\cos (x)-\sin (x))-\log (\cos (x)+\sin (x))) \sin ^6(x)+70 (200591+280858 \cos (2 x)) \sin (2 x)+\cos (x) (2858152 \sin (3 x)+16199140 \sin (5 x)+1288983 \sin (7 x)-8835165 \sin (9 x)-13422850 \sin (11 x)-7589426 \sin (13 x)+1766680 \sin (15 x)+4598336 \sin (17 x)+3590570 \sin (19 x)+1462970 \sin (21 x)-708993 \sin (23 x)-954445 \sin (25 x)-253952 \sin (27 x))\right )}{16220160 (\sin (3 x)+\sin (5 x))^6} \] Input:
Integrate[(Sin[3*x] + Sin[5*x])^(-6),x]
Output:
(Cos[2*x]*(330631741440*Cos[x]^12*Cos[2*x]^5*(Log[Cos[x] - Sin[x]] - Log[C os[x] + Sin[x]])*Sin[x]^6 + 70*(200591 + 280858*Cos[2*x])*Sin[2*x] + Cos[x ]*(2858152*Sin[3*x] + 16199140*Sin[5*x] + 1288983*Sin[7*x] - 8835165*Sin[9 *x] - 13422850*Sin[11*x] - 7589426*Sin[13*x] + 1766680*Sin[15*x] + 4598336 *Sin[17*x] + 3590570*Sin[19*x] + 1462970*Sin[21*x] - 708993*Sin[23*x] - 95 4445*Sin[25*x] - 253952*Sin[27*x])))/(16220160*(Sin[3*x] + Sin[5*x])^6)
Time = 0.45 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {3042, 4822, 27, 370, 27, 439, 439, 27, 439, 27, 439, 27, 437, 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.
| \(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^6}dx\) |
| \(\Big \downarrow \) 4822 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^{14} \cot ^6(x)}{262144 \left (1-\tan ^2(x)\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot ^6(x) \left (\tan ^2(x)+1\right )^{14}}{\left (1-\tan ^2(x)\right )^6}d\tan (x)}{262144}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {\frac {1}{10} \int \frac {4 \cot ^6(x) \left (5-8 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^{12}}{\left (1-\tan ^2(x)\right )^5}d\tan (x)+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {\cot ^6(x) \left (5-8 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^{12}}{\left (1-\tan ^2(x)\right )^5}d\tan (x)+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \int \frac {\cot ^6(x) \left (\tan ^2(x)+1\right )^{11} \left (97 \tan ^2(x)+25\right )}{\left (1-\tan ^2(x)\right )^4}d\tan (x)-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {4 \cot ^6(x) \left (190-481 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^{10}}{\left (1-\tan ^2(x)\right )^3}d\tan (x)+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \int \frac {\cot ^6(x) \left (190-481 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^{10}}{\left (1-\tan ^2(x)\right )^3}d\tan (x)+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \left (\frac {1}{4} \int -\frac {5 \cot ^6(x) \left (139-1025 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^9}{\left (1-\tan ^2(x)\right )^2}d\tan (x)-\frac {291 \left (\tan ^2(x)+1\right )^{10} \cot ^5(x)}{4 \left (1-\tan ^2(x)\right )^2}\right )+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \left (-\frac {5}{4} \int \frac {\cot ^6(x) \left (139-1025 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^9}{\left (1-\tan ^2(x)\right )^2}d\tan (x)-\frac {291 \left (\tan ^2(x)+1\right )^{10} \cot ^5(x)}{4 \left (1-\tan ^2(x)\right )^2}\right )+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \left (-\frac {5}{4} \left (\frac {1}{2} \int -\frac {12 \cot ^6(x) \left (346-983 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^8}{1-\tan ^2(x)}d\tan (x)-\frac {443 \left (\tan ^2(x)+1\right )^9 \cot ^5(x)}{1-\tan ^2(x)}\right )-\frac {291 \left (\tan ^2(x)+1\right )^{10} \cot ^5(x)}{4 \left (1-\tan ^2(x)\right )^2}\right )+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \left (-\frac {5}{4} \left (-6 \int \frac {\cot ^6(x) \left (346-983 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^8}{1-\tan ^2(x)}d\tan (x)-\frac {443 \left (\tan ^2(x)+1\right )^9 \cot ^5(x)}{1-\tan ^2(x)}\right )-\frac {291 \left (\tan ^2(x)+1\right )^{10} \cot ^5(x)}{4 \left (1-\tan ^2(x)\right )^2}\right )+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 437 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \left (-\frac {5}{4} \left (-6 \int \left (983 \tan ^{10}(x)+8501 \tan ^8(x)+33257 \tan ^6(x)+78617 \tan ^4(x)+128051 \tan ^2(x)+346 \cot ^6(x)+2131 \cot ^4(x)+3955 \cot ^2(x)+\frac {163072}{\tan ^2(x)-1}+158879\right )d\tan (x)-\frac {443 \left (\tan ^2(x)+1\right )^9 \cot ^5(x)}{1-\tan ^2(x)}\right )-\frac {291 \left (\tan ^2(x)+1\right )^{10} \cot ^5(x)}{4 \left (1-\tan ^2(x)\right )^2}\right )+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{8} \left (\frac {2}{3} \left (-\frac {5}{4} \left (-6 \left (-163072 \text {arctanh}(\tan (x))+\frac {983 \tan ^{11}(x)}{11}+\frac {8501 \tan ^9(x)}{9}+4751 \tan ^7(x)+\frac {78617 \tan ^5(x)}{5}+\frac {128051 \tan ^3(x)}{3}+158879 \tan (x)-\frac {346 \cot ^5(x)}{5}-\frac {2131 \cot ^3(x)}{3}-3955 \cot (x)\right )-\frac {443 \left (\tan ^2(x)+1\right )^9 \cot ^5(x)}{1-\tan ^2(x)}\right )-\frac {291 \left (\tan ^2(x)+1\right )^{10} \cot ^5(x)}{4 \left (1-\tan ^2(x)\right )^2}\right )+\frac {61 \left (\tan ^2(x)+1\right )^{11} \cot ^5(x)}{3 \left (1-\tan ^2(x)\right )^3}\right )-\frac {3 \left (\tan ^2(x)+1\right )^{12} \cot ^5(x)}{8 \left (1-\tan ^2(x)\right )^4}\right )+\frac {\left (\tan ^2(x)+1\right )^{13} \cot ^5(x)}{5 \left (1-\tan ^2(x)\right )^5}}{262144}\) |
Input:
Int[(Sin[3*x] + Sin[5*x])^(-6),x]
Output:
((Cot[x]^5*(1 + Tan[x]^2)^13)/(5*(1 - Tan[x]^2)^5) + (2*((-3*Cot[x]^5*(1 + Tan[x]^2)^12)/(8*(1 - Tan[x]^2)^4) + ((61*Cot[x]^5*(1 + Tan[x]^2)^11)/(3* (1 - Tan[x]^2)^3) + (2*((-291*Cot[x]^5*(1 + Tan[x]^2)^10)/(4*(1 - Tan[x]^2 )^2) - (5*((-443*Cot[x]^5*(1 + Tan[x]^2)^9)/(1 - Tan[x]^2) - 6*(-163072*Ar cTanh[Tan[x]] - 3955*Cot[x] - (2131*Cot[x]^3)/3 - (346*Cot[x]^5)/5 + 15887 9*Tan[x] + (128051*Tan[x]^3)/3 + (78617*Tan[x]^5)/5 + 4751*Tan[x]^7 + (850 1*Tan[x]^9)/9 + (983*Tan[x]^11)/11)))/4))/3)/8))/5)/262144
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f , g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Int[((a_.)*sin[(m_.)*((c_.) + (d_.)*(x_))] + (b_.)*sin[(n_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[1/d Subst[Int[Simplify[TrigExpand[a*Sin[ m*ArcTan[x]] + b*Sin[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 = 527.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(\frac {\tan \left (x \right )^{11}}{2883584}+\frac {5 \tan \left (x \right )^{9}}{589824}+\frac {7 \tan \left (x \right )^{7}}{65536}+\frac {63 \tan \left (x \right )^{5}}{65536}+\frac {1001 \tan \left (x \right )^{3}}{131072}+\frac {5691 \tan \left (x \right )}{65536}-\frac {1}{5120 \left (\tan \left (x \right )+1\right )^{5}}+\frac {5}{4096 \left (\tan \left (x \right )+1\right )^{4}}-\frac {65}{12288 \left (\tan \left (x \right )+1\right )^{3}}+\frac {35}{2048 \left (\tan \left (x \right )+1\right )^{2}}-\frac {483}{8192 \left (\tan \left (x \right )+1\right )}-\frac {637 \ln \left (\tan \left (x \right )+1\right )}{8192}-\frac {1}{1310720 \tan \left (x \right )^{5}}-\frac {5}{196608 \tan \left (x \right )^{3}}-\frac {49}{65536 \tan \left (x \right )}-\frac {1}{5120 \left (\tan \left (x \right )-1\right )^{5}}-\frac {5}{4096 \left (\tan \left (x \right )-1\right )^{4}}-\frac {65}{12288 \left (\tan \left (x \right )-1\right )^{3}}-\frac {35}{2048 \left (\tan \left (x \right )-1\right )^{2}}-\frac {483}{8192 \left (\tan \left (x \right )-1\right )}+\frac {637 \ln \left (\tan \left (x \right )-1\right )}{8192}\) | \(148\) |
| risch | \(\frac {i \left (-507904-2732109 \,{\mathrm e}^{2 i x}-5726670 \,{\mathrm e}^{4 i x}+27306940 \,{\mathrm e}^{14 i x}+28293614 \,{\mathrm e}^{12 i x}+4995770 \,{\mathrm e}^{16 i x}-55880332 \,{\mathrm e}^{20 i x}+13829334 \,{\mathrm e}^{10 i x}+596630 \,{\mathrm e}^{8 i x}-5533460 \,{\mathrm e}^{6 i x}-27964835 \,{\mathrm e}^{18 i x}+28082740 \,{\mathrm e}^{26 i x}+55213340 \,{\mathrm e}^{28 i x}-15893220 \,{\mathrm e}^{24 i x}-53648552 \,{\mathrm e}^{22 i x}+315315 \,{\mathrm e}^{50 i x}+47932248 \,{\mathrm e}^{30 i x}-12727715 \,{\mathrm e}^{34 i x}-23483460 \,{\mathrm e}^{38 i x}+1891890 \,{\mathrm e}^{48 i x}+17765748 \,{\mathrm e}^{32 i x}+4624620 \,{\mathrm e}^{46 i x}-28017990 \,{\mathrm e}^{36 i x}+1639638 \,{\mathrm e}^{42 i x}+5675670 \,{\mathrm e}^{44 i x}-9291282 \,{\mathrm e}^{40 i x}\right )}{2027520 \left ({\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}-1\right )^{5} \left ({\mathrm e}^{2 i x}+1\right )^{11}}+\frac {637 \ln \left ({\mathrm e}^{2 i x}-i\right )}{8192}-\frac {637 \ln \left ({\mathrm e}^{2 i x}+i\right )}{8192}\) | \(234\) |
Input:
int(1/(sin(3*x)+sin(5*x))^6,x,method=_RETURNVERBOSE)
Output:
0
Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=-\frac {520093696 \, \cos \left (x\right )^{26} - 2761909760 \, \cos \left (x\right )^{24} + 6252481280 \, \cos \left (x\right )^{22} - 7859968640 \, \cos \left (x\right )^{20} + 5977292480 \, \cos \left (x\right )^{18} - 2795580992 \, \cos \left (x\right )^{16} + 772785600 \, \cos \left (x\right )^{14} - 110231680 \, \cos \left (x\right )^{12} + 4862560 \, \cos \left (x\right )^{10} + 159440 \, \cos \left (x\right )^{8} + 13960 \, \cos \left (x\right )^{6} + 1820 \, \cos \left (x\right )^{4} + 5045040 \, {\left (32 \, \cos \left (x\right )^{25} - 144 \, \cos \left (x\right )^{23} + 272 \, \cos \left (x\right )^{21} - 280 \, \cos \left (x\right )^{19} + 170 \, \cos \left (x\right )^{17} - 61 \, \cos \left (x\right )^{15} + 12 \, \cos \left (x\right )^{13} - \cos \left (x\right )^{11}\right )} \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 5045040 \, {\left (32 \, \cos \left (x\right )^{25} - 144 \, \cos \left (x\right )^{23} + 272 \, \cos \left (x\right )^{21} - 280 \, \cos \left (x\right )^{19} + 170 \, \cos \left (x\right )^{17} - 61 \, \cos \left (x\right )^{15} + 12 \, \cos \left (x\right )^{13} - \cos \left (x\right )^{11}\right )} \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 290 \, \cos \left (x\right )^{2} + 45}{129761280 \, {\left (32 \, \cos \left (x\right )^{25} - 144 \, \cos \left (x\right )^{23} + 272 \, \cos \left (x\right )^{21} - 280 \, \cos \left (x\right )^{19} + 170 \, \cos \left (x\right )^{17} - 61 \, \cos \left (x\right )^{15} + 12 \, \cos \left (x\right )^{13} - \cos \left (x\right )^{11}\right )} \sin \left (x\right )} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^6,x, algorithm="fricas")
Output:
-1/129761280*(520093696*cos(x)^26 - 2761909760*cos(x)^24 + 6252481280*cos( x)^22 - 7859968640*cos(x)^20 + 5977292480*cos(x)^18 - 2795580992*cos(x)^16 + 772785600*cos(x)^14 - 110231680*cos(x)^12 + 4862560*cos(x)^10 + 159440* cos(x)^8 + 13960*cos(x)^6 + 1820*cos(x)^4 + 5045040*(32*cos(x)^25 - 144*co s(x)^23 + 272*cos(x)^21 - 280*cos(x)^19 + 170*cos(x)^17 - 61*cos(x)^15 + 1 2*cos(x)^13 - cos(x)^11)*log(2*cos(x)*sin(x) + 1)*sin(x) - 5045040*(32*cos (x)^25 - 144*cos(x)^23 + 272*cos(x)^21 - 280*cos(x)^19 + 170*cos(x)^17 - 6 1*cos(x)^15 + 12*cos(x)^13 - cos(x)^11)*log(-2*cos(x)*sin(x) + 1)*sin(x) + 290*cos(x)^2 + 45)/((32*cos(x)^25 - 144*cos(x)^23 + 272*cos(x)^21 - 280*c os(x)^19 + 170*cos(x)^17 - 61*cos(x)^15 + 12*cos(x)^13 - cos(x)^11)*sin(x) )
Timed out. \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(sin(3*x)+sin(5*x))**6,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 26212 vs. \(2 (150) = 300\).
Time = 8.04 (sec) , antiderivative size = 26212, normalized size of antiderivative = 137.96 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^6,x, algorithm="maxima")
Output:
-1/8110080*(4*(315315*sin(50*x) + 1891890*sin(48*x) + 4624620*sin(46*x) + 5675670*sin(44*x) + 1639638*sin(42*x) - 9291282*sin(40*x) - 23483460*sin(3 8*x) - 28017990*sin(36*x) - 12727715*sin(34*x) + 17765748*sin(32*x) + 4793 2248*sin(30*x) + 55213340*sin(28*x) + 28082740*sin(26*x) - 15893220*sin(24 *x) - 53648552*sin(22*x) - 55880332*sin(20*x) - 27964835*sin(18*x) + 49957 70*sin(16*x) + 27306940*sin(14*x) + 28293614*sin(12*x) + 13829334*sin(10*x ) + 596630*sin(8*x) - 5533460*sin(6*x) - 5726670*sin(4*x) - 2732109*sin(2* x))*cos(52*x) + 12*(2207205*sin(48*x) + 7147140*sin(46*x) + 10300290*sin(4 4*x) + 5801796*sin(42*x) - 10804794*sin(40*x) - 36456420*sin(38*x) - 49204 155*sin(36*x) - 28608580*sin(34*x) + 20291271*sin(32*x) + 74843496*sin(30* x) + 95711980*sin(28*x) + 56165480*sin(26*x) - 17071740*sin(24*x) - 862761 04*sin(22*x) - 96520439*sin(20*x) - 52776520*sin(18*x) + 3159715*sin(16*x) + 44103380*sin(14*x) + 48809458*sin(12*x) + 25136148*sin(10*x) + 2244310* sin(8*x) - 8964820*sin(6*x) - 9876765*sin(4*x) - 4833588*sin(2*x))*cos(50* x) + 60*(2102100*sin(46*x) + 4414410*sin(44*x) + 4666662*sin(42*x) + 42042 *sin(40*x) - 10870860*sin(38*x) - 19819800*sin(36*x) - 16511495*sin(34*x) - 522522*sin(32*x) + 22707048*sin(30*x) + 37555700*sin(28*x) + 28082740*si n(26*x) + 1764420*sin(24*x) - 28423352*sin(22*x) - 37592062*sin(20*x) - 24 181055*sin(18*x) - 3202420*sin(16*x) + 14694340*sin(14*x) + 18960290*sin(1 2*x) + 10802310*sin(10*x) + 1857890*sin(8*x) - 3010940*sin(6*x) - 38347...
Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=\frac {1}{2883584} \, \tan \left (x\right )^{11} + \frac {5}{589824} \, \tan \left (x\right )^{9} + \frac {7}{65536} \, \tan \left (x\right )^{7} + \frac {63}{65536} \, \tan \left (x\right )^{5} + \frac {1001}{131072} \, \tan \left (x\right )^{3} - \frac {466620 \, \tan \left (x\right )^{14} - 1558920 \, \tan \left (x\right )^{12} + 2086119 \, \tan \left (x\right )^{10} - 1269375 \, \tan \left (x\right )^{8} + 302690 \, \tan \left (x\right )^{6} - 2470 \, \tan \left (x\right )^{4} - 85 \, \tan \left (x\right )^{2} - 3}{3932160 \, {\left (\tan \left (x\right )^{3} - \tan \left (x\right )\right )}^{5}} - \frac {637}{8192} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) + \frac {637}{8192} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) + \frac {5691}{65536} \, \tan \left (x\right ) \] Input:
integrate(1/(sin(3*x)+sin(5*x))^6,x, algorithm="giac")
Output:
1/2883584*tan(x)^11 + 5/589824*tan(x)^9 + 7/65536*tan(x)^7 + 63/65536*tan( x)^5 + 1001/131072*tan(x)^3 - 1/3932160*(466620*tan(x)^14 - 1558920*tan(x) ^12 + 2086119*tan(x)^10 - 1269375*tan(x)^8 + 302690*tan(x)^6 - 2470*tan(x) ^4 - 85*tan(x)^2 - 3)/(tan(x)^3 - tan(x))^5 - 637/8192*log(abs(tan(x) + 1) ) + 637/8192*log(abs(tan(x) - 1)) + 5691/65536*tan(x)
Time = 20.22 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=\text {Too large to display} \] Input:
int(1/(sin(3*x) + sin(5*x))^6,x)
Output:
(1529*tan(x/2))/4194304 + (637*atanh((405769*tan(x/2))/(524288*((405769*ta n(x/2)^2)/1048576 - 405769/1048576))))/4096 - ((131*tan(x/2)^2)/62914560 + (6467*tan(x/2)^4)/25165824 - (1398771*tan(x/2)^6)/4194304 + (105697243*ta n(x/2)^8)/8388608 - (13458458059*tan(x/2)^10)/62914560 + (269257843633*tan (x/2)^12)/125829120 - (531565752613*tan(x/2)^14)/37748736 + (2688258867587 5*tan(x/2)^16)/415236096 - (4986800098581*tan(x/2)^18)/23068672 + (1118354 916224251*tan(x/2)^20)/2076180480 - (353295649020331*tan(x/2)^22)/34603008 0 + (619113558639649*tan(x/2)^24)/415236096 - (31895077087741*tan(x/2)^26) /18874368 + (618702357469885*tan(x/2)^28)/415236096 - (352813072839409*tan (x/2)^30)/346030080 + (2231900059630223*tan(x/2)^32)/4152360960 - (2982865 8098879*tan(x/2)^34)/138412032 + (53536737797231*tan(x/2)^36)/830472192 - (528472163959*tan(x/2)^38)/37748736 + (89025967259*tan(x/2)^40)/41943040 - (4433459077*tan(x/2)^42)/20971520 + (103725651*tan(x/2)^44)/8388608 - (13 46425*tan(x/2)^46)/4194304 + 1/41943040)/(tan(x/2)^5 - 41*tan(x/2)^7 + 750 *tan(x/2)^9 - 8110*tan(x/2)^11 + 58005*tan(x/2)^13 - 291533*tan(x/2)^15 + 1069928*tan(x/2)^17 - 2945000*tan(x/2)^19 + 6197490*tan(x/2)^21 - 10105730 *tan(x/2)^23 + 12877844*tan(x/2)^25 - 12877844*tan(x/2)^27 + 10105730*tan( x/2)^29 - 6197490*tan(x/2)^31 + 2945000*tan(x/2)^33 - 1069928*tan(x/2)^35 + 291533*tan(x/2)^37 - 58005*tan(x/2)^39 + 8110*tan(x/2)^41 - 750*tan(x/2) ^43 + 41*tan(x/2)^45 - tan(x/2)^47) + (77*tan(x/2)^3)/25165824 + tan(x/...
\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^6} \, dx=\int \frac {1}{\sin \left (5 x \right )^{6}+6 \sin \left (5 x \right )^{5} \sin \left (3 x \right )+15 \sin \left (5 x \right )^{4} \sin \left (3 x \right )^{2}+20 \sin \left (5 x \right )^{3} \sin \left (3 x \right )^{3}+15 \sin \left (5 x \right )^{2} \sin \left (3 x \right )^{4}+6 \sin \left (5 x \right ) \sin \left (3 x \right )^{5}+\sin \left (3 x \right )^{6}}d x \] Input:
int(1/(sin(3*x)+sin(5*x))^6,x)
Output:
int(1/(sin(5*x)**6 + 6*sin(5*x)**5*sin(3*x) + 15*sin(5*x)**4*sin(3*x)**2 + 20*sin(5*x)**3*sin(3*x)**3 + 15*sin(5*x)**2*sin(3*x)**4 + 6*sin(5*x)*sin( 3*x)**5 + sin(3*x)**6),x)