Integrand size = 11, antiderivative size = 122 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=-\frac {9}{128} \text {arctanh}(2 \cos (x) \sin (x))-\frac {37 \cot (x)}{4096}-\frac {11 \cot ^3(x)}{12288}+\frac {289 \tan (x)}{2048}+\frac {199 \tan ^3(x)}{6144}+\frac {163 \tan ^5(x)}{20480}+\frac {29 \tan ^7(x)}{28672}+\frac {\csc ^3(x) \sec ^{13}(x)}{12288 \left (1-\tan ^2(x)\right )^3}-\frac {\csc ^3(x) \sec ^{11}(x)}{12288 \left (1-\tan ^2(x)\right )^2}+\frac {5 \csc ^3(x) \sec ^9(x)}{6144 \left (1-\tan ^2(x)\right )} \] Output:
-9/128*arctanh(2*cos(x)*sin(x))-37/4096*cot(x)-11/12288*cot(x)^3+289/2048* tan(x)+199/6144*tan(x)^3+163/20480*tan(x)^5+29/28672*tan(x)^7+1/12288*csc( x)^3*sec(x)^13/(1-tan(x)^2)^3-1/12288*csc(x)^3*sec(x)^11/(1-tan(x)^2)^2+5* csc(x)^3*sec(x)^9/(6144-6144*tan(x)^2)
Time = 1.70 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=\frac {\cos (x) (\sin (x)-\sin (3 x))^4 \left (-35 \cos ^5(x) \cot ^3(x)+15 \sin (x)+228 \cos ^2(x) \sin (x)+2369 \cos ^4(x) \sin (x)+32878 \cos ^6(x) \sin (x)+70 \cos ^7(x) \left (-19 \cot (x)+4 \sec ^3(2 x) (81 \cos (2 x) (\log (\cos (x)-\sin (x))-\log (\cos (x)+\sin (x)))+27 \cos (6 x) (\log (\cos (x)-\sin (x))-\log (\cos (x)+\sin (x)))+21 \sin (2 x)-6 \sin (4 x)+17 \sin (6 x))\right )\right )}{1680 (\sin (3 x)+\sin (5 x))^4} \] Input:
Integrate[(Sin[3*x] + Sin[5*x])^(-4),x]
Output:
(Cos[x]*(Sin[x] - Sin[3*x])^4*(-35*Cos[x]^5*Cot[x]^3 + 15*Sin[x] + 228*Cos [x]^2*Sin[x] + 2369*Cos[x]^4*Sin[x] + 32878*Cos[x]^6*Sin[x] + 70*Cos[x]^7* (-19*Cot[x] + 4*Sec[2*x]^3*(81*Cos[2*x]*(Log[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x]]) + 27*Cos[6*x]*(Log[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x]]) + 21*Sin[2*x] - 6*Sin[4*x] + 17*Sin[6*x]))))/(1680*(Sin[3*x] + Sin[5*x])^4)
Time = 0.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4822, 27, 370, 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.
| \(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^4}dx\) |
| \(\Big \downarrow \) 4822 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^9 \cot ^4(x)}{4096 \left (1-\tan ^2(x)\right )^4}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot ^4(x) \left (\tan ^2(x)+1\right )^9}{\left (1-\tan ^2(x)\right )^4}d\tan (x)}{4096}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {4 \cot ^4(x) \left (3-5 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^7}{\left (1-\tan ^2(x)\right )^3}d\tan (x)+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{3} \int \frac {\cot ^4(x) \left (3-5 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^7}{\left (1-\tan ^2(x)\right )^3}d\tan (x)+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{4} \int \frac {2 \cot ^4(x) \left (\tan ^2(x)+1\right )^6 \left (17 \tan ^2(x)+3\right )}{\left (1-\tan ^2(x)\right )^2}d\tan (x)-\frac {\left (\tan ^2(x)+1\right )^7 \cot ^3(x)}{2 \left (1-\tan ^2(x)\right )^2}\right )+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} \int \frac {\cot ^4(x) \left (\tan ^2(x)+1\right )^6 \left (17 \tan ^2(x)+3\right )}{\left (1-\tan ^2(x)\right )^2}d\tan (x)-\frac {\left (\tan ^2(x)+1\right )^7 \cot ^3(x)}{2 \left (1-\tan ^2(x)\right )^2}\right )+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {6 \cot ^4(x) \left (11-29 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^5}{1-\tan ^2(x)}d\tan (x)+\frac {10 \left (\tan ^2(x)+1\right )^6 \cot ^3(x)}{1-\tan ^2(x)}\right )-\frac {\left (\tan ^2(x)+1\right )^7 \cot ^3(x)}{2 \left (1-\tan ^2(x)\right )^2}\right )+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} \left (3 \int \frac {\cot ^4(x) \left (11-29 \tan ^2(x)\right ) \left (\tan ^2(x)+1\right )^5}{1-\tan ^2(x)}d\tan (x)+\frac {10 \left (\tan ^2(x)+1\right )^6 \cot ^3(x)}{1-\tan ^2(x)}\right )-\frac {\left (\tan ^2(x)+1\right )^7 \cot ^3(x)}{2 \left (1-\tan ^2(x)\right )^2}\right )+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 437 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} \left (3 \int \left (29 \tan ^6(x)+163 \tan ^4(x)+398 \tan ^2(x)+11 \cot ^4(x)+37 \cot ^2(x)+\frac {576}{\tan ^2(x)-1}+578\right )d\tan (x)+\frac {10 \left (\tan ^2(x)+1\right )^6 \cot ^3(x)}{1-\tan ^2(x)}\right )-\frac {\left (\tan ^2(x)+1\right )^7 \cot ^3(x)}{2 \left (1-\tan ^2(x)\right )^2}\right )+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} \left (3 \left (-576 \text {arctanh}(\tan (x))+\frac {29 \tan ^7(x)}{7}+\frac {163 \tan ^5(x)}{5}+\frac {398 \tan ^3(x)}{3}+578 \tan (x)-\frac {11 \cot ^3(x)}{3}-37 \cot (x)\right )+\frac {10 \left (\tan ^2(x)+1\right )^6 \cot ^3(x)}{1-\tan ^2(x)}\right )-\frac {\left (\tan ^2(x)+1\right )^7 \cot ^3(x)}{2 \left (1-\tan ^2(x)\right )^2}\right )+\frac {\left (\tan ^2(x)+1\right )^8 \cot ^3(x)}{3 \left (1-\tan ^2(x)\right )^3}}{4096}\) |
Input:
Int[(Sin[3*x] + Sin[5*x])^(-4),x]
Output:
((Cot[x]^3*(1 + Tan[x]^2)^8)/(3*(1 - Tan[x]^2)^3) + (2*(-1/2*(Cot[x]^3*(1 + Tan[x]^2)^7)/(1 - Tan[x]^2)^2 + ((10*Cot[x]^3*(1 + Tan[x]^2)^6)/(1 - Tan [x]^2) + 3*(-576*ArcTanh[Tan[x]] - 37*Cot[x] - (11*Cot[x]^3)/3 + 578*Tan[x ] + (398*Tan[x]^3)/3 + (163*Tan[x]^5)/5 + (29*Tan[x]^7)/7))/2))/3)/4096
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 = 32.14 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.02
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(\frac {\tan \left (x \right )^{7}}{28672}+\frac {13 \tan \left (x \right )^{5}}{20480}+\frac {41 \tan \left (x \right )^{3}}{6144}+\frac {169 \tan \left (x \right )}{2048}-\frac {1}{12288 \tan \left (x \right )^{3}}-\frac {13}{4096 \tan \left (x \right )}-\frac {1}{384 \left (\tan \left (x \right )+1\right )^{3}}+\frac {3}{256 \left (\tan \left (x \right )+1\right )^{2}}-\frac {7}{128 \left (\tan \left (x \right )+1\right )}-\frac {9 \ln \left (\tan \left (x \right )+1\right )}{128}-\frac {1}{384 \left (\tan \left (x \right )-1\right )^{3}}-\frac {3}{256 \left (\tan \left (x \right )-1\right )^{2}}-\frac {7}{128 \left (\tan \left (x \right )-1\right )}+\frac {9 \ln \left (\tan \left (x \right )-1\right )}{128}\) | \(98\) |
| risch | \(\frac {i \left (945 \,{\mathrm e}^{30 i x}+3780 \,{\mathrm e}^{28 i x}+5355 \,{\mathrm e}^{26 i x}+2520 \,{\mathrm e}^{24 i x}-3591 \,{\mathrm e}^{22 i x}-11844 \,{\mathrm e}^{20 i x}-15381 \,{\mathrm e}^{18 i x}-7344 \,{\mathrm e}^{16 i x}+4587 \,{\mathrm e}^{14 i x}+18108 \,{\mathrm e}^{12 i x}+16377 \,{\mathrm e}^{10 i x}+5848 \,{\mathrm e}^{8 i x}-1301 \,{\mathrm e}^{6 i x}-6204 \,{\mathrm e}^{4 i x}-5711 \,{\mathrm e}^{2 i x}-1664\right )}{6720 \left ({\mathrm e}^{2 i x}+1\right )^{7} \left ({\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}-1\right )^{3}}-\frac {9 \ln \left ({\mathrm e}^{2 i x}+i\right )}{128}+\frac {9 \ln \left ({\mathrm e}^{2 i x}-i\right )}{128}\) | \(164\) |
Input:
int(1/(sin(3*x)+sin(5*x))^4,x,method=_RETURNVERBOSE)
Output:
0
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=-\frac {425984 \, \cos \left (x\right )^{16} - 1398912 \, \cos \left (x\right )^{14} + 1740096 \, \cos \left (x\right )^{12} - 1007552 \, \cos \left (x\right )^{10} + 260448 \, \cos \left (x\right )^{8} - 19056 \, \cos \left (x\right )^{6} - 920 \, \cos \left (x\right )^{4} + 15120 \, {\left (8 \, \cos \left (x\right )^{15} - 20 \, \cos \left (x\right )^{13} + 18 \, \cos \left (x\right )^{11} - 7 \, \cos \left (x\right )^{9} + \cos \left (x\right )^{7}\right )} \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 15120 \, {\left (8 \, \cos \left (x\right )^{15} - 20 \, \cos \left (x\right )^{13} + 18 \, \cos \left (x\right )^{11} - 7 \, \cos \left (x\right )^{9} + \cos \left (x\right )^{7}\right )} \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 108 \, \cos \left (x\right )^{2} - 15}{430080 \, {\left (8 \, \cos \left (x\right )^{15} - 20 \, \cos \left (x\right )^{13} + 18 \, \cos \left (x\right )^{11} - 7 \, \cos \left (x\right )^{9} + \cos \left (x\right )^{7}\right )} \sin \left (x\right )} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^4,x, algorithm="fricas")
Output:
-1/430080*(425984*cos(x)^16 - 1398912*cos(x)^14 + 1740096*cos(x)^12 - 1007 552*cos(x)^10 + 260448*cos(x)^8 - 19056*cos(x)^6 - 920*cos(x)^4 + 15120*(8 *cos(x)^15 - 20*cos(x)^13 + 18*cos(x)^11 - 7*cos(x)^9 + cos(x)^7)*log(2*co s(x)*sin(x) + 1)*sin(x) - 15120*(8*cos(x)^15 - 20*cos(x)^13 + 18*cos(x)^11 - 7*cos(x)^9 + cos(x)^7)*log(-2*cos(x)*sin(x) + 1)*sin(x) - 108*cos(x)^2 - 15)/((8*cos(x)^15 - 20*cos(x)^13 + 18*cos(x)^11 - 7*cos(x)^9 + cos(x)^7) *sin(x))
\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=\int \frac {1}{\left (\sin {\left (3 x \right )} + \sin {\left (5 x \right )}\right )^{4}}\, dx \] Input:
integrate(1/(sin(3*x)+sin(5*x))**4,x)
Output:
Integral((sin(3*x) + sin(5*x))**(-4), x)
Leaf count of result is larger than twice the leaf count of optimal. 10270 vs. \(2 (96) = 192\).
Time = 1.19 (sec) , antiderivative size = 10270, normalized size of antiderivative = 84.18 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(3*x)+sin(5*x))^4,x, algorithm="maxima")
Output:
-1/26880*(4*(945*sin(30*x) + 3780*sin(28*x) + 5355*sin(26*x) + 2520*sin(24 *x) - 3591*sin(22*x) - 11844*sin(20*x) - 15381*sin(18*x) - 7344*sin(16*x) + 4587*sin(14*x) + 18108*sin(12*x) + 16377*sin(10*x) + 5848*sin(8*x) - 130 1*sin(6*x) - 6204*sin(4*x) - 5711*sin(2*x))*cos(32*x) + 8*(4725*sin(28*x) + 8820*sin(26*x) + 5985*sin(24*x) - 1512*sin(22*x) - 15183*sin(20*x) - 250 92*sin(18*x) - 14688*sin(16*x) + 3504*sin(14*x) + 27711*sin(12*x) + 27084* sin(10*x) + 10751*sin(8*x) - 712*sin(6*x) - 9573*sin(4*x) - 9532*sin(2*x)) *cos(30*x) + 24*(2835*sin(26*x) + 3780*sin(24*x) + 3969*sin(22*x) - 504*si n(20*x) - 7821*sin(18*x) - 7344*sin(16*x) - 2973*sin(14*x) + 6768*sin(12*x ) + 8817*sin(10*x) + 4588*sin(8*x) + 1219*sin(6*x) - 2424*sin(4*x) - 3191* sin(2*x))*cos(28*x) + 8*(10395*sin(24*x) + 24948*sin(22*x) + 24507*sin(20* x) + 1368*sin(18*x) - 14688*sin(16*x) - 22956*sin(14*x) - 11979*sin(12*x) + 624*sin(10*x) + 6341*sin(8*x) + 8108*sin(6*x) + 3657*sin(4*x) - 712*sin( 2*x))*cos(26*x) + 8*(18711*sin(22*x) + 34524*sin(20*x) + 30501*sin(18*x) + 7344*sin(16*x) - 19707*sin(14*x) - 40788*sin(12*x) - 31497*sin(10*x) - 83 68*sin(8*x) + 6341*sin(6*x) + 13764*sin(4*x) + 10751*sin(2*x))*cos(24*x) + 24*(12915*sin(20*x) + 23580*sin(18*x) + 14688*sin(16*x) - 1992*sin(14*x) - 25443*sin(12*x) - 25572*sin(10*x) - 10499*sin(8*x) + 208*sin(6*x) + 8817 *sin(4*x) + 9028*sin(2*x))*cos(22*x) + 72*(7485*sin(18*x) + 7344*sin(16*x) + 3309*sin(14*x) - 6264*sin(12*x) - 8481*sin(10*x) - 4532*sin(8*x) - 1...
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=\frac {1}{28672} \, \tan \left (x\right )^{7} + \frac {13}{20480} \, \tan \left (x\right )^{5} + \frac {41}{6144} \, \tan \left (x\right )^{3} - \frac {1383 \, \tan \left (x\right )^{8} - 2164 \, \tan \left (x\right )^{6} + 1074 \, \tan \left (x\right )^{4} - 36 \, \tan \left (x\right )^{2} - 1}{12288 \, {\left (\tan \left (x\right )^{3} - \tan \left (x\right )\right )}^{3}} - \frac {9}{128} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) + \frac {9}{128} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) + \frac {169}{2048} \, \tan \left (x\right ) \] Input:
integrate(1/(sin(3*x)+sin(5*x))^4,x, algorithm="giac")
Output:
1/28672*tan(x)^7 + 13/20480*tan(x)^5 + 41/6144*tan(x)^3 - 1/12288*(1383*ta n(x)^8 - 2164*tan(x)^6 + 1074*tan(x)^4 - 36*tan(x)^2 - 1)/(tan(x)^3 - tan( x))^3 - 9/128*log(abs(tan(x) + 1)) + 9/128*log(abs(tan(x) - 1)) + 169/2048 *tan(x)
Time = 19.98 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=\frac {51\,\mathrm {tan}\left (\frac {x}{2}\right )}{32768}+\frac {9\,\mathrm {atanh}\left (\frac {81\,\mathrm {tan}\left (\frac {x}{2}\right )}{128\,\left (\frac {81\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{256}-\frac {81}{256}\right )}\right )}{64}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{98304}-\frac {-\frac {10579\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{28}}{32768}+\frac {14653\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{26}}{2048}-\frac {10651583\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{24}}{163840}+\frac {4505745\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{22}}{14336}-\frac {1040494093\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{20}}{1146880}+\frac {60039671\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{18}}{35840}-\frac {335936923\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}}{163840}+\frac {60310501\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}}{35840}-\frac {1050596843\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}}{1146880}+\frac {4580743\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}}{14336}-\frac {32836279\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{491520}+\frac {46097\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{6144}-\frac {11717\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{32768}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{768}+\frac {1}{98304}}{-{\mathrm {tan}\left (\frac {x}{2}\right )}^{29}+25\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{27}-258\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{25}+1442\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{23}-4871\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{21}+10623\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{19}-15548\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{17}+15548\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{15}-10623\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{13}+4871\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{11}-1442\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9+258\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7-25\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \] Input:
int(1/(sin(3*x) + sin(5*x))^4,x)
Output:
(51*tan(x/2))/32768 + (9*atanh((81*tan(x/2))/(128*((81*tan(x/2)^2)/256 - 8 1/256))))/64 + tan(x/2)^3/98304 - (tan(x/2)^2/768 - (11717*tan(x/2)^4)/327 68 + (46097*tan(x/2)^6)/6144 - (32836279*tan(x/2)^8)/491520 + (4580743*tan (x/2)^10)/14336 - (1050596843*tan(x/2)^12)/1146880 + (60310501*tan(x/2)^14 )/35840 - (335936923*tan(x/2)^16)/163840 + (60039671*tan(x/2)^18)/35840 - (1040494093*tan(x/2)^20)/1146880 + (4505745*tan(x/2)^22)/14336 - (10651583 *tan(x/2)^24)/163840 + (14653*tan(x/2)^26)/2048 - (10579*tan(x/2)^28)/3276 8 + 1/98304)/(tan(x/2)^3 - 25*tan(x/2)^5 + 258*tan(x/2)^7 - 1442*tan(x/2)^ 9 + 4871*tan(x/2)^11 - 10623*tan(x/2)^13 + 15548*tan(x/2)^15 - 15548*tan(x /2)^17 + 10623*tan(x/2)^19 - 4871*tan(x/2)^21 + 1442*tan(x/2)^23 - 258*tan (x/2)^25 + 25*tan(x/2)^27 - tan(x/2)^29)
\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^4} \, dx=\int \frac {1}{\sin \left (5 x \right )^{4}+4 \sin \left (5 x \right )^{3} \sin \left (3 x \right )+6 \sin \left (5 x \right )^{2} \sin \left (3 x \right )^{2}+4 \sin \left (5 x \right ) \sin \left (3 x \right )^{3}+\sin \left (3 x \right )^{4}}d x \] Input:
int(1/(sin(3*x)+sin(5*x))^4,x)
Output:
int(1/(sin(5*x)**4 + 4*sin(5*x)**3*sin(3*x) + 6*sin(5*x)**2*sin(3*x)**2 + 4*sin(5*x)*sin(3*x)**3 + sin(3*x)**4),x)