Integrand size = 9, antiderivative size = 221 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=-\frac {3889 \text {arctanh}(\cos (x))}{186624}-\frac {332929 \text {arctanh}(2 \cos (x))}{2916}+\frac {82683 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{512 \sqrt {2}}-\frac {1}{108 (1-2 \cos (x))^4}+\frac {19}{162 (1-2 \cos (x))^3}-\frac {749}{648 (1-2 \cos (x))^2}+\frac {71551}{5832 (1-2 \cos (x))}-\frac {1}{124416 (1-\cos (x))^2}-\frac {209}{373248 (1-\cos (x))}+\frac {1}{124416 (1+\cos (x))^2}+\frac {209}{373248 (1+\cos (x))}+\frac {1}{108 (1+2 \cos (x))^4}-\frac {19}{162 (1+2 \cos (x))^3}+\frac {749}{648 (1+2 \cos (x))^2}-\frac {71551}{5832 (1+2 \cos (x))}-\frac {11643}{512} \cos (x) \sec (2 x)+\frac {681}{256} \cos (x) \sec ^2(2 x)-\frac {21}{64} \cos (x) \sec ^3(2 x)+\frac {1}{32} \cos (x) \sec ^4(2 x) \] Output:
-3889/186624*arctanh(cos(x))-332929/2916*arctanh(2*cos(x))+82683/1024*arct anh(cos(x)*2^(1/2))*2^(1/2)-1/108/(1-2*cos(x))^4+19/162/(1-2*cos(x))^3-749 /648/(1-2*cos(x))^2+71551/(5832-11664*cos(x))-1/124416/(1-cos(x))^2-209/(3 73248-373248*cos(x))+1/124416/(1+cos(x))^2+209/(373248+373248*cos(x))+1/10 8/(1+2*cos(x))^4-19/162/(1+2*cos(x))^3+749/648/(1+2*cos(x))^2-71551/(5832+ 11664*cos(x))-11643/512*cos(x)*sec(2*x)+681/256*cos(x)*sec(2*x)^2-21/64*co s(x)*sec(2*x)^3+1/32*cos(x)*sec(2*x)^4
Time = 2.47 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\frac {-120551814 \sqrt {2} \text {arctanh}\left (\frac {-1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+120551814 \sqrt {2} \text {arctanh}\left (\frac {1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )-\frac {13824}{(1-2 \cos (x))^4}-\frac {1725696}{(1-2 \cos (x))^2}+\frac {18317056}{1-2 \cos (x)}-\frac {175104}{(-1+2 \cos (x))^3}+\frac {13824}{(1+2 \cos (x))^4}-\frac {175104}{(1+2 \cos (x))^3}+\frac {1725696}{(1+2 \cos (x))^2}-\frac {18317056}{1+2 \cos (x)}-418 \csc ^2\left (\frac {x}{2}\right )-3 \csc ^4\left (\frac {x}{2}\right )-31112 \log \left (\cos \left (\frac {x}{2}\right )\right )+85229824 \log (1-2 \cos (x))-85229824 \log (1+2 \cos (x))+31112 \log \left (\sin \left (\frac {x}{2}\right )\right )+418 \sec ^2\left (\frac {x}{2}\right )+3 \sec ^4\left (\frac {x}{2}\right )-\frac {110808}{(\cos (x)-\sin (x))^3}-\frac {15100506}{\cos (x)-\sin (x)}+\frac {11664 \sin (x)}{(\cos (x)-\sin (x))^4}+\frac {1874988 \sin (x)}{(\cos (x)-\sin (x))^2}-\frac {11664 \sin (x)}{(\cos (x)+\sin (x))^4}-\frac {110808}{(\cos (x)+\sin (x))^3}-\frac {1874988 \sin (x)}{(\cos (x)+\sin (x))^2}-\frac {15100506}{\cos (x)+\sin (x)}}{1492992} \] Input:
Integrate[(Sin[x] + Sin[5*x])^(-5),x]
Output:
(-120551814*Sqrt[2]*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] + 120551814*Sqrt[2]*A rcTanh[(1 + Tan[x/2])/Sqrt[2]] - 13824/(1 - 2*Cos[x])^4 - 1725696/(1 - 2*C os[x])^2 + 18317056/(1 - 2*Cos[x]) - 175104/(-1 + 2*Cos[x])^3 + 13824/(1 + 2*Cos[x])^4 - 175104/(1 + 2*Cos[x])^3 + 1725696/(1 + 2*Cos[x])^2 - 183170 56/(1 + 2*Cos[x]) - 418*Csc[x/2]^2 - 3*Csc[x/2]^4 - 31112*Log[Cos[x/2]] + 85229824*Log[1 - 2*Cos[x]] - 85229824*Log[1 + 2*Cos[x]] + 31112*Log[Sin[x/ 2]] + 418*Sec[x/2]^2 + 3*Sec[x/2]^4 - 110808/(Cos[x] - Sin[x])^3 - 1510050 6/(Cos[x] - Sin[x]) + (11664*Sin[x])/(Cos[x] - Sin[x])^4 + (1874988*Sin[x] )/(Cos[x] - Sin[x])^2 - (11664*Sin[x])/(Cos[x] + Sin[x])^4 - 110808/(Cos[x ] + Sin[x])^3 - (1874988*Sin[x])/(Cos[x] + Sin[x])^2 - 15100506/(Cos[x] + Sin[x]))/1492992
Time = 0.53 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4824, 27, 1567, 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.
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^5}dx\) |
| \(\Big \downarrow \) 4824 |
| \(\displaystyle -\int \frac {1}{32 \left (1-\cos ^2(x)\right )^3 \left (8 \cos ^4(x)-6 \cos ^2(x)+1\right )^5}d\cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{32} \int \frac {1}{\left (1-\cos ^2(x)\right )^3 \left (8 \cos ^4(x)-6 \cos ^2(x)+1\right )^5}d\cos (x)\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle -\frac {1}{32} \int \left (\frac {4440}{2 \cos ^2(x)-1}-\frac {5326864}{729 \left (4 \cos ^2(x)-1\right )}+\frac {209}{11664 (\cos (x)-1)^2}+\frac {209}{11664 (\cos (x)+1)^2}-\frac {572408}{729 (2 \cos (x)-1)^2}-\frac {572408}{729 (2 \cos (x)+1)^2}-\frac {1200}{\left (2 \cos ^2(x)-1\right )^2}-\frac {1}{1944 (\cos (x)-1)^3}+\frac {1}{1944 (\cos (x)+1)^3}-\frac {11984}{81 (2 \cos (x)-1)^3}+\frac {11984}{81 (2 \cos (x)+1)^3}+\frac {288}{\left (2 \cos ^2(x)-1\right )^3}-\frac {608}{27 (2 \cos (x)-1)^4}-\frac {608}{27 (2 \cos (x)+1)^4}-\frac {56}{\left (2 \cos ^2(x)-1\right )^4}-\frac {64}{27 (2 \cos (x)-1)^5}+\frac {64}{27 (2 \cos (x)+1)^5}+\frac {8}{\left (2 \cos ^2(x)-1\right )^5}-\frac {3889}{5832 \left (\cos ^2(x)-1\right )}\right )d\cos (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{32} \left (-\frac {3889 \text {arctanh}(\cos (x))}{5832}-\frac {2663432}{729} \text {arctanh}(2 \cos (x))+2574 \sqrt {2} \text {arctanh}\left (\sqrt {2} \cos (x)\right )+\frac {315 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}+\frac {11643 \cos (x)}{16 \left (1-2 \cos ^2(x)\right )}+\frac {681 \cos (x)}{8 \left (1-2 \cos ^2(x)\right )^2}+\frac {21 \cos (x)}{2 \left (1-2 \cos ^2(x)\right )^3}+\frac {\cos (x)}{\left (1-2 \cos ^2(x)\right )^4}+\frac {286204}{729 (1-2 \cos (x))}-\frac {209}{11664 (1-\cos (x))}+\frac {209}{11664 (\cos (x)+1)}-\frac {286204}{729 (2 \cos (x)+1)}-\frac {2996}{81 (1-2 \cos (x))^2}-\frac {1}{3888 (1-\cos (x))^2}+\frac {1}{3888 (\cos (x)+1)^2}+\frac {2996}{81 (2 \cos (x)+1)^2}+\frac {304}{81 (1-2 \cos (x))^3}-\frac {304}{81 (2 \cos (x)+1)^3}-\frac {8}{27 (1-2 \cos (x))^4}+\frac {8}{27 (2 \cos (x)+1)^4}\right )\) |
Input:
Int[(Sin[x] + Sin[5*x])^(-5),x]
Output:
((-3889*ArcTanh[Cos[x]])/5832 - (2663432*ArcTanh[2*Cos[x]])/729 + (315*Arc Tanh[Sqrt[2]*Cos[x]])/(16*Sqrt[2]) + 2574*Sqrt[2]*ArcTanh[Sqrt[2]*Cos[x]] - 8/(27*(1 - 2*Cos[x])^4) + 304/(81*(1 - 2*Cos[x])^3) - 2996/(81*(1 - 2*Co s[x])^2) + 286204/(729*(1 - 2*Cos[x])) - 1/(3888*(1 - Cos[x])^2) - 209/(11 664*(1 - Cos[x])) + 1/(3888*(1 + Cos[x])^2) + 209/(11664*(1 + Cos[x])) + 8 /(27*(1 + 2*Cos[x])^4) - 304/(81*(1 + 2*Cos[x])^3) + 2996/(81*(1 + 2*Cos[x ])^2) - 286204/(729*(1 + 2*Cos[x])) + Cos[x]/(1 - 2*Cos[x]^2)^4 + (21*Cos[ x])/(2*(1 - 2*Cos[x]^2)^3) + (681*Cos[x])/(8*(1 - 2*Cos[x]^2)^2) + (11643* Cos[x])/(16*(1 - 2*Cos[x]^2)))/32
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] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
Int[((a_.)*sin[(m_.)*((c_.) + (d_.)*(x_))] + (b_.)*sin[(n_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Simplify[TrigExpand[a* Sin[m*ArcCos[x]] + b*Sin[n*ArcCos[x]]]]^p/Sqrt[1 - x^2], x], x, Cos[c + d*x ]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[(m - 1)/ 2] && IntegerQ[(n - 1)/2]
Time = 115.15 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(-\frac {1}{108 \left (2 \cos \left (x \right )-1\right )^{4}}-\frac {19}{162 \left (2 \cos \left (x \right )-1\right )^{3}}-\frac {749}{648 \left (2 \cos \left (x \right )-1\right )^{2}}-\frac {71551}{5832 \left (2 \cos \left (x \right )-1\right )}+\frac {332929 \ln \left (2 \cos \left (x \right )-1\right )}{5832}+\frac {1}{108 \left (2 \cos \left (x \right )+1\right )^{4}}-\frac {19}{162 \left (2 \cos \left (x \right )+1\right )^{3}}+\frac {749}{648 \left (2 \cos \left (x \right )+1\right )^{2}}-\frac {71551}{5832 \left (2 \cos \left (x \right )+1\right )}-\frac {332929 \ln \left (2 \cos \left (x \right )+1\right )}{5832}-\frac {1}{124416 \left (\cos \left (x \right )-1\right )^{2}}+\frac {209}{373248 \left (\cos \left (x \right )-1\right )}+\frac {3889 \ln \left (\cos \left (x \right )-1\right )}{373248}-\frac {4 \left (\frac {11643 \cos \left (x \right )^{7}}{256}-\frac {36291 \cos \left (x \right )^{5}}{512}+\frac {37821 \cos \left (x \right )^{3}}{1024}-\frac {13189 \cos \left (x \right )}{2048}\right )}{\left (2 \cos \left (x \right )^{2}-1\right )^{4}}+\frac {82683 \,\operatorname {arctanh}\left (\sqrt {2}\, \cos \left (x \right )\right ) \sqrt {2}}{1024}+\frac {1}{124416 \left (1+\cos \left (x \right )\right )^{2}}+\frac {209}{373248 \left (1+\cos \left (x \right )\right )}-\frac {3889 \ln \left (1+\cos \left (x \right )\right )}{373248}\) | \(193\) |
| risch | \(-\frac {5881813 \,{\mathrm e}^{39 i x}+2770929 \,{\mathrm e}^{37 i x}+16666827 \,{\mathrm e}^{35 i x}-11603277 \,{\mathrm e}^{33 i x}+2153987 \,{\mathrm e}^{31 i x}-49799073 \,{\mathrm e}^{29 i x}-11124845 \,{\mathrm e}^{27 i x}-29440353 \,{\mathrm e}^{25 i x}+33090774 \,{\mathrm e}^{23 i x}+41444690 \,{\mathrm e}^{21 i x}+41444690 \,{\mathrm e}^{19 i x}+33090774 \,{\mathrm e}^{17 i x}-29440353 \,{\mathrm e}^{15 i x}-11124845 \,{\mathrm e}^{13 i x}-49799073 \,{\mathrm e}^{11 i x}+2153987 \,{\mathrm e}^{9 i x}-11603277 \,{\mathrm e}^{7 i x}+16666827 \,{\mathrm e}^{5 i x}+2770929 \,{\mathrm e}^{3 i x}+5881813 \,{\mathrm e}^{i x}}{124416 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}-1\right )^{4}}+\frac {3889 \ln \left ({\mathrm e}^{i x}-1\right )}{186624}-\frac {3889 \ln \left ({\mathrm e}^{i x}+1\right )}{186624}+\frac {332929 \ln \left ({\mathrm e}^{2 i x}-{\mathrm e}^{i x}+1\right )}{5832}+\frac {82683 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{2048}-\frac {82683 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{2048}-\frac {332929 \ln \left ({\mathrm e}^{2 i x}+{\mathrm e}^{i x}+1\right )}{5832}\) | \(263\) |
Input:
int(1/(sin(x)+sin(5*x))^5,x,method=_RETURNVERBOSE)
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (175) = 350\).
Time = 0.21 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.41 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(5*x))^5,x, algorithm="fricas")
Output:
-1/1492992*(144551436288*cos(x)^19 - 669594734592*cos(x)^17 + 132648439296 0*cos(x)^15 - 1475815977984*cos(x)^13 + 1017399847392*cos(x)^11 - 45122761 5984*cos(x)^9 + 128907781608*cos(x)^7 - 22906808436*cos(x)^5 + 2301482760* cos(x)^3 - 60275907*(4096*sqrt(2)*cos(x)^20 - 20480*sqrt(2)*cos(x)^18 + 44 544*sqrt(2)*cos(x)^16 - 55552*sqrt(2)*cos(x)^14 + 44048*sqrt(2)*cos(x)^12 - 23232*sqrt(2)*cos(x)^10 + 8264*sqrt(2)*cos(x)^8 - 1960*sqrt(2)*cos(x)^6 + 297*sqrt(2)*cos(x)^4 - 26*sqrt(2)*cos(x)^2 + sqrt(2))*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1)) + 15556*(4096*cos(x)^20 - 20480*c os(x)^18 + 44544*cos(x)^16 - 55552*cos(x)^14 + 44048*cos(x)^12 - 23232*cos (x)^10 + 8264*cos(x)^8 - 1960*cos(x)^6 + 297*cos(x)^4 - 26*cos(x)^2 + 1)*l og(1/2*cos(x) + 1/2) - 15556*(4096*cos(x)^20 - 20480*cos(x)^18 + 44544*cos (x)^16 - 55552*cos(x)^14 + 44048*cos(x)^12 - 23232*cos(x)^10 + 8264*cos(x) ^8 - 1960*cos(x)^6 + 297*cos(x)^4 - 26*cos(x)^2 + 1)*log(-1/2*cos(x) + 1/2 ) - 85229824*(4096*cos(x)^20 - 20480*cos(x)^18 + 44544*cos(x)^16 - 55552*c os(x)^14 + 44048*cos(x)^12 - 23232*cos(x)^10 + 8264*cos(x)^8 - 1960*cos(x) ^6 + 297*cos(x)^4 - 26*cos(x)^2 + 1)*log(-2*cos(x) + 1) + 85229824*(4096*c os(x)^20 - 20480*cos(x)^18 + 44544*cos(x)^16 - 55552*cos(x)^14 + 44048*cos (x)^12 - 23232*cos(x)^10 + 8264*cos(x)^8 - 1960*cos(x)^6 + 297*cos(x)^4 - 26*cos(x)^2 + 1)*log(-2*cos(x) - 1) - 99800124*cos(x))/(4096*cos(x)^20 - 2 0480*cos(x)^18 + 44544*cos(x)^16 - 55552*cos(x)^14 + 44048*cos(x)^12 - ...
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\int \frac {1}{\left (\sin {\left (x \right )} + \sin {\left (5 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(sin(x)+sin(5*x))**5,x)
Output:
Integral((sin(x) + sin(5*x))**(-5), x)
Exception generated. \[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(sin(x)+sin(5*x))^5,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (175) = 350\).
Time = 0.14 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.99 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(5*x))^5,x, algorithm="giac")
Output:
82683/2048*sqrt(2)*log(abs(-4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)/a bs(4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)) + 1/1492992*(424*(cos(x) - 1)/(cos(x) + 1) - 23334*(cos(x) - 1)^2/(cos(x) + 1)^2 - 3)*(cos(x) + 1)^ 2/(cos(x) - 1)^2 - 53/186624*(cos(x) - 1)/(cos(x) + 1) + 1/497664*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1/186624*(40193230365*(cos(x) - 1)/(cos(x) + 1) + 614159700129*(cos(x) - 1)^2/(cos(x) + 1)^2 + 5371001231429*(cos(x) - 1)^3/ (cos(x) + 1)^3 + 29725819078749*(cos(x) - 1)^4/(cos(x) + 1)^4 + 1087791336 74049*(cos(x) - 1)^5/(cos(x) + 1)^5 + 267371188501221*(cos(x) - 1)^6/(cos( x) + 1)^6 + 440631281631289*(cos(x) - 1)^7/(cos(x) + 1)^7 + 48095285739900 3*(cos(x) - 1)^8/(cos(x) + 1)^8 + 343468414091831*(cos(x) - 1)^9/(cos(x) + 1)^9 + 160801518474339*(cos(x) - 1)^10/(cos(x) + 1)^10 + 49418961849615*( cos(x) - 1)^11/(cos(x) + 1)^11 + 9843279601311*(cos(x) - 1)^12/(cos(x) + 1 )^12 + 1220388071083*(cos(x) - 1)^13/(cos(x) + 1)^13 + 85441358295*(cos(x) - 1)^14/(cos(x) + 1)^14 + 2577140019*(cos(x) - 1)^15/(cos(x) + 1)^15 + 11 45634921)/(28*(cos(x) - 1)/(cos(x) + 1) + 66*(cos(x) - 1)^2/(cos(x) + 1)^2 + 28*(cos(x) - 1)^3/(cos(x) + 1)^3 + 3*(cos(x) - 1)^4/(cos(x) + 1)^4 + 3) ^4 + 3889/373248*log(-(cos(x) - 1)/(cos(x) + 1)) - 332929/5832*log(abs(-(c os(x) - 1)/(cos(x) + 1) - 3)) + 332929/5832*log(abs(-3*(cos(x) - 1)/(cos(x ) + 1) - 1))
Time = 22.51 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\text {Too large to display} \] Input:
int(1/(sin(5*x) + sin(x))^5,x)
Output:
(3889*log(tan(x/2)))/186624 - (332929*atanh(529350107261172808328824126854 4485325492309354181/(865823474183712111497215443843082616832*((11878192955 51787888946106646747265840511864824769*tan(x/2)^2)/21679793289848794179523 7553242112 - 131979700816923844080997972642519976266618115117/722659776328 29313931745851080704)) - 8489643408268/8489639855729))/2916 + (82683*2^(1/ 2)*atanh((13990822351110619332532430968055105855879747*2^(1/2))/(407483448 8292557252808671232*((86490991110247218528363966682112925439090753*tan(x/2 )^2)/3056125866219417939606503424 - 55648155071144200663458342222434509839 31439/1146047199832281727352438784)) - (7339003959353096062245090263610687 21080365265*2^(1/2)*tan(x/2)^2)/(36673510394633015275278041088*((864909911 10247218528363966682112925439090753*tan(x/2)^2)/30561258662194179396065034 24 - 5564815507114420066345834222243450983931439/1146047199832281727352438 784))))/1024 - ((13*tan(x/2)^2)/62208 + (21212837*tan(x/2)^4)/279936 - (44 65642615*tan(x/2)^6)/1679616 + (136474677679*tan(x/2)^8)/3359232 - (179028 5716621*tan(x/2)^10)/5038848 + (154818945941*tan(x/2)^12)/78732 - (1087775 49985939*tan(x/2)^14)/15116544 + (356491034365231*tan(x/2)^16)/20155392 - (440627697903061*tan(x/2)^18)/15116544 + (26719440584471*tan(x/2)^20)/8398 08 - (38162979312085*tan(x/2)^22)/1679616 + (107200617808633*tan(x/2)^24)/ 10077696 - (49418813748397*tan(x/2)^26)/15116544 + (820271243347*tan(x/2)^ 28)/1259712 - (1220385450625*tan(x/2)^30)/15116544 + (75947731603*tan(x...
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^5} \, dx=\int \frac {1}{\sin \left (5 x \right )^{5}+5 \sin \left (5 x \right )^{4} \sin \left (x \right )+10 \sin \left (5 x \right )^{3} \sin \left (x \right )^{2}+10 \sin \left (5 x \right )^{2} \sin \left (x \right )^{3}+5 \sin \left (5 x \right ) \sin \left (x \right )^{4}+\sin \left (x \right )^{5}}d x \] Input:
int(1/(sin(x)+sin(5*x))^5,x)
Output:
int(1/(sin(5*x)**5 + 5*sin(5*x)**4*sin(x) + 10*sin(5*x)**3*sin(x)**2 + 10* sin(5*x)**2*sin(x)**3 + 5*sin(5*x)*sin(x)**4 + sin(x)**5),x)