Integrand size = 9, antiderivative size = 69 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=-\frac {7 \cot (x)}{1024}-\frac {\cot ^3(x)}{1536}-\frac {\cot ^5(x)}{20480}+\frac {7 \tan (x)}{512}+\frac {35 \tan ^3(x)}{6144}+\frac {7 \tan ^5(x)}{2560}+\frac {\tan ^7(x)}{1024}+\frac {\tan ^9(x)}{4608}+\frac {\tan ^{11}(x)}{45056} \] Output:
-7/1024*cot(x)-1/1536*cot(x)^3-1/20480*cot(x)^5+7/512*tan(x)+35/6144*tan(x )^3+7/2560*tan(x)^5+1/1024*tan(x)^7+1/4608*tan(x)^9+1/45056*tan(x)^11
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=-\frac {(110 \cos (2 x)+66 \cos (4 x)-34 \cos (6 x)-50 \cos (8 x)-10 \cos (10 x)+10 \cos (12 x)+6 \cos (14 x)+\cos (16 x)) \csc ^5(x) \sec ^{11}(x)}{2027520} \] Input:
Integrate[(Sin[x] + Sin[3*x])^(-6),x]
Output:
-1/2027520*((110*Cos[2*x] + 66*Cos[4*x] - 34*Cos[6*x] - 50*Cos[8*x] - 10*C os[10*x] + 10*Cos[12*x] + 6*Cos[14*x] + Cos[16*x])*Csc[x]^5*Sec[x]^11)
Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4822, 27, 244, 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.
\(\displaystyle \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(\sin (x)+\sin (3 x))^6}dx\) |
\(\Big \downarrow \) 4822 |
\(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^8 \cot ^6(x)}{4096}d\tan (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \cot ^6(x) \left (\tan ^2(x)+1\right )^8d\tan (x)}{4096}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \left (\tan ^{10}(x)+8 \tan ^8(x)+28 \tan ^6(x)+56 \tan ^4(x)+70 \tan ^2(x)+\cot ^6(x)+8 \cot ^4(x)+28 \cot ^2(x)+56\right )d\tan (x)}{4096}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\tan ^{11}(x)}{11}+\frac {8 \tan ^9(x)}{9}+4 \tan ^7(x)+\frac {56 \tan ^5(x)}{5}+\frac {70 \tan ^3(x)}{3}+56 \tan (x)-\frac {\cot ^5(x)}{5}-\frac {8 \cot ^3(x)}{3}-28 \cot (x)}{4096}\) |
Input:
Int[(Sin[x] + Sin[3*x])^(-6),x]
Output:
(-28*Cot[x] - (8*Cot[x]^3)/3 - Cot[x]^5/5 + 56*Tan[x] + (70*Tan[x]^3)/3 + (56*Tan[x]^5)/5 + 4*Tan[x]^7 + (8*Tan[x]^9)/9 + Tan[x]^11/11)/4096
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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 = 430.58 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.03
method | result | size |
parallelrisch | \(0\) | \(2\) |
risch | \(-\frac {16 i \left (110 \,{\mathrm e}^{14 i x}+66 \,{\mathrm e}^{12 i x}-34 \,{\mathrm e}^{10 i x}-50 \,{\mathrm e}^{8 i x}-10 \,{\mathrm e}^{6 i x}+10 \,{\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}+1\right )}{495 \left ({\mathrm e}^{2 i x}+1\right )^{11} \left ({\mathrm e}^{2 i x}-1\right )^{5}}\) | \(73\) |
default | \(\frac {1}{45056 \cos \left (x \right )^{11} \sin \left (x \right )^{5}}+\frac {1}{25344 \sin \left (x \right )^{5} \cos \left (x \right )^{9}}+\frac {1}{12672 \sin \left (x \right )^{5} \cos \left (x \right )^{7}}+\frac {1}{5280 \sin \left (x \right )^{5} \cos \left (x \right )^{5}}-\frac {1}{2640 \sin \left (x \right )^{5} \cos \left (x \right )^{3}}+\frac {1}{990 \sin \left (x \right )^{3} \cos \left (x \right )^{3}}-\frac {1}{495 \sin \left (x \right )^{3} \cos \left (x \right )}+\frac {4}{495 \sin \left (x \right ) \cos \left (x \right )}-\frac {8 \cot \left (x \right )}{495}\) | \(86\) |
Input:
int(1/(sin(x)+sin(3*x))^6,x,method=_RETURNVERBOSE)
Output:
0
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=-\frac {32768 \, \cos \left (x\right )^{16} - 81920 \, \cos \left (x\right )^{14} + 61440 \, \cos \left (x\right )^{12} - 10240 \, \cos \left (x\right )^{10} - 1280 \, \cos \left (x\right )^{8} - 384 \, \cos \left (x\right )^{6} - 160 \, \cos \left (x\right )^{4} - 80 \, \cos \left (x\right )^{2} - 45}{2027520 \, {\left (\cos \left (x\right )^{15} - 2 \, \cos \left (x\right )^{13} + \cos \left (x\right )^{11}\right )} \sin \left (x\right )} \] Input:
integrate(1/(sin(x)+sin(3*x))^6,x, algorithm="fricas")
Output:
-1/2027520*(32768*cos(x)^16 - 81920*cos(x)^14 + 61440*cos(x)^12 - 10240*co s(x)^10 - 1280*cos(x)^8 - 384*cos(x)^6 - 160*cos(x)^4 - 80*cos(x)^2 - 45)/ ((cos(x)^15 - 2*cos(x)^13 + cos(x)^11)*sin(x))
Timed out. \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(sin(x)+sin(3*x))**6,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2508 vs. \(2 (51) = 102\).
Time = 0.08 (sec) , antiderivative size = 2508, normalized size of antiderivative = 36.35 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(3*x))^6,x, algorithm="maxima")
Output:
16/495*(2*(55*sin(14*x) + 33*sin(12*x) - 17*sin(10*x) - 25*sin(8*x) - 5*si n(6*x) + 5*sin(4*x) + 3*sin(2*x))*cos(32*x) + 12*(55*sin(14*x) + 33*sin(12 *x) - 17*sin(10*x) - 25*sin(8*x) - 5*sin(6*x) + 5*sin(4*x) + 3*sin(2*x))*c os(30*x) + 20*(55*sin(14*x) + 33*sin(12*x) - 17*sin(10*x) - 25*sin(8*x) - 5*sin(6*x) + 5*sin(4*x) + 3*sin(2*x))*cos(28*x) - 20*(55*sin(14*x) + 33*si n(12*x) - 17*sin(10*x) - 25*sin(8*x) - 5*sin(6*x) + 5*sin(4*x) + 3*sin(2*x ))*cos(26*x) - 100*(55*sin(14*x) + 33*sin(12*x) - 17*sin(10*x) - 25*sin(8* x) - 5*sin(6*x) + 5*sin(4*x) + 3*sin(2*x))*cos(24*x) - 68*(55*sin(14*x) + 33*sin(12*x) - 17*sin(10*x) - 25*sin(8*x) - 5*sin(6*x) + 5*sin(4*x) + 3*si n(2*x))*cos(22*x) + 132*(55*sin(14*x) + 33*sin(12*x) - 17*sin(10*x) - 25*s in(8*x) - 5*sin(6*x) + 5*sin(4*x) + 3*sin(2*x))*cos(20*x) + 220*(55*sin(14 *x) + 33*sin(12*x) - 17*sin(10*x) - 25*sin(8*x) - 5*sin(6*x) + 5*sin(4*x) + 3*sin(2*x))*cos(18*x) - (110*cos(14*x) + 66*cos(12*x) - 34*cos(10*x) - 5 0*cos(8*x) - 10*cos(6*x) + 10*cos(4*x) + 6*cos(2*x) + 1)*sin(32*x) - 6*(11 0*cos(14*x) + 66*cos(12*x) - 34*cos(10*x) - 50*cos(8*x) - 10*cos(6*x) + 10 *cos(4*x) + 6*cos(2*x) + 1)*sin(30*x) - 10*(110*cos(14*x) + 66*cos(12*x) - 34*cos(10*x) - 50*cos(8*x) - 10*cos(6*x) + 10*cos(4*x) + 6*cos(2*x) + 1)* sin(28*x) + 10*(110*cos(14*x) + 66*cos(12*x) - 34*cos(10*x) - 50*cos(8*x) - 10*cos(6*x) + 10*cos(4*x) + 6*cos(2*x) + 1)*sin(26*x) + 50*(110*cos(14*x ) + 66*cos(12*x) - 34*cos(10*x) - 50*cos(8*x) - 10*cos(6*x) + 10*cos(4*...
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=\frac {1}{45056} \, \tan \left (x\right )^{11} + \frac {1}{4608} \, \tan \left (x\right )^{9} + \frac {1}{1024} \, \tan \left (x\right )^{7} + \frac {7}{2560} \, \tan \left (x\right )^{5} + \frac {35}{6144} \, \tan \left (x\right )^{3} - \frac {420 \, \tan \left (x\right )^{4} + 40 \, \tan \left (x\right )^{2} + 3}{61440 \, \tan \left (x\right )^{5}} + \frac {7}{512} \, \tan \left (x\right ) \] Input:
integrate(1/(sin(x)+sin(3*x))^6,x, algorithm="giac")
Output:
1/45056*tan(x)^11 + 1/4608*tan(x)^9 + 1/1024*tan(x)^7 + 7/2560*tan(x)^5 + 35/6144*tan(x)^3 - 1/61440*(420*tan(x)^4 + 40*tan(x)^2 + 3)/tan(x)^5 + 7/5 12*tan(x)
Time = 21.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=\frac {7\,\mathrm {tan}\left (x\right )}{512}+\frac {35\,{\mathrm {tan}\left (x\right )}^3}{6144}+\frac {7\,{\mathrm {tan}\left (x\right )}^5}{2560}+\frac {{\mathrm {tan}\left (x\right )}^7}{1024}+\frac {{\mathrm {tan}\left (x\right )}^9}{4608}+\frac {{\mathrm {tan}\left (x\right )}^{11}}{45056}-\frac {\frac {7\,{\mathrm {tan}\left (x\right )}^4}{1024}+\frac {{\mathrm {tan}\left (x\right )}^2}{1536}+\frac {1}{20480}}{{\mathrm {tan}\left (x\right )}^5} \] Input:
int(1/(sin(3*x) + sin(x))^6,x)
Output:
(7*tan(x))/512 + (35*tan(x)^3)/6144 + (7*tan(x)^5)/2560 + tan(x)^7/1024 + tan(x)^9/4608 + tan(x)^11/45056 - (tan(x)^2/1536 + (7*tan(x)^4)/1024 + 1/2 0480)/tan(x)^5
\[ \int \frac {1}{(\sin (x)+\sin (3 x))^6} \, dx=\int \frac {1}{\sin \left (3 x \right )^{6}+6 \sin \left (3 x \right )^{5} \sin \left (x \right )+15 \sin \left (3 x \right )^{4} \sin \left (x \right )^{2}+20 \sin \left (3 x \right )^{3} \sin \left (x \right )^{3}+15 \sin \left (3 x \right )^{2} \sin \left (x \right )^{4}+6 \sin \left (3 x \right ) \sin \left (x \right )^{5}+\sin \left (x \right )^{6}}d x \] Input:
int(1/(sin(x)+sin(3*x))^6,x)
Output:
int(1/(sin(3*x)**6 + 6*sin(3*x)**5*sin(x) + 15*sin(3*x)**4*sin(x)**2 + 20* sin(3*x)**3*sin(x)**3 + 15*sin(3*x)**2*sin(x)**4 + 6*sin(3*x)*sin(x)**5 + sin(x)**6),x)