Integrand size = 9, antiderivative size = 70 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=-\frac {143 \text {arctanh}(\cos (x))}{8192}+\frac {143 \sec (x)}{8192}+\frac {143 \sec ^3(x)}{24576}+\frac {143 \sec ^5(x)}{40960}+\frac {143 \sec ^7(x)}{57344}+\frac {143 \sec ^9(x)}{73728}-\frac {13 \csc ^2(x) \sec ^9(x)}{8192}-\frac {\csc ^4(x) \sec ^9(x)}{4096} \] Output:
-143/8192*arctanh(cos(x))+143/8192*sec(x)+143/24576*sec(x)^3+143/40960*sec (x)^5+143/57344*sec(x)^7+143/73728*sec(x)^9-13/8192*csc(x)^2*sec(x)^9-1/40 96*csc(x)^4*sec(x)^9
Leaf count is larger than twice the leaf count of optimal. \(231\) vs. \(2(70)=140\).
Time = 0.43 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.30 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=-\frac {\csc ^4(x) \left (1726400 \cos (2 x)+2730442 \cos (4 x)+3 \left (-404948+205920 \cos (6 x)-236236 \cos (8 x)-160160 \cos (10 x)-30030 \cos (12 x)+540540 \cos (x) \log \left (\cos \left (\frac {x}{2}\right )\right )-135135 \cos (3 x) \log \left (\cos \left (\frac {x}{2}\right )\right )-435435 \cos (5 x) \log \left (\cos \left (\frac {x}{2}\right )\right )-150150 \cos (7 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+90090 \cos (9 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+75075 \cos (11 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+15015 \cos (13 x) \log \left (\cos \left (\frac {x}{2}\right )\right )-540540 \cos (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+135135 \cos (3 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+435435 \cos (5 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+150150 \cos (7 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-90090 \cos (9 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-75075 \cos (11 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-15015 \cos (13 x) \log \left (\sin \left (\frac {x}{2}\right )\right )\right )\right ) \sec ^9(x)}{10569646080} \] Input:
Integrate[(Sin[x] + Sin[3*x])^(-5),x]
Output:
-1/10569646080*(Csc[x]^4*(1726400*Cos[2*x] + 2730442*Cos[4*x] + 3*(-404948 + 205920*Cos[6*x] - 236236*Cos[8*x] - 160160*Cos[10*x] - 30030*Cos[12*x] + 540540*Cos[x]*Log[Cos[x/2]] - 135135*Cos[3*x]*Log[Cos[x/2]] - 435435*Cos [5*x]*Log[Cos[x/2]] - 150150*Cos[7*x]*Log[Cos[x/2]] + 90090*Cos[9*x]*Log[C os[x/2]] + 75075*Cos[11*x]*Log[Cos[x/2]] + 15015*Cos[13*x]*Log[Cos[x/2]] - 540540*Cos[x]*Log[Sin[x/2]] + 135135*Cos[3*x]*Log[Sin[x/2]] + 435435*Cos[ 5*x]*Log[Sin[x/2]] + 150150*Cos[7*x]*Log[Sin[x/2]] - 90090*Cos[9*x]*Log[Si n[x/2]] - 75075*Cos[11*x]*Log[Sin[x/2]] - 15015*Cos[13*x]*Log[Sin[x/2]]))* Sec[x]^9)
Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {3042, 4824, 27, 253, 253, 264, 264, 264, 264, 264, 219}
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.
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (3 x))^5}dx\) |
| \(\Big \downarrow \) 4824 |
| \(\displaystyle -\int \frac {\sec ^{10}(x)}{1024 \left (1-\cos ^2(x)\right )^3}d\cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sec ^{10}(x)}{\left (1-\cos ^2(x)\right )^3}d\cos (x)}{1024}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {-\frac {13}{4} \int \frac {\sec ^{10}(x)}{\left (1-\cos ^2(x)\right )^2}d\cos (x)-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \int \frac {\sec ^{10}(x)}{1-\cos ^2(x)}d\cos (x)+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \left (\int \frac {\sec ^8(x)}{1-\cos ^2(x)}d\cos (x)-\frac {\sec ^9(x)}{9}\right )+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \left (\int \frac {\sec ^6(x)}{1-\cos ^2(x)}d\cos (x)-\frac {1}{9} \sec ^9(x)-\frac {\sec ^7(x)}{7}\right )+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \left (\int \frac {\sec ^4(x)}{1-\cos ^2(x)}d\cos (x)-\frac {1}{9} \sec ^9(x)-\frac {\sec ^7(x)}{7}-\frac {\sec ^5(x)}{5}\right )+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \left (\int \frac {\sec ^2(x)}{1-\cos ^2(x)}d\cos (x)-\frac {1}{9} \sec ^9(x)-\frac {\sec ^7(x)}{7}-\frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3}\right )+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \left (\int \frac {1}{1-\cos ^2(x)}d\cos (x)-\frac {1}{9} \sec ^9(x)-\frac {\sec ^7(x)}{7}-\frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3}-\sec (x)\right )+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {13}{4} \left (\frac {11}{2} \left (\text {arctanh}(\cos (x))-\frac {1}{9} \sec ^9(x)-\frac {\sec ^7(x)}{7}-\frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3}-\sec (x)\right )+\frac {\sec ^9(x)}{2 \left (1-\cos ^2(x)\right )}\right )-\frac {\sec ^9(x)}{4 \left (1-\cos ^2(x)\right )^2}}{1024}\) |
Input:
Int[(Sin[x] + Sin[3*x])^(-5),x]
Output:
(-1/4*Sec[x]^9/(1 - Cos[x]^2)^2 - (13*(Sec[x]^9/(2*(1 - Cos[x]^2)) + (11*( ArcTanh[Cos[x]] - Sec[x] - Sec[x]^3/3 - Sec[x]^5/5 - Sec[x]^7/7 - Sec[x]^9 /9))/2))/4)/1024
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 115.48 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.03
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(\frac {1}{9216 \sin \left (x \right )^{4} \cos \left (x \right )^{9}}+\frac {13}{64512 \sin \left (x \right )^{4} \cos \left (x \right )^{7}}+\frac {143}{322560 \sin \left (x \right )^{4} \cos \left (x \right )^{5}}-\frac {143}{143360 \sin \left (x \right )^{4} \cos \left (x \right )^{3}}+\frac {143}{61440 \sin \left (x \right )^{2} \cos \left (x \right )^{3}}-\frac {143}{24576 \sin \left (x \right )^{2} \cos \left (x \right )}+\frac {143}{8192 \cos \left (x \right )}+\frac {143 \ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )}{8192}\) | \(78\) |
| risch | \(\frac {45045 \,{\mathrm e}^{25 i x}+240240 \,{\mathrm e}^{23 i x}+354354 \,{\mathrm e}^{21 i x}-308880 \,{\mathrm e}^{19 i x}-1365221 \,{\mathrm e}^{17 i x}-863200 \,{\mathrm e}^{15 i x}+1214844 \,{\mathrm e}^{13 i x}-863200 \,{\mathrm e}^{11 i x}-1365221 \,{\mathrm e}^{9 i x}-308880 \,{\mathrm e}^{7 i x}+354354 \,{\mathrm e}^{5 i x}+240240 \,{\mathrm e}^{3 i x}+45045 \,{\mathrm e}^{i x}}{1290240 \left ({\mathrm e}^{2 i x}+1\right )^{9} \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\frac {143 \ln \left ({\mathrm e}^{i x}-1\right )}{8192}-\frac {143 \ln \left ({\mathrm e}^{i x}+1\right )}{8192}\) | \(134\) |
Input:
int(1/(sin(x)+sin(3*x))^5,x,method=_RETURNVERBOSE)
Output:
0
Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=\frac {90090 \, \cos \left (x\right )^{12} - 150150 \, \cos \left (x\right )^{10} + 48048 \, \cos \left (x\right )^{8} + 6864 \, \cos \left (x\right )^{6} + 2288 \, \cos \left (x\right )^{4} + 1040 \, \cos \left (x\right )^{2} - 45045 \, {\left (\cos \left (x\right )^{13} - 2 \, \cos \left (x\right )^{11} + \cos \left (x\right )^{9}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 45045 \, {\left (\cos \left (x\right )^{13} - 2 \, \cos \left (x\right )^{11} + \cos \left (x\right )^{9}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 560}{5160960 \, {\left (\cos \left (x\right )^{13} - 2 \, \cos \left (x\right )^{11} + \cos \left (x\right )^{9}\right )}} \] Input:
integrate(1/(sin(x)+sin(3*x))^5,x, algorithm="fricas")
Output:
1/5160960*(90090*cos(x)^12 - 150150*cos(x)^10 + 48048*cos(x)^8 + 6864*cos( x)^6 + 2288*cos(x)^4 + 1040*cos(x)^2 - 45045*(cos(x)^13 - 2*cos(x)^11 + co s(x)^9)*log(1/2*cos(x) + 1/2) + 45045*(cos(x)^13 - 2*cos(x)^11 + cos(x)^9) *log(-1/2*cos(x) + 1/2) + 560)/(cos(x)^13 - 2*cos(x)^11 + cos(x)^9)
Timed out. \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=\text {Timed out} \] Input:
integrate(1/(sin(x)+sin(3*x))**5,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 6347 vs. \(2 (54) = 108\).
Time = 0.60 (sec) , antiderivative size = 6347, normalized size of antiderivative = 90.67 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(3*x))^5,x, algorithm="maxima")
Output:
1/5160960*(4*(45045*cos(25*x) + 240240*cos(23*x) + 354354*cos(21*x) - 3088 80*cos(19*x) - 1365221*cos(17*x) - 863200*cos(15*x) + 1214844*cos(13*x) - 863200*cos(11*x) - 1365221*cos(9*x) - 308880*cos(7*x) + 354354*cos(5*x) + 240240*cos(3*x) + 45045*cos(x))*cos(26*x) + 180180*(5*cos(24*x) + 6*cos(22 *x) - 10*cos(20*x) - 29*cos(18*x) - 9*cos(16*x) + 36*cos(14*x) + 36*cos(12 *x) - 9*cos(10*x) - 29*cos(8*x) - 10*cos(6*x) + 6*cos(4*x) + 5*cos(2*x) + 1)*cos(25*x) + 20*(240240*cos(23*x) + 354354*cos(21*x) - 308880*cos(19*x) - 1365221*cos(17*x) - 863200*cos(15*x) + 1214844*cos(13*x) - 863200*cos(11 *x) - 1365221*cos(9*x) - 308880*cos(7*x) + 354354*cos(5*x) + 240240*cos(3* x) + 45045*cos(x))*cos(24*x) + 960960*(6*cos(22*x) - 10*cos(20*x) - 29*cos (18*x) - 9*cos(16*x) + 36*cos(14*x) + 36*cos(12*x) - 9*cos(10*x) - 29*cos( 8*x) - 10*cos(6*x) + 6*cos(4*x) + 5*cos(2*x) + 1)*cos(23*x) + 24*(354354*c os(21*x) - 308880*cos(19*x) - 1365221*cos(17*x) - 863200*cos(15*x) + 12148 44*cos(13*x) - 863200*cos(11*x) - 1365221*cos(9*x) - 308880*cos(7*x) + 354 354*cos(5*x) + 240240*cos(3*x) + 45045*cos(x))*cos(22*x) - 1417416*(10*cos (20*x) + 29*cos(18*x) + 9*cos(16*x) - 36*cos(14*x) - 36*cos(12*x) + 9*cos( 10*x) + 29*cos(8*x) + 10*cos(6*x) - 6*cos(4*x) - 5*cos(2*x) - 1)*cos(21*x) + 40*(308880*cos(19*x) + 1365221*cos(17*x) + 863200*cos(15*x) - 1214844*c os(13*x) + 863200*cos(11*x) + 1365221*cos(9*x) + 308880*cos(7*x) - 354354* cos(5*x) - 240240*cos(3*x) - 45045*cos(x))*cos(20*x) + 1235520*(29*cos(...
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (54) = 108\).
Time = 0.11 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.04 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=\frac {{\left (\frac {48 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {858 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{2}}{65536 \, {\left (\cos \left (x\right ) - 1\right )}^{2}} - \frac {3 \, {\left (\cos \left (x\right ) - 1\right )}}{4096 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {{\left (\cos \left (x\right ) - 1\right )}^{2}}{65536 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {\frac {45882 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {161478 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {326802 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {430668 \, {\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {366030 \, {\left (\cos \left (x\right ) - 1\right )}^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {204330 \, {\left (\cos \left (x\right ) - 1\right )}^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {66150 \, {\left (\cos \left (x\right ) - 1\right )}^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {11025 \, {\left (\cos \left (x\right ) - 1\right )}^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + 6323}{161280 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{9}} + \frac {143}{16384} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \] Input:
integrate(1/(sin(x)+sin(3*x))^5,x, algorithm="giac")
Output:
1/65536*(48*(cos(x) - 1)/(cos(x) + 1) - 858*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1)*(cos(x) + 1)^2/(cos(x) - 1)^2 - 3/4096*(cos(x) - 1)/(cos(x) + 1) + 1/ 65536*(cos(x) - 1)^2/(cos(x) + 1)^2 + 1/161280*(45882*(cos(x) - 1)/(cos(x) + 1) + 161478*(cos(x) - 1)^2/(cos(x) + 1)^2 + 326802*(cos(x) - 1)^3/(cos( x) + 1)^3 + 430668*(cos(x) - 1)^4/(cos(x) + 1)^4 + 366030*(cos(x) - 1)^5/( cos(x) + 1)^5 + 204330*(cos(x) - 1)^6/(cos(x) + 1)^6 + 66150*(cos(x) - 1)^ 7/(cos(x) + 1)^7 + 11025*(cos(x) - 1)^8/(cos(x) + 1)^8 + 6323)/((cos(x) - 1)/(cos(x) + 1) + 1)^9 + 143/16384*log(-(cos(x) - 1)/(cos(x) + 1))
Time = 22.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=\frac {\frac {143\,{\cos \left (x\right )}^{12}}{8192}-\frac {715\,{\cos \left (x\right )}^{10}}{24576}+\frac {143\,{\cos \left (x\right )}^8}{15360}+\frac {143\,{\cos \left (x\right )}^6}{107520}+\frac {143\,{\cos \left (x\right )}^4}{322560}+\frac {13\,{\cos \left (x\right )}^2}{64512}+\frac {1}{9216}}{{\cos \left (x\right )}^{13}-2\,{\cos \left (x\right )}^{11}+{\cos \left (x\right )}^9}-\frac {143\,\mathrm {atanh}\left (\cos \left (x\right )\right )}{8192} \] Input:
int(1/(sin(3*x) + sin(x))^5,x)
Output:
((13*cos(x)^2)/64512 + (143*cos(x)^4)/322560 + (143*cos(x)^6)/107520 + (14 3*cos(x)^8)/15360 - (715*cos(x)^10)/24576 + (143*cos(x)^12)/8192 + 1/9216) /(cos(x)^9 - 2*cos(x)^11 + cos(x)^13) - (143*atanh(cos(x)))/8192
\[ \int \frac {1}{(\sin (x)+\sin (3 x))^5} \, dx=\int \frac {1}{\sin \left (3 x \right )^{5}+5 \sin \left (3 x \right )^{4} \sin \left (x \right )+10 \sin \left (3 x \right )^{3} \sin \left (x \right )^{2}+10 \sin \left (3 x \right )^{2} \sin \left (x \right )^{3}+5 \sin \left (3 x \right ) \sin \left (x \right )^{4}+\sin \left (x \right )^{5}}d x \] Input:
int(1/(sin(x)+sin(3*x))^5,x)
Output:
int(1/(sin(3*x)**5 + 5*sin(3*x)**4*sin(x) + 10*sin(3*x)**3*sin(x)**2 + 10* sin(3*x)**2*sin(x)**3 + 5*sin(3*x)*sin(x)**4 + sin(x)**5),x)