Integrand size = 27, antiderivative size = 87 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=-\frac {28}{3} \csc (3 x)+\frac {8}{9} \csc ^3(3 x)-\frac {1}{15} \csc ^5(3 x)-\frac {56}{3} \sin (3 x)+\frac {70}{9} \sin ^3(3 x)-\frac {56}{15} \sin ^5(3 x)+\frac {4}{3} \sin ^7(3 x)-\frac {8}{27} \sin ^9(3 x)+\frac {1}{33} \sin ^{11}(3 x) \] Output:
-28/3*csc(3*x)+8/9*csc(3*x)^3-1/15*csc(3*x)^5-56/3*sin(3*x)+70/9*sin(3*x)^ 3-56/15*sin(3*x)^5+4/3*sin(3*x)^7-8/27*sin(3*x)^9+1/33*sin(3*x)^11
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=-\frac {28}{3} \csc (3 x)+\frac {8}{9} \csc ^3(3 x)-\frac {1}{15} \csc ^5(3 x)-\frac {56}{3} \sin (3 x)+\frac {70}{9} \sin ^3(3 x)-\frac {56}{15} \sin ^5(3 x)+\frac {4}{3} \sin ^7(3 x)-\frac {8}{27} \sin ^9(3 x)+\frac {1}{33} \sin ^{11}(3 x) \] Input:
Integrate[Cos[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^5,x]
Output:
(-28*Csc[3*x])/3 + (8*Csc[3*x]^3)/9 - Csc[3*x]^5/15 - (56*Sin[3*x])/3 + (7 0*Sin[3*x]^3)/9 - (56*Sin[3*x]^5)/15 + (4*Sin[3*x]^7)/3 - (8*Sin[3*x]^9)/2 7 + Sin[3*x]^11/33
Time = 0.41 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3654, 25, 3042, 4608, 3042, 3070, 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.
\(\displaystyle \int \left (1-\sin ^2(3 x)\right )^5 \cos (3 x) \left (\csc ^2(3 x)-1\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-\sin (3 x)^2\right )^5 \cos (3 x) \left (\csc (3 x)^2-1\right )^3dx\) |
\(\Big \downarrow \) 3654 |
\(\displaystyle \int -\cos ^{11}(3 x) \left (1-\csc ^2(3 x)\right )^3dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cos ^{11}(3 x) \left (1-\csc ^2(3 x)\right )^3dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \left (1-\sec \left (3 x+\frac {\pi }{2}\right )^2\right )^3 \sin \left (3 x+\frac {\pi }{2}\right )^{11}dx\) |
\(\Big \downarrow \) 4608 |
\(\displaystyle \int \cos ^{11}(3 x) \cot ^6(3 x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (3 x+\frac {\pi }{2}\right )^{11} \tan \left (3 x+\frac {\pi }{2}\right )^6dx\) |
\(\Big \downarrow \) 3070 |
\(\displaystyle -\frac {1}{3} \int \csc ^6(3 x) \left (1-\sin ^2(3 x)\right )^8d(-\sin (3 x))\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {1}{3} \int \left (\sin ^{10}(3 x)-8 \sin ^8(3 x)+28 \sin ^6(3 x)-56 \sin ^4(3 x)+70 \sin ^2(3 x)+\csc ^6(3 x)-8 \csc ^4(3 x)+28 \csc ^2(3 x)-56\right )d(-\sin (3 x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{11} \sin ^{11}(3 x)-\frac {8}{9} \sin ^9(3 x)+4 \sin ^7(3 x)-\frac {56}{5} \sin ^5(3 x)+\frac {70}{3} \sin ^3(3 x)-56 \sin (3 x)-\frac {1}{5} \csc ^5(3 x)+\frac {8}{3} \csc ^3(3 x)-28 \csc (3 x)\right )\) |
Input:
Int[Cos[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^5,x]
Output:
(-28*Csc[3*x] + (8*Csc[3*x]^3)/3 - Csc[3*x]^5/5 - 56*Sin[3*x] + (70*Sin[3* x]^3)/3 - (56*Sin[3*x]^5)/5 + 4*Sin[3*x]^7 - (8*Sin[3*x]^9)/9 + Sin[3*x]^1 1/11)/3
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[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ a^p Int[ActivateTrig[u*cos[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ b^p Int[ActivateTrig[u*tan[e + f*x]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]
Time = 0.69 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\sin \left (3 x \right )^{11}}{33}-\frac {8 \sin \left (3 x \right )^{9}}{27}+\frac {4 \sin \left (3 x \right )^{7}}{3}-\frac {56 \sin \left (3 x \right )^{5}}{15}+\frac {70 \sin \left (3 x \right )^{3}}{9}-\frac {56 \sin \left (3 x \right )}{3}-\frac {28}{3 \sin \left (3 x \right )}+\frac {8}{9 \sin \left (3 x \right )^{3}}-\frac {1}{15 \sin \left (3 x \right )^{5}}\) | \(72\) |
default | \(\frac {\sin \left (3 x \right )^{11}}{33}-\frac {8 \sin \left (3 x \right )^{9}}{27}+\frac {4 \sin \left (3 x \right )^{7}}{3}-\frac {56 \sin \left (3 x \right )^{5}}{15}+\frac {70 \sin \left (3 x \right )^{3}}{9}-\frac {56 \sin \left (3 x \right )}{3}-\frac {28}{3 \sin \left (3 x \right )}+\frac {8}{9 \sin \left (3 x \right )^{3}}-\frac {1}{15 \sin \left (3 x \right )^{5}}\) | \(72\) |
risch | \(\frac {23 i {\mathrm e}^{27 i x}}{55296}+\frac {37 i {\mathrm e}^{21 i x}}{6144}+\frac {1909 i {\mathrm e}^{15 i x}}{30720}+\frac {5197 i {\mathrm e}^{9 i x}}{9216}+\frac {22379 i {\mathrm e}^{3 i x}}{3072}-\frac {22379 i {\mathrm e}^{-3 i x}}{3072}-\frac {5197 i {\mathrm e}^{-9 i x}}{9216}-\frac {1909 i {\mathrm e}^{-15 i x}}{30720}-\frac {37 i {\mathrm e}^{-21 i x}}{6144}-\frac {23 i {\mathrm e}^{-27 i x}}{55296}-\frac {8 i \left (105 \,{\mathrm e}^{27 i x}-380 \,{\mathrm e}^{21 i x}+562 \,{\mathrm e}^{15 i x}-380 \,{\mathrm e}^{9 i x}+105 \,{\mathrm e}^{3 i x}\right )}{45 \left ({\mathrm e}^{6 i x}-1\right )^{5}}-\frac {\sin \left (33 x \right )}{33792}\) | \(136\) |
parallelrisch | \(\frac {-99 \tan \left (\frac {3 x}{2}\right )^{27}+3696 \tan \left (\frac {3 x}{2}\right )^{25}-159720 \tan \left (\frac {3 x}{2}\right )^{23}-4010160 \tan \left (\frac {3 x}{2}\right )^{21}-27678420 \tan \left (\frac {3 x}{2}\right )^{19}-105790608 \tan \left (\frac {3 x}{2}\right )^{17}-256044888 \tan \left (\frac {3 x}{2}\right )^{15}-427811120 \tan \left (\frac {3 x}{2}\right )^{13}-504501010 \tan \left (\frac {3 x}{2}\right )^{11}-427811120 \tan \left (\frac {3 x}{2}\right )^{9}-256044888 \tan \left (\frac {3 x}{2}\right )^{7}-105790608 \tan \left (\frac {3 x}{2}\right )^{5}-99 \cot \left (\frac {3 x}{2}\right )^{5}-27678420 \tan \left (\frac {3 x}{2}\right )^{3}+3696 \cot \left (\frac {3 x}{2}\right )^{3}-4010160 \tan \left (\frac {3 x}{2}\right )-159720 \cot \left (\frac {3 x}{2}\right )}{47520 \left (1+\tan \left (\frac {3 x}{2}\right )^{2}\right )^{11}}\) | \(146\) |
Input:
int(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x,method=_RETURNVERBOSE)
Output:
1/33*sin(3*x)^11-8/27*sin(3*x)^9+4/3*sin(3*x)^7-56/15*sin(3*x)^5+70/9*sin( 3*x)^3-56/3*sin(3*x)-28/3/sin(3*x)+8/9/sin(3*x)^3-1/15/sin(3*x)^5
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=\frac {45 \, \cos \left (3 \, x\right )^{16} + 80 \, \cos \left (3 \, x\right )^{14} + 160 \, \cos \left (3 \, x\right )^{12} + 384 \, \cos \left (3 \, x\right )^{10} + 1280 \, \cos \left (3 \, x\right )^{8} + 10240 \, \cos \left (3 \, x\right )^{6} - 61440 \, \cos \left (3 \, x\right )^{4} + 81920 \, \cos \left (3 \, x\right )^{2} - 32768}{1485 \, {\left (\cos \left (3 \, x\right )^{4} - 2 \, \cos \left (3 \, x\right )^{2} + 1\right )} \sin \left (3 \, x\right )} \] Input:
integrate(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x, algorithm="fricas ")
Output:
1/1485*(45*cos(3*x)^16 + 80*cos(3*x)^14 + 160*cos(3*x)^12 + 384*cos(3*x)^1 0 + 1280*cos(3*x)^8 + 10240*cos(3*x)^6 - 61440*cos(3*x)^4 + 81920*cos(3*x) ^2 - 32768)/((cos(3*x)^4 - 2*cos(3*x)^2 + 1)*sin(3*x))
Timed out. \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=\text {Timed out} \] Input:
integrate(cos(3*x)*(-1+csc(3*x)**2)**3*(1-sin(3*x)**2)**5,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=\frac {1}{33} \, \sin \left (3 \, x\right )^{11} - \frac {8}{27} \, \sin \left (3 \, x\right )^{9} + \frac {4}{3} \, \sin \left (3 \, x\right )^{7} - \frac {56}{15} \, \sin \left (3 \, x\right )^{5} + \frac {70}{9} \, \sin \left (3 \, x\right )^{3} - \frac {420 \, \sin \left (3 \, x\right )^{4} - 40 \, \sin \left (3 \, x\right )^{2} + 3}{45 \, \sin \left (3 \, x\right )^{5}} - \frac {56}{3} \, \sin \left (3 \, x\right ) \] Input:
integrate(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x, algorithm="maxima ")
Output:
1/33*sin(3*x)^11 - 8/27*sin(3*x)^9 + 4/3*sin(3*x)^7 - 56/15*sin(3*x)^5 + 7 0/9*sin(3*x)^3 - 1/45*(420*sin(3*x)^4 - 40*sin(3*x)^2 + 3)/sin(3*x)^5 - 56 /3*sin(3*x)
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=\frac {1}{33} \, \sin \left (3 \, x\right )^{11} - \frac {8}{27} \, \sin \left (3 \, x\right )^{9} + \frac {4}{3} \, \sin \left (3 \, x\right )^{7} - \frac {56}{15} \, \sin \left (3 \, x\right )^{5} + \frac {70}{9} \, \sin \left (3 \, x\right )^{3} - \frac {420 \, \sin \left (3 \, x\right )^{4} - 40 \, \sin \left (3 \, x\right )^{2} + 3}{45 \, \sin \left (3 \, x\right )^{5}} - \frac {56}{3} \, \sin \left (3 \, x\right ) \] Input:
integrate(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x, algorithm="giac")
Output:
1/33*sin(3*x)^11 - 8/27*sin(3*x)^9 + 4/3*sin(3*x)^7 - 56/15*sin(3*x)^5 + 7 0/9*sin(3*x)^3 - 1/45*(420*sin(3*x)^4 - 40*sin(3*x)^2 + 3)/sin(3*x)^5 - 56 /3*sin(3*x)
Time = 15.72 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=-\frac {-45\,{\sin \left (3\,x\right )}^{16}+440\,{\sin \left (3\,x\right )}^{14}-1980\,{\sin \left (3\,x\right )}^{12}+5544\,{\sin \left (3\,x\right )}^{10}-11550\,{\sin \left (3\,x\right )}^8+27720\,{\sin \left (3\,x\right )}^6+13860\,{\sin \left (3\,x\right )}^4-1320\,{\sin \left (3\,x\right )}^2+99}{1485\,{\sin \left (3\,x\right )}^5} \] Input:
int(-cos(3*x)*(1/sin(3*x)^2 - 1)^3*(sin(3*x)^2 - 1)^5,x)
Output:
-(13860*sin(3*x)^4 - 1320*sin(3*x)^2 + 27720*sin(3*x)^6 - 11550*sin(3*x)^8 + 5544*sin(3*x)^10 - 1980*sin(3*x)^12 + 440*sin(3*x)^14 - 45*sin(3*x)^16 + 99)/(1485*sin(3*x)^5)
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx=\frac {45 \sin \left (3 x \right )^{16}-440 \sin \left (3 x \right )^{14}+1980 \sin \left (3 x \right )^{12}-5544 \sin \left (3 x \right )^{10}+11550 \sin \left (3 x \right )^{8}-27720 \sin \left (3 x \right )^{6}-13860 \sin \left (3 x \right )^{4}+1320 \sin \left (3 x \right )^{2}-99}{1485 \sin \left (3 x \right )^{5}} \] Input:
int(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x)
Output:
(45*sin(3*x)**16 - 440*sin(3*x)**14 + 1980*sin(3*x)**12 - 5544*sin(3*x)**1 0 + 11550*sin(3*x)**8 - 27720*sin(3*x)**6 - 13860*sin(3*x)**4 + 1320*sin(3 *x)**2 - 99)/(1485*sin(3*x)**5)