3.896 \(\int \cos (3 x) (-1+\csc ^2(3 x))^3 (1-\sin ^2(3 x))^5 \, dx\)

Optimal. Leaf size=87 \[ \frac {1}{33} \sin ^{11}(3 x)-\frac {8}{27} \sin ^9(3 x)+\frac {4}{3} \sin ^7(3 x)-\frac {56}{15} \sin ^5(3 x)+\frac {70}{9} \sin ^3(3 x)-\frac {56}{3} \sin (3 x)-\frac {1}{15} \csc ^5(3 x)+\frac {8}{9} \csc ^3(3 x)-\frac {28}{3} \csc (3 x) \]

[Out]

-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

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3175, 4120, 2590, 270} \[ \frac {1}{33} \sin ^{11}(3 x)-\frac {8}{27} \sin ^9(3 x)+\frac {4}{3} \sin ^7(3 x)-\frac {56}{15} \sin ^5(3 x)+\frac {70}{9} \sin ^3(3 x)-\frac {56}{3} \sin (3 x)-\frac {1}{15} \csc ^5(3 x)+\frac {8}{9} \csc ^3(3 x)-\frac {28}{3} \csc (3 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^5,x]

[Out]

(-28*Csc[3*x])/3 + (8*Csc[3*x]^3)/9 - Csc[3*x]^5/15 - (56*Sin[3*x])/3 + (70*Sin[3*x]^3)/9 - (56*Sin[3*x]^5)/15
+ (4*Sin[3*x]^7)/3 - (8*Sin[3*x]^9)/27 + Sin[3*x]^11/33

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[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]

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[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]

Rule 4120

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \cos (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^5 \, dx &=\int \cos ^{11}(3 x) \left (-1+\csc ^2(3 x)\right )^3 \, dx\\ &=\int \cos ^{11}(3 x) \cot ^6(3 x) \, dx\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^8}{x^6} \, dx,x,-\sin (3 x)\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (-56+\frac {1}{x^6}-\frac {8}{x^4}+\frac {28}{x^2}+70 x^2-56 x^4+28 x^6-8 x^8+x^{10}\right ) \, dx,x,-\sin (3 x)\right )\right )\\ &=-\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)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 87, normalized size = 1.00 \[ \frac {1}{33} \sin ^{11}(3 x)-\frac {8}{27} \sin ^9(3 x)+\frac {4}{3} \sin ^7(3 x)-\frac {56}{15} \sin ^5(3 x)+\frac {70}{9} \sin ^3(3 x)-\frac {56}{3} \sin (3 x)-\frac {1}{15} \csc ^5(3 x)+\frac {8}{9} \csc ^3(3 x)-\frac {28}{3} \csc (3 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^5,x]

[Out]

(-28*Csc[3*x])/3 + (8*Csc[3*x]^3)/9 - Csc[3*x]^5/15 - (56*Sin[3*x])/3 + (70*Sin[3*x]^3)/9 - (56*Sin[3*x]^5)/15
+ (4*Sin[3*x]^7)/3 - (8*Sin[3*x]^9)/27 + Sin[3*x]^11/33

________________________________________________________________________________________

fricas [A]  time = 1.05, size = 92, normalized size = 1.06 \[ \frac {45 \, \cos \left (3 \, x\right )^{16} + 80 \, \cos \left (3 \, x\right )^{14} + 160 \, \cos \left (3 \, x\right )^{12} + 384 \, \cos \left (3 \, x\right )^{10} + 1280 \, \cos \left (3 \, x\right )^{8} + 10240 \, \cos \left (3 \, x\right )^{6} - 61440 \, \cos \left (3 \, x\right )^{4} + 81920 \, \cos \left (3 \, x\right )^{2} - 32768}{1485 \, {\left (\cos \left (3 \, x\right )^{4} - 2 \, \cos \left (3 \, x\right )^{2} + 1\right )} \sin \left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x, algorithm="fricas")

[Out]

1/1485*(45*cos(3*x)^16 + 80*cos(3*x)^14 + 160*cos(3*x)^12 + 384*cos(3*x)^10 + 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))

________________________________________________________________________________________

giac [A]  time = 0.49, size = 73, normalized size = 0.84 \[ \frac {1}{33} \, \sin \left (3 \, x\right )^{11} - \frac {8}{27} \, \sin \left (3 \, x\right )^{9} + \frac {4}{3} \, \sin \left (3 \, x\right )^{7} - \frac {56}{15} \, \sin \left (3 \, x\right )^{5} + \frac {70}{9} \, \sin \left (3 \, x\right )^{3} - \frac {420 \, \sin \left (3 \, x\right )^{4} - 40 \, \sin \left (3 \, x\right )^{2} + 3}{45 \, \sin \left (3 \, x\right )^{5}} - \frac {56}{3} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x, algorithm="giac")

[Out]

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 - 1/45*(420*sin(3*x)^
4 - 40*sin(3*x)^2 + 3)/sin(3*x)^5 - 56/3*sin(3*x)

________________________________________________________________________________________

maple [A]  time = 0.16, size = 72, normalized size = 0.83 \[ \frac {\left (\sin ^{11}\left (3 x \right )\right )}{33}-\frac {8 \left (\sin ^{9}\left (3 x \right )\right )}{27}+\frac {4 \left (\sin ^{7}\left (3 x \right )\right )}{3}-\frac {56 \left (\sin ^{5}\left (3 x \right )\right )}{15}+\frac {70 \left (\sin ^{3}\left (3 x \right )\right )}{9}-\frac {56 \sin \left (3 x \right )}{3}-\frac {28}{3 \sin \left (3 x \right )}+\frac {8}{9 \sin \left (3 x \right )^{3}}-\frac {1}{15 \sin \left (3 x \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 73, normalized size = 0.84 \[ \frac {1}{33} \, \sin \left (3 \, x\right )^{11} - \frac {8}{27} \, \sin \left (3 \, x\right )^{9} + \frac {4}{3} \, \sin \left (3 \, x\right )^{7} - \frac {56}{15} \, \sin \left (3 \, x\right )^{5} + \frac {70}{9} \, \sin \left (3 \, x\right )^{3} - \frac {420 \, \sin \left (3 \, x\right )^{4} - 40 \, \sin \left (3 \, x\right )^{2} + 3}{45 \, \sin \left (3 \, x\right )^{5}} - \frac {56}{3} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*(-1+csc(3*x)^2)^3*(1-sin(3*x)^2)^5,x, algorithm="maxima")

[Out]

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 - 1/45*(420*sin(3*x)^
4 - 40*sin(3*x)^2 + 3)/sin(3*x)^5 - 56/3*sin(3*x)

________________________________________________________________________________________

mupad [B]  time = 2.97, size = 74, normalized size = 0.85 \[ -\frac {-45\,{\sin \left (3\,x\right )}^{16}+440\,{\sin \left (3\,x\right )}^{14}-1980\,{\sin \left (3\,x\right )}^{12}+5544\,{\sin \left (3\,x\right )}^{10}-11550\,{\sin \left (3\,x\right )}^8+27720\,{\sin \left (3\,x\right )}^6+13860\,{\sin \left (3\,x\right )}^4-1320\,{\sin \left (3\,x\right )}^2+99}{1485\,{\sin \left (3\,x\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(3*x)*(1/sin(3*x)^2 - 1)^3*(sin(3*x)^2 - 1)^5,x)

[Out]

-(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)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*(-1+csc(3*x)**2)**3*(1-sin(3*x)**2)**5,x)

[Out]

Timed out

________________________________________________________________________________________