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*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

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Rubi [A]  time = 0.130178, antiderivative size = 87, normalized size of antiderivative = 1., 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 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]

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 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]

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 = 0.0604997, size = 87, normalized size = 1. \[ \frac{1}{33} \sin ^{11}(3 x)-\frac{8}{27} \sin ^9(3 x)+\frac{4}{3} \sin ^7(3 x)-\frac{56}{15} \sin ^5(3 x)+\frac{70}{9} \sin ^3(3 x)-\frac{56}{3} \sin (3 x)-\frac{1}{15} \csc ^5(3 x)+\frac{8}{9} \csc ^3(3 x)-\frac{28}{3} \csc (3 x) \]

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

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Maple [A]  time = 0.056, size = 72, normalized size = 0.8 \begin{align*}{\frac{ \left ( \sin \left ( 3\,x \right ) \right ) ^{11}}{33}}-{\frac{8\, \left ( \sin \left ( 3\,x \right ) \right ) ^{9}}{27}}+{\frac{4\, \left ( \sin \left ( 3\,x \right ) \right ) ^{7}}{3}}-{\frac{56\, \left ( \sin \left ( 3\,x \right ) \right ) ^{5}}{15}}+{\frac{70\, \left ( \sin \left ( 3\,x \right ) \right ) ^{3}}{9}}-{\frac{56\,\sin \left ( 3\,x \right ) }{3}}-{\frac{28}{3\,\sin \left ( 3\,x \right ) }}+{\frac{8}{9\, \left ( \sin \left ( 3\,x \right ) \right ) ^{3}}}-{\frac{1}{15\, \left ( \sin \left ( 3\,x \right ) \right ) ^{5}}} \end{align*}

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

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Maxima [A]  time = 0.943058, size = 99, normalized size = 1.14 \begin{align*} \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 ) \end{align*}

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)

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Fricas [A]  time = 2.70848, size = 275, normalized size = 3.16 \begin{align*} \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 )} \end{align*}

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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [A]  time = 1.18115, size = 99, normalized size = 1.14 \begin{align*} \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 ) \end{align*}

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)