3.895 \(\int \cos ^5(2 x) \cot ^4(2 x) \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{10} \sin ^5(2 x)-\frac{2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac{1}{6} \csc ^3(2 x)+2 \csc (2 x) \]

[Out]

2*Csc[2*x] - Csc[2*x]^3/6 + 3*Sin[2*x] - (2*Sin[2*x]^3)/3 + Sin[2*x]^5/10

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Rubi [A]  time = 0.0360787, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2590, 270} \[ \frac{1}{10} \sin ^5(2 x)-\frac{2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac{1}{6} \csc ^3(2 x)+2 \csc (2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]^5*Cot[2*x]^4,x]

[Out]

2*Csc[2*x] - Csc[2*x]^3/6 + 3*Sin[2*x] - (2*Sin[2*x]^3)/3 + Sin[2*x]^5/10

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

Mathematica [A]  time = 0.0259009, size = 43, normalized size = 1. \[ \frac{1}{10} \sin ^5(2 x)-\frac{2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac{1}{6} \csc ^3(2 x)+2 \csc (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*x]^5*Cot[2*x]^4,x]

[Out]

2*Csc[2*x] - Csc[2*x]^3/6 + 3*Sin[2*x] - (2*Sin[2*x]^3)/3 + Sin[2*x]^5/10

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Maple [A]  time = 0.054, size = 68, normalized size = 1.6 \begin{align*} -{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{10}}{6\, \left ( \sin \left ( 2\,x \right ) \right ) ^{3}}}+{\frac{7\, \left ( \cos \left ( 2\,x \right ) \right ) ^{10}}{6\,\sin \left ( 2\,x \right ) }}+{\frac{7\,\sin \left ( 2\,x \right ) }{6} \left ({\frac{128}{35}}+ \left ( \cos \left ( 2\,x \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( 2\,x \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( 2\,x \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( 2\,x \right ) \right ) ^{2}}{35}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)^5*cot(2*x)^4,x)

[Out]

-1/6/sin(2*x)^3*cos(2*x)^10+7/6/sin(2*x)*cos(2*x)^10+7/6*(128/35+cos(2*x)^8+8/7*cos(2*x)^6+48/35*cos(2*x)^4+64
/35*cos(2*x)^2)*sin(2*x)

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Maxima [A]  time = 0.939942, size = 55, normalized size = 1.28 \begin{align*} \frac{1}{10} \, \sin \left (2 \, x\right )^{5} - \frac{2}{3} \, \sin \left (2 \, x\right )^{3} + \frac{12 \, \sin \left (2 \, x\right )^{2} - 1}{6 \, \sin \left (2 \, x\right )^{3}} + 3 \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^5*cot(2*x)^4,x, algorithm="maxima")

[Out]

1/10*sin(2*x)^5 - 2/3*sin(2*x)^3 + 1/6*(12*sin(2*x)^2 - 1)/sin(2*x)^3 + 3*sin(2*x)

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Fricas [A]  time = 2.41084, size = 140, normalized size = 3.26 \begin{align*} -\frac{3 \, \cos \left (2 \, x\right )^{8} + 8 \, \cos \left (2 \, x\right )^{6} + 48 \, \cos \left (2 \, x\right )^{4} - 192 \, \cos \left (2 \, x\right )^{2} + 128}{30 \,{\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^5*cot(2*x)^4,x, algorithm="fricas")

[Out]

-1/30*(3*cos(2*x)^8 + 8*cos(2*x)^6 + 48*cos(2*x)^4 - 192*cos(2*x)^2 + 128)/((cos(2*x)^2 - 1)*sin(2*x))

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Sympy [A]  time = 0.098547, size = 42, normalized size = 0.98 \begin{align*} \frac{12 \sin ^{2}{\left (2 x \right )} - 1}{6 \sin ^{3}{\left (2 x \right )}} + \frac{\sin ^{5}{\left (2 x \right )}}{10} - \frac{2 \sin ^{3}{\left (2 x \right )}}{3} + 3 \sin{\left (2 x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)**5*cot(2*x)**4,x)

[Out]

(12*sin(2*x)**2 - 1)/(6*sin(2*x)**3) + sin(2*x)**5/10 - 2*sin(2*x)**3/3 + 3*sin(2*x)

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Giac [A]  time = 1.08401, size = 55, normalized size = 1.28 \begin{align*} \frac{1}{10} \, \sin \left (2 \, x\right )^{5} - \frac{2}{3} \, \sin \left (2 \, x\right )^{3} + \frac{12 \, \sin \left (2 \, x\right )^{2} - 1}{6 \, \sin \left (2 \, x\right )^{3}} + 3 \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^5*cot(2*x)^4,x, algorithm="giac")

[Out]

1/10*sin(2*x)^5 - 2/3*sin(2*x)^3 + 1/6*(12*sin(2*x)^2 - 1)/sin(2*x)^3 + 3*sin(2*x)