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

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Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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*}

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Mathematica [A]  time = 0.03, size = 43, normalized size = 1.00 \[ \frac {1}{10} \sin ^5(2 x)-\frac {2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac {1}{6} \csc ^3(2 x)+2 \csc (2 x) \]

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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fricas [A]  time = 0.96, size = 52, normalized size = 1.21 \[ -\frac {3 \, \cos \left (2 \, x\right )^{8} + 8 \, \cos \left (2 \, x\right )^{6} + 48 \, \cos \left (2 \, x\right )^{4} - 192 \, \cos \left (2 \, x\right )^{2} + 128}{30 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right )} \]

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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giac [A]  time = 0.16, size = 41, normalized size = 0.95 \[ \frac {1}{10} \, \sin \left (2 \, x\right )^{5} - \frac {2}{3} \, \sin \left (2 \, x\right )^{3} + \frac {12 \, \sin \left (2 \, x\right )^{2} - 1}{6 \, \sin \left (2 \, x\right )^{3}} + 3 \, \sin \left (2 \, x\right ) \]

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)

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maple [A]  time = 0.17, size = 68, normalized size = 1.58 \[ -\frac {\cos ^{10}\left (2 x \right )}{6 \sin \left (2 x \right )^{3}}+\frac {7 \left (\cos ^{10}\left (2 x \right )\right )}{6 \sin \left (2 x \right )}+\frac {7 \left (\frac {128}{35}+\cos ^{8}\left (2 x \right )+\frac {8 \left (\cos ^{6}\left (2 x \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (2 x \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (2 x \right )\right )}{35}\right ) \sin \left (2 x \right )}{6} \]

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 = 0.31, size = 41, normalized size = 0.95 \[ \frac {1}{10} \, \sin \left (2 \, x\right )^{5} - \frac {2}{3} \, \sin \left (2 \, x\right )^{3} + \frac {12 \, \sin \left (2 \, x\right )^{2} - 1}{6 \, \sin \left (2 \, x\right )^{3}} + 3 \, \sin \left (2 \, x\right ) \]

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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mupad [B]  time = 3.06, size = 42, normalized size = 0.98 \[ \frac {3\,{\sin \left (2\,x\right )}^8-20\,{\sin \left (2\,x\right )}^6+90\,{\sin \left (2\,x\right )}^4+60\,{\sin \left (2\,x\right )}^2-5}{30\,{\sin \left (2\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(60*sin(2*x)^2 + 90*sin(2*x)^4 - 20*sin(2*x)^6 + 3*sin(2*x)^8 - 5)/(30*sin(2*x)^3)

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sympy [A]  time = 0.09, size = 42, normalized size = 0.98 \[ \frac {12 \sin ^{2}{\left (2 x \right )} - 1}{6 \sin ^{3}{\left (2 x \right )}} + \frac {\sin ^{5}{\left (2 x \right )}}{10} - \frac {2 \sin ^{3}{\left (2 x \right )}}{3} + 3 \sin {\left (2 x \right )} \]

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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