3.314 \(\int \cot ^3(2 x) \csc ^3(2 x) \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{6} \csc ^3(2 x)-\frac{1}{10} \csc ^5(2 x) \]

[Out]

Csc[2*x]^3/6 - Csc[2*x]^5/10

________________________________________________________________________________________

Rubi [A]  time = 0.0294961, antiderivative size = 21, 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 = {2606, 14} \[ \frac{1}{6} \csc ^3(2 x)-\frac{1}{10} \csc ^5(2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cot[2*x]^3*Csc[2*x]^3,x]

[Out]

Csc[2*x]^3/6 - Csc[2*x]^5/10

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0233481, size = 21, normalized size = 1. \[ \frac{1}{6} \csc ^3(2 x)-\frac{1}{10} \csc ^5(2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[2*x]^3*Csc[2*x]^3,x]

[Out]

Csc[2*x]^3/6 - Csc[2*x]^5/10

________________________________________________________________________________________

Maple [B]  time = 0.028, size = 58, normalized size = 2.8 \begin{align*} -{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{4}}{10\, \left ( \sin \left ( 2\,x \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{4}}{30\, \left ( \sin \left ( 2\,x \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{4}}{30\,\sin \left ( 2\,x \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( 2\,x \right ) \right ) ^{2} \right ) \sin \left ( 2\,x \right ) }{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(2*x)^3*csc(2*x)^3,x)

[Out]

-1/10/sin(2*x)^5*cos(2*x)^4-1/30/sin(2*x)^3*cos(2*x)^4+1/30/sin(2*x)*cos(2*x)^4+1/30*(2+cos(2*x)^2)*sin(2*x)

________________________________________________________________________________________

Maxima [A]  time = 0.931975, size = 24, normalized size = 1.14 \begin{align*} \frac{5 \, \sin \left (2 \, x\right )^{2} - 3}{30 \, \sin \left (2 \, x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(2*x)^3*csc(2*x)^3,x, algorithm="maxima")

[Out]

1/30*(5*sin(2*x)^2 - 3)/sin(2*x)^5

________________________________________________________________________________________

Fricas [B]  time = 2.32517, size = 93, normalized size = 4.43 \begin{align*} -\frac{5 \, \cos \left (2 \, x\right )^{2} - 2}{30 \,{\left (\cos \left (2 \, x\right )^{4} - 2 \, \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(cot(2*x)^3*csc(2*x)^3,x, algorithm="fricas")

[Out]

-1/30*(5*cos(2*x)^2 - 2)/((cos(2*x)^4 - 2*cos(2*x)^2 + 1)*sin(2*x))

________________________________________________________________________________________

Sympy [A]  time = 0.099444, size = 17, normalized size = 0.81 \begin{align*} \frac{5 \sin ^{2}{\left (2 x \right )} - 3}{30 \sin ^{5}{\left (2 x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(2*x)**3*csc(2*x)**3,x)

[Out]

(5*sin(2*x)**2 - 3)/(30*sin(2*x)**5)

________________________________________________________________________________________

Giac [A]  time = 1.08467, size = 24, normalized size = 1.14 \begin{align*} \frac{5 \, \sin \left (2 \, x\right )^{2} - 3}{30 \, \sin \left (2 \, x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(2*x)^3*csc(2*x)^3,x, algorithm="giac")

[Out]

1/30*(5*sin(2*x)^2 - 3)/sin(2*x)^5