3.355 \(\int \cot ^3(x) \csc (x) \, dx\)

Optimal. Leaf size=11 \[ \csc (x)-\frac{\csc ^3(x)}{3} \]

[Out]

Csc[x] - Csc[x]^3/3

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Rubi [A]  time = 0.0156247, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2606} \[ \csc (x)-\frac{\csc ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csc[x] - Csc[x]^3/3

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])

Rubi steps

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

Mathematica [A]  time = 0.00721, size = 11, normalized size = 1. \[ \csc (x)-\frac{\csc ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csc[x] - Csc[x]^3/3

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Maple [B]  time = 0.009, size = 32, normalized size = 2.9 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{3\,\sin \left ( x \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.943858, size = 19, normalized size = 1.73 \begin{align*} \frac{3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

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Fricas [B]  time = 1.86833, size = 62, normalized size = 5.64 \begin{align*} \frac{3 \, \cos \left (x\right )^{2} - 2}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.089148, size = 14, normalized size = 1.27 \begin{align*} \frac{3 \sin ^{2}{\left (x \right )} - 1}{3 \sin ^{3}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*sin(x)**2 - 1)/(3*sin(x)**3)

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Giac [A]  time = 1.08197, size = 19, normalized size = 1.73 \begin{align*} \frac{3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3