\(\int \cot ^3(x) \csc (x) \, dx\) [355]

Optimal result

Integrand size = 7, antiderivative size = 11 \[ \int \cot ^3(x) \csc (x) \, dx=\csc (x)-\frac {\csc ^3(x)}{3} \]

[Out]

csc(x)-1/3*csc(x)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2686} \[ \int \cot ^3(x) \csc (x) \, dx=\csc (x)-\frac {\csc ^3(x)}{3} \]

[In]

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

[Out]

Csc[x] - Csc[x]^3/3

Rule 2686

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cot ^3(x) \csc (x) \, dx=\csc (x)-\frac {\csc ^3(x)}{3} \]

[In]

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

[Out]

Csc[x] - Csc[x]^3/3

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\csc \left (x \right )-\frac {\left (\csc ^{3}\left (x \right )\right )}{3}\)	\(10\)
default	\(\csc \left (x \right )-\frac {\left (\csc ^{3}\left (x \right )\right )}{3}\)	\(10\)
risch	\(\frac {2 i \left (3 \,{\mathrm e}^{5 i x}-2 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\)	\(35\)

[In]

int(cot(x)^3*csc(x),x,method=_RETURNVERBOSE)

[Out]

csc(x)-1/3*csc(x)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (9) = 18\).

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.00 \[ \int \cot ^3(x) \csc (x) \, dx=\frac {3 \, \cos \left (x\right )^{2} - 2}{3 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \cot ^3(x) \csc (x) \, dx=- \frac {1 - 3 \sin ^{2}{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \]

[In]

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

[Out]

-(1 - 3*sin(x)**2)/(3*sin(x)**3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \cot ^3(x) \csc (x) \, dx=\frac {3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \]

[In]

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

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \cot ^3(x) \csc (x) \, dx=\frac {3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \]

[In]

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

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cot ^3(x) \csc (x) \, dx=\frac {{\sin \left (x\right )}^2-\frac {1}{3}}{{\sin \left (x\right )}^3} \]

[In]

int(cot(x)^3/sin(x),x)

[Out]

(sin(x)^2 - 1/3)/sin(x)^3