\(\int \cot ^3(x) \csc (x) \, dx\) [35]

Optimal result

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

[Out]

-1/3/sin(x)^3+1/sin(x)

Rubi [A] (verified)

Time = 0.01 (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 = 0.01 (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 = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36

method	result	size
parallelrisch	\(-\frac {\left (\csc ^{3}\left (x \right )\right ) \left (-1+3 \cos \left (2 x \right )\right )}{6}\)	\(15\)
default	\(-\frac {\cos ^{4}\left (x \right )}{3 \sin \left (x \right )^{3}}+\frac {\cos ^{4}\left (x \right )}{3 \sin \left (x \right )}+\frac {\left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{3}\)	\(32\)
norman	\(\frac {-\frac {1}{24}+\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}-\frac {\left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3}}\)	\(34\)
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(cos(x)^3/sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

-1/6*csc(x)^3*(-1+3*cos(2*x))

Fricas [A] (verification not implemented)

none

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(cos(x)^3/sin(x)^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (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(cos(x)**3/sin(x)**4,x)

[Out]

-(1 - 3*sin(x)**2)/(3*sin(x)**3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (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(cos(x)^3/sin(x)^4,x, algorithm="maxima")

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

Giac [A] (verification not implemented)

none

Time = 0.29 (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(cos(x)^3/sin(x)^4,x, algorithm="giac")

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

Mupad [B] (verification not implemented)

Time = 0.10 (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(cos(x)^3/sin(x)^4,x)

[Out]

(sin(x)^2 - 1/3)/sin(x)^3