\(\int \cot (x) \csc ^2(x) \, dx\) [8]

Optimal result

Integrand size = 7, antiderivative size = 8 \[ \int \cot (x) \csc ^2(x) \, dx=-\frac {1}{2} \csc ^2(x) \]

[Out]

-1/2*csc(x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 30} \[ \int \cot (x) \csc ^2(x) \, dx=-\frac {1}{2} \csc ^2(x) \]

[In]

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

[Out]

-1/2*Csc[x]^2

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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}(\int x \, dx,x,\csc (x)) \\ & = -\frac {1}{2} \csc ^2(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \cot (x) \csc ^2(x) \, dx=-\frac {1}{2} \csc ^2(x) \]

[In]

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

[Out]

-1/2*Csc[x]^2

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(-\frac {1}{2 \sin \left (x \right )^{2}}\)	\(7\)
default	\(-\frac {1}{2 \sin \left (x \right )^{2}}\)	\(7\)
risch	\(\frac {2 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\)	\(17\)
norman	\(\frac {-\frac {1}{8}-\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}}{\tan \left (\frac {x}{2}\right )^{2}}\)	\(18\)
parallelrisch	\(\frac {-1-\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 \tan \left (\frac {x}{2}\right )^{2}}\)	\(19\)

[In]

int(cos(x)/sin(x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2/sin(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.25 \[ \int \cot (x) \csc ^2(x) \, dx=\frac {1}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]

[In]

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

[Out]

1/2/(cos(x)^2 - 1)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \cot (x) \csc ^2(x) \, dx=- \frac {1}{2 \sin ^{2}{\left (x \right )}} \]

[In]

integrate(cos(x)/sin(x)**3,x)

[Out]

-1/(2*sin(x)**2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cot (x) \csc ^2(x) \, dx=-\frac {1}{2 \, \sin \left (x\right )^{2}} \]

[In]

integrate(cos(x)/sin(x)^3,x, algorithm="maxima")

[Out]

-1/2/sin(x)^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cot (x) \csc ^2(x) \, dx=-\frac {1}{2 \, \sin \left (x\right )^{2}} \]

[In]

integrate(cos(x)/sin(x)^3,x, algorithm="giac")

[Out]

-1/2/sin(x)^2

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \cot (x) \csc ^2(x) \, dx=-\frac {{\mathrm {cot}\left (x\right )}^2}{2} \]

[In]

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

[Out]

-cot(x)^2/2