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

Optimal. Leaf size=8 \[ -\frac {1}{2} \csc ^2(x) \]

[Out]

-1/2*csc(x)^2

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2686, 30} \begin {gather*} -\frac {1}{2} \csc ^2(x) \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.04, size = 8, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \csc ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Csc[x]^2

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Maple [A]
time = 0.02, size = 7, normalized size = 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	\(\frac {-\frac {1}{8}-\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}}{\tan \left (\frac {x}{2}\right )^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sin(x)^2

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Maxima [A]
time = 4.89, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2 \, \sin \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sin(x)^2

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Fricas [A]
time = 0.70, size = 10, normalized size = 1.25 \begin {gather*} \frac {1}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(cos(x)^2 - 1)

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Sympy [A]
time = 0.03, size = 8, normalized size = 1.00 \begin {gather*} - \frac {1}{2 \sin ^{2}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.69, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2 \, \sin \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sin(x)^2

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Mupad [B]
time = 0.42, size = 6, normalized size = 0.75 \begin {gather*} -\frac {{\mathrm {cot}\left (x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cot(x)^2/2

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