3.8 \(\int \cot (x) \csc ^2(x) \, dx\)

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

[Out]

-Csc[x]^2/2

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Rubi [A]  time = 0.0130718, antiderivative size = 8, normalized size of antiderivative = 1., 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 = {2606, 30} \[ -\frac{1}{2} \csc ^2(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[x]^2/2

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])

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.004811, size = 8, normalized size = 1. \[ -\frac{1}{2} \csc ^2(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[x]^2/2

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Maple [A]  time = 0.007, size = 7, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/sin(x)^2

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Maxima [A]  time = 0.942057, size = 8, normalized size = 1. \begin{align*} -\frac{1}{2 \, \sin \left (x\right )^{2}} \end{align*}

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 = 0.43484, size = 27, normalized size = 3.38 \begin{align*} \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

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 = 0.059657, size = 8, normalized size = 1. \begin{align*} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \end{align*}

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 = 1.08131, size = 14, normalized size = 1.75 \begin{align*} \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(cos(x)^2 - 1)