3.9 \(\int \cot (x) \csc ^3(x) \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{3} \csc ^3(x) \]

[Out]

-Csc[x]^3/3

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Rubi [A]  time = 0.0124362, 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} \[ -\frac{1}{3} \csc ^3(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[x]^3/3

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

Mathematica [A]  time = 0.0040954, size = 8, normalized size = 1. \[ -\frac{1}{3} \csc ^3(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Csc[x]^3/3

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Maple [A]  time = 0.013, size = 7, normalized size = 0.9 \begin{align*} -{\frac{1}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*csc(x)^2/sin(x)^2,x)

[Out]

-1/3/sin(x)^3

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Maxima [A]  time = 0.93987, size = 8, normalized size = 1. \begin{align*} -\frac{1}{3 \, \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)^2/sin(x)^2,x, algorithm="maxima")

[Out]

-1/3/sin(x)^3

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Fricas [B]  time = 1.60745, size = 39, normalized size = 4.88 \begin{align*} \frac{1}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)^2/sin(x)^2,x, algorithm="fricas")

[Out]

1/3/((cos(x)^2 - 1)*sin(x))

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Sympy [A]  time = 0.064042, size = 8, normalized size = 1. \begin{align*} - \frac{1}{3 \sin ^{3}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)**2/sin(x)**2,x)

[Out]

-1/(3*sin(x)**3)

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Giac [A]  time = 1.07199, size = 8, normalized size = 1. \begin{align*} -\frac{1}{3 \, \sin \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*csc(x)^2/sin(x)^2,x, algorithm="giac")

[Out]

-1/3/sin(x)^3