3.1.9 \(\int \cot (x) \csc ^3(x) \, dx\) [9]

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

[Out]

-1/3*csc(x)^3

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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}{3} \csc ^3(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Csc[x]^3

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

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \csc ^3(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Csc[x]^3

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Mathics [A]
time = 1.73, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{3 \text {Sin}\left [x\right ]^3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Csc[x]^2*(Cos[x]/Sin[x]^2),x]')

[Out]

-1 / (3 Sin[x] ^ 3)

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Maple [A]
time = 0.03, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(-\frac {1}{3 \sin \left (x \right )^{3}}\)	\(7\)
default	\(-\frac {1}{3 \sin \left (x \right )^{3}}\)	\(7\)
risch	\(\frac {8 i {\mathrm e}^{3 i x}}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\)	\(18\)
norman	\(\frac {-\frac {1}{24}-\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}-\frac {\left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/sin(x)^3

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Maxima [A]
time = 0.25, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{3 \, \sin \left (x\right )^{3}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 0.30, size = 14, normalized size = 1.75 \begin {gather*} \frac {1}{3 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end {gather*}

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 = 0.03, size = 8, normalized size = 1.00 \begin {gather*} - \frac {1}{3 \sin ^{3}{\left (x \right )}} \end {gather*}

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 = 0.00, size = 9, normalized size = 1.12 \begin {gather*} -\frac {1}{3 \sin ^{3}x} \end {gather*}

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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Mupad [B]
time = 0.05, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{3\,{\sin \left (x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/sin(x)^4,x)

[Out]

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

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