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

Optimal. Leaf size=11 \[ \csc (x)-\frac {\csc ^3(x)}{3} \]

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2606} \[ \csc (x)-\frac {\csc ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csc[x] - 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])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ \csc (x)-\frac {\csc ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csc[x] - Csc[x]^3/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^3(x) \csc (x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 0.84, size = 22, normalized size = 2.00 \[ \frac {3 \, \cos \relax (x)^{2} - 2}{3 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*csc(x),x, algorithm="fricas")

[Out]

1/3*(3*cos(x)^2 - 2)/((cos(x)^2 - 1)*sin(x))

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giac [A]  time = 0.60, size = 14, normalized size = 1.27 \[ \frac {3 \, \sin \relax (x)^{2} - 1}{3 \, \sin \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*csc(x),x, algorithm="giac")

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

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maple [B]  time = 0.31, size = 32, normalized size = 2.91

method	result	size
default	\(-\frac {\cos ^{4}\relax (x )}{3 \sin \relax (x )^{3}}+\frac {\cos ^{4}\relax (x )}{3 \sin \relax (x )}+\frac {\left (2+\cos ^{2}\relax (x )\right ) \sin \relax (x )}{3}\)	\(32\)
risch	\(\frac {2 i \left (3 \,{\mathrm e}^{5 i x}-2 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^3*csc(x),x,method=_RETURNVERBOSE)

[Out]

-1/3/sin(x)^3*cos(x)^4+1/3/sin(x)*cos(x)^4+1/3*(2+cos(x)^2)*sin(x)

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maxima [A]  time = 0.56, size = 14, normalized size = 1.27 \[ \frac {3 \, \sin \relax (x)^{2} - 1}{3 \, \sin \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*csc(x),x, algorithm="maxima")

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

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mupad [B]  time = 0.32, size = 11, normalized size = 1.00 \[ \frac {{\sin \relax (x)}^2-\frac {1}{3}}{{\sin \relax (x)}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sin(x)^2 - 1/3)/sin(x)^3

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sympy [A]  time = 0.10, size = 15, normalized size = 1.36 \[ - \frac {1 - 3 \sin ^{2}{\relax (x )}}{3 \sin ^{3}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**3*csc(x),x)

[Out]

-(1 - 3*sin(x)**2)/(3*sin(x)**3)

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