3.337 \(\int \csc ^6(x) \, dx\)

Optimal. Leaf size=21 \[ -\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x) \]

[Out]

-Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5

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Rubi [A]  time = 0.0089103, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767} \[ -\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^6,x]

[Out]

-Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \csc ^6(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (x)\right )\\ &=-\cot (x)-\frac{2 \cot ^3(x)}{3}-\frac{\cot ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0030074, size = 27, normalized size = 1.29 \[ -\frac{8 \cot (x)}{15}-\frac{1}{5} \cot (x) \csc ^4(x)-\frac{4}{15} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^6,x]

[Out]

(-8*Cot[x])/15 - (4*Cot[x]*Csc[x]^2)/15 - (Cot[x]*Csc[x]^4)/5

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Maple [A]  time = 0.028, size = 18, normalized size = 0.9 \begin{align*} \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( x \right ) \right ) ^{2}}{15}} \right ) \cot \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^6,x)

[Out]

(-8/15-1/5*csc(x)^4-4/15*csc(x)^2)*cot(x)

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Maxima [A]  time = 0.936538, size = 27, normalized size = 1.29 \begin{align*} -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^6,x, algorithm="maxima")

[Out]

-1/15*(15*tan(x)^4 + 10*tan(x)^2 + 3)/tan(x)^5

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Fricas [B]  time = 1.58287, size = 112, normalized size = 5.33 \begin{align*} -\frac{8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )}{15 \,{\left (\cos \left (x\right )^{4} - 2 \, \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(1/sin(x)^6,x, algorithm="fricas")

[Out]

-1/15*(8*cos(x)^5 - 20*cos(x)^3 + 15*cos(x))/((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x))

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Sympy [A]  time = 0.058264, size = 32, normalized size = 1.52 \begin{align*} - \frac{8 \cos{\left (x \right )}}{15 \sin{\left (x \right )}} - \frac{4 \cos{\left (x \right )}}{15 \sin ^{3}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{5 \sin ^{5}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)**6,x)

[Out]

-8*cos(x)/(15*sin(x)) - 4*cos(x)/(15*sin(x)**3) - cos(x)/(5*sin(x)**5)

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Giac [A]  time = 1.06628, size = 27, normalized size = 1.29 \begin{align*} -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^6,x, algorithm="giac")

[Out]

-1/15*(15*tan(x)^4 + 10*tan(x)^2 + 3)/tan(x)^5