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

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

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^6(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Csc[x]^6,x]

[Out]

Could not integrate

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fricas [B]  time = 0.48, size = 37, normalized size = 1.76 \[ -\frac {8 \, \cos \relax (x)^{5} - 20 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)}{15 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)} \]

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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giac [A]  time = 0.59, size = 20, normalized size = 0.95 \[ -\frac {15 \, \tan \relax (x)^{4} + 10 \, \tan \relax (x)^{2} + 3}{15 \, \tan \relax (x)^{5}} \]

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

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maple [A]  time = 0.31, size = 18, normalized size = 0.86

method	result	size
default	\(\left (-\frac {8}{15}-\frac {\left (\csc ^{4}\relax (x )\right )}{5}-\frac {4 \left (\csc ^{2}\relax (x )\right )}{15}\right ) \cot \relax (x )\)	\(18\)
risch	\(-\frac {16 i \left (10 \,{\mathrm e}^{4 i x}-5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}-1\right )^{5}}\)	\(29\)
norman	\(\frac {-\frac {1}{160}-\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{96}-\frac {5 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {5 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {5 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{96}+\frac {\left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{160}}{\tan \left (\frac {x}{2}\right )^{5}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^6,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 20, normalized size = 0.95 \[ -\frac {15 \, \tan \relax (x)^{4} + 10 \, \tan \relax (x)^{2} + 3}{15 \, \tan \relax (x)^{5}} \]

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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mupad [B]  time = 0.20, size = 27, normalized size = 1.29 \[ -\frac {8\,\cos \relax (x)\,{\sin \relax (x)}^4+4\,\cos \relax (x)\,{\sin \relax (x)}^2+3\,\cos \relax (x)}{15\,{\sin \relax (x)}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.06, size = 32, normalized size = 1.52 \[ - \frac {8 \cos {\relax (x )}}{15 \sin {\relax (x )}} - \frac {4 \cos {\relax (x )}}{15 \sin ^{3}{\relax (x )}} - \frac {\cos {\relax (x )}}{5 \sin ^{5}{\relax (x )}} \]

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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