3.4.37 \(\int \csc ^6(x) \, dx\) [337]

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

[Out]

-cot(x)-2/3*cot(x)^3-1/5*cot(x)^5

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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 = {3852} \begin {gather*} -\frac {1}{5} \cot ^5(x)-\frac {2 \cot ^3(x)}{3}-\cot (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^6,x]

[Out]

-Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5

Rule 3852

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 &=-\text {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 \begin {gather*} -\frac {8 \cot (x)}{15}-\frac {4}{15} \cot (x) \csc ^2(x)-\frac {1}{5} \cot (x) \csc ^4(x) \end {gather*}

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 = 0.08, size = 18, normalized size = 0.86

method	result	size
default	\(\left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (x \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (x \right )\right )}{15}\right ) \cot \left (x \right )\)	\(18\)
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 = 2.85, size = 20, normalized size = 0.95 \begin {gather*} -\frac {15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.84, size = 37, normalized size = 1.76 \begin {gather*} -\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 {gather*}

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 = 0.01, size = 32, normalized size = 1.52 \begin {gather*} - \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 {gather*}

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 = 1.57, size = 20, normalized size = 0.95 \begin {gather*} -\frac {15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \end {gather*}

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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Mupad [B]
time = 0.20, size = 27, normalized size = 1.29 \begin {gather*} -\frac {8\,\cos \left (x\right )\,{\sin \left (x\right )}^4+4\,\cos \left (x\right )\,{\sin \left (x\right )}^2+3\,\cos \left (x\right )}{15\,{\sin \left (x\right )}^5} \end {gather*}

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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