3.1.27 \(\int \frac {1}{(1-\cos ^2(x))^3} \, dx\) [27]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3254, 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[(1 - Cos[x]^2)^(-3),x]

[Out]

-Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5

Rule 3254

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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 \frac {1}{\left (1-\cos ^2(x)\right )^3} \, dx &=\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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, 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[(1 - Cos[x]^2)^(-3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 20, normalized size = 0.95

method	result	size
default	\(-\frac {1}{\tan \left (x \right )}-\frac {1}{5 \tan \left (x \right )^{5}}-\frac {2}{3 \tan \left (x \right )^{3}}\)	\(20\)
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/(1-cos(x)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/tan(x)-1/5/tan(x)^5-2/3/tan(x)^3

________________________________________________________________________________________

Maxima [A]
time = 0.27, 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/(1-cos(x)^2)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.41, 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/(1-cos(x)^2)^3,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))

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).
time = 1.28, size = 54, normalized size = 2.57 \begin {gather*} \frac {\tan ^{5}{\left (\frac {x}{2} \right )}}{160} + \frac {5 \tan ^{3}{\left (\frac {x}{2} \right )}}{96} + \frac {5 \tan {\left (\frac {x}{2} \right )}}{16} - \frac {5}{16 \tan {\left (\frac {x}{2} \right )}} - \frac {5}{96 \tan ^{3}{\left (\frac {x}{2} \right )}} - \frac {1}{160 \tan ^{5}{\left (\frac {x}{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(x)**2)**3,x)

[Out]

tan(x/2)**5/160 + 5*tan(x/2)**3/96 + 5*tan(x/2)/16 - 5/(16*tan(x/2)) - 5/(96*tan(x/2)**3) - 1/(160*tan(x/2)**5
)

________________________________________________________________________________________

Giac [A]
time = 0.41, 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/(1-cos(x)^2)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 2.24, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\mathrm {cot}\left (x\right )}^5}{5}-\frac {2\,{\mathrm {cot}\left (x\right )}^3}{3}-\mathrm {cot}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cos(x)^2 - 1)^3,x)

[Out]

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

________________________________________________________________________________________