3.309 \(\int \frac {1}{(\csc (x)-\sin (x))^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3} \]

[Out]

-1/3*sec(x)^3+1/5*sec(x)^5

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4397, 2606, 14} \[ \frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-3),x]

[Out]

-Sec[x]^3/3 + Sec[x]^5/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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])

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.00 \[ \frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-3),x]

[Out]

-1/3*Sec[x]^3 + Sec[x]^5/5

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fricas [A]  time = 1.20, size = 14, normalized size = 0.82 \[ -\frac {5 \, \cos \relax (x)^{2} - 3}{15 \, \cos \relax (x)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^3,x, algorithm="fricas")

[Out]

-1/15*(5*cos(x)^2 - 3)/cos(x)^5

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giac [B]  time = 0.13, size = 59, normalized size = 3.47 \[ -\frac {4 \, {\left (\frac {5 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - \frac {5 \, {\left (\cos \relax (x) - 1\right )}^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {15 \, {\left (\cos \relax (x) - 1\right )}^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + 1\right )}}{15 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} + 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 14, normalized size = 0.82 \[ -\frac {1}{3 \cos \relax (x )^{3}}+\frac {1}{5 \cos \relax (x )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)-sin(x))^3,x)

[Out]

-1/3/cos(x)^3+1/5/cos(x)^5

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maxima [B]  time = 0.31, size = 103, normalized size = 6.06 \[ -\frac {4 \, {\left (\frac {5 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {5 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {15 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - 1\right )}}{15 \, {\left (\frac {5 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {10 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {10 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {5 \, \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} + \frac {\sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^3,x, algorithm="maxima")

[Out]

-4/15*(5*sin(x)^2/(cos(x) + 1)^2 + 5*sin(x)^4/(cos(x) + 1)^4 + 15*sin(x)^6/(cos(x) + 1)^6 - 1)/(5*sin(x)^2/(co
s(x) + 1)^2 - 10*sin(x)^4/(cos(x) + 1)^4 + 10*sin(x)^6/(cos(x) + 1)^6 - 5*sin(x)^8/(cos(x) + 1)^8 + sin(x)^10/
(cos(x) + 1)^10 - 1)

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mupad [B]  time = 2.55, size = 13, normalized size = 0.76 \[ \frac {1}{5\,{\cos \relax (x)}^5}-\frac {1}{3\,{\cos \relax (x)}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sin(x) - 1/sin(x))^3,x)

[Out]

1/(5*cos(x)^5) - 1/(3*cos(x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \sin {\relax (x )} + \csc {\relax (x )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))**3,x)

[Out]

Integral((-sin(x) + csc(x))**(-3), x)

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