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]

-Sec[x]^3/3 + Sec[x]^5/5

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Rubi [A]  time = 0.0380005, antiderivative size = 17, normalized size of antiderivative = 1., 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 4397

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

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 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]]

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*}

Mathematica [A]  time = 0.0201522, size = 17, normalized size = 1. \[ \frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-3),x]

[Out]

-Sec[x]^3/3 + Sec[x]^5/5

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Maple [A]  time = 0.046, size = 14, normalized size = 0.8 \begin{align*} -{\frac{1}{3\, \left ( \cos \left ( x \right ) \right ) ^{3}}}+{\frac{1}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}} \end{align*}

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 = 0.999292, size = 139, normalized size = 8.18 \begin{align*} -\frac{4 \,{\left (\frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{5 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{15 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 1\right )}}{15 \,{\left (\frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{10 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{10 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{5 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{\sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - 1\right )}} \end{align*}

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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Fricas [A]  time = 1.93792, size = 45, normalized size = 2.65 \begin{align*} -\frac{5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \sin{\left (x \right )} + \csc{\left (x \right )}\right )^{3}}\, dx \end{align*}

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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Giac [B]  time = 1.17964, size = 80, normalized size = 4.71 \begin{align*} -\frac{4 \,{\left (\frac{5 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac{5 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{15 \,{\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 1\right )}}{15 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{5}} \end{align*}

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