3.311 \(\int \frac{1}{(\csc (x)-\sin (x))^5} \, dx\)

Optimal. Leaf size=25 \[ \frac{\sec ^9(x)}{9}-\frac{2 \sec ^7(x)}{7}+\frac{\sec ^5(x)}{5} \]

[Out]

Sec[x]^5/5 - (2*Sec[x]^7)/7 + Sec[x]^9/9

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Rubi [A]  time = 0.0398882, antiderivative size = 25, 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, 270} \[ \frac{\sec ^9(x)}{9}-\frac{2 \sec ^7(x)}{7}+\frac{\sec ^5(x)}{5} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-5),x]

[Out]

Sec[x]^5/5 - (2*Sec[x]^7)/7 + Sec[x]^9/9

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{1}{(\csc (x)-\sin (x))^5} \, dx &=\int \sec ^5(x) \tan ^5(x) \, dx\\ &=\operatorname{Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (x)\right )\\ &=\frac{\sec ^5(x)}{5}-\frac{2 \sec ^7(x)}{7}+\frac{\sec ^9(x)}{9}\\ \end{align*}

Mathematica [A]  time = 0.0160423, size = 25, normalized size = 1. \[ \frac{\sec ^9(x)}{9}-\frac{2 \sec ^7(x)}{7}+\frac{\sec ^5(x)}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-5),x]

[Out]

Sec[x]^5/5 - (2*Sec[x]^7)/7 + Sec[x]^9/9

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Maple [A]  time = 0.05, size = 20, normalized size = 0.8 \begin{align*}{\frac{1}{9\, \left ( \cos \left ( x \right ) \right ) ^{9}}}-{\frac{2}{7\, \left ( \cos \left ( x \right ) \right ) ^{7}}}+{\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))^5,x)

[Out]

1/9/cos(x)^9-2/7/cos(x)^7+1/5/cos(x)^5

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Maxima [B]  time = 1.04796, size = 252, normalized size = 10.08 \begin{align*} \frac{16 \,{\left (\frac{9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{36 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{126 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{441 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} - \frac{315 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac{210 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} - 1\right )}}{315 \,{\left (\frac{9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{36 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{84 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{126 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac{126 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac{84 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac{36 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - \frac{9 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac{\sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^5,x, algorithm="maxima")

[Out]

16/315*(9*sin(x)^2/(cos(x) + 1)^2 - 36*sin(x)^4/(cos(x) + 1)^4 - 126*sin(x)^6/(cos(x) + 1)^6 - 441*sin(x)^8/(c
os(x) + 1)^8 - 315*sin(x)^10/(cos(x) + 1)^10 - 210*sin(x)^12/(cos(x) + 1)^12 - 1)/(9*sin(x)^2/(cos(x) + 1)^2 -
36*sin(x)^4/(cos(x) + 1)^4 + 84*sin(x)^6/(cos(x) + 1)^6 - 126*sin(x)^8/(cos(x) + 1)^8 + 126*sin(x)^10/(cos(x)
+ 1)^10 - 84*sin(x)^12/(cos(x) + 1)^12 + 36*sin(x)^14/(cos(x) + 1)^14 - 9*sin(x)^16/(cos(x) + 1)^16 + sin(x)^
18/(cos(x) + 1)^18 - 1)

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Fricas [A]  time = 1.99351, size = 66, normalized size = 2.64 \begin{align*} \frac{63 \, \cos \left (x\right )^{4} - 90 \, \cos \left (x\right )^{2} + 35}{315 \, \cos \left (x\right )^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^5,x, algorithm="fricas")

[Out]

1/315*(63*cos(x)^4 - 90*cos(x)^2 + 35)/cos(x)^9

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))**5,x)

[Out]

Timed out

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Giac [B]  time = 1.13144, size = 136, normalized size = 5.44 \begin{align*} \frac{16 \,{\left (\frac{9 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{36 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{126 \,{\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{441 \,{\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{315 \,{\left (\cos \left (x\right ) - 1\right )}^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{210 \,{\left (\cos \left (x\right ) - 1\right )}^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}}{315 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x))^5,x, algorithm="giac")

[Out]

16/315*(9*(cos(x) - 1)/(cos(x) + 1) + 36*(cos(x) - 1)^2/(cos(x) + 1)^2 - 126*(cos(x) - 1)^3/(cos(x) + 1)^3 + 4
41*(cos(x) - 1)^4/(cos(x) + 1)^4 - 315*(cos(x) - 1)^5/(cos(x) + 1)^5 + 210*(cos(x) - 1)^6/(cos(x) + 1)^6 + 1)/
((cos(x) - 1)/(cos(x) + 1) + 1)^9