\(\int \frac {1}{(\csc (x)-\sin (x))^5} \, dx\) [311]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {\sec ^5(x)}{5}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^9(x)}{9} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4482, 2686, 276} \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {\sec ^9(x)}{9}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^5(x)}{5} \]

[In]

Int[(Csc[x] - Sin[x])^(-5),x]

[Out]

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

Rule 276

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]

Rule 2686

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 4482

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

Rubi steps \begin{align*} \text {integral}& = \int \sec ^5(x) \tan ^5(x) \, dx \\ & = \text {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right ) \\ & = \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {\sec ^5(x)}{5}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^9(x)}{9} \]

[In]

Integrate[(Csc[x] - Sin[x])^(-5),x]

[Out]

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

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {1}{9 \cos \left (x \right )^{9}}-\frac {2}{7 \cos \left (x \right )^{7}}+\frac {1}{5 \cos \left (x \right )^{5}}\)	\(20\)
parallelrisch	\(\frac {32}{315}+\frac {\sec \left (x \right )^{5}}{5}-\frac {2 \sec \left (x \right )^{7}}{7}+\frac {\sec \left (x \right )^{9}}{9}\)	\(21\)
risch	\(\frac {\frac {32 \,{\mathrm e}^{13 i x}}{5}-\frac {384 \,{\mathrm e}^{11 i x}}{35}+\frac {6976 \,{\mathrm e}^{9 i x}}{315}-\frac {384 \,{\mathrm e}^{7 i x}}{35}+\frac {32 \,{\mathrm e}^{5 i x}}{5}}{\left ({\mathrm e}^{2 i x}+1\right )^{9}}\)	\(48\)

[In]

int(1/(csc(x)-sin(x))^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {63 \, \cos \left (x\right )^{4} - 90 \, \cos \left (x\right )^{2} + 35}{315 \, \cos \left (x\right )^{9}} \]

[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

Sympy [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\int \frac {1}{\left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{5}}\, dx \]

[In]

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

[Out]

Integral((-sin(x) + csc(x))**(-5), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (19) = 38\).

Time = 0.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 7.48 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {16 \, {\left (\frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {36 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {126 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {441 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} - \frac {315 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac {210 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} - 1\right )}}{315 \, {\left (\frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {36 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {84 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {126 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {126 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac {84 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac {36 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - \frac {9 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac {\sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - 1\right )}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (19) = 38\).

Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.04 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {16 \, {\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {36 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {126 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {441 \, {\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {315 \, {\left (\cos \left (x\right ) - 1\right )}^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {210 \, {\left (\cos \left (x\right ) - 1\right )}^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}}{315 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{9}} \]

[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

Mupad [B] (verification not implemented)

Time = 29.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {1}{5\,{\cos \left (x\right )}^5}-\frac {2}{7\,{\cos \left (x\right )}^7}+\frac {1}{9\,{\cos \left (x\right )}^9} \]

[In]

int(-1/(sin(x) - 1/sin(x))^5,x)

[Out]

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