\(\int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx\) [330]

Optimal result

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

[Out]

-1/5*csc(x)^5+2/7*csc(x)^7-1/9*csc(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} \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=-\frac {1}{9} \csc ^9(x)+\frac {2 \csc ^7(x)}{7}-\frac {\csc ^5(x)}{5} \]

[In]

Int[(-Cos[x] + Sec[x])^(-5),x]

[Out]

-1/5*Csc[x]^5 + (2*Csc[x]^7)/7 - Csc[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 \cot ^5(x) \csc ^5(x) \, dx \\ & = -\text {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\csc (x)\right ) \\ & = -\text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (x)\right ) \\ & = -\frac {1}{5} \csc ^5(x)+\frac {2 \csc ^7(x)}{7}-\frac {\csc ^9(x)}{9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=-\frac {1}{5} \csc ^5(x)+\frac {2 \csc ^7(x)}{7}-\frac {\csc ^9(x)}{9} \]

[In]

Integrate[(-Cos[x] + Sec[x])^(-5),x]

[Out]

-1/5*Csc[x]^5 + (2*Csc[x]^7)/7 - Csc[x]^9/9

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {2}{7 \sin \left (x \right )^{7}}-\frac {1}{9 \sin \left (x \right )^{9}}-\frac {1}{5 \sin \left (x \right )^{5}}\)	\(20\)
parallelrisch	\(-\frac {\csc \left (x \right )^{9} \left (109+63 \cos \left (4 x \right )+108 \cos \left (2 x \right )\right )}{2520}\)	\(21\)
risch	\(-\frac {32 i \left (63 \,{\mathrm e}^{13 i x}+108 \,{\mathrm e}^{11 i x}+218 \,{\mathrm e}^{9 i x}+108 \,{\mathrm e}^{7 i x}+63 \,{\mathrm e}^{5 i x}\right )}{315 \left ({\mathrm e}^{2 i x}-1\right )^{9}}\)	\(49\)

[In]

int(1/(-cos(x)+sec(x))^5,x,method=_RETURNVERBOSE)

[Out]

2/7/sin(x)^7-1/9/sin(x)^9-1/5/sin(x)^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=-\frac {63 \, \cos \left (x\right )^{4} - 36 \, \cos \left (x\right )^{2} + 8}{315 \, {\left (\cos \left (x\right )^{8} - 4 \, \cos \left (x\right )^{6} + 6 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]

[In]

integrate(1/(-cos(x)+sec(x))^5,x, algorithm="fricas")

[Out]

-1/315*(63*cos(x)^4 - 36*cos(x)^2 + 8)/((cos(x)^8 - 4*cos(x)^6 + 6*cos(x)^4 - 4*cos(x)^2 + 1)*sin(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=\text {Timed out} \]

[In]

integrate(1/(-cos(x)+sec(x))**5,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (19) = 38\).

Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.84 \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=\frac {{\left (\frac {45 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {252 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {420 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {1890 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (x\right ) + 1\right )}^{9}}{161280 \, \sin \left (x\right )^{9}} - \frac {3 \, \sin \left (x\right )}{256 \, {\left (\cos \left (x\right ) + 1\right )}} - \frac {\sin \left (x\right )^{3}}{384 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {\sin \left (x\right )^{5}}{640 \, {\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {\sin \left (x\right )^{7}}{3584 \, {\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac {\sin \left (x\right )^{9}}{4608 \, {\left (\cos \left (x\right ) + 1\right )}^{9}} \]

[In]

integrate(1/(-cos(x)+sec(x))^5,x, algorithm="maxima")

[Out]

1/161280*(45*sin(x)^2/(cos(x) + 1)^2 + 252*sin(x)^4/(cos(x) + 1)^4 - 420*sin(x)^6/(cos(x) + 1)^6 - 1890*sin(x)
^8/(cos(x) + 1)^8 - 35)*(cos(x) + 1)^9/sin(x)^9 - 3/256*sin(x)/(cos(x) + 1) - 1/384*sin(x)^3/(cos(x) + 1)^3 +
1/640*sin(x)^5/(cos(x) + 1)^5 + 1/3584*sin(x)^7/(cos(x) + 1)^7 - 1/4608*sin(x)^9/(cos(x) + 1)^9

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=-\frac {63 \, \sin \left (x\right )^{4} - 90 \, \sin \left (x\right )^{2} + 35}{315 \, \sin \left (x\right )^{9}} \]

[In]

integrate(1/(-cos(x)+sec(x))^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 28.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(-\cos (x)+\sec (x))^5} \, dx=-\frac {63\,{\sin \left (x\right )}^4-90\,{\sin \left (x\right )}^2+35}{315\,{\sin \left (x\right )}^9} \]

[In]

int(-1/(cos(x) - 1/cos(x))^5,x)

[Out]

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