\(\int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx\) [332]

Optimal result

Integrand size = 9, antiderivative size = 33 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {\csc ^7(x)}{7}-\frac {\csc ^9(x)}{3}+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^{13}(x)}{13} \]

[Out]

1/7*csc(x)^7-1/3*csc(x)^9+3/11*csc(x)^11-1/13*csc(x)^13

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, 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} \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=-\frac {1}{13} \csc ^{13}(x)+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^9(x)}{3}+\frac {\csc ^7(x)}{7} \]

[In]

Int[(-Cos[x] + Sec[x])^(-7),x]

[Out]

Csc[x]^7/7 - Csc[x]^9/3 + (3*Csc[x]^11)/11 - Csc[x]^13/13

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 ^7(x) \csc ^7(x) \, dx \\ & = -\text {Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (x)\right ) \\ & = -\text {Subst}\left (\int \left (-x^6+3 x^8-3 x^{10}+x^{12}\right ) \, dx,x,\csc (x)\right ) \\ & = \frac {\csc ^7(x)}{7}-\frac {\csc ^9(x)}{3}+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^{13}(x)}{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {\csc ^7(x)}{7}-\frac {\csc ^9(x)}{3}+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^{13}(x)}{13} \]

[In]

Integrate[(-Cos[x] + Sec[x])^(-7),x]

[Out]

Csc[x]^7/7 - Csc[x]^9/3 + (3*Csc[x]^11)/11 - Csc[x]^13/13

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {1}{13 \sin \left (x \right )^{13}}+\frac {1}{7 \sin \left (x \right )^{7}}+\frac {3}{11 \sin \left (x \right )^{11}}-\frac {1}{3 \sin \left (x \right )^{9}}\)	\(26\)
parallelrisch	\(-\frac {\csc \left (x \right )^{13} \left (2010+429 \cos \left (6 x \right )+1430 \cos \left (4 x \right )+3523 \cos \left (2 x \right )\right )}{96096}\)	\(27\)
risch	\(-\frac {128 i \left (429 \,{\mathrm e}^{19 i x}+1430 \,{\mathrm e}^{17 i x}+3523 \,{\mathrm e}^{15 i x}+4020 \,{\mathrm e}^{13 i x}+3523 \,{\mathrm e}^{11 i x}+1430 \,{\mathrm e}^{9 i x}+429 \,{\mathrm e}^{7 i x}\right )}{3003 \left ({\mathrm e}^{2 i x}-1\right )^{13}}\)	\(63\)

[In]

int(1/(-cos(x)+sec(x))^7,x,method=_RETURNVERBOSE)

[Out]

-1/13/sin(x)^13+1/7/sin(x)^7+3/11/sin(x)^11-1/3/sin(x)^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=-\frac {429 \, \cos \left (x\right )^{6} - 286 \, \cos \left (x\right )^{4} + 104 \, \cos \left (x\right )^{2} - 16}{3003 \, {\left (\cos \left (x\right )^{12} - 6 \, \cos \left (x\right )^{10} + 15 \, \cos \left (x\right )^{8} - 20 \, \cos \left (x\right )^{6} + 15 \, \cos \left (x\right )^{4} - 6 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]

[In]

integrate(1/(-cos(x)+sec(x))^7,x, algorithm="fricas")

[Out]

-1/3003*(429*cos(x)^6 - 286*cos(x)^4 + 104*cos(x)^2 - 16)/((cos(x)^12 - 6*cos(x)^10 + 15*cos(x)^8 - 20*cos(x)^
6 + 15*cos(x)^4 - 6*cos(x)^2 + 1)*sin(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\text {Timed out} \]

[In]

integrate(1/(-cos(x)+sec(x))**7,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (25) = 50\).

Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.12 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {{\left (\frac {273 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2002 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {2574 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {9009 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {15015 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac {60060 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} - 231\right )} {\left (\cos \left (x\right ) + 1\right )}^{13}}{24600576 \, \sin \left (x\right )^{13}} + \frac {5 \, \sin \left (x\right )}{2048 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {5 \, \sin \left (x\right )^{3}}{8192 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {3 \, \sin \left (x\right )^{5}}{8192 \, {\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (x\right )^{7}}{28672 \, {\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {\sin \left (x\right )^{9}}{12288 \, {\left (\cos \left (x\right ) + 1\right )}^{9}} + \frac {\sin \left (x\right )^{11}}{90112 \, {\left (\cos \left (x\right ) + 1\right )}^{11}} - \frac {\sin \left (x\right )^{13}}{106496 \, {\left (\cos \left (x\right ) + 1\right )}^{13}} \]

[In]

integrate(1/(-cos(x)+sec(x))^7,x, algorithm="maxima")

[Out]

1/24600576*(273*sin(x)^2/(cos(x) + 1)^2 + 2002*sin(x)^4/(cos(x) + 1)^4 - 2574*sin(x)^6/(cos(x) + 1)^6 - 9009*s
in(x)^8/(cos(x) + 1)^8 + 15015*sin(x)^10/(cos(x) + 1)^10 + 60060*sin(x)^12/(cos(x) + 1)^12 - 231)*(cos(x) + 1)
^13/sin(x)^13 + 5/2048*sin(x)/(cos(x) + 1) + 5/8192*sin(x)^3/(cos(x) + 1)^3 - 3/8192*sin(x)^5/(cos(x) + 1)^5 -
3/28672*sin(x)^7/(cos(x) + 1)^7 + 1/12288*sin(x)^9/(cos(x) + 1)^9 + 1/90112*sin(x)^11/(cos(x) + 1)^11 - 1/106
496*sin(x)^13/(cos(x) + 1)^13

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {429 \, \sin \left (x\right )^{6} - 1001 \, \sin \left (x\right )^{4} + 819 \, \sin \left (x\right )^{2} - 231}{3003 \, \sin \left (x\right )^{13}} \]

[In]

integrate(1/(-cos(x)+sec(x))^7,x, algorithm="giac")

[Out]

1/3003*(429*sin(x)^6 - 1001*sin(x)^4 + 819*sin(x)^2 - 231)/sin(x)^13

Mupad [B] (verification not implemented)

Time = 27.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.30 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=-\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^{13}}{106496}+\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^{11}}{90112}+\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^9}{12288}-\frac {3\,{\mathrm {cot}\left (\frac {x}{2}\right )}^7}{28672}-\frac {3\,{\mathrm {cot}\left (\frac {x}{2}\right )}^5}{8192}+\frac {5\,{\mathrm {cot}\left (\frac {x}{2}\right )}^3}{8192}+\frac {5\,\mathrm {cot}\left (\frac {x}{2}\right )}{2048}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^{13}}{106496}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^{11}}{90112}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^9}{12288}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{28672}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8192}+\frac {5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{8192}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2048} \]

[In]

int(-1/(cos(x) - 1/cos(x))^7,x)

[Out]

(5*cot(x/2))/2048 + (5*tan(x/2))/2048 + (5*cot(x/2)^3)/8192 - (3*cot(x/2)^5)/8192 - (3*cot(x/2)^7)/28672 + cot
(x/2)^9/12288 + cot(x/2)^11/90112 - cot(x/2)^13/106496 + (5*tan(x/2)^3)/8192 - (3*tan(x/2)^5)/8192 - (3*tan(x/
2)^7)/28672 + tan(x/2)^9/12288 + tan(x/2)^11/90112 - tan(x/2)^13/106496