3.331 \(\int \frac {1}{(-\cos (x)+\sec (x))^6} \, dx\)

Optimal. Leaf size=25 \[ -\frac {1}{11} \cot ^{11}(x)-\frac {2 \cot ^9(x)}{9}-\frac {\cot ^7(x)}{7} \]

[Out]

-1/7*cot(x)^7-2/9*cot(x)^9-1/11*cot(x)^11

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {270} \[ -\frac {1}{11} \cot ^{11}(x)-\frac {2 \cot ^9(x)}{9}-\frac {\cot ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Int[(-Cos[x] + Sec[x])^(-6),x]

[Out]

-Cot[x]^7/7 - (2*Cot[x]^9)/9 - Cot[x]^11/11

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}{(-\cos (x)+\sec (x))^6} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^{12}} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{x^{12}}+\frac {2}{x^{10}}+\frac {1}{x^8}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac {1}{7} \cot ^7(x)-\frac {2 \cot ^9(x)}{9}-\frac {\cot ^{11}(x)}{11}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.02, size = 57, normalized size = 2.28 \[ \frac {8 \cot (x)}{693}-\frac {1}{11} \cot (x) \csc ^{10}(x)+\frac {23}{99} \cot (x) \csc ^8(x)-\frac {113}{693} \cot (x) \csc ^6(x)+\frac {1}{231} \cot (x) \csc ^4(x)+\frac {4}{693} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cos[x] + Sec[x])^(-6),x]

[Out]

(8*Cot[x])/693 + (4*Cot[x]*Csc[x]^2)/693 + (Cot[x]*Csc[x]^4)/231 - (113*Cot[x]*Csc[x]^6)/693 + (23*Cot[x]*Csc[
x]^8)/99 - (Cot[x]*Csc[x]^10)/11

________________________________________________________________________________________

fricas [B]  time = 1.47, size = 57, normalized size = 2.28 \[ \frac {8 \, \cos \relax (x)^{11} - 44 \, \cos \relax (x)^{9} + 99 \, \cos \relax (x)^{7}}{693 \, {\left (\cos \relax (x)^{10} - 5 \, \cos \relax (x)^{8} + 10 \, \cos \relax (x)^{6} - 10 \, \cos \relax (x)^{4} + 5 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^6,x, algorithm="fricas")

[Out]

1/693*(8*cos(x)^11 - 44*cos(x)^9 + 99*cos(x)^7)/((cos(x)^10 - 5*cos(x)^8 + 10*cos(x)^6 - 10*cos(x)^4 + 5*cos(x
)^2 - 1)*sin(x))

________________________________________________________________________________________

giac [A]  time = 0.12, size = 20, normalized size = 0.80 \[ -\frac {99 \, \tan \relax (x)^{4} + 154 \, \tan \relax (x)^{2} + 63}{693 \, \tan \relax (x)^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^6,x, algorithm="giac")

[Out]

-1/693*(99*tan(x)^4 + 154*tan(x)^2 + 63)/tan(x)^11

________________________________________________________________________________________

maple [A]  time = 0.11, size = 20, normalized size = 0.80 \[ -\frac {2}{9 \tan \relax (x )^{9}}-\frac {1}{7 \tan \relax (x )^{7}}-\frac {1}{11 \tan \relax (x )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-cos(x)+sec(x))^6,x)

[Out]

-2/9/tan(x)^9-1/7/tan(x)^7-1/11/tan(x)^11

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 20, normalized size = 0.80 \[ -\frac {99 \, \tan \relax (x)^{4} + 154 \, \tan \relax (x)^{2} + 63}{693 \, \tan \relax (x)^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^6,x, algorithm="maxima")

[Out]

-1/693*(99*tan(x)^4 + 154*tan(x)^2 + 63)/tan(x)^11

________________________________________________________________________________________

mupad [B]  time = 2.45, size = 46, normalized size = 1.84 \[ -\frac {80\,{\cos \relax (x)}^7-18\,{\cos \relax (x)}^7\,\left (2\,{\cos \relax (x)}^2-1\right )+{\cos \relax (x)}^7\,\left (2\,{\left (2\,{\cos \relax (x)}^2-1\right )}^2-1\right )}{693\,{\sin \relax (x)}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x) - 1/cos(x))^6,x)

[Out]

-(80*cos(x)^7 - 18*cos(x)^7*(2*cos(x)^2 - 1) + cos(x)^7*(2*(2*cos(x)^2 - 1)^2 - 1))/(693*sin(x)^11)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))**6,x)

[Out]

Timed out

________________________________________________________________________________________