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]

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

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Rubi [A]  time = 0.0205889, antiderivative size = 25, normalized size of antiderivative = 1., 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 = 0.0200594, 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

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Maple [A]  time = 0.039, size = 20, normalized size = 0.8 \begin{align*} -{\frac{1}{7\, \left ( \tan \left ( x \right ) \right ) ^{7}}}-{\frac{2}{9\, \left ( \tan \left ( x \right ) \right ) ^{9}}}-{\frac{1}{11\, \left ( \tan \left ( x \right ) \right ) ^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.990126, size = 27, normalized size = 1.08 \begin{align*} -\frac{99 \, \tan \left (x\right )^{4} + 154 \, \tan \left (x\right )^{2} + 63}{693 \, \tan \left (x\right )^{11}} \end{align*}

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

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Fricas [B]  time = 2.45493, size = 173, normalized size = 6.92 \begin{align*} \frac{8 \, \cos \left (x\right )^{11} - 44 \, \cos \left (x\right )^{9} + 99 \, \cos \left (x\right )^{7}}{693 \,{\left (\cos \left (x\right )^{10} - 5 \, \cos \left (x\right )^{8} + 10 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + 5 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \end{align*}

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))

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

[Out]

Timed out

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Giac [A]  time = 1.14656, size = 27, normalized size = 1.08 \begin{align*} -\frac{99 \, \tan \left (x\right )^{4} + 154 \, \tan \left (x\right )^{2} + 63}{693 \, \tan \left (x\right )^{11}} \end{align*}

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