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

Optimal. Leaf size=33 \[ -\frac{1}{13} \csc ^{13}(x)+\frac{3 \csc ^{11}(x)}{11}-\frac{\csc ^9(x)}{3}+\frac{\csc ^7(x)}{7} \]

[Out]

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

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Rubi [A]  time = 0.0422955, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4397, 2606, 270} \[ -\frac{1}{13} \csc ^{13}(x)+\frac{3 \csc ^{11}(x)}{11}-\frac{\csc ^9(x)}{3}+\frac{\csc ^7(x)}{7} \]

Antiderivative was successfully verified.

[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 4397

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

Rule 2606

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 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))^7} \, dx &=\int \cot ^7(x) \csc ^7(x) \, dx\\ &=-\operatorname{Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (x)\right )\\ &=-\operatorname{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]  time = 0.0133771, size = 33, normalized size = 1. \[ -\frac{1}{13} \csc ^{13}(x)+\frac{3 \csc ^{11}(x)}{11}-\frac{\csc ^9(x)}{3}+\frac{\csc ^7(x)}{7} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.044, size = 26, normalized size = 0.8 \begin{align*} -{\frac{1}{13\, \left ( \sin \left ( x \right ) \right ) ^{13}}}+{\frac{1}{7\, \left ( \sin \left ( x \right ) \right ) ^{7}}}-{\frac{1}{3\, \left ( \sin \left ( x \right ) \right ) ^{9}}}+{\frac{3}{11\, \left ( \sin \left ( x \right ) \right ) ^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04915, size = 228, normalized size = 6.91 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.30183, size = 207, normalized size = 6.27 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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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))**7,x)

[Out]

Timed out

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Giac [A]  time = 1.16044, size = 35, normalized size = 1.06 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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