3.327 \(\int \frac{1}{(-\cos (x)+\sec (x))^2} \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{3} \cot ^3(x) \]

[Out]

-Cot[x]^3/3

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Rubi [A]  time = 0.0140351, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {30} \[ -\frac{1}{3} \cot ^3(x) \]

Antiderivative was successfully verified.

[In]

Int[(-Cos[x] + Sec[x])^(-2),x]

[Out]

-Cot[x]^3/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(-\cos (x)+\sec (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^4} \, dx,x,\tan (x)\right )\\ &=-\frac{1}{3} \cot ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0025405, size = 8, normalized size = 1. \[ -\frac{1}{3} \cot ^3(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cos[x] + Sec[x])^(-2),x]

[Out]

-Cot[x]^3/3

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Maple [A]  time = 0.03, size = 7, normalized size = 0.9 \begin{align*} -{\frac{1}{3\, \left ( \tan \left ( x \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-cos(x)+sec(x))^2,x)

[Out]

-1/3/tan(x)^3

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Maxima [A]  time = 0.980036, size = 8, normalized size = 1. \begin{align*} -\frac{1}{3 \, \tan \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^2,x, algorithm="maxima")

[Out]

-1/3/tan(x)^3

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Fricas [B]  time = 2.03443, size = 51, normalized size = 6.38 \begin{align*} \frac{\cos \left (x\right )^{3}}{3 \,{\left (\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))^2,x, algorithm="fricas")

[Out]

1/3*cos(x)^3/((cos(x)^2 - 1)*sin(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \cos{\left (x \right )} + \sec{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))**2,x)

[Out]

Integral((-cos(x) + sec(x))**(-2), x)

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Giac [A]  time = 1.14738, size = 8, normalized size = 1. \begin{align*} -\frac{1}{3 \, \tan \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-cos(x)+sec(x))^2,x, algorithm="giac")

[Out]

-1/3/tan(x)^3