3.326 \(\int \frac{1}{-\cos (x)+\sec (x)} \, dx\)

Optimal. Leaf size=4 \[ -\csc (x) \]

[Out]

-Csc[x]

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Rubi [A]  time = 0.0175251, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4397, 2606, 8} \[ -\csc (x) \]

Antiderivative was successfully verified.

[In]

Int[(-Cos[x] + Sec[x])^(-1),x]

[Out]

-Csc[x]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{-\cos (x)+\sec (x)} \, dx &=\int \cot (x) \csc (x) \, dx\\ &=-\operatorname{Subst}(\int 1 \, dx,x,\csc (x))\\ &=-\csc (x)\\ \end{align*}

Mathematica [A]  time = 0.0026784, size = 4, normalized size = 1. \[ -\csc (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cos[x] + Sec[x])^(-1),x]

[Out]

-Csc[x]

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Maple [A]  time = 0.03, size = 7, normalized size = 1.8 \begin{align*} - \left ( \sin \left ( x \right ) \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sin(x)

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Maxima [B]  time = 0.996598, size = 28, normalized size = 7. \begin{align*} -\frac{\cos \left (x\right ) + 1}{2 \, \sin \left (x\right )} - \frac{\sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(cos(x) + 1)/sin(x) - 1/2*sin(x)/(cos(x) + 1)

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Fricas [A]  time = 1.95337, size = 15, normalized size = 3.75 \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sin(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(cos(x) - sec(x)), x)

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Giac [A]  time = 1.10473, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sin(x)