3.307 \(\int \frac {1}{\csc (x)-\sin (x)} \, dx\)

Optimal. Leaf size=2 \[ \sec (x) \]

[Out]

sec(x)

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Rubi [A]  time = 0.02, antiderivative size = 2, normalized size of antiderivative = 1.00, 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} \[ \sec (x) \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-1),x]

[Out]

Sec[x]

Rule 8

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

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 4397

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \[ \sec (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-1),x]

[Out]

Sec[x]

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fricas [A]  time = 2.20, size = 4, normalized size = 2.00 \[ \frac {1}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x)),x, algorithm="fricas")

[Out]

1/cos(x)

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giac [B]  time = 0.14, size = 17, normalized size = 8.50 \[ \frac {2}{\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x)),x, algorithm="giac")

[Out]

2/((cos(x) - 1)/(cos(x) + 1) + 1)

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maple [A]  time = 0.10, size = 5, normalized size = 2.50 \[ \frac {1}{\cos \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(csc(x)-sin(x)),x)

[Out]

1/cos(x)

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maxima [B]  time = 0.31, size = 17, normalized size = 8.50 \[ -\frac {2}{\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x)),x, algorithm="maxima")

[Out]

-2/(sin(x)^2/(cos(x) + 1)^2 - 1)

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mupad [B]  time = 2.46, size = 12, normalized size = 6.00 \[ -\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sin(x) - 1/sin(x)),x)

[Out]

-2/(tan(x/2)^2 - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{- \sin {\relax (x )} + \csc {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(csc(x)-sin(x)),x)

[Out]

Integral(1/(-sin(x) + csc(x)), x)

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