3.493 \(\int \frac{1}{\cot ^2(x)-\csc ^2(x)} \, dx\)

Optimal. Leaf size=3 \[ -x \]

[Out]

-x

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Rubi [A]  time = 0.0131306, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4382, 8} \[ -x \]

Antiderivative was successfully verified.

[In]

Int[(Cot[x]^2 - Csc[x]^2)^(-1),x]

[Out]

-x

Rule 4382

Int[((a_.) + cot[(d_.) + (e_.)*(x_)]^2*(b_.) + csc[(d_.) + (e_.)*(x_)]^2*(c_.))^(p_.)*(u_.), x_Symbol] :> Dist
[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{1}{\cot ^2(x)-\csc ^2(x)} \, dx &=-\int 1 \, dx\\ &=-x\\ \end{align*}

Mathematica [A]  time = 0.0004945, size = 3, normalized size = 1. \[ -x \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[x]^2 - Csc[x]^2)^(-1),x]

[Out]

-x

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Maple [C]  time = 0.023, size = 6, normalized size = 2. \begin{align*} -\arctan \left ( \tan \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cot(x)^2-csc(x)^2),x)

[Out]

-arctan(tan(x))

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Maxima [A]  time = 1.48515, size = 4, normalized size = 1.33 \begin{align*} -x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)^2-csc(x)^2),x, algorithm="maxima")

[Out]

-x

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Fricas [A]  time = 1.64396, size = 5, normalized size = 1.67 \begin{align*} -x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)^2-csc(x)^2),x, algorithm="fricas")

[Out]

-x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)**2-csc(x)**2),x)

[Out]

Integral(1/((cot(x) - csc(x))*(cot(x) + csc(x))), x)

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Giac [A]  time = 1.12555, size = 4, normalized size = 1.33 \begin{align*} -x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cot(x)^2-csc(x)^2),x, algorithm="giac")

[Out]

-x