\(\int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx\) [493]

Optimal result

Integrand size = 13, antiderivative size = 3 \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[Out]

-x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4467, 8} \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[In]

Int[(Cot[x]^2 - Csc[x]^2)^(-1),x]

[Out]

-x

Rule 8

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

Rule 4467

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]

Rubi steps \begin{align*} \text {integral}& = -\int 1 \, dx \\ & = -x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[In]

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

[Out]

-x

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33

method	result	size
risch	\(-x\)	\(4\)
default	\(-\arctan \left (\tan \left (x \right )\right )\)	\(6\)

[In]

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

[Out]

-x

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[In]

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

[Out]

-x

Sympy [F]

\[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=\int \frac {1}{\left (\cot {\left (x \right )} - \csc {\left (x \right )}\right ) \left (\cot {\left (x \right )} + \csc {\left (x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[In]

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

[Out]

-x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[In]

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

[Out]

-x

Mupad [B] (verification not implemented)

Time = 26.31 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cot ^2(x)-\csc ^2(x)} \, dx=-x \]

[In]

int(1/(cot(x)^2 - 1/sin(x)^2),x)

[Out]

-x