\(\int \frac {1}{(\sec ^2(x)-\tan ^2(x))^2} \, dx\) [488]

Optimal result

Integrand size = 13, antiderivative size = 1 \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[Out]

x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 1, 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 = {4466, 8} \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[In]

Int[(Sec[x]^2 - Tan[x]^2)^(-2),x]

[Out]

x

Rule 8

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

Rule 4466

Int[(u_.)*((a_.) + (c_.)*sec[(d_.) + (e_.)*(x_)]^2 + (b_.)*tan[(d_.) + (e_.)*(x_)]^2)^(p_.), 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 = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[In]

Integrate[(Sec[x]^2 - Tan[x]^2)^(-2),x]

[Out]

x

Maple [A] (verified)

Time = 5.35 (sec) , antiderivative size = 2, normalized size of antiderivative = 2.00

method	result	size
risch	\(x\)	\(2\)
default	\(\arctan \left (\tan \left (x \right )\right )\)	\(4\)

[In]

int(1/(sec(x)^2-tan(x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

x

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[In]

integrate(1/(sec(x)^2-tan(x)^2)^2,x, algorithm="fricas")

[Out]

x

Sympy [F]

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

[In]

integrate(1/(sec(x)**2-tan(x)**2)**2,x)

[Out]

Integral(1/((-tan(x) + sec(x))**2*(tan(x) + sec(x))**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[In]

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

[Out]

x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[In]

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

[Out]

x

Mupad [B] (verification not implemented)

Time = 27.11 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^2} \, dx=x \]

[In]

int(1/(1/cos(x)^2 - tan(x)^2)^2,x)

[Out]

x