3.489 \(\int \frac {1}{(\sec ^2(x)-\tan ^2(x))^3} \, dx\)

Optimal. Leaf size=1 \[ x \]

[Out]

x

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Rubi [A]  time = 0.01, antiderivative size = 1, normalized size of antiderivative = 1.00, 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 = {4381, 8} \[ x \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]^2 - Tan[x]^2)^(-3),x]

[Out]

x

Rule 8

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

Rule 4381

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*} \int \frac {1}{\left (\sec ^2(x)-\tan ^2(x)\right )^3} \, dx &=\int 1 \, dx\\ &=x\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 1, normalized size = 1.00 \[ x \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]^2 - Tan[x]^2)^(-3),x]

[Out]

x

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fricas [A]  time = 1.53, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sec(x)^2-tan(x)^2)^3,x, algorithm="fricas")

[Out]

x

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giac [A]  time = 0.14, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sec(x)^2-tan(x)^2)^3,x, algorithm="giac")

[Out]

x

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maple [C]  time = 0.07, size = 4, normalized size = 4.00 \[ \arctan \left (\tan \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sec(x)^2-tan(x)^2)^3,x)

[Out]

arctan(tan(x))

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maxima [A]  time = 0.41, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sec(x)^2-tan(x)^2)^3,x, algorithm="maxima")

[Out]

x

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mupad [B]  time = 2.57, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/cos(x)^2 - tan(x)^2)^3,x)

[Out]

x

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \tan {\relax (x )} + \sec {\relax (x )}\right )^{3} \left (\tan {\relax (x )} + \sec {\relax (x )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sec(x)**2-tan(x)**2)**3,x)

[Out]

Integral(1/((-tan(x) + sec(x))**3*(tan(x) + sec(x))**3), x)

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