3.476 \(\int \frac {1}{(\cos ^2(x)-\sin ^2(x))^2} \, dx\)

Optimal. Leaf size=13 \[ \frac {\tan (x)}{1-\tan ^2(x)} \]

[Out]

tan(x)/(1-tan(x)^2)

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {383} \[ \frac {\tan (x)}{1-\tan ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^2 - Sin[x]^2)^(-2),x]

[Out]

Tan[x]/(1 - Tan[x]^2)

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (\cos ^2(x)-\sin ^2(x)\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{\left (1-x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{1-\tan ^2(x)}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.62 \[ \frac {1}{2} \tan (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^2 - Sin[x]^2)^(-2),x]

[Out]

Tan[2*x]/2

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fricas [A]  time = 1.31, size = 15, normalized size = 1.15 \[ \frac {\cos \relax (x) \sin \relax (x)}{2 \, \cos \relax (x)^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2-sin(x)^2)^2,x, algorithm="fricas")

[Out]

cos(x)*sin(x)/(2*cos(x)^2 - 1)

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giac [A]  time = 0.12, size = 6, normalized size = 0.46 \[ \frac {1}{2} \, \tan \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2-sin(x)^2)^2,x, algorithm="giac")

[Out]

1/2*tan(2*x)

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maple [A]  time = 0.10, size = 18, normalized size = 1.38 \[ -\frac {1}{2 \left (1+\tan \relax (x )\right )}-\frac {1}{2 \left (\tan \relax (x )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2-sin(x)^2)^2,x)

[Out]

-1/2/(1+tan(x))-1/2/(tan(x)-1)

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maxima [A]  time = 0.30, size = 12, normalized size = 0.92 \[ -\frac {\tan \relax (x)}{\tan \relax (x)^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2-sin(x)^2)^2,x, algorithm="maxima")

[Out]

-tan(x)/(tan(x)^2 - 1)

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mupad [B]  time = 2.63, size = 6, normalized size = 0.46 \[ \frac {\mathrm {tan}\left (2\,x\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2 - sin(x)^2)^2,x)

[Out]

tan(2*x)/2

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sympy [B]  time = 1.34, size = 48, normalized size = 3.69 \[ - \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{\tan ^{4}{\left (\frac {x}{2} \right )} - 6 \tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {2 \tan {\left (\frac {x}{2} \right )}}{\tan ^{4}{\left (\frac {x}{2} \right )} - 6 \tan ^{2}{\left (\frac {x}{2} \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)**2-sin(x)**2)**2,x)

[Out]

-2*tan(x/2)**3/(tan(x/2)**4 - 6*tan(x/2)**2 + 1) + 2*tan(x/2)/(tan(x/2)**4 - 6*tan(x/2)**2 + 1)

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