3.474 \(\int \frac {1}{(\cos ^2(x)+\sin ^2(x))^3} \, dx\)

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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4380, 8} \[ x \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^2 + Sin[x]^2)^(-3),x]

[Out]

x

Rule 8

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

Rule 4380

Int[(u_.)*((a_.) + cos[(d_.) + (e_.)*(x_)]^2*(b_.) + (c_.)*sin[(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 (\cos ^2(x)+\sin ^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[(Cos[x]^2 + Sin[x]^2)^(-3),x]

[Out]

x

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fricas [A]  time = 0.99, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2+sin(x)^2)^3,x, algorithm="fricas")

[Out]

x

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giac [A]  time = 0.13, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2+sin(x)^2)^3,x, algorithm="giac")

[Out]

x

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maple [A]  time = 0.05, size = 2, normalized size = 2.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2+sin(x)^2)^3,x)

[Out]

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/(cos(x)^2+sin(x)^2)^3,x, algorithm="maxima")

[Out]

x

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mupad [B]  time = 2.59, size = 1, normalized size = 1.00 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2 + sin(x)^2)^3,x)

[Out]

x

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sympy [B]  time = 2.10, size = 34, normalized size = 34.00 \[ \frac {x}{\sin ^{6}{\relax (x )} + 3 \sin ^{4}{\relax (x )} \cos ^{2}{\relax (x )} + 3 \sin ^{2}{\relax (x )} \cos ^{4}{\relax (x )} + \cos ^{6}{\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)**2+sin(x)**2)**3,x)

[Out]

x/(sin(x)**6 + 3*sin(x)**4*cos(x)**2 + 3*sin(x)**2*cos(x)**4 + cos(x)**6)

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