3.704 \(\int \frac{\sec ^2(x) \tan ^2(x)}{(2+\tan ^3(x))^2} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{3 \left (\tan ^3(x)+2\right )} \]

[Out]

-1/(3*(2 + Tan[x]^3))

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Rubi [A]  time = 0.0760194, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4342, 261} \[ -\frac{1}{3 \left (\tan ^3(x)+2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]^2*Tan[x]^2)/(2 + Tan[x]^3)^2,x]

[Out]

-1/(3*(2 + Tan[x]^3))

Rule 4342

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/
(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0357123, size = 12, normalized size = 1. \[ -\frac{1}{3 \left (\tan ^3(x)+2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]^2*Tan[x]^2)/(2 + Tan[x]^3)^2,x]

[Out]

-1/(3*(2 + Tan[x]^3))

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Maple [A]  time = 0.072, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{6+3\, \left ( \tan \left ( x \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*tan(x)^2/(2+tan(x)^3)^2,x)

[Out]

-1/3/(2+tan(x)^3)

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Maxima [A]  time = 0.97347, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{3 \,{\left (\tan \left (x\right )^{3} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^2/(2+tan(x)^3)^2,x, algorithm="maxima")

[Out]

-1/3/(tan(x)^3 + 2)

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Fricas [B]  time = 2.02154, size = 109, normalized size = 9.08 \begin{align*} -\frac{\cos \left (x\right )^{3} + 2 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{15 \,{\left (2 \, \cos \left (x\right )^{3} -{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^2/(2+tan(x)^3)^2,x, algorithm="fricas")

[Out]

-1/15*(cos(x)^3 + 2*(cos(x)^2 - 1)*sin(x))/(2*cos(x)^3 - (cos(x)^2 - 1)*sin(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*tan(x)**2/(2+tan(x)**3)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.10688, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{3 \,{\left (\tan \left (x\right )^{3} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^2/(2+tan(x)^3)^2,x, algorithm="giac")

[Out]

-1/3/(tan(x)^3 + 2)