3.354 \(\int \sec ^2(x) \tan ^2(x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\tan ^3(x)}{3} \]

[Out]

Tan[x]^3/3

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Rubi [A]  time = 0.021766, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2607, 30} \[ \frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2*Tan[x]^2,x]

[Out]

Tan[x]^3/3

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0027176, size = 8, normalized size = 1. \[ \frac{\tan ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2*Tan[x]^2,x]

[Out]

Tan[x]^3/3

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Maple [A]  time = 0.01, size = 11, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3\, \left ( \cos \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,x)

[Out]

1/3*sin(x)^3/cos(x)^3

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Maxima [A]  time = 0.932575, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)^3

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Fricas [B]  time = 1.74863, size = 50, normalized size = 6.25 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(cos(x)^2 - 1)*sin(x)/cos(x)^3

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Sympy [B]  time = 0.062806, size = 17, normalized size = 2.12 \begin{align*} - \frac{\sin{\left (x \right )}}{3 \cos{\left (x \right )}} + \frac{\sin{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*tan(x)**2,x)

[Out]

-sin(x)/(3*cos(x)) + sin(x)/(3*cos(x)**3)

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Giac [A]  time = 1.07314, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(x)^3