3.4.54 \(\int \sec ^2(x) \tan ^2(x) \, dx\) [354]

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

[Out]

1/3*tan(x)^3

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {2687, 30} \begin {gather*} \frac {\tan ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x]^3/3

Rule 30

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

Rule 2687

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\tan ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x]^3/3

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 11, normalized size = 1.38

method	result	size
default	\(\frac {\sin ^{3}\left (x \right )}{3 \cos \left (x \right )^{3}}\)	\(11\)
risch	\(-\frac {2 i \left (3 \,{\mathrm e}^{4 i x}+1\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*tan(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 1.71, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 1.01, size = 14, normalized size = 1.75 \begin {gather*} -\frac {{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \end {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
time = 0.01, size = 17, normalized size = 2.12 \begin {gather*} - \frac {\sin {\left (x \right )}}{3 \cos {\left (x \right )}} + \frac {\sin {\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.73, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 6, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^2/cos(x)^2,x)

[Out]

tan(x)^3/3

________________________________________________________________________________________