### 3.86 $$\int \sec ^2(x) \tan ^4(x) \, dx$$

Optimal. Leaf size=8 $\frac{\tan ^5(x)}{5}$

[Out]

Tan[x]^5/5

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Rubi [A]  time = 0.0197036, 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} $\frac{\tan ^5(x)}{5}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sec[x]^2*Tan[x]^4,x]

[Out]

Tan[x]^5/5

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 ^4(x) \, dx &=\operatorname{Subst}\left (\int x^4 \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0021575, size = 8, normalized size = 1. $\frac{\tan ^5(x)}{5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sec[x]^2*Tan[x]^4,x]

[Out]

Tan[x]^5/5

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Maple [A]  time = 0.011, size = 11, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*tan(x)^4,x)

[Out]

1/5*sin(x)^5/cos(x)^5

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Maxima [A]  time = 0.92368, size = 8, normalized size = 1. \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^4,x, algorithm="maxima")

[Out]

1/5*tan(x)^5

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Fricas [B]  time = 1.97748, size = 66, normalized size = 8.25 \begin{align*} \frac{{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{5 \, \cos \left (x\right )^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^4,x, algorithm="fricas")

[Out]

1/5*(cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)/cos(x)^5

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Sympy [B]  time = 0.063451, size = 29, normalized size = 3.62 \begin{align*} \frac{\sin{\left (x \right )}}{5 \cos{\left (x \right )}} - \frac{2 \sin{\left (x \right )}}{5 \cos ^{3}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*tan(x)**4,x)

[Out]

sin(x)/(5*cos(x)) - 2*sin(x)/(5*cos(x)**3) + sin(x)/(5*cos(x)**5)

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Giac [A]  time = 1.05643, size = 8, normalized size = 1. \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*tan(x)^4,x, algorithm="giac")

[Out]

1/5*tan(x)^5