### 3.85 $$\int \sec ^6(x) \, dx$$

Optimal. Leaf size=19 $\frac{\tan ^5(x)}{5}+\frac{2 \tan ^3(x)}{3}+\tan (x)$

[Out]

Tan[x] + (2*Tan[x]^3)/3 + Tan[x]^5/5

________________________________________________________________________________________

Rubi [A]  time = 0.0088159, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3767} $\frac{\tan ^5(x)}{5}+\frac{2 \tan ^3(x)}{3}+\tan (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sec[x]^6,x]

[Out]

Tan[x] + (2*Tan[x]^3)/3 + Tan[x]^5/5

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \sec ^6(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )\\ &=\tan (x)+\frac{2 \tan ^3(x)}{3}+\frac{\tan ^5(x)}{5}\\ \end{align*}

Mathematica [A]  time = 0.0027936, size = 27, normalized size = 1.42 $\frac{8 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)+\frac{4}{15} \tan (x) \sec ^2(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sec[x]^6,x]

[Out]

(8*Tan[x])/15 + (4*Sec[x]^2*Tan[x])/15 + (Sec[x]^4*Tan[x])/5

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 19, normalized size = 1. \begin{align*} - \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( x \right ) \right ) ^{2}}{15}} \right ) \tan \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^6,x)

[Out]

-(-8/15-1/5*sec(x)^4-4/15*sec(x)^2)*tan(x)

________________________________________________________________________________________

Maxima [A]  time = 0.926333, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{2}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x, algorithm="maxima")

[Out]

1/5*tan(x)^5 + 2/3*tan(x)^3 + tan(x)

________________________________________________________________________________________

Fricas [A]  time = 1.94805, size = 70, normalized size = 3.68 \begin{align*} \frac{{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, \cos \left (x\right )^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x, algorithm="fricas")

[Out]

1/15*(8*cos(x)^4 + 4*cos(x)^2 + 3)*sin(x)/cos(x)^5

________________________________________________________________________________________

Sympy [A]  time = 0.062246, size = 31, normalized size = 1.63 \begin{align*} \frac{8 \sin{\left (x \right )}}{15 \cos{\left (x \right )}} + \frac{4 \sin{\left (x \right )}}{15 \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)**6,x)

[Out]

8*sin(x)/(15*cos(x)) + 4*sin(x)/(15*cos(x)**3) + sin(x)/(5*cos(x)**5)

________________________________________________________________________________________

Giac [A]  time = 1.05634, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{2}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^6,x, algorithm="giac")

[Out]

1/5*tan(x)^5 + 2/3*tan(x)^3 + tan(x)