### 3.84 $$\int \sec ^4(x) \, dx$$

Optimal. Leaf size=11 $\frac{\tan ^3(x)}{3}+\tan (x)$

[Out]

Tan[x] + Tan[x]^3/3

________________________________________________________________________________________

Rubi [A]  time = 0.006882, antiderivative size = 11, 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} $\frac{\tan ^3(x)}{3}+\tan (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sec[x]^4,x]

[Out]

Tan[x] + Tan[x]^3/3

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

Mathematica [A]  time = 0.0024551, size = 17, normalized size = 1.55 $\frac{2 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sec[x]^4,x]

[Out]

(2*Tan[x])/3 + (Sec[x]^2*Tan[x])/3

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 13, normalized size = 1.2 \begin{align*} - \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{2}}{3}} \right ) \tan \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^4,x)

[Out]

-(-2/3-1/3*sec(x)^2)*tan(x)

________________________________________________________________________________________

Maxima [A]  time = 0.923523, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{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)^4,x, algorithm="maxima")

[Out]

1/3*tan(x)^3 + tan(x)

________________________________________________________________________________________

Fricas [A]  time = 1.96521, size = 51, normalized size = 4.64 \begin{align*} \frac{{\left (2 \, \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)^4,x, algorithm="fricas")

[Out]

1/3*(2*cos(x)^2 + 1)*sin(x)/cos(x)^3

________________________________________________________________________________________

Sympy [B]  time = 0.061384, size = 19, normalized size = 1.73 \begin{align*} \frac{2 \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)**4,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.07935, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{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)^4,x, algorithm="giac")

[Out]

1/3*tan(x)^3 + tan(x)