### 3.9 $$\int \sec (x) \tan (x) \, dx$$

Optimal. Leaf size=2 $\sec (x)$

[Out]

Sec[x]

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Rubi [A]  time = 0.0065987, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 5, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {2606, 8} $\sec (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sec[x]

Rule 2606

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sec (x) \tan (x) \, dx &=\operatorname{Subst}(\int 1 \, dx,x,\sec (x))\\ &=\sec (x)\\ \end{align*}

Mathematica [A]  time = 0.0012405, size = 2, normalized size = 1. $\sec (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sec[x]

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Maple [A]  time = 0.006, size = 3, normalized size = 1.5 \begin{align*} \sec \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x),x)

[Out]

sec(x)

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Maxima [A]  time = 0.927725, size = 5, normalized size = 2.5 \begin{align*} \frac{1}{\cos \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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Fricas [A]  time = 1.85484, size = 14, normalized size = 7. \begin{align*} \frac{1}{\cos \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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Sympy [A]  time = 0.061392, size = 3, normalized size = 1.5 \begin{align*} \frac{1}{\cos{\left (x \right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)

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Giac [A]  time = 1.06429, size = 5, normalized size = 2.5 \begin{align*} \frac{1}{\cos \left (x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(x)