3.730 \(\int e^{\sec (x)} \sec (x) \tan (x) \, dx\)

Optimal. Leaf size=4 \[ e^{\sec (x)} \]

[Out]

E^Sec[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0217493, antiderivative size = 4, 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 = {4339, 2209} \[ e^{\sec (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^Sec[x]

Rule 4339

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*
c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{\sec (x)} \sec (x) \tan (x) \, dx &=-\operatorname{Subst}\left (\int \frac{e^{\frac{1}{x}}}{x^2} \, dx,x,\cos (x)\right )\\ &=e^{\sec (x)}\\ \end{align*}

Mathematica [A]  time = 0.0073632, size = 4, normalized size = 1. \[ e^{\sec (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^Sec[x]

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 4, normalized size = 1. \begin{align*}{{\rm e}^{\sec \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(sec(x))

________________________________________________________________________________________

Maxima [A]  time = 0.955632, size = 4, normalized size = 1. \begin{align*} e^{\sec \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^sec(x)

________________________________________________________________________________________

Fricas [A]  time = 2.29366, size = 19, normalized size = 4.75 \begin{align*} e^{\frac{1}{\cos \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(1/cos(x))

________________________________________________________________________________________

Sympy [A]  time = 0.828806, size = 3, normalized size = 0.75 \begin{align*} e^{\sec{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(sec(x))

________________________________________________________________________________________

Giac [A]  time = 1.11965, size = 7, normalized size = 1.75 \begin{align*} e^{\frac{1}{\cos \left (x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(1/cos(x))