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

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

[Out]

exp(sec(x))

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Rubi [A]  time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ e^{\sec (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^Sec[x]

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fricas [A]  time = 0.92, size = 5, normalized size = 1.25 \[ e^{\frac {1}{\cos \relax (x)}} \]

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))

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giac [A]  time = 0.14, size = 5, normalized size = 1.25 \[ e^{\frac {1}{\cos \relax (x)}} \]

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))

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maple [A]  time = 0.02, size = 4, normalized size = 1.00 \[ {\mathrm e}^{\sec \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(sec(x))

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maxima [A]  time = 0.33, size = 3, normalized size = 0.75 \[ e^{\sec \relax (x)} \]

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)

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mupad [B]  time = 3.09, size = 5, normalized size = 1.25 \[ {\mathrm {e}}^{\frac {1}{\cos \relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/cos(x))*tan(x))/cos(x),x)

[Out]

exp(1/cos(x))

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sympy [A]  time = 0.61, size = 3, normalized size = 0.75 \[ e^{\sec {\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(sec(x))

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