3.731 \(\int 2^{\sec (x)} \sec (x) \tan (x) \, dx\)

Optimal. Leaf size=9 \[ \frac{2^{\sec (x)}}{\log (2)} \]

[Out]

2^Sec[x]/Log[2]

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Rubi [A]  time = 0.0216664, antiderivative size = 9, 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} \[ \frac{2^{\sec (x)}}{\log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^Sec[x]*Sec[x]*Tan[x],x]

[Out]

2^Sec[x]/Log[2]

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

Mathematica [A]  time = 0.0081172, size = 9, normalized size = 1. \[ \frac{2^{\sec (x)}}{\log (2)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^Sec[x]*Sec[x]*Tan[x],x]

[Out]

2^Sec[x]/Log[2]

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Maple [A]  time = 0.006, size = 10, normalized size = 1.1 \begin{align*}{\frac{{2}^{\sec \left ( x \right ) }}{\ln \left ( 2 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^sec(x)*sec(x)*tan(x),x)

[Out]

2^sec(x)/ln(2)

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Maxima [A]  time = 0.956561, size = 12, normalized size = 1.33 \begin{align*} \frac{2^{\sec \left (x\right )}}{\log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^sec(x)*sec(x)*tan(x),x, algorithm="maxima")

[Out]

2^sec(x)/log(2)

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Fricas [A]  time = 2.31062, size = 28, normalized size = 3.11 \begin{align*} \frac{2^{\left (\frac{1}{\cos \left (x\right )}\right )}}{\log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^sec(x)*sec(x)*tan(x),x, algorithm="fricas")

[Out]

2^(1/cos(x))/log(2)

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Sympy [A]  time = 0.927133, size = 7, normalized size = 0.78 \begin{align*} \frac{2^{\sec{\left (x \right )}}}{\log{\left (2 \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**sec(x)*sec(x)*tan(x),x)

[Out]

2**sec(x)/log(2)

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Giac [A]  time = 1.0824, size = 15, normalized size = 1.67 \begin{align*} \frac{2^{\left (\frac{1}{\cos \left (x\right )}\right )}}{\log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^sec(x)*sec(x)*tan(x),x, algorithm="giac")

[Out]

2^(1/cos(x))/log(2)