3.71 \(\int e^x \sec (e^x) \tan (e^x) \, dx\)

Optimal. Leaf size=4 \[ \sec \left (e^x\right ) \]

[Out]

Sec[E^x]

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Rubi [A]  time = 0.0200626, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 2606, 8} \[ \sec \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[E^x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[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 e^x \sec \left (e^x\right ) \tan \left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \sec (x) \tan (x) \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int 1 \, dx,x,\sec \left (e^x\right )\right )\\ &=\sec \left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0102617, size = 4, normalized size = 1. \[ \sec \left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[E^x]

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Maple [A]  time = 0.008, size = 4, normalized size = 1. \begin{align*} \sec \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sec(exp(x))

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Maxima [A]  time = 1.01238, size = 7, normalized size = 1.75 \begin{align*} \frac{1}{\cos \left (e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(e^x)

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Fricas [A]  time = 0.464941, size = 16, normalized size = 4. \begin{align*} \frac{1}{\cos \left (e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(e^x)

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Sympy [A]  time = 1.05962, size = 3, normalized size = 0.75 \begin{align*} \sec{\left (e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sec(exp(x))

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Giac [A]  time = 1.18569, size = 7, normalized size = 1.75 \begin{align*} \frac{1}{\cos \left (e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/cos(e^x)