3.70 \(\int e^x \sec (e^x) \, dx\)

Optimal. Leaf size=5 \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]

[Out]

ArcTanh[Sin[E^x]]

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Rubi [A]  time = 0.0097277, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 3770} \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.005083, size = 5, normalized size = 1. \[ \tanh ^{-1}\left (\sin \left (e^x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[E^x]]

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Maple [A]  time = 0.002, size = 9, normalized size = 1.8 \begin{align*} \ln \left ( \sec \left ({{\rm e}^{x}} \right ) +\tan \left ({{\rm e}^{x}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(sec(exp(x))+tan(exp(x)))

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Maxima [B]  time = 1.0065, size = 11, normalized size = 2.2 \begin{align*} \log \left (\sec \left (e^{x}\right ) + \tan \left (e^{x}\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sec(e^x) + tan(e^x))

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Fricas [B]  time = 0.482003, size = 65, normalized size = 13. \begin{align*} \frac{1}{2} \, \log \left (\sin \left (e^{x}\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (e^{x}\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(sin(e^x) + 1) - 1/2*log(-sin(e^x) + 1)

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Sympy [A]  time = 3.5652, size = 10, normalized size = 2. \begin{align*} \log{\left (\tan{\left (e^{x} \right )} + \sec{\left (e^{x} \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(exp(x)) + sec(exp(x)))

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Giac [B]  time = 1.18458, size = 39, normalized size = 7.8 \begin{align*} \frac{1}{4} \, \log \left ({\left | \frac{1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) + 2 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | \frac{1}{\sin \left (e^{x}\right )} + \sin \left (e^{x}\right ) - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(1/sin(e^x) + sin(e^x) + 2)) - 1/4*log(abs(1/sin(e^x) + sin(e^x) - 2))