3.351 $$\int e^x \text{sech}(e^x) \, dx$$

Optimal. Leaf size=5 $\tan ^{-1}\left (\sinh \left (e^x\right )\right )$

[Out]

ArcTan[Sinh[E^x]]

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Rubi [A]  time = 0.009588, 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} $\tan ^{-1}\left (\sinh \left (e^x\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[Sinh[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 \text{sech}\left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \text{sech}(x) \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (\sinh \left (e^x\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0049734, size = 5, normalized size = 1. $\tan ^{-1}\left (\sinh \left (e^x\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[Sinh[E^x]]

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Maple [A]  time = 0.002, size = 5, normalized size = 1. \begin{align*} \arctan \left ( \sinh \left ({{\rm e}^{x}} \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sech(exp(x)),x)

[Out]

arctan(sinh(exp(x)))

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Maxima [A]  time = 0.953006, size = 5, normalized size = 1. \begin{align*} \arctan \left (\sinh \left (e^{x}\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sinh(e^x))

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Fricas [B]  time = 1.92324, size = 82, normalized size = 16.4 \begin{align*} 2 \, \arctan \left (\cosh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sinh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(cosh(cosh(x) + sinh(x)) + sinh(cosh(x) + sinh(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \operatorname{sech}{\left (e^{x} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(exp(x)),x)

[Out]

Integral(exp(x)*sech(exp(x)), x)

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Giac [A]  time = 1.06723, size = 8, normalized size = 1.6 \begin{align*} 2 \, \arctan \left (e^{\left (e^{x}\right )}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(e^(e^x))