\(\int e^x \text {sech}(e^x) \, dx\) [681]

Optimal result

Integrand size = 8, antiderivative size = 5 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=\arctan \left (\sinh \left (e^x\right )\right ) \]

[Out]

arctan(sinh(exp(x)))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 3855} \[ \int e^x \text {sech}\left (e^x\right ) \, dx=\arctan \left (\sinh \left (e^x\right )\right ) \]

[In]

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

[Out]

ArcTan[Sinh[E^x]]

Rule 2320

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \text {sech}(x) \, dx,x,e^x\right ) \\ & = \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=\arctan \left (\sinh \left (e^x\right )\right ) \]

[In]

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

[Out]

ArcTan[Sinh[E^x]]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right )\)	\(5\)
default	\(\arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right )\)	\(5\)
risch	\(i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}+i\right )-i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-i\right )\)	\(22\)
parallelrisch	\(-i \left (\ln \left (\tanh \left (\frac {{\mathrm e}^{x}}{2}\right )-i\right )-\ln \left (\tanh \left (\frac {{\mathrm e}^{x}}{2}\right )+i\right )\right )\)	\(25\)

[In]

int(exp(x)*sech(exp(x)),x,method=_RETURNVERBOSE)

[Out]

arctan(sinh(exp(x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (4) = 8\).

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 3.20 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=2 \, \arctan \left (\cosh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sinh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.60 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=2 \operatorname {atan}{\left (\tanh {\left (\frac {e^{x}}{2} \right )} \right )} \]

[In]

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

[Out]

2*atan(tanh(exp(x)/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.80 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=\arctan \left (\sinh \left (e^{x}\right )\right ) \]

[In]

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

[Out]

arctan(sinh(e^x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=2 \, \arctan \left (e^{\left (e^{x}\right )}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20 \[ \int e^x \text {sech}\left (e^x\right ) \, dx=2\,\mathrm {atan}\left ({\mathrm {e}}^{{\mathrm {e}}^x}\right ) \]

[In]

int(exp(x)/cosh(exp(x)),x)

[Out]

2*atan(exp(exp(x)))