3.66 \(\int e^{-2 x} \cos (e^{-2 x}) \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{2} \sin \left (e^{-2 x}\right ) \]

[Out]

-Sin[E^(-2*x)]/2

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Rubi [A]  time = 0.0104177, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 2637} \[ -\frac{1}{2} \sin \left (e^{-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[E^(-2*x)]/E^(2*x),x]

[Out]

-Sin[E^(-2*x)]/2

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0112636, size = 10, normalized size = 1. \[ -\frac{1}{2} \sin \left (e^{-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[E^(-2*x)]/E^(2*x),x]

[Out]

-Sin[E^(-2*x)]/2

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Maple [A]  time = 0.007, size = 8, normalized size = 0.8 \begin{align*} -{\frac{\sin \left ({{\rm e}^{-2\,x}} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(exp(-2*x))/exp(2*x),x)

[Out]

-1/2*sin(exp(-2*x))

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Maxima [A]  time = 0.998854, size = 9, normalized size = 0.9 \begin{align*} -\frac{1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(exp(-2*x))/exp(2*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.463653, size = 27, normalized size = 2.7 \begin{align*} -\frac{1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(exp(-2*x))/exp(2*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.492375, size = 10, normalized size = 1. \begin{align*} - \frac{\sin{\left (e^{- 2 x} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(exp(-2*x))/exp(2*x),x)

[Out]

-sin(exp(-2*x))/2

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Giac [A]  time = 1.09451, size = 9, normalized size = 0.9 \begin{align*} -\frac{1}{2} \, \sin \left (e^{\left (-2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(exp(-2*x))/exp(2*x),x, algorithm="giac")

[Out]

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