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

Optimal. Leaf size=37 \[ \frac{e^{-2 \pi x} \sin (2 \pi x)}{4 \pi }-\frac{e^{-2 \pi x} \cos (2 \pi x)}{4 \pi } \]

[Out]

-Cos[2*Pi*x]/(4*E^(2*Pi*x)*Pi) + Sin[2*Pi*x]/(4*E^(2*Pi*x)*Pi)

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Rubi [A]  time = 0.0135448, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4433} \[ \frac{e^{-2 \pi x} \sin (2 \pi x)}{4 \pi }-\frac{e^{-2 \pi x} \cos (2 \pi x)}{4 \pi } \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[2*Pi*x]/(4*E^(2*Pi*x)*Pi) + Sin[2*Pi*x]/(4*E^(2*Pi*x)*Pi)

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{-2 \pi x} \cos (2 \pi x) \, dx &=-\frac{e^{-2 \pi x} \cos (2 \pi x)}{4 \pi }+\frac{e^{-2 \pi x} \sin (2 \pi x)}{4 \pi }\\ \end{align*}

Mathematica [A]  time = 0.0281835, size = 26, normalized size = 0.7 \[ \frac{e^{-2 \pi x} (\sin (2 \pi x)-\cos (2 \pi x))}{4 \pi } \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-Cos[2*Pi*x] + Sin[2*Pi*x])/(4*E^(2*Pi*x)*Pi)

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Maple [A]  time = 0.01, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{2\,\pi } \left ( -{\frac{{{\rm e}^{-2\,\pi \,x}}\cos \left ( 2\,\pi \,x \right ) }{2}}+{\frac{{{\rm e}^{-2\,\pi \,x}}\sin \left ( 2\,\pi \,x \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/Pi*(-1/2*exp(-2*Pi*x)*cos(2*Pi*x)+1/2*exp(-2*Pi*x)*sin(2*Pi*x))

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Maxima [A]  time = 0.967941, size = 35, normalized size = 0.95 \begin{align*} -\frac{{\left (\pi \cos \left (2 \, \pi x\right ) - \pi \sin \left (2 \, \pi x\right )\right )} e^{\left (-2 \, \pi x\right )}}{4 \, \pi ^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(pi*cos(2*pi*x) - pi*sin(2*pi*x))*e^(-2*pi*x)/pi^2

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Fricas [A]  time = 1.82783, size = 82, normalized size = 2.22 \begin{align*} -\frac{\cos \left (2 \, \pi x\right ) e^{\left (-2 \, \pi x\right )} - e^{\left (-2 \, \pi x\right )} \sin \left (2 \, \pi x\right )}{4 \, \pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(cos(2*pi*x)*e^(-2*pi*x) - e^(-2*pi*x)*sin(2*pi*x))/pi

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Sympy [A]  time = 0.486224, size = 32, normalized size = 0.86 \begin{align*} \frac{e^{- 2 \pi x} \sin{\left (2 \pi x \right )}}{4 \pi } - \frac{e^{- 2 \pi x} \cos{\left (2 \pi x \right )}}{4 \pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-2*pi*x)*sin(2*pi*x)/(4*pi) - exp(-2*pi*x)*cos(2*pi*x)/(4*pi)

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Giac [A]  time = 1.12773, size = 36, normalized size = 0.97 \begin{align*} -\frac{1}{4} \,{\left (\frac{\cos \left (2 \, \pi x\right )}{\pi } - \frac{\sin \left (2 \, \pi x\right )}{\pi }\right )} e^{\left (-2 \, \pi x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(cos(2*pi*x)/pi - sin(2*pi*x)/pi)*e^(-2*pi*x)