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]

-1/4*cos(2*Pi*x)/exp(2*Pi*x)/Pi+1/4*sin(2*Pi*x)/exp(2*Pi*x)/Pi

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 26, normalized size = 0.70 \[ \frac {e^{-2 \pi x} (\sin (2 \pi x)-\cos (2 \pi x))}{4 \pi } \]

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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fricas [A]  time = 0.70, size = 29, normalized size = 0.78 \[ -\frac {\cos \left (2 \, \pi x\right ) e^{\left (-2 \, \pi x\right )} - e^{\left (-2 \, \pi x\right )} \sin \left (2 \, \pi x\right )}{4 \, \pi } \]

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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giac [A]  time = 0.13, size = 27, normalized size = 0.73 \[ -\frac {1}{4} \, {\left (\frac {\cos \left (2 \, \pi x\right )}{\pi } - \frac {\sin \left (2 \, \pi x\right )}{\pi }\right )} e^{\left (-2 \, \pi x\right )} \]

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)

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maple [A]  time = 0.06, size = 31, normalized size = 0.84 \[ \frac {-\frac {{\mathrm e}^{-2 \pi x} \cos \left (2 \pi x \right )}{2}+\frac {{\mathrm e}^{-2 \pi x} \sin \left (2 \pi x \right )}{2}}{2 \pi } \]

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 = 0.32, size = 26, normalized size = 0.70 \[ -\frac {{\left (\pi \cos \left (2 \, \pi x\right ) - \pi \sin \left (2 \, \pi x\right )\right )} e^{\left (-2 \, \pi x\right )}}{4 \, \pi ^{2}} \]

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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mupad [B]  time = 2.90, size = 25, normalized size = 0.68 \[ -\frac {{\mathrm {e}}^{-2\,\Pi \,x}\,\left (2\,\cos \left (2\,\Pi \,x\right )-2\,\sin \left (2\,\Pi \,x\right )\right )}{8\,\Pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(-2*Pi*x)*(2*cos(2*Pi*x) - 2*sin(2*Pi*x)))/(8*Pi)

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sympy [A]  time = 0.42, size = 32, normalized size = 0.86 \[ \frac {e^{- 2 \pi x} \sin {\left (2 \pi x \right )}}{4 \pi } - \frac {e^{- 2 \pi x} \cos {\left (2 \pi x \right )}}{4 \pi } \]

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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