\(\int e^{-2 \pi x} \cos (2 \pi x) \, dx\) [758]

Optimal result

Integrand size = 12, antiderivative size = 37 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=-\frac {e^{-2 \pi x} \cos (2 \pi x)}{4 \pi }+\frac {e^{-2 \pi x} \sin (2 \pi x)}{4 \pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4518} \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=\frac {e^{-2 \pi x} \sin (2 \pi x)}{4 \pi }-\frac {e^{-2 \pi x} \cos (2 \pi x)}{4 \pi } \]

[In]

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

[Out]

-1/4*Cos[2*Pi*x]/(E^(2*Pi*x)*Pi) + Sin[2*Pi*x]/(4*E^(2*Pi*x)*Pi)

Rule 4518

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*} \text {integral}& = -\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] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=\frac {e^{-2 \pi x} (-\cos (2 \pi x)+\sin (2 \pi x))}{4 \pi } \]

[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)

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{-2 \pi x} \left (-\cos \left (2 \pi x \right )+\sin \left (2 \pi x \right )\right )}{4 \pi }\)	\(24\)
derivativedivides	\(\frac {-\frac {{\mathrm e}^{-2 \pi x} \cos \left (2 \pi x \right )}{2}+\frac {{\mathrm e}^{-2 \pi x} \sin \left (2 \pi x \right )}{2}}{2 \pi }\)	\(31\)
default	\(\frac {-\frac {{\mathrm e}^{-2 \pi x} \cos \left (2 \pi x \right )}{2}+\frac {{\mathrm e}^{-2 \pi x} \sin \left (2 \pi x \right )}{2}}{2 \pi }\)	\(31\)
norman	\(\frac {\left (-\frac {1}{4 \pi }+\frac {\tan \left (\pi x \right )}{2 \pi }+\frac {\tan \left (\pi x \right )^{2}}{4 \pi }\right ) {\mathrm e}^{-2 \pi x}}{1+\tan \left (\pi x \right )^{2}}\)	\(45\)
risch	\(-\frac {{\mathrm e}^{\left (-2+2 i\right ) \pi x}}{8 \pi }-\frac {i {\mathrm e}^{\left (-2+2 i\right ) \pi x}}{8 \pi }-\frac {{\mathrm e}^{\left (-2-2 i\right ) \pi x}}{8 \pi }+\frac {i {\mathrm e}^{\left (-2-2 i\right ) \pi x}}{8 \pi }\)	\(52\)

[In]

int(cos(2*Pi*x)/exp(2*Pi*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=-\frac {\cos \left (2 \, \pi x\right ) e^{\left (-2 \, \pi x\right )} - e^{\left (-2 \, \pi x\right )} \sin \left (2 \, \pi x\right )}{4 \, \pi } \]

[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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=\frac {e^{- 2 \pi x} \sin {\left (2 \pi x \right )}}{4 \pi } - \frac {e^{- 2 \pi x} \cos {\left (2 \pi x \right )}}{4 \pi } \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=-\frac {{\left (\pi \cos \left (2 \, \pi x\right ) - \pi \sin \left (2 \, \pi x\right )\right )} e^{\left (-2 \, \pi x\right )}}{4 \, \pi ^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.62 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=-\frac {{\left (\cos \left (2 \, \pi x\right ) - \sin \left (2 \, \pi x\right )\right )} e^{\left (-2 \, \pi x\right )}}{4 \, \pi } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 26.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int e^{-2 \pi x} \cos (2 \pi x) \, dx=-\frac {{\mathrm {e}}^{-2\,\Pi \,x}\,\left (2\,\cos \left (2\,\Pi \,x\right )-2\,\sin \left (2\,\Pi \,x\right )\right )}{8\,\Pi } \]

[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)