\(\int (-e^{-x}+e^x) \, dx\) [496]

Optimal result

Integrand size = 11, antiderivative size = 9 \[ \int \left (-e^{-x}+e^x\right ) \, dx=e^{-x}+e^x \]

[Out]

exp(-x)+exp(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2225} \[ \int \left (-e^{-x}+e^x\right ) \, dx=e^{-x}+e^x \]

[In]

Int[-E^(-x) + E^x,x]

[Out]

E^(-x) + E^x

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = -\int e^{-x} \, dx+\int e^x \, dx \\ & = e^{-x}+e^x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \left (-e^{-x}+e^x\right ) \, dx=e^{-x}+e^x \]

[In]

Integrate[-E^(-x) + E^x,x]

[Out]

E^(-x) + E^x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\({\mathrm e}^{-x}+{\mathrm e}^{x}\)	\(8\)
default	\({\mathrm e}^{-x}+{\mathrm e}^{x}\)	\(8\)
risch	\({\mathrm e}^{-x}+{\mathrm e}^{x}\)	\(8\)
parts	\({\mathrm e}^{-x}+{\mathrm e}^{x}\)	\(8\)
meijerg	\(-2+{\mathrm e}^{-x}+{\mathrm e}^{x}\)	\(9\)
norman	\(\left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\)	\(12\)
parallelrisch	\(\left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\)	\(12\)

[In]

int(-1/exp(x)+exp(x),x,method=_RETURNVERBOSE)

[Out]

1/exp(x)+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.22 \[ \int \left (-e^{-x}+e^x\right ) \, dx={\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \]

[In]

integrate(-1/exp(x)+exp(x),x, algorithm="fricas")

[Out]

(e^(2*x) + 1)*e^(-x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \left (-e^{-x}+e^x\right ) \, dx=e^{x} + e^{- x} \]

[In]

integrate(-1/exp(x)+exp(x),x)

[Out]

exp(x) + exp(-x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \left (-e^{-x}+e^x\right ) \, dx=e^{\left (-x\right )} + e^{x} \]

[In]

integrate(-1/exp(x)+exp(x),x, algorithm="maxima")

[Out]

e^(-x) + e^x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \left (-e^{-x}+e^x\right ) \, dx=e^{\left (-x\right )} + e^{x} \]

[In]

integrate(-1/exp(x)+exp(x),x, algorithm="giac")

[Out]

e^(-x) + e^x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.44 \[ \int \left (-e^{-x}+e^x\right ) \, dx=2\,\mathrm {cosh}\left (x\right ) \]

[In]

int(exp(x) - exp(-x),x)

[Out]

2*cosh(x)