3.496 \(\int (-e^{-x}+e^x) \, dx\)

Optimal. Leaf size=9 \[ e^{-x}+e^x \]

[Out]

E^(-x) + E^x

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Rubi [A]  time = 0.0034646, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2194} \[ e^{-x}+e^x \]

Antiderivative was successfully verified.

[In]

Int[-E^(-x) + E^x,x]

[Out]

E^(-x) + E^x

Rule 2194

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*} \int \left (-e^{-x}+e^x\right ) \, dx &=-\int e^{-x} \, dx+\int e^x \, dx\\ &=e^{-x}+e^x\\ \end{align*}

Mathematica [A]  time = 0.0030865, size = 9, normalized size = 1. \[ e^{-x}+e^x \]

Antiderivative was successfully verified.

[In]

Integrate[-E^(-x) + E^x,x]

[Out]

E^(-x) + E^x

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Maple [A]  time = 0.002, size = 8, normalized size = 0.9 \begin{align*} \left ({{\rm e}^{x}} \right ) ^{-1}+{{\rm e}^{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/exp(x)+exp(x)

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Maxima [A]  time = 0.927173, size = 9, normalized size = 1. \begin{align*} e^{\left (-x\right )} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x) + e^x

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Fricas [A]  time = 1.68799, size = 30, normalized size = 3.33 \begin{align*}{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^(2*x) + 1)*e^(-x)

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Sympy [A]  time = 0.08325, size = 7, normalized size = 0.78 \begin{align*} e^{x} + e^{- x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x) + exp(-x)

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Giac [A]  time = 1.08643, size = 9, normalized size = 1. \begin{align*} e^{\left (-x\right )} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x) + e^x