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

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

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ e^{-x}+e^x \]

Antiderivative was successfully verified.

[In]

Integrate[-E^(-x) + E^x,x]

[Out]

E^(-x) + E^x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (-e^{-x}+e^x\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[-E^(-x) + E^x,x]

[Out]

Could not integrate

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fricas [A]  time = 1.11, size = 11, normalized size = 1.22 \[ {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \]

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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giac [A]  time = 0.60, size = 7, normalized size = 0.78 \[ e^{\left (-x\right )} + e^{x} \]

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

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maple [A]  time = 0.03, size = 8, normalized size = 0.89

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/exp(x)+exp(x)

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maxima [A]  time = 0.55, size = 7, normalized size = 0.78 \[ e^{\left (-x\right )} + e^{x} \]

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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mupad [B]  time = 0.05, size = 4, normalized size = 0.44 \[ 2\,\mathrm {cosh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) - exp(-x),x)

[Out]

2*cosh(x)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.78 \[ e^{x} + e^{- x} \]

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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