3.646 \(\int (e^x-x^e) \, dx\)

Optimal. Leaf size=16 \[ e^x-\frac{x^{1+e}}{1+e} \]

[Out]

E^x - x^(1 + E)/(1 + E)

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Rubi [A]  time = 0.004769, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2194} \[ e^x-\frac{x^{1+e}}{1+e} \]

Antiderivative was successfully verified.

[In]

Int[E^x - x^E,x]

[Out]

E^x - x^(1 + E)/(1 + E)

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-x^e\right ) \, dx &=-\frac{x^{1+e}}{1+e}+\int e^x \, dx\\ &=e^x-\frac{x^{1+e}}{1+e}\\ \end{align*}

Mathematica [A]  time = 0.0053215, size = 16, normalized size = 1. \[ e^x-\frac{x^{1+e}}{1+e} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x - x^E,x]

[Out]

E^x - x^(1 + E)/(1 + E)

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Maple [A]  time = 0.036, size = 16, normalized size = 1. \begin{align*}{{\rm e}^{x}}-{\frac{{x}^{1+E}}{1+E}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)-x^E,x)

[Out]

exp(x)-x^(1+E)/(1+E)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)-x^E,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.86158, size = 43, normalized size = 2.69 \begin{align*} -\frac{x x^{E} -{\left (E + 1\right )} e^{x}}{E + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)-x^E,x, algorithm="fricas")

[Out]

-(x*x^E - (E + 1)*e^x)/(E + 1)

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Sympy [A]  time = 0.080443, size = 10, normalized size = 0.62 \begin{align*} - \frac{x^{1 + e}}{1 + e} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)-x**E,x)

[Out]

-x**(1 + E)/(1 + E) + exp(x)

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Giac [A]  time = 1.16478, size = 20, normalized size = 1.25 \begin{align*} -\frac{x^{E + 1}}{E + 1} + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)-x^E,x, algorithm="giac")

[Out]

-x^(E + 1)/(E + 1) + e^x