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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = 0.01, size = 16, normalized size = 1.00 \[ e^x-\frac {x^{1+e}}{1+e} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x - x^E,x]

[Out]

E^x - x^(1 + E)/(1 + E)

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 20, normalized size = 1.25 \[ -\frac {x x^{E} - {\left (E + 1\right )} e^{x}}{E + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 15, normalized size = 0.94 \[ -\frac {x^{E + 1}}{E + 1} + e^{x} \]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 16, normalized size = 1.00 \[ -\frac {x^{E +1}}{E +1}+{\mathrm e}^{x} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.81, size = 15, normalized size = 0.94 \[ -\frac {x^{E + 1}}{E + 1} + e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^(E + 1)/(E + 1) + e^x

________________________________________________________________________________________

mupad [B]  time = 3.33, size = 16, normalized size = 1.00 \[ {\mathrm {e}}^x-\frac {x\,x^{\mathrm {e}}}{\mathrm {e}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) - x^exp(1),x)

[Out]

exp(x) - (x*x^exp(1))/(exp(1) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 14, normalized size = 0.88 \[ - \frac {x^{1 + e}}{1 + e} + e^{x} \]

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)

________________________________________________________________________________________