\(\int (e^x-x^e) \, dx\) [646]

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int \left (e^x-x^e\right ) \, dx=e^x-\frac {x^{1+e}}{1+e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2225} \[ \int \left (e^x-x^e\right ) \, dx=e^x-\frac {x^{1+e}}{1+e} \]

[In]

Int[E^x - x^E,x]

[Out]

E^x - x^(1 + E)/(1 + E)

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (e^x-x^e\right ) \, dx=e^x-\frac {x^{1+e}}{1+e} \]

[In]

Integrate[E^x - x^E,x]

[Out]

E^x - x^(1 + E)/(1 + E)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
risch	\(-\frac {x \,x^{E}}{1+E}+{\mathrm e}^{x}\)	\(15\)
default	\({\mathrm e}^{x}-\frac {x^{1+E}}{1+E}\)	\(16\)
parts	\({\mathrm e}^{x}-\frac {x^{1+E}}{1+E}\)	\(16\)
norman	\(-\frac {x \,{\mathrm e}^{E \ln \left (x \right )}}{1+E}+{\mathrm e}^{x}\)	\(17\)
parallelrisch	\(\frac {{\mathrm e}^{x} E -x \,x^{E}+{\mathrm e}^{x}}{1+E}\)	\(20\)

[In]

int(exp(x)-x^E,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \left (e^x-x^e\right ) \, dx=-\frac {x x^{E} - {\left (E + 1\right )} e^{x}}{E + 1} \]

[In]

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

[Out]

-(x*x^E - (E + 1)*e^x)/(E + 1)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (e^x-x^e\right ) \, dx=- \frac {x^{1 + e}}{1 + e} + e^{x} \]

[In]

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

[Out]

-x**(1 + E)/(1 + E) + exp(x)

Maxima [F(-2)]

Exception generated. \[ \int \left (e^x-x^e\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(E>0)', see `assume?` for more
details)Is E

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \left (e^x-x^e\right ) \, dx=-\frac {x^{E + 1}}{E + 1} + e^{x} \]

[In]

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

[Out]

-x^(E + 1)/(E + 1) + e^x

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (e^x-x^e\right ) \, dx={\mathrm {e}}^x-\frac {x\,x^{\mathrm {e}}}{\mathrm {e}+1} \]

[In]

int(exp(x) - x^exp(1),x)

[Out]

exp(x) - (x*x^exp(1))/(exp(1) + 1)