3.12.2 \(\int \frac {1+e^3+e^{5+e^2 x}}{e^3} \, dx\)

Optimal. Leaf size=15 \[ 17+e^{e^2 x}+x+\frac {x}{e^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2194} \begin {gather*} \left (1+\frac {1}{e^3}\right ) x+e^{e^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^3 + E^(5 + E^2*x))/E^3,x]

[Out]

E^(E^2*x) + (1 + E^(-3))*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (1+e^3+e^{5+e^2 x}\right ) \, dx}{e^3}\\ &=\left (1+\frac {1}{e^3}\right ) x+\frac {\int e^{5+e^2 x} \, dx}{e^3}\\ &=e^{e^2 x}+\left (1+\frac {1}{e^3}\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.93 \begin {gather*} e^{e^2 x}+x+\frac {x}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^3 + E^(5 + E^2*x))/E^3,x]

[Out]

E^(E^2*x) + x + x/E^3

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fricas [A]  time = 0.64, size = 19, normalized size = 1.27 \begin {gather*} {\left (x e^{5} + x e^{2} + e^{\left (x e^{2} + 5\right )}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3/2)^2*exp(2)*exp(exp(2)*x)+exp(3/2)^2+1)/exp(3/2)^2,x, algorithm="fricas")

[Out]

(x*e^5 + x*e^2 + e^(x*e^2 + 5))*e^(-5)

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giac [A]  time = 0.34, size = 16, normalized size = 1.07 \begin {gather*} {\left (x e^{3} + x + e^{\left (x e^{2} + 3\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3/2)^2*exp(2)*exp(exp(2)*x)+exp(3/2)^2+1)/exp(3/2)^2,x, algorithm="giac")

[Out]

(x*e^3 + x + e^(x*e^2 + 3))*e^(-3)

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maple [A]  time = 0.03, size = 17, normalized size = 1.13

method	result	size
risch	\({\mathrm e}^{-3} x \,{\mathrm e}^{3}+x \,{\mathrm e}^{-3}+{\mathrm e}^{{\mathrm e}^{2} x}\)	\(17\)
default	\({\mathrm e}^{-3} \left (x +x \,{\mathrm e}^{3}+{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{2} x}\right )\)	\(24\)
norman	\(\left ({\mathrm e}^{\frac {3}{2}} {\mathrm e}^{{\mathrm e}^{2} x}+\left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-\frac {3}{2}} x \right ) {\mathrm e}^{-\frac {3}{2}}\)	\(27\)
derivativedivides	\({\mathrm e}^{-3} {\mathrm e}^{-2} \left ({\mathrm e}^{{\mathrm e}^{2} x} {\mathrm e}^{5}+\left ({\mathrm e}^{3}+1\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{2} x}\right )\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3/2)^2*exp(2)*exp(exp(2)*x)+exp(3/2)^2+1)/exp(3/2)^2,x,method=_RETURNVERBOSE)

[Out]

exp(-3)*x*exp(3)+x*exp(-3)+exp(exp(2)*x)

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maxima [A]  time = 0.40, size = 16, normalized size = 1.07 \begin {gather*} {\left (x e^{3} + x + e^{\left (x e^{2} + 3\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3/2)^2*exp(2)*exp(exp(2)*x)+exp(3/2)^2+1)/exp(3/2)^2,x, algorithm="maxima")

[Out]

(x*e^3 + x + e^(x*e^2 + 3))*e^(-3)

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mupad [B]  time = 0.73, size = 12, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^2}+x\,\left ({\mathrm {e}}^{-3}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-3)*(exp(3) + exp(5)*exp(x*exp(2)) + 1),x)

[Out]

exp(x*exp(2)) + x*(exp(-3) + 1)

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sympy [A]  time = 0.10, size = 15, normalized size = 1.00 \begin {gather*} \frac {x \left (1 + e^{3}\right )}{e^{3}} + e^{x e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3/2)**2*exp(2)*exp(exp(2)*x)+exp(3/2)**2+1)/exp(3/2)**2,x)

[Out]

x*(1 + exp(3))*exp(-3) + exp(x*exp(2))

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