3.74.86 \(\int \frac {e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} (6+2 e^4)}{3 e^2+e^6} \, dx\)

Optimal. Leaf size=25 \[ 28+e^{e^{\frac {2 x}{e^2}}}+\frac {e^x+x}{3+e^4} \]

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 2194, 2282} \begin {gather*} \frac {x}{3+e^4}+e^{e^{\frac {2 x}{e^2}}}+\frac {e^x}{3+e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^2 + E^(2 + x) + E^(E^((2*x)/E^2) + (2*x)/E^2)*(6 + 2*E^4))/(3*E^2 + E^6),x]

[Out]

E^E^((2*x)/E^2) + E^x/(3 + E^4) + x/(3 + E^4)

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]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (e^2+e^{2+x}+e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \left (6+2 e^4\right )\right ) \, dx}{e^2 \left (3+e^4\right )}\\ &=\frac {x}{3+e^4}+\frac {2 \int e^{e^{\frac {2 x}{e^2}}+\frac {2 x}{e^2}} \, dx}{e^2}+\frac {\int e^{2+x} \, dx}{e^2 \left (3+e^4\right )}\\ &=\frac {e^x}{3+e^4}+\frac {x}{3+e^4}+\operatorname {Subst}\left (\int e^x \, dx,x,e^{\frac {2 x}{e^2}}\right )\\ &=e^{e^{\frac {2 x}{e^2}}}+\frac {e^x}{3+e^4}+\frac {x}{3+e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 29, normalized size = 1.16 \begin {gather*} \frac {e^x+e^{e^{\frac {2 x}{e^2}}} \left (3+e^4\right )+x}{3+e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2 + E^(2 + x) + E^(E^((2*x)/E^2) + (2*x)/E^2)*(6 + 2*E^4))/(3*E^2 + E^6),x]

[Out]

(E^x + E^E^((2*x)/E^2)*(3 + E^4) + x)/(3 + E^4)

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fricas [B]  time = 0.94, size = 60, normalized size = 2.40 \begin {gather*} \frac {{\left ({\left (e^{6} + 3 \, e^{2}\right )} e^{\left ({\left (2 \, x + e^{\left (2 \, x e^{\left (-2\right )} + 2\right )}\right )} e^{\left (-2\right )}\right )} + x e^{\left (2 \, x e^{\left (-2\right )} + 2\right )} + e^{\left (2 \, x e^{\left (-2\right )} + x + 2\right )}\right )} e^{\left (-2 \, x e^{\left (-2\right )}\right )}}{e^{6} + 3 \, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(4)+6)*exp(2*x/exp(2))*exp(exp(2*x/exp(2)))+exp(2)*exp(x)+exp(2))/(exp(2)*exp(4)+3*exp(2)),x,
algorithm="fricas")

[Out]

((e^6 + 3*e^2)*e^((2*x + e^(2*x*e^(-2) + 2))*e^(-2)) + x*e^(2*x*e^(-2) + 2) + e^(2*x*e^(-2) + x + 2))*e^(-2*x*
e^(-2))/(e^6 + 3*e^2)

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giac [A]  time = 0.18, size = 33, normalized size = 1.32 \begin {gather*} \frac {x e^{2} + {\left (e^{4} + 3\right )} e^{\left (e^{\left (2 \, x e^{\left (-2\right )}\right )} + 2\right )} + e^{\left (x + 2\right )}}{e^{6} + 3 \, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(4)+6)*exp(2*x/exp(2))*exp(exp(2*x/exp(2)))+exp(2)*exp(x)+exp(2))/(exp(2)*exp(4)+3*exp(2)),x,
algorithm="giac")

[Out]

(x*e^2 + (e^4 + 3)*e^(e^(2*x*e^(-2)) + 2) + e^(x + 2))/(e^6 + 3*e^2)

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maple [A]  time = 0.06, size = 28, normalized size = 1.12

method	result	size
norman	\(\frac {x}{{\mathrm e}^{4}+3}+\frac {{\mathrm e}^{x}}{{\mathrm e}^{4}+3}+{\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}\)	\(28\)
default	\(\frac {{\mathrm e}^{2} {\mathrm e}^{x}+\frac {\left (2 \,{\mathrm e}^{4}+6\right ) {\mathrm e}^{{\mathrm e}^{2 x \,{\mathrm e}^{-2}}} {\mathrm e}^{2}}{2}+{\mathrm e}^{2} x}{{\mathrm e}^{2} {\mathrm e}^{4}+3 \,{\mathrm e}^{2}}\)	\(43\)
risch	\(\frac {{\mathrm e}^{2} x}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}+\frac {{\mathrm e}^{2+x}}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}+\frac {{\mathrm e}^{2+{\mathrm e}^{2 x \,{\mathrm e}^{-2}}} {\mathrm e}^{4}}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}+\frac {3 \,{\mathrm e}^{2+{\mathrm e}^{2 x \,{\mathrm e}^{-2}}}}{{\mathrm e}^{6}+3 \,{\mathrm e}^{2}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(4)+6)*exp(2*x/exp(2))*exp(exp(2*x/exp(2)))+exp(2)*exp(x)+exp(2))/(exp(2)*exp(4)+3*exp(2)),x,method
=_RETURNVERBOSE)

[Out]

1/(exp(4)+3)*x+1/(exp(4)+3)*exp(x)+exp(exp(2*x/exp(2)))

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maxima [A]  time = 0.44, size = 33, normalized size = 1.32 \begin {gather*} \frac {x e^{2} + {\left (e^{4} + 3\right )} e^{\left (e^{\left (2 \, x e^{\left (-2\right )}\right )} + 2\right )} + e^{\left (x + 2\right )}}{e^{6} + 3 \, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(4)+6)*exp(2*x/exp(2))*exp(exp(2*x/exp(2)))+exp(2)*exp(x)+exp(2))/(exp(2)*exp(4)+3*exp(2)),x,
algorithm="maxima")

[Out]

(x*e^2 + (e^4 + 3)*e^(e^(2*x*e^(-2)) + 2) + e^(x + 2))/(e^6 + 3*e^2)

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mupad [B]  time = 4.77, size = 25, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-2}}}+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^4+3}+\frac {x}{{\mathrm {e}}^4+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2) + exp(2)*exp(x) + exp(exp(2*x*exp(-2)))*exp(2*x*exp(-2))*(2*exp(4) + 6))/(3*exp(2) + exp(6)),x)

[Out]

exp(exp(2*x*exp(-2))) + exp(x)/(exp(4) + 3) + x/(exp(4) + 3)

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sympy [B]  time = 0.33, size = 48, normalized size = 1.92 \begin {gather*} \frac {x e^{2} + e^{2} e^{x} + 3 e^{2} e^{e^{\frac {2 x}{e^{2}}}} + e^{6} e^{e^{\frac {2 x}{e^{2}}}}}{3 e^{2} + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(4)+6)*exp(2*x/exp(2))*exp(exp(2*x/exp(2)))+exp(2)*exp(x)+exp(2))/(exp(2)*exp(4)+3*exp(2)),x)

[Out]

(x*exp(2) + exp(2)*exp(x) + 3*exp(2)*exp(exp(2*x*exp(-2))) + exp(6)*exp(exp(2*x*exp(-2))))/(3*exp(2) + exp(6))

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