3.46.34 \(\int \frac {1}{3} (3+e^{e^{x/3}-x} (-3+e^{x/3})+30 x) \, dx\)

Optimal. Leaf size=23 \[ -3+e^{e^{x/3}-x}+x+5 x^2+\log (4) \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {12, 2282, 2197} \begin {gather*} 5 x^2+x+e^{e^{x/3}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + E^(E^(x/3) - x)*(-3 + E^(x/3)) + 30*x)/3,x]

[Out]

E^(E^(x/3) - x) + x + 5*x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, 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 {1}{3} \int \left (3+e^{e^{x/3}-x} \left (-3+e^{x/3}\right )+30 x\right ) \, dx\\ &=x+5 x^2+\frac {1}{3} \int e^{e^{x/3}-x} \left (-3+e^{x/3}\right ) \, dx\\ &=x+5 x^2+\operatorname {Subst}\left (\int \frac {e^x (-3+x)}{x^4} \, dx,x,e^{x/3}\right )\\ &=e^{e^{x/3}-x}+x+5 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 20, normalized size = 0.87 \begin {gather*} e^{e^{x/3}-x}+x+5 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^(E^(x/3) - x)*(-3 + E^(x/3)) + 30*x)/3,x]

[Out]

E^(E^(x/3) - x) + x + 5*x^2

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fricas [A]  time = 0.52, size = 16, normalized size = 0.70 \begin {gather*} 5 \, x^{2} + x + e^{\left (-x + e^{\left (\frac {1}{3} \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(1/3*x)-3)*exp(exp(1/3*x)-x)+10*x+1,x, algorithm="fricas")

[Out]

5*x^2 + x + e^(-x + e^(1/3*x))

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giac [A]  time = 0.13, size = 16, normalized size = 0.70 \begin {gather*} 5 \, x^{2} + x + e^{\left (-x + e^{\left (\frac {1}{3} \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(1/3*x)-3)*exp(exp(1/3*x)-x)+10*x+1,x, algorithm="giac")

[Out]

5*x^2 + x + e^(-x + e^(1/3*x))

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maple [A]  time = 0.04, size = 17, normalized size = 0.74

method	result	size
default	\(5 x^{2}+x +{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}-x}\)	\(17\)
norman	\(5 x^{2}+x +{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}-x}\)	\(17\)
risch	\(5 x^{2}+x +{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}-x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(exp(1/3*x)-3)*exp(exp(1/3*x)-x)+10*x+1,x,method=_RETURNVERBOSE)

[Out]

5*x^2+x+exp(exp(1/3*x)-x)

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maxima [A]  time = 0.39, size = 16, normalized size = 0.70 \begin {gather*} 5 \, x^{2} + x + e^{\left (-x + e^{\left (\frac {1}{3} \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(1/3*x)-3)*exp(exp(1/3*x)-x)+10*x+1,x, algorithm="maxima")

[Out]

5*x^2 + x + e^(-x + e^(1/3*x))

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mupad [B]  time = 3.29, size = 17, normalized size = 0.74 \begin {gather*} x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\left ({\mathrm {e}}^x\right )}^{1/3}}+5\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*x + (exp(exp(x/3) - x)*(exp(x/3) - 3))/3 + 1,x)

[Out]

x + exp(-x)*exp(exp(x)^(1/3)) + 5*x^2

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sympy [A]  time = 0.13, size = 14, normalized size = 0.61 \begin {gather*} 5 x^{2} + x + e^{- x + e^{\frac {x}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(exp(1/3*x)-3)*exp(exp(1/3*x)-x)+10*x+1,x)

[Out]

5*x**2 + x + exp(-x + exp(x/3))

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