3.92.9 \(\int \frac {-2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}+x^9}{x^9} \, dx\)

Optimal. Leaf size=13 \[ e^{e^{\frac {1}{4 x^8}}}+x \]

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Rubi [A]  time = 0.08, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {14, 6715, 2282, 2194} \begin {gather*} e^{e^{\frac {1}{4 x^8}}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^(E^(1/(4*x^8)) + 1/(4*x^8)) + x^9)/x^9,x]

[Out]

E^E^(1/(4*x^8)) + x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]]]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {2 e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}}{x^9}\right ) \, dx\\ &=x-2 \int \frac {e^{e^{\frac {1}{4 x^8}}+\frac {1}{4 x^8}}}{x^9} \, dx\\ &=x+\frac {1}{4} \operatorname {Subst}\left (\int e^{e^{x/4}+\frac {x}{4}} \, dx,x,\frac {1}{x^8}\right )\\ &=x+\operatorname {Subst}\left (\int e^x \, dx,x,e^{\frac {1}{4 x^8}}\right )\\ &=e^{e^{\frac {1}{4 x^8}}}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} e^{e^{\frac {1}{4 x^8}}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^(E^(1/(4*x^8)) + 1/(4*x^8)) + x^9)/x^9,x]

[Out]

E^E^(1/(4*x^8)) + x

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fricas [B]  time = 0.90, size = 35, normalized size = 2.69 \begin {gather*} {\left (x e^{\left (\frac {1}{4 \, x^{8}}\right )} + e^{\left (\frac {4 \, x^{8} e^{\left (\frac {1}{4 \, x^{8}}\right )} + 1}{4 \, x^{8}}\right )}\right )} e^{\left (-\frac {1}{4 \, x^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/4/x^8)*exp(exp(1/4/x^8))+x^9)/x^9,x, algorithm="fricas")

[Out]

(x*e^(1/4/x^8) + e^(1/4*(4*x^8*e^(1/4/x^8) + 1)/x^8))*e^(-1/4/x^8)

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giac [B]  time = 0.20, size = 35, normalized size = 2.69 \begin {gather*} {\left (x e^{\left (\frac {1}{4 \, x^{8}}\right )} + e^{\left (\frac {4 \, x^{8} e^{\left (\frac {1}{4 \, x^{8}}\right )} + 1}{4 \, x^{8}}\right )}\right )} e^{\left (-\frac {1}{4 \, x^{8}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/4/x^8)*exp(exp(1/4/x^8))+x^9)/x^9,x, algorithm="giac")

[Out]

(x*e^(1/4/x^8) + e^(1/4*(4*x^8*e^(1/4/x^8) + 1)/x^8))*e^(-1/4/x^8)

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maple [A]  time = 0.02, size = 10, normalized size = 0.77

method	result	size
default	\({\mathrm e}^{{\mathrm e}^{\frac {1}{4 x^{8}}}}+x\)	\(10\)
risch	\({\mathrm e}^{{\mathrm e}^{\frac {1}{4 x^{8}}}}+x\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(1/4/x^8)*exp(exp(1/4/x^8))+x^9)/x^9,x,method=_RETURNVERBOSE)

[Out]

exp(exp(1/4/x^8))+x

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maxima [A]  time = 0.36, size = 9, normalized size = 0.69 \begin {gather*} x + e^{\left (e^{\left (\frac {1}{4 \, x^{8}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/4/x^8)*exp(exp(1/4/x^8))+x^9)/x^9,x, algorithm="maxima")

[Out]

x + e^(e^(1/4/x^8))

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mupad [B]  time = 6.99, size = 9, normalized size = 0.69 \begin {gather*} x+{\mathrm {e}}^{{\mathrm {e}}^{\frac {1}{4\,x^8}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^9 - 2*exp(exp(1/(4*x^8)))*exp(1/(4*x^8)))/x^9,x)

[Out]

x + exp(exp(1/(4*x^8)))

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sympy [A]  time = 0.26, size = 10, normalized size = 0.77 \begin {gather*} x + e^{e^{\frac {1}{4 x^{8}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(1/4/x**8)*exp(exp(1/4/x**8))+x**9)/x**9,x)

[Out]

x + exp(exp(1/(4*x**8)))

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