3.1.47 \(\int e^{e^{x^2}} x \, dx\) [47]

Optimal. Leaf size=10 \[ \frac {\operatorname {ExpIntegralEi}\left (e^{x^2}\right )}{2} \]

[Out]

1/2*Ei(exp(x^2))

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Rubi [A]
time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6847, 2320, 2209} \begin {gather*} \frac {\operatorname {ExpIntegralEi}\left (e^{x^2}\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^E^x^2*x,x]

[Out]

ExpIntegralEi[E^x^2]/2

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2320

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 6847

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} &=\frac {1}{2} \text {Subst}\left (\int e^{e^x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^{x^2}\right )\\ &=\frac {\operatorname {ExpIntegralEi}\left (e^{x^2}\right )}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} \frac {\operatorname {ExpIntegralEi}\left (e^{x^2}\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^E^x^2*x,x]

[Out]

ExpIntegralEi[E^x^2]/2

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Maple [A]
time = 0.03, size = 11, normalized size = 1.10

method	result	size
derivativedivides	\(-\frac {\expIntegral _{1}\left (-{\mathrm e}^{x^{2}}\right )}{2}\)	\(11\)
default	\(-\frac {\expIntegral _{1}\left (-{\mathrm e}^{x^{2}}\right )}{2}\)	\(11\)
risch	\(-\frac {\expIntegral _{1}\left (-{\mathrm e}^{x^{2}}\right )}{2}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(exp(x^2)),x,method=_RETURNVERBOSE)

[Out]

-1/2*Ei(1,-exp(x^2))

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Maxima [A]
time = 0.35, size = 7, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, {\rm Ei}\left (e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(exp(x^2)),x, algorithm="maxima")

[Out]

1/2*Ei(e^(x^2))

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Fricas [A]
time = 0.57, size = 7, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, {\rm Ei}\left (e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(exp(x^2)),x, algorithm="fricas")

[Out]

1/2*Ei(e^(x^2))

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Sympy [A]
time = 0.92, size = 7, normalized size = 0.70 \begin {gather*} \frac {\operatorname {Ei}{\left (e^{x^{2}} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(exp(x**2)),x)

[Out]

Ei(exp(x**2))/2

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Giac [A]
time = 0.40, size = 7, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, {\rm Ei}\left (e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*exp(exp(x^2)),x, algorithm="giac")

[Out]

1/2*Ei(e^(x^2))

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Mupad [B]
time = 0.05, size = 7, normalized size = 0.70 \begin {gather*} \frac {\mathrm {ei}\left ({\mathrm {e}}^{x^2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(exp(x^2)),x)

[Out]

ei(exp(x^2))/2

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x*exp(exp(x^2)),x)

[Out]

not solved

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