3.93.2 \(\int \frac {2 e^{x^2} x^4-2 e^{e^{\frac {\log ^2(\frac {174}{7})}{x^2}}+\frac {\log ^2(\frac {174}{7})}{x^2}} \log ^2(\frac {174}{7})}{x^3} \, dx\)

Optimal. Leaf size=20 \[ e^{e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}}+e^{x^2} \]

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Rubi [A]  time = 0.39, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {14, 2209, 6715, 2282, 2194} \begin {gather*} e^{x^2}+e^{e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*E^x^2*x^4 - 2*E^(E^(Log[174/7]^2/x^2) + Log[174/7]^2/x^2)*Log[174/7]^2)/x^3,x]

[Out]

E^E^(Log[174/7]^2/x^2) + E^x^2

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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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 (2 e^{x^2} x-\frac {2 e^{e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}+\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}} \log ^2\left (\frac {174}{7}\right )}{x^3}\right ) \, dx\\ &=2 \int e^{x^2} x \, dx-\left (2 \log ^2\left (\frac {174}{7}\right )\right ) \int \frac {e^{e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}+\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}}{x^3} \, dx\\ &=e^{x^2}+\log ^2\left (\frac {174}{7}\right ) \operatorname {Subst}\left (\int e^{e^{x \log ^2\left (\frac {174}{7}\right )}+x \log ^2\left (\frac {174}{7}\right )} \, dx,x,\frac {1}{x^2}\right )\\ &=e^{x^2}+\operatorname {Subst}\left (\int e^x \, dx,x,e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}\right )\\ &=e^{e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}}+e^{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 20, normalized size = 1.00 \begin {gather*} e^{e^{\frac {\log ^2\left (\frac {174}{7}\right )}{x^2}}}+e^{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^x^2*x^4 - 2*E^(E^(Log[174/7]^2/x^2) + Log[174/7]^2/x^2)*Log[174/7]^2)/x^3,x]

[Out]

E^E^(Log[174/7]^2/x^2) + E^x^2

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fricas [B]  time = 0.68, size = 48, normalized size = 2.40 \begin {gather*} {\left (e^{\left (x^{2} + \frac {\log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )} + e^{\left (\frac {x^{2} e^{\left (\frac {\log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )} + \log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )}\right )} e^{\left (-\frac {\log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(174/7)^2*exp(log(174/7)^2/x^2)*exp(exp(log(174/7)^2/x^2))+2*x^4*exp(x^2))/x^3,x, algorithm="
fricas")

[Out]

(e^(x^2 + log(174/7)^2/x^2) + e^((x^2*e^(log(174/7)^2/x^2) + log(174/7)^2)/x^2))*e^(-log(174/7)^2/x^2)

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giac [B]  time = 0.30, size = 48, normalized size = 2.40 \begin {gather*} {\left (e^{\left (x^{2} + \frac {\log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )} + e^{\left (\frac {x^{2} e^{\left (\frac {\log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )} + \log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )}\right )} e^{\left (-\frac {\log \left (\frac {174}{7}\right )^{2}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(174/7)^2*exp(log(174/7)^2/x^2)*exp(exp(log(174/7)^2/x^2))+2*x^4*exp(x^2))/x^3,x, algorithm="
giac")

[Out]

(e^(x^2 + log(174/7)^2/x^2) + e^((x^2*e^(log(174/7)^2/x^2) + log(174/7)^2)/x^2))*e^(-log(174/7)^2/x^2)

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maple [A]  time = 0.12, size = 25, normalized size = 1.25

method	result	size
risch	\({\mathrm e}^{x^{2}}+{\mathrm e}^{{\mathrm e}^{\frac {\left (\ln \relax (2)+\ln \relax (3)+\ln \left (29\right )-\ln \relax (7)\right )^{2}}{x^{2}}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(174/7)^2*exp(ln(174/7)^2/x^2)*exp(exp(ln(174/7)^2/x^2))+2*x^4*exp(x^2))/x^3,x,method=_RETURNVERBOSE
)

[Out]

exp(x^2)+exp(exp((ln(2)+ln(3)+ln(29)-ln(7))^2/x^2))

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maxima [B]  time = 0.52, size = 148, normalized size = 7.40 \begin {gather*} \frac {e^{\left (e^{\left (\frac {\log \left (29\right )^{2}}{x^{2}} - \frac {2 \, \log \left (29\right ) \log \relax (7)}{x^{2}} + \frac {\log \relax (7)^{2}}{x^{2}} + \frac {2 \, \log \left (29\right ) \log \relax (3)}{x^{2}} - \frac {2 \, \log \relax (7) \log \relax (3)}{x^{2}} + \frac {\log \relax (3)^{2}}{x^{2}} + \frac {2 \, \log \left (29\right ) \log \relax (2)}{x^{2}} - \frac {2 \, \log \relax (7) \log \relax (2)}{x^{2}} + \frac {2 \, \log \relax (3) \log \relax (2)}{x^{2}} + \frac {\log \relax (2)^{2}}{x^{2}}\right )}\right )} \log \left (\frac {174}{7}\right )^{2}}{\log \left (29\right )^{2} - 2 \, \log \left (29\right ) \log \relax (7) + \log \relax (7)^{2} + 2 \, {\left (\log \left (29\right ) - \log \relax (7)\right )} \log \relax (3) + \log \relax (3)^{2} + 2 \, {\left (\log \left (29\right ) - \log \relax (7) + \log \relax (3)\right )} \log \relax (2) + \log \relax (2)^{2}} + e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(174/7)^2*exp(log(174/7)^2/x^2)*exp(exp(log(174/7)^2/x^2))+2*x^4*exp(x^2))/x^3,x, algorithm="
maxima")

[Out]

e^(e^(log(29)^2/x^2 - 2*log(29)*log(7)/x^2 + log(7)^2/x^2 + 2*log(29)*log(3)/x^2 - 2*log(7)*log(3)/x^2 + log(3
)^2/x^2 + 2*log(29)*log(2)/x^2 - 2*log(7)*log(2)/x^2 + 2*log(3)*log(2)/x^2 + log(2)^2/x^2))*log(174/7)^2/(log(
29)^2 - 2*log(29)*log(7) + log(7)^2 + 2*(log(29) - log(7))*log(3) + log(3)^2 + 2*(log(29) - log(7) + log(3))*l
og(2) + log(2)^2) + e^(x^2)

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mupad [B]  time = 7.84, size = 36, normalized size = 1.80 \begin {gather*} {\mathrm {e}}^{x^2}+{\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {{\ln \relax (7)}^2}{x^2}}\,{\mathrm {e}}^{\frac {{\ln \left (174\right )}^2}{x^2}}}{7^{\frac {2\,\ln \left (174\right )}{x^2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4*exp(x^2) - 2*exp(log(174/7)^2/x^2)*exp(exp(log(174/7)^2/x^2))*log(174/7)^2)/x^3,x)

[Out]

exp(x^2) + exp((exp(log(7)^2/x^2)*exp(log(174)^2/x^2))/7^((2*log(174))/x^2))

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sympy [A]  time = 1.16, size = 17, normalized size = 0.85 \begin {gather*} e^{x^{2}} + e^{e^{\frac {\log {\left (\frac {174}{7} \right )}^{2}}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(174/7)**2*exp(ln(174/7)**2/x**2)*exp(exp(ln(174/7)**2/x**2))+2*x**4*exp(x**2))/x**3,x)

[Out]

exp(x**2) + exp(exp(log(174/7)**2/x**2))

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