3.16.40 \(\int e^{\frac {e^6+e^3 (6+16 x^4)+16 \log ^2(3)}{64 e^3}} x^3 \, dx\)

Optimal. Leaf size=27 \[ e^{\frac {1}{4} \left (\frac {1}{16} \left (6+e^3\right )+x^4+\frac {\log ^2(3)}{e^3}\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2225, 2209} \begin {gather*} e^{\frac {x^4}{4}+\frac {6 e^3+e^6+16 \log ^2(3)}{64 e^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((E^6 + E^3*(6 + 16*x^4) + 16*Log[3]^2)/(64*E^3))*x^3,x]

[Out]

E^(x^4/4 + (6*E^3 + E^6 + 16*Log[3]^2)/(64*E^3))

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 2225

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && BinomialQ[v, x] &&  !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{\frac {x^4}{4}+\frac {6 e^3+e^6+16 \log ^2(3)}{64 e^3}} x^3 \, dx\\ &=e^{\frac {x^4}{4}+\frac {6 e^3+e^6+16 \log ^2(3)}{64 e^3}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 25, normalized size = 0.93 \begin {gather*} e^{\frac {1}{64} \left (6+e^3+16 x^4+\frac {16 \log ^2(3)}{e^3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((E^6 + E^3*(6 + 16*x^4) + 16*Log[3]^2)/(64*E^3))*x^3,x]

[Out]

E^((6 + E^3 + 16*x^4 + (16*Log[3]^2)/E^3)/64)

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 25, normalized size = 0.93 \begin {gather*} e^{\left (\frac {1}{64} \, {\left (2 \, {\left (8 \, x^{4} + 3\right )} e^{3} + 16 \, \log \relax (3)^{2} + e^{6}\right )} e^{\left (-3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*exp(1/64*(16*log(3)^2+exp(3)^2+(16*x^4+6)*exp(3))/exp(3)),x, algorithm="fricas")

[Out]

e^(1/64*(2*(8*x^4 + 3)*e^3 + 16*log(3)^2 + e^6)*e^(-3))

________________________________________________________________________________________

giac [A]  time = 0.22, size = 20, normalized size = 0.74 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} + \frac {1}{4} \, e^{\left (-3\right )} \log \relax (3)^{2} + \frac {1}{64} \, e^{3} + \frac {3}{32}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*exp(1/64*(16*log(3)^2+exp(3)^2+(16*x^4+6)*exp(3))/exp(3)),x, algorithm="giac")

[Out]

e^(1/4*x^4 + 1/4*e^(-3)*log(3)^2 + 1/64*e^3 + 3/32)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 26, normalized size = 0.96

method	result	size
risch	\({\mathrm e}^{\frac {\left (16 x^{4} {\mathrm e}^{3}+16 \ln \relax (3)^{2}+6 \,{\mathrm e}^{3}+{\mathrm e}^{6}\right ) {\mathrm e}^{-3}}{64}}\)	\(26\)
derivativedivides	\({\mathrm e}^{\frac {\left (16 \ln \relax (3)^{2}+{\mathrm e}^{6}+\left (16 x^{4}+6\right ) {\mathrm e}^{3}\right ) {\mathrm e}^{-3}}{64}}\)	\(29\)
default	\({\mathrm e}^{\frac {\left (16 \ln \relax (3)^{2}+{\mathrm e}^{6}+\left (16 x^{4}+6\right ) {\mathrm e}^{3}\right ) {\mathrm e}^{-3}}{64}}\)	\(29\)
norman	\({\mathrm e}^{\frac {\left (16 \ln \relax (3)^{2}+{\mathrm e}^{6}+\left (16 x^{4}+6\right ) {\mathrm e}^{3}\right ) {\mathrm e}^{-3}}{64}}\)	\(29\)
gosper	\({\mathrm e}^{\frac {\left (16 x^{4} {\mathrm e}^{3}+16 \ln \relax (3)^{2}+6 \,{\mathrm e}^{3}+{\mathrm e}^{6}\right ) {\mathrm e}^{-3}}{64}}\)	\(30\)
meijerg	\(-{\mathrm e}^{{\mathrm e}^{-3} \left (\frac {\ln \relax (3)^{2}}{4}+\frac {{\mathrm e}^{6}}{64}+\frac {3 \,{\mathrm e}^{3}}{32}\right )} \left (1-{\mathrm e}^{\frac {x^{4}}{4}}\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(1/64*(16*ln(3)^2+exp(3)^2+(16*x^4+6)*exp(3))/exp(3)),x,method=_RETURNVERBOSE)

[Out]

exp(1/64*(16*x^4*exp(3)+16*ln(3)^2+6*exp(3)+exp(6))*exp(-3))

________________________________________________________________________________________

maxima [A]  time = 0.73, size = 25, normalized size = 0.93 \begin {gather*} e^{\left (\frac {1}{64} \, {\left (2 \, {\left (8 \, x^{4} + 3\right )} e^{3} + 16 \, \log \relax (3)^{2} + e^{6}\right )} e^{\left (-3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*exp(1/64*(16*log(3)^2+exp(3)^2+(16*x^4+6)*exp(3))/exp(3)),x, algorithm="maxima")

[Out]

e^(1/64*(2*(8*x^4 + 3)*e^3 + 16*log(3)^2 + e^6)*e^(-3))

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 23, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^3}{64}}\,{\mathrm {e}}^{3/32}\,{\mathrm {e}}^{\frac {x^4}{4}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-3}\,{\ln \relax (3)}^2}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(exp(-3)*(exp(6)/64 + (exp(3)*(16*x^4 + 6))/64 + log(3)^2/4)),x)

[Out]

exp(exp(3)/64)*exp(3/32)*exp(x^4/4)*exp((exp(-3)*log(3)^2)/4)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 27, normalized size = 1.00 \begin {gather*} e^{\frac {\frac {\left (16 x^{4} + 6\right ) e^{3}}{64} + \frac {\log {\relax (3 )}^{2}}{4} + \frac {e^{6}}{64}}{e^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*exp(1/64*(16*ln(3)**2+exp(3)**2+(16*x**4+6)*exp(3))/exp(3)),x)

[Out]

exp(((16*x**4 + 6)*exp(3)/64 + log(3)**2/4 + exp(6)/64)*exp(-3))

________________________________________________________________________________________