3.85.28 \(\int \frac {e^{\frac {2 e^{4 x^2}-21 x-x^2}{6 x}} (-x^2+e^{4 x^2} (-2+16 x^2))}{6 x^2} \, dx\)

Optimal. Leaf size=30 \[ e^{\frac {1-x}{2}+\frac {1}{3} \left (-12+\frac {e^{4 x^2}}{x}+x\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.35, antiderivative size = 27, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12, 6706} \begin {gather*} e^{\frac {-x^2+2 e^{4 x^2}-21 x}{6 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*E^(4*x^2) - 21*x - x^2)/(6*x))*(-x^2 + E^(4*x^2)*(-2 + 16*x^2)))/(6*x^2),x]

[Out]

E^((2*E^(4*x^2) - 21*x - x^2)/(6*x))

Rule 12

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{6} \int \frac {e^{\frac {2 e^{4 x^2}-21 x-x^2}{6 x}} \left (-x^2+e^{4 x^2} \left (-2+16 x^2\right )\right )}{x^2} \, dx\\ &=e^{\frac {2 e^{4 x^2}-21 x-x^2}{6 x}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.26, size = 25, normalized size = 0.83 \begin {gather*} e^{-\frac {7}{2}+\frac {e^{4 x^2}}{3 x}-\frac {x}{6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*E^(4*x^2) - 21*x - x^2)/(6*x))*(-x^2 + E^(4*x^2)*(-2 + 16*x^2)))/(6*x^2),x]

[Out]

E^(-7/2 + E^(4*x^2)/(3*x) - x/6)

________________________________________________________________________________________

fricas [A]  time = 1.08, size = 21, normalized size = 0.70 \begin {gather*} e^{\left (-\frac {x^{2} + 21 \, x - 2 \, e^{\left (4 \, x^{2}\right )}}{6 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*((16*x^2-2)*exp(4*x^2)-x^2)*exp(1/6*(2*exp(4*x^2)-x^2-21*x)/x)/x^2,x, algorithm="fricas")

[Out]

e^(-1/6*(x^2 + 21*x - 2*e^(4*x^2))/x)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 17, normalized size = 0.57 \begin {gather*} e^{\left (-\frac {1}{6} \, x + \frac {e^{\left (4 \, x^{2}\right )}}{3 \, x} - \frac {7}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*((16*x^2-2)*exp(4*x^2)-x^2)*exp(1/6*(2*exp(4*x^2)-x^2-21*x)/x)/x^2,x, algorithm="giac")

[Out]

e^(-1/6*x + 1/3*e^(4*x^2)/x - 7/2)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 22, normalized size = 0.73

method	result	size
risch	\({\mathrm e}^{-\frac {-2 \,{\mathrm e}^{4 x^{2}}+x^{2}+21 x}{6 x}}\)	\(22\)
norman	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{4 x^{2}}-x^{2}-21 x}{6 x}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6*((16*x^2-2)*exp(4*x^2)-x^2)*exp(1/6*(2*exp(4*x^2)-x^2-21*x)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(-1/6*(-2*exp(4*x^2)+x^2+21*x)/x)

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 17, normalized size = 0.57 \begin {gather*} e^{\left (-\frac {1}{6} \, x + \frac {e^{\left (4 \, x^{2}\right )}}{3 \, x} - \frac {7}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*((16*x^2-2)*exp(4*x^2)-x^2)*exp(1/6*(2*exp(4*x^2)-x^2-21*x)/x)/x^2,x, algorithm="maxima")

[Out]

e^(-1/6*x + 1/3*e^(4*x^2)/x - 7/2)

________________________________________________________________________________________

mupad [B]  time = 5.27, size = 19, normalized size = 0.63 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{4\,x^2}}{3\,x}}\,{\mathrm {e}}^{-\frac {x}{6}}\,{\mathrm {e}}^{-\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-((7*x)/2 - exp(4*x^2)/3 + x^2/6)/x)*(exp(4*x^2)*(16*x^2 - 2) - x^2))/(6*x^2),x)

[Out]

exp(exp(4*x^2)/(3*x))*exp(-x/6)*exp(-7/2)

________________________________________________________________________________________

sympy [A]  time = 0.35, size = 20, normalized size = 0.67 \begin {gather*} e^{\frac {- \frac {x^{2}}{6} - \frac {7 x}{2} + \frac {e^{4 x^{2}}}{3}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*((16*x**2-2)*exp(4*x**2)-x**2)*exp(1/6*(2*exp(4*x**2)-x**2-21*x)/x)/x**2,x)

[Out]

exp((-x**2/6 - 7*x/2 + exp(4*x**2)/3)/x)

________________________________________________________________________________________