3.61.100 \(\int e^{13-16 x+4 x^2} (16-3 e^{-13+16 x-4 x^2}-8 x) \, dx\)

Optimal. Leaf size=24 \[ \frac {3}{25}-e^{-3+(-4+2 x)^2}+3 (5-x) \]

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Rubi [A]  time = 0.12, antiderivative size = 18, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6688, 2236} \begin {gather*} -e^{4 x^2-16 x+13}-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(13 - 16*x + 4*x^2)*(16 - 3*E^(-13 + 16*x - 4*x^2) - 8*x),x]

[Out]

-E^(13 - 16*x + 4*x^2) - 3*x

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3-8 e^{13-16 x+4 x^2} (-2+x)\right ) \, dx\\ &=-3 x-8 \int e^{13-16 x+4 x^2} (-2+x) \, dx\\ &=-e^{13-16 x+4 x^2}-3 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 0.75 \begin {gather*} -e^{13-16 x+4 x^2}-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(13 - 16*x + 4*x^2)*(16 - 3*E^(-13 + 16*x - 4*x^2) - 8*x),x]

[Out]

-E^(13 - 16*x + 4*x^2) - 3*x

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fricas [A]  time = 0.49, size = 17, normalized size = 0.71 \begin {gather*} -3 \, x - e^{\left (4 \, x^{2} - 16 \, x + 13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(-4*x^2+16*x-13)-8*x+16)/exp(-4*x^2+16*x-13),x, algorithm="fricas")

[Out]

-3*x - e^(4*x^2 - 16*x + 13)

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giac [A]  time = 0.13, size = 17, normalized size = 0.71 \begin {gather*} -3 \, x - e^{\left (4 \, x^{2} - 16 \, x + 13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(-4*x^2+16*x-13)-8*x+16)/exp(-4*x^2+16*x-13),x, algorithm="giac")

[Out]

-3*x - e^(4*x^2 - 16*x + 13)

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maple [A]  time = 0.04, size = 18, normalized size = 0.75

method	result	size
default	\(-3 x -{\mathrm e}^{4 x^{2}-16 x +13}\)	\(18\)
risch	\(-3 x -{\mathrm e}^{4 x^{2}-16 x +13}\)	\(18\)
norman	\(\left (-1-3 x \,{\mathrm e}^{-4 x^{2}+16 x -13}\right ) {\mathrm e}^{4 x^{2}-16 x +13}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*exp(-4*x^2+16*x-13)-8*x+16)/exp(-4*x^2+16*x-13),x,method=_RETURNVERBOSE)

[Out]

-3*x-exp(4*x^2-16*x+13)

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maxima [C]  time = 1.95, size = 61, normalized size = 2.54 \begin {gather*} -4 i \, \sqrt {\pi } \operatorname {erf}\left (2 i \, x - 4 i\right ) e^{\left (-3\right )} - {\left (\frac {4 \, \sqrt {\pi } {\left (x - 2\right )} {\left (\operatorname {erf}\left (2 \, \sqrt {-{\left (x - 2\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 2\right )}^{2}}} + e^{\left (4 \, {\left (x - 2\right )}^{2}\right )}\right )} e^{\left (-3\right )} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(-4*x^2+16*x-13)-8*x+16)/exp(-4*x^2+16*x-13),x, algorithm="maxima")

[Out]

-4*I*sqrt(pi)*erf(2*I*x - 4*I)*e^(-3) - (4*sqrt(pi)*(x - 2)*(erf(2*sqrt(-(x - 2)^2)) - 1)/sqrt(-(x - 2)^2) + e
^(4*(x - 2)^2))*e^(-3) - 3*x

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mupad [B]  time = 0.09, size = 18, normalized size = 0.75 \begin {gather*} -3\,x-{\mathrm {e}}^{-16\,x}\,{\mathrm {e}}^{13}\,{\mathrm {e}}^{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(4*x^2 - 16*x + 13)*(8*x + 3*exp(16*x - 4*x^2 - 13) - 16),x)

[Out]

- 3*x - exp(-16*x)*exp(13)*exp(4*x^2)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.62 \begin {gather*} - 3 x - e^{4 x^{2} - 16 x + 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*exp(-4*x**2+16*x-13)-8*x+16)/exp(-4*x**2+16*x-13),x)

[Out]

-3*x - exp(4*x**2 - 16*x + 13)

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