3.61.68 \(\int (e^{6 x-4 x^2} (6-8 x)+e^{3 x-2 x^2} (66-88 x+(6-8 x) \log (16))) \, dx\)

Optimal. Leaf size=27 \[ \left (-11-e^{\left (-1+\frac {3-x}{x}\right ) x^2}-\log (16)\right )^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2236, 2244} \begin {gather*} e^{6 x-4 x^2}+2 e^{3 x-2 x^2} (11+\log (16)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(6*x - 4*x^2)*(6 - 8*x) + E^(3*x - 2*x^2)*(66 - 88*x + (6 - 8*x)*Log[16]),x]

[Out]

E^(6*x - 4*x^2) + 2*E^(3*x - 2*x^2)*(11 + Log[16])

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 2244

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{6 x-4 x^2} (6-8 x) \, dx+\int e^{3 x-2 x^2} (66-88 x+(6-8 x) \log (16)) \, dx\\ &=e^{6 x-4 x^2}+\int e^{3 x-2 x^2} (6 (11+\log (16))-8 x (11+\log (16))) \, dx\\ &=e^{6 x-4 x^2}+2 e^{3 x-2 x^2} (11+\log (16))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 29, normalized size = 1.07 \begin {gather*} e^{(3-4 x) x} \left (e^{3 x}+2 e^{2 x^2} (11+\log (16))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(6*x - 4*x^2)*(6 - 8*x) + E^(3*x - 2*x^2)*(66 - 88*x + (6 - 8*x)*Log[16]),x]

[Out]

E^((3 - 4*x)*x)*(E^(3*x) + 2*E^(2*x^2)*(11 + Log[16]))

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fricas [A]  time = 0.58, size = 29, normalized size = 1.07 \begin {gather*} 2 \, {\left (4 \, \log \relax (2) + 11\right )} e^{\left (-2 \, x^{2} + 3 \, x\right )} + e^{\left (-4 \, x^{2} + 6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+6)*exp(-2*x^2+3*x)^2+(4*(-8*x+6)*log(2)-88*x+66)*exp(-2*x^2+3*x),x, algorithm="fricas")

[Out]

2*(4*log(2) + 11)*e^(-2*x^2 + 3*x) + e^(-4*x^2 + 6*x)

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giac [A]  time = 0.22, size = 29, normalized size = 1.07 \begin {gather*} 2 \, {\left (4 \, \log \relax (2) + 11\right )} e^{\left (-2 \, x^{2} + 3 \, x\right )} + e^{\left (-4 \, x^{2} + 6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+6)*exp(-2*x^2+3*x)^2+(4*(-8*x+6)*log(2)-88*x+66)*exp(-2*x^2+3*x),x, algorithm="giac")

[Out]

2*(4*log(2) + 11)*e^(-2*x^2 + 3*x) + e^(-4*x^2 + 6*x)

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maple [A]  time = 0.04, size = 31, normalized size = 1.15

method	result	size
norman	\({\mathrm e}^{-4 x^{2}+6 x}+\left (8 \ln \relax (2)+22\right ) {\mathrm e}^{-2 x^{2}+3 x}\)	\(31\)
risch	\({\mathrm e}^{-2 x \left (2 x -3\right )}+8 \,{\mathrm e}^{-x \left (2 x -3\right )} \ln \relax (2)+22 \,{\mathrm e}^{-x \left (2 x -3\right )}\)	\(35\)
default	\({\mathrm e}^{-4 x^{2}+6 x}+22 \,{\mathrm e}^{-2 x^{2}+3 x}+8 \ln \relax (2) {\mathrm e}^{-2 x^{2}+3 x}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x+6)*exp(-2*x^2+3*x)^2+(4*(-8*x+6)*ln(2)-88*x+66)*exp(-2*x^2+3*x),x,method=_RETURNVERBOSE)

[Out]

exp(-2*x^2+3*x)^2+(8*ln(2)+22)*exp(-2*x^2+3*x)

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maxima [C]  time = 0.50, size = 189, normalized size = 7.00 \begin {gather*} 6 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x - \frac {3}{4} \, \sqrt {2}\right ) e^{\frac {9}{8}} \log \relax (2) + 2 i \, \sqrt {2} {\left (\frac {3 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (4 \, x - 3\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {{\left (4 \, x - 3\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (4 \, x - 3\right )}^{2}}} - 2 i \, \sqrt {2} e^{\left (-\frac {1}{8} \, {\left (4 \, x - 3\right )}^{2}\right )}\right )} e^{\frac {9}{8}} \log \relax (2) + \frac {33}{2} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x - \frac {3}{4} \, \sqrt {2}\right ) e^{\frac {9}{8}} + \frac {11}{2} i \, \sqrt {2} {\left (\frac {3 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (4 \, x - 3\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {{\left (4 \, x - 3\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (4 \, x - 3\right )}^{2}}} - 2 i \, \sqrt {2} e^{\left (-\frac {1}{8} \, {\left (4 \, x - 3\right )}^{2}\right )}\right )} e^{\frac {9}{8}} + e^{\left (-4 \, x^{2} + 6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+6)*exp(-2*x^2+3*x)^2+(4*(-8*x+6)*log(2)-88*x+66)*exp(-2*x^2+3*x),x, algorithm="maxima")

[Out]

6*sqrt(2)*sqrt(pi)*erf(sqrt(2)*x - 3/4*sqrt(2))*e^(9/8)*log(2) + 2*I*sqrt(2)*(3*I*sqrt(2)*sqrt(1/2)*sqrt(pi)*(
4*x - 3)*(erf(1/2*sqrt(1/2)*sqrt((4*x - 3)^2)) - 1)/sqrt((4*x - 3)^2) - 2*I*sqrt(2)*e^(-1/8*(4*x - 3)^2))*e^(9
/8)*log(2) + 33/2*sqrt(2)*sqrt(pi)*erf(sqrt(2)*x - 3/4*sqrt(2))*e^(9/8) + 11/2*I*sqrt(2)*(3*I*sqrt(2)*sqrt(1/2
)*sqrt(pi)*(4*x - 3)*(erf(1/2*sqrt(1/2)*sqrt((4*x - 3)^2)) - 1)/sqrt((4*x - 3)^2) - 2*I*sqrt(2)*e^(-1/8*(4*x -
3)^2))*e^(9/8) + e^(-4*x^2 + 6*x)

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mupad [B]  time = 0.12, size = 26, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{6\,x-4\,x^2}+{\mathrm {e}}^{3\,x-2\,x^2}\,\left (\ln \left (256\right )+22\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- exp(6*x - 4*x^2)*(8*x - 6) - exp(3*x - 2*x^2)*(88*x + 4*log(2)*(8*x - 6) - 66),x)

[Out]

exp(6*x - 4*x^2) + exp(3*x - 2*x^2)*(log(256) + 22)

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sympy [A]  time = 0.14, size = 26, normalized size = 0.96 \begin {gather*} e^{- 4 x^{2} + 6 x} + \left (8 \log {\relax (2 )} + 22\right ) e^{- 2 x^{2} + 3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x+6)*exp(-2*x**2+3*x)**2+(4*(-8*x+6)*ln(2)-88*x+66)*exp(-2*x**2+3*x),x)

[Out]

exp(-4*x**2 + 6*x) + (8*log(2) + 22)*exp(-2*x**2 + 3*x)

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