3.32.17 \(\int e^{x+\frac {1}{2} (4 x-x^2-2 x^3-2 x \log (16))} (1+3 x-x^2-3 x^3-x \log (16)) \, dx\)

Optimal. Leaf size=24 \[ e^{2 x-x \left (\frac {1}{2} (-2+x)+x^2+\log (16)\right )} x \]

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Rubi [B]  time = 0.13, antiderivative size = 63, normalized size of antiderivative = 2.62, number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6, 2288} \begin {gather*} -\frac {\left (3 x^3+x^2-x (3-\log (16))\right ) \exp \left (\frac {1}{2} \left (-2 x^3-x^2+4 x-2 x \log (16)\right )+x\right )}{-3 x^2-x+3-\log (16)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(x + (4*x - x^2 - 2*x^3 - 2*x*Log[16])/2)*(1 + 3*x - x^2 - 3*x^3 - x*Log[16]),x]

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{x+\frac {1}{2} \left (4 x-x^2-2 x^3-2 x \log (16)\right )} \left (1-x^2-3 x^3+x (3-\log (16))\right ) \, dx\\ &=-\frac {e^{x+\frac {1}{2} \left (4 x-x^2-2 x^3-2 x \log (16)\right )} \left (x^2+3 x^3-x (3-\log (16))\right )}{3-x-3 x^2-\log (16)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 22, normalized size = 0.92 \begin {gather*} 16^{-x} e^{-\frac {1}{2} x \left (-6+x+2 x^2\right )} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(x + (4*x - x^2 - 2*x^3 - 2*x*Log[16])/2)*(1 + 3*x - x^2 - 3*x^3 - x*Log[16]),x]

[Out]

x/(16^x*E^((x*(-6 + x + 2*x^2))/2))

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fricas [A]  time = 1.41, size = 22, normalized size = 0.92 \begin {gather*} x e^{\left (-x^{3} - \frac {1}{2} \, x^{2} - 4 \, x \log \relax (2) + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.27, size = 22, normalized size = 0.92 \begin {gather*} x e^{\left (-x^{3} - \frac {1}{2} \, x^{2} - 4 \, x \log \relax (2) + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 18, normalized size = 0.75

method	result	size
risch	\(x \left (\frac {1}{16}\right )^{x} {\mathrm e}^{-\frac {x \left (2+x \right ) \left (2 x -3\right )}{2}}\)	\(18\)
gosper	\({\mathrm e}^{3 x -4 x \ln \relax (2)-x^{3}-\frac {x^{2}}{2}} x\)	\(23\)
norman	\({\mathrm e}^{x} {\mathrm e}^{-4 x \ln \relax (2)-x^{3}-\frac {x^{2}}{2}+2 x} x\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(1/16)^x*exp(-1/2*x*(2+x)*(2*x-3))

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maxima [A]  time = 0.58, size = 22, normalized size = 0.92 \begin {gather*} x e^{\left (-x^{3} - \frac {1}{2} \, x^{2} - 4 \, x \log \relax (2) + 3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -{\mathrm {e}}^{2\,x-4\,x\,\ln \relax (2)-\frac {x^2}{2}-x^3}\,{\mathrm {e}}^x\,\left (4\,x\,\ln \relax (2)-3\,x+x^2+3\,x^3-1\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(2*x - 4*x*log(2) - x^2/2 - x^3)*exp(x)*(4*x*log(2) - 3*x + x^2 + 3*x^3 - 1),x)

[Out]

int(-exp(2*x - 4*x*log(2) - x^2/2 - x^3)*exp(x)*(4*x*log(2) - 3*x + x^2 + 3*x^3 - 1), x)

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sympy [A]  time = 0.21, size = 24, normalized size = 1.00 \begin {gather*} x e^{x} e^{- x^{3} - \frac {x^{2}}{2} - 4 x \log {\relax (2 )} + 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*ln(2)-3*x**3-x**2+3*x+1)*exp(-4*x*ln(2)-x**3-1/2*x**2+2*x)*exp(x),x)

[Out]

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

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