3.23.90 \(\int \frac {1}{2} e^{6 x+e^{6 x} (e^{-6 x} (4-2 x)+x)} (-x-5 x^2+6 x^3+e^{-6 x} (-1+4 x-2 x^2)) \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{2} e^{4-2 x+e^{6 x} x} (-1+x) x \]

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Rubi [F]  time = 1.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{2} \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) \left (-x-5 x^2+6 x^3+e^{-6 x} \left (-1+4 x-2 x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(6*x + E^(6*x)*((4 - 2*x)/E^(6*x) + x))*(-x - 5*x^2 + 6*x^3 + (-1 + 4*x - 2*x^2)/E^(6*x)))/2,x]

[Out]

-1/2*Defer[Int][E^(4 + (-2 + E^(6*x))*x), x] + 2*Defer[Int][E^(4 + (-2 + E^(6*x))*x)*x, x] - Defer[Int][E^(4 +
(4 + E^(6*x))*x)*x, x]/2 - Defer[Int][E^(4 + (-2 + E^(6*x))*x)*x^2, x] - (5*Defer[Int][E^(4 + (4 + E^(6*x))*x
)*x^2, x])/2 + 3*Defer[Int][E^(4 + (4 + E^(6*x))*x)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) \left (-x-5 x^2+6 x^3+e^{-6 x} \left (-1+4 x-2 x^2\right )\right ) \, dx\\ &=\frac {1}{2} \int \left (-\exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) x-5 \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) x^2+6 \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) x^3-e^{e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )} \left (1-4 x+2 x^2\right )\right ) \, dx\\ &=-\left (\frac {1}{2} \int \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) x \, dx\right )-\frac {1}{2} \int e^{e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )} \left (1-4 x+2 x^2\right ) \, dx-\frac {5}{2} \int \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) x^2 \, dx+3 \int \exp \left (6 x+e^{6 x} \left (e^{-6 x} (4-2 x)+x\right )\right ) x^3 \, dx\\ &=-\left (\frac {1}{2} \int e^{4+\left (4+e^{6 x}\right ) x} x \, dx\right )-\frac {1}{2} \int e^{4+\left (-2+e^{6 x}\right ) x} \left (1-4 x+2 x^2\right ) \, dx-\frac {5}{2} \int e^{4+\left (4+e^{6 x}\right ) x} x^2 \, dx+3 \int e^{4+\left (4+e^{6 x}\right ) x} x^3 \, dx\\ &=-\left (\frac {1}{2} \int e^{4+\left (4+e^{6 x}\right ) x} x \, dx\right )-\frac {1}{2} \int \left (e^{4+\left (-2+e^{6 x}\right ) x}-4 e^{4+\left (-2+e^{6 x}\right ) x} x+2 e^{4+\left (-2+e^{6 x}\right ) x} x^2\right ) \, dx-\frac {5}{2} \int e^{4+\left (4+e^{6 x}\right ) x} x^2 \, dx+3 \int e^{4+\left (4+e^{6 x}\right ) x} x^3 \, dx\\ &=-\left (\frac {1}{2} \int e^{4+\left (-2+e^{6 x}\right ) x} \, dx\right )-\frac {1}{2} \int e^{4+\left (4+e^{6 x}\right ) x} x \, dx+2 \int e^{4+\left (-2+e^{6 x}\right ) x} x \, dx-\frac {5}{2} \int e^{4+\left (4+e^{6 x}\right ) x} x^2 \, dx+3 \int e^{4+\left (4+e^{6 x}\right ) x} x^3 \, dx-\int e^{4+\left (-2+e^{6 x}\right ) x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{2} e^{4+\left (-2+e^{6 x}\right ) x} (-1+x) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(6*x + E^(6*x)*((4 - 2*x)/E^(6*x) + x))*(-x - 5*x^2 + 6*x^3 + (-1 + 4*x - 2*x^2)/E^(6*x)))/2,x]

[Out]

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

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fricas [A]  time = 0.83, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - x\right )} e^{\left (x e^{\left (6 \, x\right )} - 2 \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2+4*x-1)*exp(-6*x)+6*x^3-5*x^2-x)*exp(((4-2*x)*exp(-6*x)+x)/exp(-6*x))/exp(-6*x),x, algor
ithm="fricas")

[Out]

1/2*(x^2 - x)*e^(x*e^(6*x) - 2*x + 4)

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giac [B]  time = 0.35, size = 38, normalized size = 1.73 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e^{\left (x e^{\left (6 \, x\right )} + 4 \, x + 4\right )} - x e^{\left (x e^{\left (6 \, x\right )} + 4 \, x + 4\right )}\right )} e^{\left (-6 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2+4*x-1)*exp(-6*x)+6*x^3-5*x^2-x)*exp(((4-2*x)*exp(-6*x)+x)/exp(-6*x))/exp(-6*x),x, algor
ithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 31, normalized size = 1.41

method	result	size
risch	\(\frac {x \left (x -1\right ) {\mathrm e}^{-\left (2 \,{\mathrm e}^{-6 x} x -4 \,{\mathrm e}^{-6 x}-x \right ) {\mathrm e}^{6 x}}}{2}\)	\(31\)
norman	\(\left (-\frac {{\mathrm e}^{-6 x} x \,{\mathrm e}^{\left (\left (4-2 x \right ) {\mathrm e}^{-6 x}+x \right ) {\mathrm e}^{6 x}}}{2}+\frac {{\mathrm e}^{-6 x} x^{2} {\mathrm e}^{\left (\left (4-2 x \right ) {\mathrm e}^{-6 x}+x \right ) {\mathrm e}^{6 x}}}{2}\right ) {\mathrm e}^{6 x}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-2*x^2+4*x-1)*exp(-6*x)+6*x^3-5*x^2-x)*exp(((4-2*x)*exp(-6*x)+x)/exp(-6*x))/exp(-6*x),x,method=_RETU
RNVERBOSE)

[Out]

1/2*x*(x-1)*exp(-(2*exp(-6*x)*x-4*exp(-6*x)-x)*exp(6*x))

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maxima [A]  time = 0.75, size = 25, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e^{4} - x e^{4}\right )} e^{\left (x e^{\left (6 \, x\right )} - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2+4*x-1)*exp(-6*x)+6*x^3-5*x^2-x)*exp(((4-2*x)*exp(-6*x)+x)/exp(-6*x))/exp(-6*x),x, algor
ithm="maxima")

[Out]

1/2*(x^2*e^4 - x*e^4)*e^(x*e^(6*x) - 2*x)

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mupad [B]  time = 1.40, size = 19, normalized size = 0.86 \begin {gather*} \frac {x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{x\,{\mathrm {e}}^{6\,x}}\,\left (x-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(6*x)*exp(exp(6*x)*(x - exp(-6*x)*(2*x - 4)))*(x + exp(-6*x)*(2*x^2 - 4*x + 1) + 5*x^2 - 6*x^3))/2,x)

[Out]

(x*exp(-2*x)*exp(4)*exp(x*exp(6*x))*(x - 1))/2

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sympy [A]  time = 0.23, size = 24, normalized size = 1.09 \begin {gather*} \frac {\left (x^{2} - x\right ) e^{\left (x + \left (4 - 2 x\right ) e^{- 6 x}\right ) e^{6 x}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x**2+4*x-1)*exp(-6*x)+6*x**3-5*x**2-x)*exp(((4-2*x)*exp(-6*x)+x)/exp(-6*x))/exp(-6*x),x)

[Out]

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

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