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

Optimal. Leaf size=19 \[ e^{\left (-e^x+x\right ) \left (-3-\frac {1}{x}+\log (2)\right )} \]

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Rubi [F]  time = 2.50, 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 {\exp \left (\frac {-x-3 x^2+x^2 \log (2)+e^x (1+3 x-x \log (2))}{x}\right ) \left (-3 x^2+x^2 \log (2)+e^x \left (-1+x+3 x^2-x^2 \log (2)\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3 - Log[2])*Defer[Int][E^(x + ((-E^x + x)*(-1 - x*(3 - Log[2])))/x), x] - (3 - Log[2])*Defer[Int][E^(((-E^x +
x)*(-1 - x*(3 - Log[2])))/x), x] - Defer[Int][E^(x + ((-E^x + x)*(-1 - x*(3 - Log[2])))/x)/x^2, x] + Defer[In
t][E^(x + ((-E^x + x)*(-1 - x*(3 - Log[2])))/x)/x, x]

Rubi steps

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

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Mathematica [A]  time = 1.52, size = 24, normalized size = 1.26 \begin {gather*} 2^x e^{-1-3 x+e^x \left (3+\frac {1}{x}-\log (2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2^x*E^(-1 - 3*x + E^x*(3 + x^(-1) - Log[2]))

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fricas [A]  time = 0.79, size = 33, normalized size = 1.74 \begin {gather*} e^{\left (\frac {x^{2} \log \relax (2) - 3 \, x^{2} - {\left (x \log \relax (2) - 3 \, x - 1\right )} e^{x} - x}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 26, normalized size = 1.37 \begin {gather*} e^{\left (x \log \relax (2) - e^{x} \log \relax (2) - 3 \, x + \frac {e^{x}}{x} + 3 \, e^{x} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 22, normalized size = 1.16

method	result	size
risch	\({\mathrm e}^{-\frac {\left (x \ln \relax (2)-3 x -1\right ) \left ({\mathrm e}^{x}-x \right )}{x}}\)	\(22\)
norman	\({\mathrm e}^{\frac {\left (-x \ln \relax (2)+3 x +1\right ) {\mathrm e}^{x}+x^{2} \ln \relax (2)-3 x^{2}-x}{x}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-(x*ln(2)-3*x-1)*(exp(x)-x)/x)

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maxima [A]  time = 0.56, size = 26, normalized size = 1.37 \begin {gather*} e^{\left (x \log \relax (2) - e^{x} \log \relax (2) - 3 \, x + \frac {e^{x}}{x} + 3 \, e^{x} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.94, size = 27, normalized size = 1.42 \begin {gather*} 2^{x-{\mathrm {e}}^x}\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{3\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.32, size = 29, normalized size = 1.53 \begin {gather*} e^{\frac {- 3 x^{2} + x^{2} \log {\relax (2 )} - x + \left (- x \log {\relax (2 )} + 3 x + 1\right ) e^{x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp((-3*x**2 + x**2*log(2) - x + (-x*log(2) + 3*x + 1)*exp(x))/x)

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