3.59.99 \(\int \frac {e^{-2 e^x+2 x} (24 x+24 x^2-24 e^x x^2+e^{2 e^x-2 x} \log (4))}{\log (4)} \, dx\)

Optimal. Leaf size=23 \[ -3+x+\frac {12 e^{-2 e^x+2 x} x^2}{\log (4)} \]

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Rubi [F]  time = 0.86, 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 {e^{-2 e^x+2 x} \left (24 x+24 x^2-24 e^x x^2+e^{2 e^x-2 x} \log (4)\right )}{\log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-2*E^x + 2*x)*(24*x + 24*x^2 - 24*E^x*x^2 + E^(2*E^x - 2*x)*Log[4]))/Log[4],x]

[Out]

x + (24*Defer[Int][x/E^(2*(E^x - x)), x])/Log[4] + (24*Defer[Int][x^2/E^(2*(E^x - x)), x])/Log[4] - (24*Defer[
Int][E^(-2*(E^x - x) + x)*x^2, x])/Log[4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{-2 e^x+2 x} \left (24 x+24 x^2-24 e^x x^2+e^{2 e^x-2 x} \log (4)\right ) \, dx}{\log (4)}\\ &=\frac {\int e^{-2 \left (e^x-x\right )} \left (24 x+24 x^2-24 e^x x^2+e^{2 e^x-2 x} \log (4)\right ) \, dx}{\log (4)}\\ &=\frac {\int \left (24 e^{-2 \left (e^x-x\right )} x+24 e^{-2 \left (e^x-x\right )} x^2-24 e^{-2 \left (e^x-x\right )+x} x^2+\log (4)\right ) \, dx}{\log (4)}\\ &=x+\frac {24 \int e^{-2 \left (e^x-x\right )} x \, dx}{\log (4)}+\frac {24 \int e^{-2 \left (e^x-x\right )} x^2 \, dx}{\log (4)}-\frac {24 \int e^{-2 \left (e^x-x\right )+x} x^2 \, dx}{\log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 23, normalized size = 1.00 \begin {gather*} \frac {x \left (12 e^{-2 e^x+2 x} x+\log (4)\right )}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2*E^x + 2*x)*(24*x + 24*x^2 - 24*E^x*x^2 + E^(2*E^x - 2*x)*Log[4]))/Log[4],x]

[Out]

(x*(12*E^(-2*E^x + 2*x)*x + Log[4]))/Log[4]

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fricas [A]  time = 0.84, size = 33, normalized size = 1.43 \begin {gather*} \frac {{\left (x e^{\left (-2 \, x + 2 \, e^{x}\right )} \log \relax (2) + 6 \, x^{2}\right )} e^{\left (2 \, x - 2 \, e^{x}\right )}}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*log(2)*exp(exp(x)-x)^2-24*exp(x)*x^2+24*x^2+24*x)/log(2)/exp(exp(x)-x)^2,x, algorithm="fricas
")

[Out]

(x*e^(-2*x + 2*e^x)*log(2) + 6*x^2)*e^(2*x - 2*e^x)/log(2)

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giac [A]  time = 0.33, size = 24, normalized size = 1.04 \begin {gather*} \frac {6 \, x^{2} e^{\left (2 \, x - 2 \, e^{x}\right )} + x \log \relax (2)}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*log(2)*exp(exp(x)-x)^2-24*exp(x)*x^2+24*x^2+24*x)/log(2)/exp(exp(x)-x)^2,x, algorithm="giac")

[Out]

(6*x^2*e^(2*x - 2*e^x) + x*log(2))/log(2)

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maple [A]  time = 0.09, size = 21, normalized size = 0.91

method	result	size
risch	\(x +\frac {6 x^{2} {\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x}}{\ln \relax (2)}\)	\(21\)
norman	\(\left ({\mathrm e}^{2 \,{\mathrm e}^{x}-2 x} x +\frac {6 x^{2}}{\ln \relax (2)}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{x}+2 x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*ln(2)*exp(exp(x)-x)^2-24*exp(x)*x^2+24*x^2+24*x)/ln(2)/exp(exp(x)-x)^2,x,method=_RETURNVERBOSE)

[Out]

x+6*x^2*exp(-2*exp(x)+2*x)/ln(2)

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maxima [A]  time = 0.43, size = 24, normalized size = 1.04 \begin {gather*} \frac {6 \, x^{2} e^{\left (2 \, x - 2 \, e^{x}\right )} + x \log \relax (2)}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*log(2)*exp(exp(x)-x)^2-24*exp(x)*x^2+24*x^2+24*x)/log(2)/exp(exp(x)-x)^2,x, algorithm="maxima
")

[Out]

(6*x^2*e^(2*x - 2*e^x) + x*log(2))/log(2)

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mupad [B]  time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} x+\frac {6\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}}{\ln \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x - 2*exp(x))*(12*x - 12*x^2*exp(x) + exp(2*exp(x) - 2*x)*log(2) + 12*x^2))/log(2),x)

[Out]

x + (6*x^2*exp(2*x)*exp(-2*exp(x)))/log(2)

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sympy [A]  time = 0.19, size = 19, normalized size = 0.83 \begin {gather*} \frac {6 x^{2} e^{2 x - 2 e^{x}}}{\log {\relax (2 )}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*ln(2)*exp(exp(x)-x)**2-24*exp(x)*x**2+24*x**2+24*x)/ln(2)/exp(exp(x)-x)**2,x)

[Out]

6*x**2*exp(2*x - 2*exp(x))/log(2) + x

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