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

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

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Rubi [F]  time = 0.61, 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 {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \left (x+\left (e^{-2 (1+x)} x^4\right )^{\frac {1}{x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.76, size = 23, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, \left (x^{4} e^{\left (-2 \, x - 2\right )}\right )^{\left (\frac {1}{x^{2}}\right )} x + \frac {1}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} + {\left (x^{2} - 2 \, x - 2 \, \log \left (x^{4} e^{\left (-2 \, x - 2\right )}\right ) + 4\right )} \left (x^{4} e^{\left (-2 \, x - 2\right )}\right )^{\left (\frac {1}{x^{2}}\right )}}{2 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 26, normalized size = 1.18

method	result	size
default	\(\frac {{\mathrm e}^{\frac {\ln \left (x^{4} {\mathrm e}^{-2 x -2}\right )}{x^{2}}} x}{2}+\frac {x^{2}}{2}\)	\(26\)
risch	\(\frac {x^{2}}{2}+\frac {x \,x^{\frac {4}{x^{2}}} \left ({\mathrm e}^{x +1}\right )^{-\frac {2}{x^{2}}} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i x^{2}\right )^{3}-2 \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )-\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4} {\mathrm e}^{-2 x -2}\right )^{3}-\mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x^{4} {\mathrm e}^{-2 x -2}\right )^{2}-\mathrm {csgn}\left (i x^{4} {\mathrm e}^{-2 x -2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-2 x -2}\right )+\mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x^{4} {\mathrm e}^{-2 x -2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x -2}\right )+\mathrm {csgn}\left (i x^{3}\right )^{3}-\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4}\right )^{3}-\mathrm {csgn}\left (i {\mathrm e}^{2 x +2}\right )^{3}+2 \mathrm {csgn}\left (i {\mathrm e}^{2 x +2}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x +1}\right )-\mathrm {csgn}\left (i {\mathrm e}^{2 x +2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x +1}\right )^{2}\right )}{2 x^{2}}}}{2}\)	\(357\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*exp(ln(x^4/exp(x+1)^2)/x^2)*x+1/2*x^2

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maxima [A]  time = 0.54, size = 28, normalized size = 1.27 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{\left (-\frac {2}{x} + \frac {4 \, \log \relax (x)}{x^{2}} - \frac {2}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.68, size = 28, normalized size = 1.27 \begin {gather*} \frac {x^2}{2}+\frac {x\,{\mathrm {e}}^{-\frac {2}{x}}\,{\mathrm {e}}^{-\frac {2}{x^2}}\,{\left (x^4\right )}^{\frac {1}{x^2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2/2 + (x*exp(-2/x)*exp(-2/x^2)*(x^4)^(1/x^2))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int 2 x\, dx + \int e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}}\, dx + \int \frac {8 e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}}}{x^{2}}\, dx + \int \left (- \frac {2 e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}}}{x}\right )\, dx + \int \left (- \frac {2 e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}} \log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}\right )\, dx}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(2*x, x) + Integral(exp(-2/x**2)*exp(log(x**4*exp(-2*x))/x**2), x) + Integral(8*exp(-2/x**2)*exp(log(
x**4*exp(-2*x))/x**2)/x**2, x) + Integral(-2*exp(-2/x**2)*exp(log(x**4*exp(-2*x))/x**2)/x, x) + Integral(-2*ex
p(-2/x**2)*exp(log(x**4*exp(-2*x))/x**2)*log(x**4*exp(-2*x))/x**2, x))/2

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