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

Optimal. Leaf size=31 \[ \frac {e^{6-2 x-\frac {2 \left (x+\log \left (\frac {x}{2}\right )\right )^2}{x^2}} \left (e^4+x\right )}{x} \]

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Rubi [F]  time = 11.30, 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 {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*Defer[Int][E^((2*(2*x^2 - x^3 - 2*x*Log[x/2] - Log[x/2]^2))/x^2), x] - 4*Defer[Int][1/(E^((2*(-4*x^2 + x^3
+ 2*x*Log[x/2] + Log[x/2]^2))/x^2)*x^3), x] - (4 + E^4)*Defer[Int][E^((2*(2*x^2 - x^3 - 2*x*Log[x/2] - Log[x/2
]^2))/x^2)/x^2, x] - 2*Defer[Int][1/(E^((2*(-4*x^2 + x^3 + 2*x*Log[x/2] + Log[x/2]^2))/x^2)*x), x] - 4*Defer[I
nt][Log[x/2]/(E^((2*(-4*x^2 + x^3 + 2*x*Log[x/2] + Log[x/2]^2))/x^2)*x^4), x] - 4*(1 - E^4)*Defer[Int][(E^((2*
(2*x^2 - x^3 - 2*x*Log[x/2] - Log[x/2]^2))/x^2)*Log[x/2])/x^3, x] + 4*Defer[Int][(E^((2*(2*x^2 - x^3 - 2*x*Log
[x/2] - Log[x/2]^2))/x^2)*Log[x/2])/x^2, x] + 4*Defer[Int][Log[x/2]^2/(E^((2*(-4*x^2 + x^3 + 2*x*Log[x/2] + Lo
g[x/2]^2))/x^2)*x^4), x] + 4*Defer[Int][(E^((2*(2*x^2 - x^3 - 2*x*Log[x/2] - Log[x/2]^2))/x^2)*Log[x/2]^2)/x^3
, x]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 41, normalized size = 1.32 \begin {gather*} 16^{\frac {1}{x}} e^{4-2 x-\frac {2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^{-\frac {4+x}{x}} \left (e^4+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 36, normalized size = 1.16 \begin {gather*} \frac {{\left (x + e^{4}\right )} e^{\left (-\frac {2 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.58, size = 63, normalized size = 2.03 \begin {gather*} \frac {x e^{\left (-\frac {2 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )} + e^{\left (-\frac {2 \, {\left (x^{3} - 4 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 37, normalized size = 1.19

method	result	size
risch	\(\frac {\left (x +{\mathrm e}^{4}\right ) {\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}+2 x \ln \left (\frac {x}{2}\right )-2 x^{2}\right )}{x^{2}}}}{x}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.56, size = 57, normalized size = 1.84 \begin {gather*} \frac {{\left (x e^{4} + e^{8}\right )} e^{\left (-2 \, x + \frac {4 \, \log \relax (2)}{x} - \frac {2 \, \log \relax (2)^{2}}{x^{2}} - \frac {4 \, \log \relax (x)}{x} + \frac {4 \, \log \relax (2) \log \relax (x)}{x^{2}} - \frac {2 \, \log \relax (x)^{2}}{x^{2}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^4 + e^8)*e^(-2*x + 4*log(2)/x - 2*log(2)^2/x^2 - 4*log(x)/x + 4*log(2)*log(x)/x^2 - 2*log(x)^2/x^2)/x

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mupad [B]  time = 4.64, size = 57, normalized size = 1.84 \begin {gather*} \frac {2^{4/x}\,x^{\frac {4\,\ln \relax (2)}{x^2}}\,{\mathrm {e}}^{4-\frac {2\,{\ln \relax (2)}^2}{x^2}-\frac {2\,{\ln \relax (x)}^2}{x^2}-2\,x}\,\left (x+{\mathrm {e}}^4\right )}{x^{4/x}\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.49, size = 37, normalized size = 1.19 \begin {gather*} \frac {\left (x + e^{4}\right ) e^{\frac {2 \left (- x^{3} + 2 x^{2} - 2 x \log {\left (\frac {x}{2} \right )} - \log {\left (\frac {x}{2} \right )}^{2}\right )}{x^{2}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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