3.40.49 \(\int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} (-8 \log (x)+(-4+x) \log ^2(x))}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} (-16 x+8 x^2)} \, dx\)

Optimal. Leaf size=31 \[ 3+\frac {\log ^2(x)}{2-\sqrt {\frac {e^{x/2}}{x^2}}-x} \]

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Rubi [F]  time = 14.38, 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 {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((16 - 8*x)*Log[x] + 4*x*Log[x]^2 + Sqrt[E^(x/2)/x^2]*(-8*Log[x] + (-4 + x)*Log[x]^2))/((4*E^(x/2))/x + 16
*x - 16*x^2 + 4*x^3 + Sqrt[E^(x/2)/x^2]*(-16*x + 8*x^2)),x]

[Out]

(-2*Sqrt[E^(x/2)/x^2]*x*Log[x]*Defer[Int][E^(x/4)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4), x])/E^(x/4) + 4*Log[x]*Defe
r[Int][x/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4), x] - 2*Log[x]*Defer[Int][x^2/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4), x] -
(8*Sqrt[E^(x/2)/x^2]*x*Defer[Int][(E^(x/4)*x^2*Log[x]^2)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4)^2, x])/E^(x/4) + (14
*Sqrt[E^(x/2)/x^2]*x*Defer[Int][(E^(x/4)*x^3*Log[x]^2)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4)^2, x])/E^(x/4) - (6*Sqr
t[E^(x/2)/x^2]*x*Defer[Int][(E^(x/4)*x^4*Log[x]^2)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4)^2, x])/E^(x/4) + (Sqrt[E^(x
/2)/x^2]*x*Defer[Int][(E^(x/4)*x^5*Log[x]^2)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4)^2, x])/(2*E^(x/4)) - (Sqrt[E^(x/2
)/x^2]*x*Defer[Int][(E^(x/4)*Log[x]^2)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4), x])/E^(x/4) + (Sqrt[E^(x/2)/x^2]*x*Def
er[Int][(E^(x/4)*x*Log[x]^2)/(E^(x/2) - 4*x^2 + 4*x^3 - x^4), x])/(4*E^(x/4)) - 16*Defer[Int][(x^3*Log[x]^2)/(
-E^(x/2) + 4*x^2 - 4*x^3 + x^4)^2, x] + 36*Defer[Int][(x^4*Log[x]^2)/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4)^2, x] -
26*Defer[Int][(x^5*Log[x]^2)/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4)^2, x] + 7*Defer[Int][(x^6*Log[x]^2)/(-E^(x/2) +
4*x^2 - 4*x^3 + x^4)^2, x] - Defer[Int][(x^7*Log[x]^2)/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4)^2, x]/2 + 4*Defer[Int]
[(x*Log[x]^2)/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4), x] - 4*Defer[Int][(x^2*Log[x]^2)/(-E^(x/2) + 4*x^2 - 4*x^3 + x
^4), x] + Defer[Int][(x^3*Log[x]^2)/(-E^(x/2) + 4*x^2 - 4*x^3 + x^4), x]/2 + (2*Sqrt[E^(x/2)/x^2]*x*Defer[Int]
[Defer[Int][E^(x/4)/(E^(x/2) - (-2 + x)^2*x^2), x]/x, x])/E^(x/4) - 4*Defer[Int][Defer[Int][x/(-E^(x/2) + (-2
+ x)^2*x^2), x]/x, x] + 2*Defer[Int][Defer[Int][x^2/(-E^(x/2) + (-2 + x)^2*x^2), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \sqrt {\frac {e^{x/2}}{x^2}} x \log (x)}{e^{x/2}-4 x^2+4 x^3-x^4}-\frac {8 \sqrt {\frac {e^{x/2}}{x^2}} x^3 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2}+\frac {14 \sqrt {\frac {e^{x/2}}{x^2}} x^4 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2}-\frac {6 \sqrt {\frac {e^{x/2}}{x^2}} x^5 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} x^6 \log ^2(x)}{2 \left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2}-\frac {\sqrt {\frac {e^{x/2}}{x^2}} x \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4}+\frac {\sqrt {\frac {e^{x/2}}{x^2}} x^2 \log ^2(x)}{4 \left (e^{x/2}-4 x^2+4 x^3-x^4\right )}-\frac {(-2+x)^2 x^3 \left (8-10 x+x^2\right ) \log ^2(x)}{2 \left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2}+\frac {x \log (x) \left (8-4 x+8 \log (x)-8 x \log (x)+x^2 \log (x)\right )}{2 \left (-e^{x/2}+4 x^2-4 x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x^2 \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x^6 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-\frac {1}{2} \int \frac {(-2+x)^2 x^3 \left (8-10 x+x^2\right ) \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx+\frac {1}{2} \int \frac {x \log (x) \left (8-4 x+8 \log (x)-8 x \log (x)+x^2 \log (x)\right )}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx-2 \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x \log (x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx-6 \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x^5 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-8 \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x^3 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx+14 \int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x^4 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-\int \frac {\sqrt {\frac {e^{x/2}}{x^2}} x \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {32 x^3 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2}-\frac {72 x^4 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2}+\frac {52 x^5 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2}-\frac {14 x^6 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2}+\frac {x^7 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {8 x \log (x)}{-e^{x/2}+4 x^2-4 x^3+x^4}-\frac {4 x^2 \log (x)}{-e^{x/2}+4 x^2-4 x^3+x^4}+\frac {8 x \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4}-\frac {8 x^2 \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4}+\frac {x^3 \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4}\right ) \, dx+2 \int e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} \int \frac {e^{x/4}}{e^{x/2}-(-2+x)^2 x^2} \, dx \, dx+\frac {\left (\sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\sqrt {e^{x/2}} x \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx}{4 \sqrt {e^{x/2}}}+\frac {\left (\sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\sqrt {e^{x/2}} x^5 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx}{2 \sqrt {e^{x/2}}}-\frac {\left (\sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\sqrt {e^{x/2}} \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx}{\sqrt {e^{x/2}}}-\frac {\left (6 \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\sqrt {e^{x/2}} x^4 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx}{\sqrt {e^{x/2}}}-\frac {\left (8 \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\sqrt {e^{x/2}} x^2 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx}{\sqrt {e^{x/2}}}+\frac {\left (14 \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\sqrt {e^{x/2}} x^3 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx}{\sqrt {e^{x/2}}}-\left (2 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x \log (x)\right ) \int \frac {e^{x/4}}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^7 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx\right )+\frac {1}{2} \int \frac {x^3 \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx-2 \int \frac {x^2 \log (x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx+4 \int \frac {x \log (x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx+4 \int \frac {x \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx-4 \int \frac {x^2 \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx+7 \int \frac {x^6 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx-16 \int \frac {x^3 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx-26 \int \frac {x^5 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx+36 \int \frac {x^4 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx+\frac {1}{4} \left (e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx+\frac {1}{2} \left (e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^5 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-\left (e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx-\left (6 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^4 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-\left (8 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^2 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx+\left (14 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^3 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx+\frac {\left (2 \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{-x/4} \sqrt {e^{x/2}} \int \frac {e^{x/4}}{e^{x/2}-(-2+x)^2 x^2} \, dx}{x} \, dx}{\sqrt {e^{x/2}}}-\left (2 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x \log (x)\right ) \int \frac {e^{x/4}}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^7 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx\right )+\frac {1}{2} \int \frac {x^3 \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx+2 \int \frac {\int \frac {x^2}{-e^{x/2}+(-2+x)^2 x^2} \, dx}{x} \, dx+4 \int \frac {x \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx-4 \int \frac {x^2 \log ^2(x)}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx-4 \int \frac {\int \frac {x}{-e^{x/2}+(-2+x)^2 x^2} \, dx}{x} \, dx+7 \int \frac {x^6 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx-16 \int \frac {x^3 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx-26 \int \frac {x^5 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx+36 \int \frac {x^4 \log ^2(x)}{\left (-e^{x/2}+4 x^2-4 x^3+x^4\right )^2} \, dx+\frac {1}{4} \left (e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx+\frac {1}{2} \left (e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^5 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-\left (e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} \log ^2(x)}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx+\left (2 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {\int \frac {e^{x/4}}{e^{x/2}-(-2+x)^2 x^2} \, dx}{x} \, dx-\left (6 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^4 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-\left (8 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^2 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx+\left (14 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x\right ) \int \frac {e^{x/4} x^3 \log ^2(x)}{\left (e^{x/2}-4 x^2+4 x^3-x^4\right )^2} \, dx-(2 \log (x)) \int \frac {x^2}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx+(4 \log (x)) \int \frac {x}{-e^{x/2}+4 x^2-4 x^3+x^4} \, dx-\left (2 e^{-x/4} \sqrt {\frac {e^{x/2}}{x^2}} x \log (x)\right ) \int \frac {e^{x/4}}{e^{x/2}-4 x^2+4 x^3-x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(16-8 x) \log (x)+4 x \log ^2(x)+\sqrt {\frac {e^{x/2}}{x^2}} \left (-8 \log (x)+(-4+x) \log ^2(x)\right )}{\frac {4 e^{x/2}}{x}+16 x-16 x^2+4 x^3+\sqrt {\frac {e^{x/2}}{x^2}} \left (-16 x+8 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((16 - 8*x)*Log[x] + 4*x*Log[x]^2 + Sqrt[E^(x/2)/x^2]*(-8*Log[x] + (-4 + x)*Log[x]^2))/((4*E^(x/2))/
x + 16*x - 16*x^2 + 4*x^3 + Sqrt[E^(x/2)/x^2]*(-16*x + 8*x^2)),x]

[Out]

Integrate[((16 - 8*x)*Log[x] + 4*x*Log[x]^2 + Sqrt[E^(x/2)/x^2]*(-8*Log[x] + (-4 + x)*Log[x]^2))/((4*E^(x/2))/
x + 16*x - 16*x^2 + 4*x^3 + Sqrt[E^(x/2)/x^2]*(-16*x + 8*x^2)), x]

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fricas [A]  time = 1.00, size = 20, normalized size = 0.65 \begin {gather*} -\frac {x \log \relax (x)^{2}}{x^{2} - 2 \, x + e^{\left (\frac {1}{4} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+(-8*x+16)*log(x))/(4*exp(1/2*x)/x+(8*
x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16*x),x, algorithm="fricas")

[Out]

-x*log(x)^2/(x^2 - 2*x + e^(1/4*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, x \log \relax (x)^{2} + {\left ({\left (x - 4\right )} \log \relax (x)^{2} - 8 \, \log \relax (x)\right )} \sqrt {\frac {1}{x^{2}}} e^{\left (\frac {1}{4} \, x\right )} - 8 \, {\left (x - 2\right )} \log \relax (x)}{4 \, {\left (x^{3} + 2 \, {\left (x^{2} - 2 \, x\right )} \sqrt {\frac {1}{x^{2}}} e^{\left (\frac {1}{4} \, x\right )} - 4 \, x^{2} + 4 \, x + \frac {e^{\left (\frac {1}{2} \, x\right )}}{x}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+(-8*x+16)*log(x))/(4*exp(1/2*x)/x+(8*
x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16*x),x, algorithm="giac")

[Out]

integrate(1/4*(4*x*log(x)^2 + ((x - 4)*log(x)^2 - 8*log(x))*sqrt(x^(-2))*e^(1/4*x) - 8*(x - 2)*log(x))/(x^3 +
2*(x^2 - 2*x)*sqrt(x^(-2))*e^(1/4*x) - 4*x^2 + 4*x + e^(1/2*x)/x), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (x -4\right ) \ln \relax (x )^{2}-8 \ln \relax (x )\right ) \sqrt {\frac {{\mathrm e}^{\frac {x}{2}}}{x^{2}}}+4 x \ln \relax (x )^{2}+\left (-8 x +16\right ) \ln \relax (x )}{\frac {4 \,{\mathrm e}^{\frac {x}{2}}}{x}+\left (8 x^{2}-16 x \right ) \sqrt {\frac {{\mathrm e}^{\frac {x}{2}}}{x^{2}}}+4 x^{3}-16 x^{2}+16 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-4)*ln(x)^2-8*ln(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*ln(x)^2+(-8*x+16)*ln(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*
(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16*x),x)

[Out]

int((((x-4)*ln(x)^2-8*ln(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*ln(x)^2+(-8*x+16)*ln(x))/(4*exp(1/2*x)/x+(8*x^2-16*x)*
(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16*x),x)

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maxima [A]  time = 0.50, size = 20, normalized size = 0.65 \begin {gather*} -\frac {x \log \relax (x)^{2}}{x^{2} - 2 \, x + e^{\left (\frac {1}{4} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*log(x)^2-8*log(x))*(exp(1/2*x)/x^2)^(1/2)+4*x*log(x)^2+(-8*x+16)*log(x))/(4*exp(1/2*x)/x+(8*
x^2-16*x)*(exp(1/2*x)/x^2)^(1/2)+4*x^3-16*x^2+16*x),x, algorithm="maxima")

[Out]

-x*log(x)^2/(x^2 - 2*x + e^(1/4*x))

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mupad [B]  time = 3.26, size = 564, normalized size = 18.19 \begin {gather*} \frac {x^2\,{\left (x-2\right )}^2\,\left (\frac {x^3\,\ln \relax (x)\,\left (x^3-24\,x^2+68\,x-48\right )\,\left (8\,\ln \relax (x)-4\,x+x^2\,\ln \relax (x)-8\,x\,\ln \relax (x)+8\right )-2\,x^3\,\ln \relax (x)\,{\left (x-2\right )}^2\,\left (x+4\right )\,\left (x\,\ln \relax (x)-2\,x+4\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}+\frac {x^3\,\ln \relax (x)\,{\left (x-2\right )}^2\,\left (x+4\right )\,\left (8\,\ln \relax (x)-4\,x+x^2\,\ln \relax (x)-8\,x\,\ln \relax (x)+8\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}\right )}{{\mathrm {e}}^{x/2}-x^2\,{\left (x-2\right )}^2}-\frac {\sqrt {\frac {{\mathrm {e}}^{x/2}}{x^2}}\,\left (x^2\,\left (\frac {x^4\,\ln \relax (x)\,\left (x-2\right )\,\left (8\,x+8\,\ln \relax (x)+x^2\,\ln \relax (x)-14\,x\,\ln \relax (x)-16\right )+8\,x^3\,\ln \relax (x)\,\left (x^2-3\,x+2\right )\,\left (4\,\ln \relax (x)-x\,\ln \relax (x)+8\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}-\frac {x^4\,\ln \relax (x)\,{\left (x-2\right )}^2\,\left (4\,\ln \relax (x)-x\,\ln \relax (x)+8\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}\right )\,{\left (x-2\right )}^2-\frac {8\,x^5\,\ln \relax (x)\,\left (x-2\right )\,\left (x^2-3\,x+2\right )\,\left (8\,x+8\,\ln \relax (x)+x^2\,\ln \relax (x)-14\,x\,\ln \relax (x)-16\right )}{2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3}\right )}{{\mathrm {e}}^{x/2}-x^2\,{\left (x-2\right )}^2}-\frac {2\,x^5\,\ln \relax (x)\,{\left (x-2\right )}^2\,\left (x\,\ln \relax (x)-2\,x+4\right )\,\left (x^3-24\,x^2+68\,x-48\right )}{\left ({\mathrm {e}}^{x/2}-x^2\,{\left (x-2\right )}^2\right )\,\left (2\,x^9-48\,x^8+400\,x^7-1408\,x^6+2336\,x^5-1792\,x^4+512\,x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((8*log(x) - log(x)^2*(x - 4))*(exp(x/2)/x^2)^(1/2) - 4*x*log(x)^2 + log(x)*(8*x - 16))/(16*x + (4*exp(x/
2))/x - (16*x - 8*x^2)*(exp(x/2)/x^2)^(1/2) - 16*x^2 + 4*x^3),x)

[Out]

(x^2*(x - 2)^2*((x^3*log(x)*(68*x - 24*x^2 + x^3 - 48)*(8*log(x) - 4*x + x^2*log(x) - 8*x*log(x) + 8) - 2*x^3*
log(x)*(x - 2)^2*(x + 4)*(x*log(x) - 2*x + 4))/(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 +
2*x^9) + (x^3*log(x)*(x - 2)^2*(x + 4)*(8*log(x) - 4*x + x^2*log(x) - 8*x*log(x) + 8))/(512*x^3 - 1792*x^4 + 2
336*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 + 2*x^9)))/(exp(x/2) - x^2*(x - 2)^2) - ((exp(x/2)/x^2)^(1/2)*(x^2*((x^4
*log(x)*(x - 2)*(8*x + 8*log(x) + x^2*log(x) - 14*x*log(x) - 16) + 8*x^3*log(x)*(x^2 - 3*x + 2)*(4*log(x) - x*
log(x) + 8))/(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 + 2*x^9) - (x^4*log(x)*(x - 2)^2*(4*
log(x) - x*log(x) + 8))/(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 + 2*x^9))*(x - 2)^2 - (8*
x^5*log(x)*(x - 2)*(x^2 - 3*x + 2)*(8*x + 8*log(x) + x^2*log(x) - 14*x*log(x) - 16))/(512*x^3 - 1792*x^4 + 233
6*x^5 - 1408*x^6 + 400*x^7 - 48*x^8 + 2*x^9)))/(exp(x/2) - x^2*(x - 2)^2) - (2*x^5*log(x)*(x - 2)^2*(x*log(x)
- 2*x + 4)*(68*x - 24*x^2 + x^3 - 48))/((exp(x/2) - x^2*(x - 2)^2)*(512*x^3 - 1792*x^4 + 2336*x^5 - 1408*x^6 +
400*x^7 - 48*x^8 + 2*x^9))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-4)*ln(x)**2-8*ln(x))*(exp(1/2*x)/x**2)**(1/2)+4*x*ln(x)**2+(-8*x+16)*ln(x))/(4*exp(1/2*x)/x+(8*
x**2-16*x)*(exp(1/2*x)/x**2)**(1/2)+4*x**3-16*x**2+16*x),x)

[Out]

Timed out

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