3.38.57 \(\int \frac {(4-8 x) \log ^2(\frac {x^2}{4})+e^{\frac {x-x \log (\frac {x^2}{4})}{\log (\frac {x^2}{4})}} (-2 x+2 x^2+(x-x^2) \log (\frac {x^2}{4})+(-1+x+x^2) \log ^2(\frac {x^2}{4}))}{(9 x^2-18 x^3+9 x^4) \log ^2(\frac {x^2}{4})} \, dx\)

Optimal. Leaf size=33 \[ \frac {-4+e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{9 \left (x-x^2\right )} \]

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Rubi [F]  time = 7.28, 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 {(4-8 x) \log ^2\left (\frac {x^2}{4}\right )+e^{\frac {x-x \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )}} \left (-2 x+2 x^2+\left (x-x^2\right ) \log \left (\frac {x^2}{4}\right )+\left (-1+x+x^2\right ) \log ^2\left (\frac {x^2}{4}\right )\right )}{\left (9 x^2-18 x^3+9 x^4\right ) \log ^2\left (\frac {x^2}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 - 8*x)*Log[x^2/4]^2 + E^((x - x*Log[x^2/4])/Log[x^2/4])*(-2*x + 2*x^2 + (x - x^2)*Log[x^2/4] + (-1 + x
+ x^2)*Log[x^2/4]^2))/((9*x^2 - 18*x^3 + 9*x^4)*Log[x^2/4]^2),x]

[Out]

-4/(9*(1 - x)*x) + Defer[Int][E^(-x + x/Log[x^2/4])/(-1 + x)^2, x]/9 + Defer[Int][E^(-x + x/Log[x^2/4])/(-1 +
x), x]/9 - Defer[Int][E^(-x + x/Log[x^2/4])/x^2, x]/9 - Defer[Int][E^(-x + x/Log[x^2/4])/x, x]/9 + (2*Defer[In
t][E^(-x + x/Log[x^2/4])/((-1 + x)*Log[x^2/4]^2), x])/9 - (2*Defer[Int][E^(-x + x/Log[x^2/4])/(x*Log[x^2/4]^2)
, x])/9 - Defer[Int][E^(-x + x/Log[x^2/4])/((-1 + x)*Log[x^2/4]), x]/9 + Defer[Int][E^(-x + x/Log[x^2/4])/(x*L
og[x^2/4]), x]/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(4-8 x) \log ^2\left (\frac {x^2}{4}\right )+e^{\frac {x-x \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )}} \left (-2 x+2 x^2+\left (x-x^2\right ) \log \left (\frac {x^2}{4}\right )+\left (-1+x+x^2\right ) \log ^2\left (\frac {x^2}{4}\right )\right )}{x^2 \left (9-18 x+9 x^2\right ) \log ^2\left (\frac {x^2}{4}\right )} \, dx\\ &=\int \frac {(4-8 x) \log ^2\left (\frac {x^2}{4}\right )+e^{\frac {x-x \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )}} \left (-2 x+2 x^2+\left (x-x^2\right ) \log \left (\frac {x^2}{4}\right )+\left (-1+x+x^2\right ) \log ^2\left (\frac {x^2}{4}\right )\right )}{9 (-1+x)^2 x^2 \log ^2\left (\frac {x^2}{4}\right )} \, dx\\ &=\frac {1}{9} \int \frac {(4-8 x) \log ^2\left (\frac {x^2}{4}\right )+e^{\frac {x-x \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )}} \left (-2 x+2 x^2+\left (x-x^2\right ) \log \left (\frac {x^2}{4}\right )+\left (-1+x+x^2\right ) \log ^2\left (\frac {x^2}{4}\right )\right )}{(-1+x)^2 x^2 \log ^2\left (\frac {x^2}{4}\right )} \, dx\\ &=\frac {1}{9} \int \left (-\frac {4 (-1+2 x)}{(-1+x)^2 x^2}+\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}} \left (-2 x+2 x^2+x \log \left (\frac {x^2}{4}\right )-x^2 \log \left (\frac {x^2}{4}\right )-\log ^2\left (\frac {x^2}{4}\right )+x \log ^2\left (\frac {x^2}{4}\right )+x^2 \log ^2\left (\frac {x^2}{4}\right )\right )}{(1-x)^2 x^2 \log ^2\left (\frac {x^2}{4}\right )}\right ) \, dx\\ &=\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}} \left (-2 x+2 x^2+x \log \left (\frac {x^2}{4}\right )-x^2 \log \left (\frac {x^2}{4}\right )-\log ^2\left (\frac {x^2}{4}\right )+x \log ^2\left (\frac {x^2}{4}\right )+x^2 \log ^2\left (\frac {x^2}{4}\right )\right )}{(1-x)^2 x^2 \log ^2\left (\frac {x^2}{4}\right )} \, dx-\frac {4}{9} \int \frac {-1+2 x}{(-1+x)^2 x^2} \, dx\\ &=-\frac {4}{9 (1-x) x}+\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}} \left (2 (-1+x) x-(-1+x) x \log \left (\frac {x^2}{4}\right )+\left (-1+x+x^2\right ) \log ^2\left (\frac {x^2}{4}\right )\right )}{(1-x)^2 x^2 \log ^2\left (\frac {x^2}{4}\right )} \, dx\\ &=-\frac {4}{9 (1-x) x}+\frac {1}{9} \int \left (\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}} \left (-1+x+x^2\right )}{(-1+x)^2 x^2}+\frac {2 e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) x \log ^2\left (\frac {x^2}{4}\right )}-\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) x \log \left (\frac {x^2}{4}\right )}\right ) \, dx\\ &=-\frac {4}{9 (1-x) x}+\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}} \left (-1+x+x^2\right )}{(-1+x)^2 x^2} \, dx-\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) x \log \left (\frac {x^2}{4}\right )} \, dx+\frac {2}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) x \log ^2\left (\frac {x^2}{4}\right )} \, dx\\ &=-\frac {4}{9 (1-x) x}+\frac {1}{9} \int \left (\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x)^2}+\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{-1+x}-\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x^2}-\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x}\right ) \, dx-\frac {1}{9} \int \left (\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) \log \left (\frac {x^2}{4}\right )}-\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x \log \left (\frac {x^2}{4}\right )}\right ) \, dx+\frac {2}{9} \int \left (\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) \log ^2\left (\frac {x^2}{4}\right )}-\frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x \log ^2\left (\frac {x^2}{4}\right )}\right ) \, dx\\ &=-\frac {4}{9 (1-x) x}+\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x)^2} \, dx+\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{-1+x} \, dx-\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x^2} \, dx-\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x} \, dx-\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) \log \left (\frac {x^2}{4}\right )} \, dx+\frac {1}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x \log \left (\frac {x^2}{4}\right )} \, dx+\frac {2}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{(-1+x) \log ^2\left (\frac {x^2}{4}\right )} \, dx-\frac {2}{9} \int \frac {e^{-x+\frac {x}{\log \left (\frac {x^2}{4}\right )}}}{x \log ^2\left (\frac {x^2}{4}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 39, normalized size = 1.18 \begin {gather*} \frac {e^{-x} \left (4 e^x-e^{\frac {x}{\log \left (\frac {x^2}{4}\right )}}\right )}{9 (-1+x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((4 - 8*x)*Log[x^2/4]^2 + E^((x - x*Log[x^2/4])/Log[x^2/4])*(-2*x + 2*x^2 + (x - x^2)*Log[x^2/4] + (
-1 + x + x^2)*Log[x^2/4]^2))/((9*x^2 - 18*x^3 + 9*x^4)*Log[x^2/4]^2),x]

[Out]

(4*E^x - E^(x/Log[x^2/4]))/(9*E^x*(-1 + x)*x)

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fricas [A]  time = 0.86, size = 36, normalized size = 1.09 \begin {gather*} -\frac {e^{\left (-\frac {x \log \left (\frac {1}{4} \, x^{2}\right ) - x}{\log \left (\frac {1}{4} \, x^{2}\right )}\right )} - 4}{9 \, {\left (x^{2} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x-1)*log(1/4*x^2)^2+(-x^2+x)*log(1/4*x^2)+2*x^2-2*x)*exp((-x*log(1/4*x^2)+x)/log(1/4*x^2))+(-
8*x+4)*log(1/4*x^2)^2)/(9*x^4-18*x^3+9*x^2)/log(1/4*x^2)^2,x, algorithm="fricas")

[Out]

-1/9*(e^(-(x*log(1/4*x^2) - x)/log(1/4*x^2)) - 4)/(x^2 - x)

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giac [A]  time = 2.29, size = 36, normalized size = 1.09 \begin {gather*} -\frac {e^{\left (-\frac {x \log \left (\frac {1}{4} \, x^{2}\right ) - x}{\log \left (\frac {1}{4} \, x^{2}\right )}\right )} - 4}{9 \, {\left (x^{2} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x-1)*log(1/4*x^2)^2+(-x^2+x)*log(1/4*x^2)+2*x^2-2*x)*exp((-x*log(1/4*x^2)+x)/log(1/4*x^2))+(-
8*x+4)*log(1/4*x^2)^2)/(9*x^4-18*x^3+9*x^2)/log(1/4*x^2)^2,x, algorithm="giac")

[Out]

-1/9*(e^(-(x*log(1/4*x^2) - x)/log(1/4*x^2)) - 4)/(x^2 - x)

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maple [A]  time = 0.61, size = 42, normalized size = 1.27

method	result	size
risch	\(\frac {4}{9 x \left (x -1\right )}-\frac {{\mathrm e}^{-\frac {x \left (\ln \left (\frac {x^{2}}{4}\right )-1\right )}{\ln \left (\frac {x^{2}}{4}\right )}}}{9 x \left (x -1\right )}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2+x-1)*ln(1/4*x^2)^2+(-x^2+x)*ln(1/4*x^2)+2*x^2-2*x)*exp((-x*ln(1/4*x^2)+x)/ln(1/4*x^2))+(-8*x+4)*ln(
1/4*x^2)^2)/(9*x^4-18*x^3+9*x^2)/ln(1/4*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

4/9/x/(x-1)-1/9/x/(x-1)*exp(-x*(ln(1/4*x^2)-1)/ln(1/4*x^2))

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maxima [A]  time = 0.51, size = 35, normalized size = 1.06 \begin {gather*} \frac {{\left (4 \, e^{x} - e^{\left (-\frac {x}{2 \, {\left (\log \relax (2) - \log \relax (x)\right )}}\right )}\right )} e^{\left (-x\right )}}{9 \, {\left (x^{2} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+x-1)*log(1/4*x^2)^2+(-x^2+x)*log(1/4*x^2)+2*x^2-2*x)*exp((-x*log(1/4*x^2)+x)/log(1/4*x^2))+(-
8*x+4)*log(1/4*x^2)^2)/(9*x^4-18*x^3+9*x^2)/log(1/4*x^2)^2,x, algorithm="maxima")

[Out]

1/9*(4*e^x - e^(-1/2*x/(log(2) - log(x))))*e^(-x)/(x^2 - x)

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mupad [B]  time = 2.30, size = 65, normalized size = 1.97 \begin {gather*} \frac {2^{\frac {2\,x}{\ln \left (\frac {x^2}{4}\right )}}\,{\mathrm {e}}^{\frac {x}{\ln \left (\frac {x^2}{4}\right )}}}{9\,\left (x-x^2\right )\,{\left (x^2\right )}^{\frac {x}{\ln \left (\frac {x^2}{4}\right )}}}-\frac {4}{9\,\left (x-x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x - x*log(x^2/4))/log(x^2/4))*(log(x^2/4)*(x - x^2) - 2*x + log(x^2/4)^2*(x + x^2 - 1) + 2*x^2) - lo
g(x^2/4)^2*(8*x - 4))/(log(x^2/4)^2*(9*x^2 - 18*x^3 + 9*x^4)),x)

[Out]

(2^((2*x)/log(x^2/4))*exp(x/log(x^2/4)))/(9*(x - x^2)*(x^2)^(x/log(x^2/4))) - 4/(9*(x - x^2))

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sympy [A]  time = 0.55, size = 36, normalized size = 1.09 \begin {gather*} - \frac {e^{\frac {- x \log {\left (\frac {x^{2}}{4} \right )} + x}{\log {\left (\frac {x^{2}}{4} \right )}}}}{9 x^{2} - 9 x} + \frac {4}{9 x^{2} - 9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2+x-1)*ln(1/4*x**2)**2+(-x**2+x)*ln(1/4*x**2)+2*x**2-2*x)*exp((-x*ln(1/4*x**2)+x)/ln(1/4*x**2)
)+(-8*x+4)*ln(1/4*x**2)**2)/(9*x**4-18*x**3+9*x**2)/ln(1/4*x**2)**2,x)

[Out]

-exp((-x*log(x**2/4) + x)/log(x**2/4))/(9*x**2 - 9*x) + 4/(9*x**2 - 9*x)

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