3.23.75 \(\int \frac {-384+(-384+3 x) \log (\frac {16 x}{-128+x})+e^{16/x} (2048-16 x) \log ^2(\frac {16 x}{-128+x})}{-1152+9 x+e^{16/x} (768 x-6 x^2) \log (\frac {16 x}{-128+x})+e^{32/x} (-128 x^2+x^3) \log ^2(\frac {16 x}{-128+x})} \, dx\)

Optimal. Leaf size=31 \[ \frac {x}{-e^{16/x} x+\frac {3}{\log \left (\frac {x}{-8+\frac {x}{16}}\right )}} \]

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Rubi [F]  time = 7.70, 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 {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-384 + (-384 + 3*x)*Log[(16*x)/(-128 + x)] + E^(16/x)*(2048 - 16*x)*Log[(16*x)/(-128 + x)]^2)/(-1152 + 9*
x + E^(16/x)*(768*x - 6*x^2)*Log[(16*x)/(-128 + x)] + E^(32/x)*(-128*x^2 + x^3)*Log[(16*x)/(-128 + x)]^2),x]

[Out]

-384*Defer[Int][1/((-128 + x)*(-3 + E^(16/x)*x*Log[(16*x)/(-128 + x)])^2), x] + 3*Defer[Int][Log[(16*x)/(-128
+ x)]/(-3 + E^(16/x)*x*Log[(16*x)/(-128 + x)])^2, x] - 48*Defer[Int][Log[(16*x)/(-128 + x)]/(x*(-3 + E^(16/x)*
x*Log[(16*x)/(-128 + x)])^2), x] - 16*Defer[Int][Log[(16*x)/(-128 + x)]/(x*(-3 + E^(16/x)*x*Log[(16*x)/(-128 +
x)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {384-3 (-128+x) \log \left (\frac {16 x}{-128+x}\right )+16 e^{16/x} (-128+x) \log ^2\left (\frac {16 x}{-128+x}\right )}{(128-x) \left (3-e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx\\ &=\int \left (-\frac {16 \log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )}+\frac {3 \left (-128 x+2048 \log \left (\frac {16 x}{-128+x}\right )-144 x \log \left (\frac {16 x}{-128+x}\right )+x^2 \log \left (\frac {16 x}{-128+x}\right )\right )}{(-128+x) x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx\\ &=3 \int \frac {-128 x+2048 \log \left (\frac {16 x}{-128+x}\right )-144 x \log \left (\frac {16 x}{-128+x}\right )+x^2 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx\\ &=3 \int \left (\frac {-128 x+2048 \log \left (\frac {16 x}{-128+x}\right )-144 x \log \left (\frac {16 x}{-128+x}\right )+x^2 \log \left (\frac {16 x}{-128+x}\right )}{128 (-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}-\frac {-128 x+2048 \log \left (\frac {16 x}{-128+x}\right )-144 x \log \left (\frac {16 x}{-128+x}\right )+x^2 \log \left (\frac {16 x}{-128+x}\right )}{128 x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx\\ &=\frac {3}{128} \int \frac {-128 x+2048 \log \left (\frac {16 x}{-128+x}\right )-144 x \log \left (\frac {16 x}{-128+x}\right )+x^2 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-\frac {3}{128} \int \frac {-128 x+2048 \log \left (\frac {16 x}{-128+x}\right )-144 x \log \left (\frac {16 x}{-128+x}\right )+x^2 \log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx\\ &=-\left (\frac {3}{128} \int \frac {-128 x+\left (2048-144 x+x^2\right ) \log \left (\frac {16 x}{-128+x}\right )}{x \left (3-e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx\right )+\frac {3}{128} \int \left (-\frac {128 x}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {2048 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}-\frac {144 x \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {x^2 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx\\ &=\frac {3}{128} \int \frac {x^2 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-\frac {3}{128} \int \left (-\frac {128}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}-\frac {144 \log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {2048 \log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {x \log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx-3 \int \frac {x}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-\frac {27}{8} \int \frac {x \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx+48 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx\\ &=-\left (\frac {3}{128} \int \frac {x \log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx\right )+\frac {3}{128} \int \left (\frac {128 \log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {16384 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {x \log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx+3 \int \frac {1}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-3 \int \left (\frac {1}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {128}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx+\frac {27}{8} \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-\frac {27}{8} \int \left (\frac {\log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}+\frac {128 \log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2}\right ) \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx+48 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-48 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx\\ &=3 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{\left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-16 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )} \, dx+48 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-48 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{x \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-384 \int \frac {1}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx+384 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx-432 \int \frac {\log \left (\frac {16 x}{-128+x}\right )}{(-128+x) \left (-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.72, size = 34, normalized size = 1.10 \begin {gather*} -\frac {x \log \left (\frac {16 x}{-128+x}\right )}{-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-384 + (-384 + 3*x)*Log[(16*x)/(-128 + x)] + E^(16/x)*(2048 - 16*x)*Log[(16*x)/(-128 + x)]^2)/(-115
2 + 9*x + E^(16/x)*(768*x - 6*x^2)*Log[(16*x)/(-128 + x)] + E^(32/x)*(-128*x^2 + x^3)*Log[(16*x)/(-128 + x)]^2
),x]

[Out]

-((x*Log[(16*x)/(-128 + x)])/(-3 + E^(16/x)*x*Log[(16*x)/(-128 + x)]))

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fricas [A]  time = 0.69, size = 33, normalized size = 1.06 \begin {gather*} -\frac {x \log \left (\frac {16 \, x}{x - 128}\right )}{x e^{\frac {16}{x}} \log \left (\frac {16 \, x}{x - 128}\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x+2048)*exp(16/x)*log(16*x/(x-128))^2+(3*x-384)*log(16*x/(x-128))-384)/((x^3-128*x^2)*exp(16/x
)^2*log(16*x/(x-128))^2+(-6*x^2+768*x)*exp(16/x)*log(16*x/(x-128))+9*x-1152),x, algorithm="fricas")

[Out]

-x*log(16*x/(x - 128))/(x*e^(16/x)*log(16*x/(x - 128)) - 3)

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giac [A]  time = 0.62, size = 39, normalized size = 1.26 \begin {gather*} -\frac {3}{x e^{\frac {32}{x}} \log \left (\frac {16 \, x}{x - 128}\right ) - 3 \, e^{\frac {16}{x}}} - e^{\left (-\frac {16}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x+2048)*exp(16/x)*log(16*x/(x-128))^2+(3*x-384)*log(16*x/(x-128))-384)/((x^3-128*x^2)*exp(16/x
)^2*log(16*x/(x-128))^2+(-6*x^2+768*x)*exp(16/x)*log(16*x/(x-128))+9*x-1152),x, algorithm="giac")

[Out]

-3/(x*e^(32/x)*log(16*x/(x - 128)) - 3*e^(16/x)) - e^(-16/x)

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maple [C]  time = 0.20, size = 174, normalized size = 5.61

method	result	size
risch	\(-{\mathrm e}^{-\frac {16}{x}}+\frac {6 i {\mathrm e}^{-\frac {16}{x}}}{\pi x \,\mathrm {csgn}\left (\frac {i}{x -128}\right ) \mathrm {csgn}\left (\frac {i x}{x -128}\right )^{2} {\mathrm e}^{\frac {16}{x}}-\pi x \,\mathrm {csgn}\left (\frac {i}{x -128}\right ) \mathrm {csgn}\left (\frac {i x}{x -128}\right ) \mathrm {csgn}\left (i x \right ) {\mathrm e}^{\frac {16}{x}}-\pi x \mathrm {csgn}\left (\frac {i x}{x -128}\right )^{3} {\mathrm e}^{\frac {16}{x}}+\pi x \mathrm {csgn}\left (\frac {i x}{x -128}\right )^{2} \mathrm {csgn}\left (i x \right ) {\mathrm e}^{\frac {16}{x}}-8 i \ln \relax (2) x \,{\mathrm e}^{\frac {16}{x}}-2 i {\mathrm e}^{\frac {16}{x}} x \ln \relax (x )+2 i {\mathrm e}^{\frac {16}{x}} x \ln \left (x -128\right )+6 i}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x+2048)*exp(16/x)*ln(16*x/(x-128))^2+(3*x-384)*ln(16*x/(x-128))-384)/((x^3-128*x^2)*exp(16/x)^2*ln(1
6*x/(x-128))^2+(-6*x^2+768*x)*exp(16/x)*ln(16*x/(x-128))+9*x-1152),x,method=_RETURNVERBOSE)

[Out]

-exp(-16/x)+6*I*exp(-16/x)/(Pi*x*csgn(I/(x-128))*csgn(I*x/(x-128))^2*exp(16/x)-Pi*x*csgn(I/(x-128))*csgn(I*x/(
x-128))*csgn(I*x)*exp(16/x)-Pi*x*csgn(I*x/(x-128))^3*exp(16/x)+Pi*x*csgn(I*x/(x-128))^2*csgn(I*x)*exp(16/x)-8*
I*ln(2)*x*exp(16/x)-2*I*exp(16/x)*x*ln(x)+2*I*exp(16/x)*x*ln(x-128)+6*I)

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maxima [A]  time = 0.79, size = 47, normalized size = 1.52 \begin {gather*} -\frac {4 \, x \log \relax (2) - x \log \left (x - 128\right ) + x \log \relax (x)}{{\left (4 \, x \log \relax (2) - x \log \left (x - 128\right ) + x \log \relax (x)\right )} e^{\frac {16}{x}} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x+2048)*exp(16/x)*log(16*x/(x-128))^2+(3*x-384)*log(16*x/(x-128))-384)/((x^3-128*x^2)*exp(16/x
)^2*log(16*x/(x-128))^2+(-6*x^2+768*x)*exp(16/x)*log(16*x/(x-128))+9*x-1152),x, algorithm="maxima")

[Out]

-(4*x*log(2) - x*log(x - 128) + x*log(x))/((4*x*log(2) - x*log(x - 128) + x*log(x))*e^(16/x) - 3)

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mupad [B]  time = 1.86, size = 33, normalized size = 1.06 \begin {gather*} -\frac {x\,\ln \left (\frac {16\,x}{x-128}\right )}{x\,{\mathrm {e}}^{16/x}\,\ln \left (\frac {16\,x}{x-128}\right )-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(16/x)*log((16*x)/(x - 128))^2*(16*x - 2048) - log((16*x)/(x - 128))*(3*x - 384) + 384)/(9*x - exp(32
/x)*log((16*x)/(x - 128))^2*(128*x^2 - x^3) + exp(16/x)*log((16*x)/(x - 128))*(768*x - 6*x^2) - 1152),x)

[Out]

-(x*log((16*x)/(x - 128)))/(x*exp(16/x)*log((16*x)/(x - 128)) - 3)

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sympy [A]  time = 0.42, size = 27, normalized size = 0.87 \begin {gather*} - \frac {x \log {\left (\frac {16 x}{x - 128} \right )}}{x e^{\frac {16}{x}} \log {\left (\frac {16 x}{x - 128} \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x+2048)*exp(16/x)*ln(16*x/(x-128))**2+(3*x-384)*ln(16*x/(x-128))-384)/((x**3-128*x**2)*exp(16/
x)**2*ln(16*x/(x-128))**2+(-6*x**2+768*x)*exp(16/x)*ln(16*x/(x-128))+9*x-1152),x)

[Out]

-x*log(16*x/(x - 128))/(x*exp(16/x)*log(16*x/(x - 128)) - 3)

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