3.37.67 \(\int \frac {-48+48 x-12 x^2+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} (4 x+8 x^2)+(24-12 x) \log (x)-3 \log ^2(x)}{48 x-48 x^2+12 x^3+(-24 x+12 x^2) \log (x)+3 x \log ^2(x)+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} (16 x^2-16 x^3+4 x^4+(-8 x^2+4 x^3) \log (x)+x^2 \log ^2(x))} \, dx\) [3667]

Optimal. Leaf size=26 \[ 9+\log \left (e^{2+\frac {4}{4-2 x-\log (x)}}+\frac {3}{x}\right ) \]

[Out]

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

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Rubi [F]
time = 23.13, 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 {-48+48 x-12 x^2+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} \left (4 x+8 x^2\right )+(24-12 x) \log (x)-3 \log ^2(x)}{48 x-48 x^2+12 x^3+\left (-24 x+12 x^2\right ) \log (x)+3 x \log ^2(x)+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} \left (16 x^2-16 x^3+4 x^4+\left (-8 x^2+4 x^3\right ) \log (x)+x^2 \log ^2(x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-48 + 48*x - 12*x^2 + E^((-12 + 4*x + 2*Log[x])/(-4 + 2*x + Log[x]))*(4*x + 8*x^2) + (24 - 12*x)*Log[x] -
3*Log[x]^2)/(48*x - 48*x^2 + 12*x^3 + (-24*x + 12*x^2)*Log[x] + 3*x*Log[x]^2 + E^((-12 + 4*x + 2*Log[x])/(-4
+ 2*x + Log[x]))*(16*x^2 - 16*x^3 + 4*x^4 + (-8*x^2 + 4*x^3)*Log[x] + x^2*Log[x]^2)),x]

[Out]

4*Defer[Int][(E^((4*x)/(-4 + 2*x + Log[x]))*x^(2/(-4 + 2*x + Log[x])))/((3*E^(12/(-4 + 2*x + Log[x])) + E^((4*
x)/(-4 + 2*x + Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))*(4 - 2*x - Log[x])^2), x] + 8*Defer[Int][(E^((4*x)/(-4
+ 2*x + Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))/((3*E^(12/(-4 + 2*x + Log[x])) + E^((4*x)/(-4 + 2*x + Log[x]))
*x^(1 + 2/(-4 + 2*x + Log[x])))*(4 - 2*x - Log[x])^2), x] + 48*Defer[Int][E^(12/(-4 + 2*x + Log[x]))/((3*E^(12
/(-4 + 2*x + Log[x])) + E^((4*x)/(-4 + 2*x + Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))*(-4 + 2*x + Log[x])^2), x
] - 48*Defer[Int][E^(12/(-4 + 2*x + Log[x]))/(x*(3*E^(12/(-4 + 2*x + Log[x])) + E^((4*x)/(-4 + 2*x + Log[x]))*
x^(1 + 2/(-4 + 2*x + Log[x])))*(-4 + 2*x + Log[x])^2), x] - 12*Defer[Int][(E^(12/(-4 + 2*x + Log[x]))*x)/((3*E
^(12/(-4 + 2*x + Log[x])) + E^((4*x)/(-4 + 2*x + Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))*(-4 + 2*x + Log[x])^2
), x] - 12*Defer[Int][(E^(12/(-4 + 2*x + Log[x]))*Log[x])/((3*E^(12/(-4 + 2*x + Log[x])) + E^((4*x)/(-4 + 2*x
+ Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))*(-4 + 2*x + Log[x])^2), x] + 24*Defer[Int][(E^(12/(-4 + 2*x + Log[x]
))*Log[x])/(x*(3*E^(12/(-4 + 2*x + Log[x])) + E^((4*x)/(-4 + 2*x + Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))*(-4
+ 2*x + Log[x])^2), x] - 3*Defer[Int][(E^(12/(-4 + 2*x + Log[x]))*Log[x]^2)/(x*(3*E^(12/(-4 + 2*x + Log[x]))
+ E^((4*x)/(-4 + 2*x + Log[x]))*x^(1 + 2/(-4 + 2*x + Log[x])))*(-4 + 2*x + Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \left (-48+48 x-12 x^2+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} \left (4 x+8 x^2\right )+(24-12 x) \log (x)-3 \log ^2(x)\right )}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2} \, dx\\ &=\int \left (\frac {4 \exp \left (\frac {12}{-4+2 x+\log (x)}+\frac {4 (-3+x)}{-4+2 x+\log (x)}\right ) x^{\frac {2}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2}+\frac {8 \exp \left (\frac {12}{-4+2 x+\log (x)}+\frac {4 (-3+x)}{-4+2 x+\log (x)}\right ) x^{1+\frac {2}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2}+\frac {48 e^{\frac {12}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2}-\frac {48 e^{\frac {12}{-4+2 x+\log (x)}}}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2}-\frac {12 e^{\frac {12}{-4+2 x+\log (x)}} x}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2}-\frac {12 e^{\frac {12}{-4+2 x+\log (x)}} \log (x)}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2}+\frac {24 e^{\frac {12}{-4+2 x+\log (x)}} \log (x)}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2}-\frac {3 e^{\frac {12}{-4+2 x+\log (x)}} \log ^2(x)}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \log ^2(x)}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx\right )+4 \int \frac {\exp \left (\frac {12}{-4+2 x+\log (x)}+\frac {4 (-3+x)}{-4+2 x+\log (x)}\right ) x^{\frac {2}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2} \, dx+8 \int \frac {\exp \left (\frac {12}{-4+2 x+\log (x)}+\frac {4 (-3+x)}{-4+2 x+\log (x)}\right ) x^{1+\frac {2}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2} \, dx-12 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} x}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx-12 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \log (x)}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx+24 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \log (x)}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx+48 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx-48 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}}}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx\\ &=-\left (3 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \log ^2(x)}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx\right )+4 \int \frac {e^{\frac {4 x}{-4+2 x+\log (x)}} x^{\frac {2}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2} \, dx+8 \int \frac {e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (4-2 x-\log (x))^2} \, dx-12 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} x}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx-12 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \log (x)}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx+24 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}} \log (x)}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx+48 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}}}{\left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx-48 \int \frac {e^{\frac {12}{-4+2 x+\log (x)}}}{x \left (3 e^{\frac {12}{-4+2 x+\log (x)}}+e^{\frac {4 x}{-4+2 x+\log (x)}} x^{1+\frac {2}{-4+2 x+\log (x)}}\right ) (-4+2 x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]
time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-48+48 x-12 x^2+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} \left (4 x+8 x^2\right )+(24-12 x) \log (x)-3 \log ^2(x)}{48 x-48 x^2+12 x^3+\left (-24 x+12 x^2\right ) \log (x)+3 x \log ^2(x)+e^{\frac {-12+4 x+2 \log (x)}{-4+2 x+\log (x)}} \left (16 x^2-16 x^3+4 x^4+\left (-8 x^2+4 x^3\right ) \log (x)+x^2 \log ^2(x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-48 + 48*x - 12*x^2 + E^((-12 + 4*x + 2*Log[x])/(-4 + 2*x + Log[x]))*(4*x + 8*x^2) + (24 - 12*x)*Lo
g[x] - 3*Log[x]^2)/(48*x - 48*x^2 + 12*x^3 + (-24*x + 12*x^2)*Log[x] + 3*x*Log[x]^2 + E^((-12 + 4*x + 2*Log[x]
)/(-4 + 2*x + Log[x]))*(16*x^2 - 16*x^3 + 4*x^4 + (-8*x^2 + 4*x^3)*Log[x] + x^2*Log[x]^2)),x]

[Out]

Integrate[(-48 + 48*x - 12*x^2 + E^((-12 + 4*x + 2*Log[x])/(-4 + 2*x + Log[x]))*(4*x + 8*x^2) + (24 - 12*x)*Lo
g[x] - 3*Log[x]^2)/(48*x - 48*x^2 + 12*x^3 + (-24*x + 12*x^2)*Log[x] + 3*x*Log[x]^2 + E^((-12 + 4*x + 2*Log[x]
)/(-4 + 2*x + Log[x]))*(16*x^2 - 16*x^3 + 4*x^4 + (-8*x^2 + 4*x^3)*Log[x] + x^2*Log[x]^2)), x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(25)=50\).
time = 0.15, size = 59, normalized size = 2.27

method	result	size
risch	\(-\frac {4}{\ln \left (x \right )+2 x -4}-\frac {2 \ln \left (x \right )+4 x -12}{\ln \left (x \right )+2 x -4}+\ln \left ({\mathrm e}^{\frac {2 \ln \left (x \right )+4 x -12}{\ln \left (x \right )+2 x -4}}+\frac {3}{x}\right )\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2+4*x)*exp((2*ln(x)+4*x-12)/(ln(x)+2*x-4))-3*ln(x)^2+(-12*x+24)*ln(x)-12*x^2+48*x-48)/((x^2*ln(x)^2+
(4*x^3-8*x^2)*ln(x)+4*x^4-16*x^3+16*x^2)*exp((2*ln(x)+4*x-12)/(ln(x)+2*x-4))+3*x*ln(x)^2+(12*x^2-24*x)*ln(x)+1
2*x^3-48*x^2+48*x),x,method=_RETURNVERBOSE)

[Out]

-4/(ln(x)+2*x-4)-(2*ln(x)+4*x-12)/(ln(x)+2*x-4)+ln(exp(2*(ln(x)+2*x-6)/(ln(x)+2*x-4))+3/x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (23) = 46\).
time = 0.36, size = 92, normalized size = 3.54 \begin {gather*} \frac {4 \, {\left (x - 2\right )}}{2 \, x + \log \left (x\right ) - 4} + \log \left (\frac {1}{3} \, {\left (x e^{\left (\frac {2 \, \log \left (x\right )}{2 \, x + \log \left (x\right ) - 4} + \frac {8}{2 \, x + \log \left (x\right ) - 4} + 2\right )} + 3 \, e^{\left (\frac {2 \, \log \left (x\right )}{2 \, x + \log \left (x\right ) - 4} + \frac {12}{2 \, x + \log \left (x\right ) - 4}\right )}\right )} e^{\left (-\frac {12}{2 \, x + \log \left (x\right ) - 4}\right )}\right ) - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+4*x)*exp((2*log(x)+4*x-12)/(log(x)+2*x-4))-3*log(x)^2+(-12*x+24)*log(x)-12*x^2+48*x-48)/((x^
2*log(x)^2+(4*x^3-8*x^2)*log(x)+4*x^4-16*x^3+16*x^2)*exp((2*log(x)+4*x-12)/(log(x)+2*x-4))+3*x*log(x)^2+(12*x^
2-24*x)*log(x)+12*x^3-48*x^2+48*x),x, algorithm="maxima")

[Out]

4*(x - 2)/(2*x + log(x) - 4) + log(1/3*(x*e^(2*log(x)/(2*x + log(x) - 4) + 8/(2*x + log(x) - 4) + 2) + 3*e^(2*
log(x)/(2*x + log(x) - 4) + 12/(2*x + log(x) - 4)))*e^(-12/(2*x + log(x) - 4))) - log(x)

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Fricas [A]
time = 0.35, size = 28, normalized size = 1.08 \begin {gather*} \log \left (\frac {x e^{\left (\frac {2 \, {\left (2 \, x + \log \left (x\right ) - 6\right )}}{2 \, x + \log \left (x\right ) - 4}\right )} + 3}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+4*x)*exp((2*log(x)+4*x-12)/(log(x)+2*x-4))-3*log(x)^2+(-12*x+24)*log(x)-12*x^2+48*x-48)/((x^
2*log(x)^2+(4*x^3-8*x^2)*log(x)+4*x^4-16*x^3+16*x^2)*exp((2*log(x)+4*x-12)/(log(x)+2*x-4))+3*x*log(x)^2+(12*x^
2-24*x)*log(x)+12*x^3-48*x^2+48*x),x, algorithm="fricas")

[Out]

log((x*e^(2*(2*x + log(x) - 6)/(2*x + log(x) - 4)) + 3)/x)

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Sympy [A]
time = 0.60, size = 24, normalized size = 0.92 \begin {gather*} \log {\left (e^{\frac {4 x + 2 \log {\left (x \right )} - 12}{2 x + \log {\left (x \right )} - 4}} + \frac {3}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2+4*x)*exp((2*ln(x)+4*x-12)/(ln(x)+2*x-4))-3*ln(x)**2+(-12*x+24)*ln(x)-12*x**2+48*x-48)/((x**
2*ln(x)**2+(4*x**3-8*x**2)*ln(x)+4*x**4-16*x**3+16*x**2)*exp((2*ln(x)+4*x-12)/(ln(x)+2*x-4))+3*x*ln(x)**2+(12*
x**2-24*x)*ln(x)+12*x**3-48*x**2+48*x),x)

[Out]

log(exp((4*x + 2*log(x) - 12)/(2*x + log(x) - 4)) + 3/x)

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Giac [A]
time = 0.63, size = 30, normalized size = 1.15 \begin {gather*} \log \left (x e^{\left (-\frac {2 \, x + \log \left (x\right )}{2 \, x + \log \left (x\right ) - 4} + 3\right )} + 3\right ) - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+4*x)*exp((2*log(x)+4*x-12)/(log(x)+2*x-4))-3*log(x)^2+(-12*x+24)*log(x)-12*x^2+48*x-48)/((x^
2*log(x)^2+(4*x^3-8*x^2)*log(x)+4*x^4-16*x^3+16*x^2)*exp((2*log(x)+4*x-12)/(log(x)+2*x-4))+3*x*log(x)^2+(12*x^
2-24*x)*log(x)+12*x^3-48*x^2+48*x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.69, size = 46, normalized size = 1.77 \begin {gather*} \ln \left (\frac {3}{x}+x^{\frac {2}{2\,x+\ln \left (x\right )-4}}\,{\mathrm {e}}^{-\frac {12}{2\,x+\ln \left (x\right )-4}}\,{\mathrm {e}}^{\frac {4\,x}{2\,x+\ln \left (x\right )-4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*log(x)^2 - 48*x + log(x)*(12*x - 24) - exp((4*x + 2*log(x) - 12)/(2*x + log(x) - 4))*(4*x + 8*x^2) + 1
2*x^2 + 48)/(48*x + 3*x*log(x)^2 + exp((4*x + 2*log(x) - 12)/(2*x + log(x) - 4))*(x^2*log(x)^2 - log(x)*(8*x^2
- 4*x^3) + 16*x^2 - 16*x^3 + 4*x^4) - log(x)*(24*x - 12*x^2) - 48*x^2 + 12*x^3),x)

[Out]

log(3/x + x^(2/(2*x + log(x) - 4))*exp(-12/(2*x + log(x) - 4))*exp((4*x)/(2*x + log(x) - 4)))

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