3.87.9 \(\int \frac {4 e^{\frac {1}{6} (2+3 x)} x^2+e^{\frac {1}{12} (2+3 x)} (-6 x+x^2+x^3)+e^{\frac {1}{12} (2+3 x)} (8-14 x-2 x^2) \log (x)}{16-16 x+4 x^2+4 e^{\frac {1}{6} (2+3 x)} x^2+e^{\frac {1}{12} (2+3 x)} (-16 x+8 x^2)+(32-16 x-16 e^{\frac {1}{12} (2+3 x)} x) \log (x)+16 \log ^2(x)} \, dx\) [8609]

Optimal. Leaf size=36 \[ \frac {1-x}{-1+\frac {e^{\frac {1}{4} \left (-\frac {2}{3}-x\right )} (2-x+2 \log (x))}{x}} \]

[Out]

(1-x)/((2+2*ln(x)-x)/x/exp(1/4*x+1/6)-1)

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Rubi [F]
time = 4.59, 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 e^{\frac {1}{6} (2+3 x)} x^2+e^{\frac {1}{12} (2+3 x)} \left (-6 x+x^2+x^3\right )+e^{\frac {1}{12} (2+3 x)} \left (8-14 x-2 x^2\right ) \log (x)}{16-16 x+4 x^2+4 e^{\frac {1}{6} (2+3 x)} x^2+e^{\frac {1}{12} (2+3 x)} \left (-16 x+8 x^2\right )+\left (32-16 x-16 e^{\frac {1}{12} (2+3 x)} x\right ) \log (x)+16 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E^((2 + 3*x)/6)*x^2 + E^((2 + 3*x)/12)*(-6*x + x^2 + x^3) + E^((2 + 3*x)/12)*(8 - 14*x - 2*x^2)*Log[x])
/(16 - 16*x + 4*x^2 + 4*E^((2 + 3*x)/6)*x^2 + E^((2 + 3*x)/12)*(-16*x + 8*x^2) + (32 - 16*x - 16*E^((2 + 3*x)/
12)*x)*Log[x] + 16*Log[x]^2),x]

[Out]

Defer[Int][(E^(1/6 + x/4)*x)/(-2 + x + E^(1/6 + x/4)*x - 2*Log[x])^2, x]/2 - (3*Defer[Int][(E^(1/6 + x/4)*x^2)
/(-2 + x + E^(1/6 + x/4)*x - 2*Log[x])^2, x])/4 + Defer[Int][(E^(1/6 + x/4)*x^3)/(-2 + x + E^(1/6 + x/4)*x - 2
*Log[x])^2, x]/4 + Defer[Int][(E^(1/6 + x/4)*x)/(-2 + x + E^(1/6 + x/4)*x - 2*Log[x]), x] + 2*Defer[Int][(E^(1
/6 + x/4)*Log[x])/(-2 + x + E^(1/6 + x/4)*x - 2*Log[x])^2, x] - (3*Defer[Int][(E^(1/6 + x/4)*x*Log[x])/(-2 + x
+ E^(1/6 + x/4)*x - 2*Log[x])^2, x])/2 - Defer[Int][(E^(1/6 + x/4)*x^2*Log[x])/(-2 + x + E^(1/6 + x/4)*x - 2*
Log[x])^2, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{6}+\frac {x}{4}} \left (x \left (-6+x+4 e^{\frac {1}{6}+\frac {x}{4}} x+x^2\right )-2 \left (-4+7 x+x^2\right ) \log (x)\right )}{4 \left (2-x-e^{\frac {1}{6}+\frac {x}{4}} x+2 \log (x)\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} \left (x \left (-6+x+4 e^{\frac {1}{6}+\frac {x}{4}} x+x^2\right )-2 \left (-4+7 x+x^2\right ) \log (x)\right )}{\left (2-x-e^{\frac {1}{6}+\frac {x}{4}} x+2 \log (x)\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {4 e^{\frac {1}{6}+\frac {x}{4}} x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)}+\frac {e^{\frac {1}{6}+\frac {x}{4}} (-1+x) \left (-2 x+x^2-8 \log (x)-2 x \log (x)\right )}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} (-1+x) \left (-2 x+x^2-8 \log (x)-2 x \log (x)\right )}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx+\int \frac {e^{\frac {1}{6}+\frac {x}{4}} x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)} \, dx\\ &=\frac {1}{4} \int \left (-\frac {e^{\frac {1}{6}+\frac {x}{4}} \left (-2 x+x^2-8 \log (x)-2 x \log (x)\right )}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}+\frac {e^{\frac {1}{6}+\frac {x}{4}} x \left (-2 x+x^2-8 \log (x)-2 x \log (x)\right )}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}\right ) \, dx+\int \frac {e^{\frac {1}{6}+\frac {x}{4}} x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)} \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} \left (-2 x+x^2-8 \log (x)-2 x \log (x)\right )}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx\right )+\frac {1}{4} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x \left (-2 x+x^2-8 \log (x)-2 x \log (x)\right )}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx+\int \frac {e^{\frac {1}{6}+\frac {x}{4}} x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)} \, dx\\ &=-\left (\frac {1}{4} \int \left (-\frac {2 e^{\frac {1}{6}+\frac {x}{4}} x}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}+\frac {e^{\frac {1}{6}+\frac {x}{4}} x^2}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}-\frac {8 e^{\frac {1}{6}+\frac {x}{4}} \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}-\frac {2 e^{\frac {1}{6}+\frac {x}{4}} x \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}\right ) \, dx\right )+\frac {1}{4} \int \left (-\frac {2 e^{\frac {1}{6}+\frac {x}{4}} x^2}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}+\frac {e^{\frac {1}{6}+\frac {x}{4}} x^3}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}-\frac {8 e^{\frac {1}{6}+\frac {x}{4}} x \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}-\frac {2 e^{\frac {1}{6}+\frac {x}{4}} x^2 \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2}\right ) \, dx+\int \frac {e^{\frac {1}{6}+\frac {x}{4}} x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)} \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x^2}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx\right )+\frac {1}{4} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x^3}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx-\frac {1}{2} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x^2}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx-\frac {1}{2} \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x^2 \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx+2 \int \frac {e^{\frac {1}{6}+\frac {x}{4}} \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx-2 \int \frac {e^{\frac {1}{6}+\frac {x}{4}} x \log (x)}{\left (-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)\right )^2} \, dx+\int \frac {e^{\frac {1}{6}+\frac {x}{4}} x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.35, size = 38, normalized size = 1.06 \begin {gather*} \frac {e^{\frac {1}{6}+\frac {x}{4}} (-1+x) x}{-2+x+e^{\frac {1}{6}+\frac {x}{4}} x-2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^((2 + 3*x)/6)*x^2 + E^((2 + 3*x)/12)*(-6*x + x^2 + x^3) + E^((2 + 3*x)/12)*(8 - 14*x - 2*x^2)*L
og[x])/(16 - 16*x + 4*x^2 + 4*E^((2 + 3*x)/6)*x^2 + E^((2 + 3*x)/12)*(-16*x + 8*x^2) + (32 - 16*x - 16*E^((2 +
3*x)/12)*x)*Log[x] + 16*Log[x]^2),x]

[Out]

(E^(1/6 + x/4)*(-1 + x)*x)/(-2 + x + E^(1/6 + x/4)*x - 2*Log[x])

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Maple [A]
time = 0.03, size = 29, normalized size = 0.81

method	result	size
risch	\(\frac {\left (x -1\right ) x \,{\mathrm e}^{\frac {x}{4}+\frac {1}{6}}}{x \,{\mathrm e}^{\frac {x}{4}+\frac {1}{6}}+x -2 \ln \left (x \right )-2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-14*x+8)*exp(1/4*x+1/6)*ln(x)+4*x^2*exp(1/4*x+1/6)^2+(x^3+x^2-6*x)*exp(1/4*x+1/6))/(16*ln(x)^2+(-1
6*x*exp(1/4*x+1/6)-16*x+32)*ln(x)+4*x^2*exp(1/4*x+1/6)^2+(8*x^2-16*x)*exp(1/4*x+1/6)+4*x^2-16*x+16),x,method=_
RETURNVERBOSE)

[Out]

(x-1)*x*exp(1/4*x+1/6)/(x*exp(1/4*x+1/6)+x-2*ln(x)-2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-14*x+8)*exp(1/4*x+1/6)*log(x)+4*x^2*exp(1/4*x+1/6)^2+(x^3+x^2-6*x)*exp(1/4*x+1/6))/(16*log(
x)^2+(-16*x*exp(1/4*x+1/6)-16*x+32)*log(x)+4*x^2*exp(1/4*x+1/6)^2+(8*x^2-16*x)*exp(1/4*x+1/6)+4*x^2-16*x+16),x
, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [A]
time = 0.39, size = 31, normalized size = 0.86 \begin {gather*} \frac {{\left (x^{2} - x\right )} e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )}}{x e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )} + x - 2 \, \log \left (x\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-14*x+8)*exp(1/4*x+1/6)*log(x)+4*x^2*exp(1/4*x+1/6)^2+(x^3+x^2-6*x)*exp(1/4*x+1/6))/(16*log(
x)^2+(-16*x*exp(1/4*x+1/6)-16*x+32)*log(x)+4*x^2*exp(1/4*x+1/6)^2+(8*x^2-16*x)*exp(1/4*x+1/6)+4*x^2-16*x+16),x
, algorithm="fricas")

[Out]

(x^2 - x)*e^(1/4*x + 1/6)/(x*e^(1/4*x + 1/6) + x - 2*log(x) - 2)

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Sympy [A]
time = 0.12, size = 39, normalized size = 1.08 \begin {gather*} x + \frac {- x^{2} + 2 x \log {\left (x \right )} + 3 x - 2 \log {\left (x \right )} - 2}{x e^{\frac {x}{4} + \frac {1}{6}} + x - 2 \log {\left (x \right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-14*x+8)*exp(1/4*x+1/6)*ln(x)+4*x**2*exp(1/4*x+1/6)**2+(x**3+x**2-6*x)*exp(1/4*x+1/6))/(16*
ln(x)**2+(-16*x*exp(1/4*x+1/6)-16*x+32)*ln(x)+4*x**2*exp(1/4*x+1/6)**2+(8*x**2-16*x)*exp(1/4*x+1/6)+4*x**2-16*
x+16),x)

[Out]

x + (-x**2 + 2*x*log(x) + 3*x - 2*log(x) - 2)/(x*exp(x/4 + 1/6) + x - 2*log(x) - 2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
time = 0.51, size = 79, normalized size = 2.19 \begin {gather*} \frac {{\left (3 \, x + 2\right )}^{2} e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )} - 7 \, {\left (3 \, x + 2\right )} e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )} + 10 \, e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )}}{3 \, {\left ({\left (3 \, x + 2\right )} e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )} + 3 \, x - 2 \, e^{\left (\frac {1}{4} \, x + \frac {1}{6}\right )} + 6 \, \log \left (3\right ) - 6 \, \log \left (2\right ) - 6 \, \log \left (\frac {3}{2} \, x\right ) - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-14*x+8)*exp(1/4*x+1/6)*log(x)+4*x^2*exp(1/4*x+1/6)^2+(x^3+x^2-6*x)*exp(1/4*x+1/6))/(16*log(
x)^2+(-16*x*exp(1/4*x+1/6)-16*x+32)*log(x)+4*x^2*exp(1/4*x+1/6)^2+(8*x^2-16*x)*exp(1/4*x+1/6)+4*x^2-16*x+16),x
, algorithm="giac")

[Out]

1/3*((3*x + 2)^2*e^(1/4*x + 1/6) - 7*(3*x + 2)*e^(1/4*x + 1/6) + 10*e^(1/4*x + 1/6))/((3*x + 2)*e^(1/4*x + 1/6
) + 3*x - 2*e^(1/4*x + 1/6) + 6*log(3) - 6*log(2) - 6*log(3/2*x) - 6)

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Mupad [B]
time = 5.72, size = 39, normalized size = 1.08 \begin {gather*} -\frac {x\,{\mathrm {e}}^{\frac {x}{4}+\frac {1}{6}}-x^2\,{\mathrm {e}}^{\frac {x}{4}+\frac {1}{6}}}{x-2\,\ln \left (x\right )+x\,{\mathrm {e}}^{\frac {x}{4}+\frac {1}{6}}-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*exp(x/2 + 1/3) + exp(x/4 + 1/6)*(x^2 - 6*x + x^3) - exp(x/4 + 1/6)*log(x)*(14*x + 2*x^2 - 8))/(16*l
og(x)^2 - 16*x - exp(x/4 + 1/6)*(16*x - 8*x^2) + 4*x^2*exp(x/2 + 1/3) + 4*x^2 - log(x)*(16*x + 16*x*exp(x/4 +
1/6) - 32) + 16),x)

[Out]

-(x*exp(x/4 + 1/6) - x^2*exp(x/4 + 1/6))/(x - 2*log(x) + x*exp(x/4 + 1/6) - 2)

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