3.78.62 \(\int \frac {3^{-\frac {2 (x^3-x^4)}{1+4 x^3 \log (\frac {x}{3})}} x^{\frac {2 (x^3-x^4)}{1+4 x^3 \log (\frac {x}{3})}} (2 x^2-2 x^3+(6 x^2-8 x^3) \log (\frac {x}{3})-8 x^6 \log ^2(\frac {x}{3}))}{1+8 x^3 \log (\frac {x}{3})+16 x^6 \log ^2(\frac {x}{3})} \, dx\)

Optimal. Leaf size=27 \[ 1+e^{\frac {2 (1-x)}{4+\frac {1}{x^3 \log \left (\frac {x}{3}\right )}}} \]

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Rubi [F]  time = 16.45, 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 {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]^2))/(
3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)),x]

[Out]

-1/2*Defer[Int][x^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/3^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3])), x] + (3*Def
er[Int][x^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/(3^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3]
)^2), x])/2 + 2*(1 - Log[27])*Defer[Int][x^(2 + (2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/(3^((2*(1 - x)*x^3)/(1 +
4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3])^2), x] - 2*Defer[Int][x^(3 + (2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/(3^(
(2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3])^2), x] - Defer[Int][x^((2*(1 - x)*x^3)/(1 + 4*x^3*L
og[x/3]))/(3^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3])), x] + 2*Defer[Int][(3^(1 - (2*(1 - x
)*x^3)/(1 + 4*x^3*Log[x/3]))*x^(2 + (2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*Log[x])/(1 + 4*x^3*Log[x/3])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2\ 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (1-x-\log (27)-4 x \log \left (\frac {x}{3}\right )-4 x^4 \log ^2\left (\frac {x}{3}\right )+3 \log (x)\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\\ &=2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (1-x-\log (27)-4 x \log \left (\frac {x}{3}\right )-4 x^4 \log ^2\left (\frac {x}{3}\right )+3 \log (x)\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\\ &=2 \int \left (-\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}+\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} (1-\log (27))}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}-\frac {4\ 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log \left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}-\frac {4\ 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{6+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log ^2\left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}+\frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\right )+2 \int \frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-8 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log \left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-8 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{6+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log ^2\left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+(2 (1-\log (27))) \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\right )+2 \int \frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-8 \int \left (\frac {1}{16} 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}+\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{16 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}-\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{8 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )}\right ) \, dx-8 \int \left (-\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{4 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}+\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{4 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )}\right ) \, dx+(2 (1-\log (27))) \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \, dx\right )-\frac {1}{2} \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{1+4 x^3 \log \left (\frac {x}{3}\right )} \, dx+2 \int \frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+(2 (1-\log (27))) \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+\int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{1+4 x^3 \log \left (\frac {x}{3}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]
^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)),x]

[Out]

Integrate[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]
^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)), x]

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fricas [A]  time = 1.04, size = 30, normalized size = 1.11 \begin {gather*} \frac {1}{\left (\frac {1}{3} \, x\right )^{\frac {2 \, {\left (x^{4} - x^{3}\right )}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.35, size = 47, normalized size = 1.74 \begin {gather*} \frac {\left (\frac {1}{3} \, x\right )^{\frac {2 \, x^{3}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}}{\left (\frac {1}{3} \, x\right )^{\frac {2 \, x^{4}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/3*x)^(2*x^3/(4*x^3*log(1/3*x) + 1))/(1/3*x)^(2*x^4/(4*x^3*log(1/3*x) + 1))

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maple [A]  time = 0.11, size = 33, normalized size = 1.22

method	result	size
risch	\(\left (\frac {x}{3}\right )^{-\frac {2 x^{3} \left (x -1\right )}{4 x^{3} \ln \left (\frac {x}{3}\right )+1}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1/3*x)^(-x^3*(x-1)/(4*x^3*ln(x)-4*x^3*ln(3)+1)))^2

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maxima [B]  time = 0.61, size = 47, normalized size = 1.74 \begin {gather*} e^{\left (-\frac {1}{2} \, x - \frac {x}{2 \, {\left (4 \, x^{3} \log \relax (3) - 4 \, x^{3} \log \relax (x) - 1\right )}} + \frac {1}{2 \, {\left (4 \, x^{3} \log \relax (3) - 4 \, x^{3} \log \relax (x) - 1\right )}} + \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.52, size = 62, normalized size = 2.30 \begin {gather*} {\left (\frac {1}{9}\right )}^{\frac {x^3-x^4}{4\,x^3\,\ln \relax (x)-4\,x^3\,\ln \relax (3)+1}}\,x^{\frac {2\,\left (x^3-x^4\right )}{4\,x^3\,\ln \relax (x)-4\,x^3\,\ln \relax (3)+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/9)^((x^3 - x^4)/(4*x^3*log(x) - 4*x^3*log(3) + 1))*x^((2*(x^3 - x^4))/(4*x^3*log(x) - 4*x^3*log(3) + 1))

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sympy [A]  time = 0.62, size = 26, normalized size = 0.96 \begin {gather*} e^{\frac {2 \left (- x^{4} + x^{3}\right ) \log {\left (\frac {x}{3} \right )}}{4 x^{3} \log {\left (\frac {x}{3} \right )} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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