3.48.26 \(\int \frac {e^{\frac {3}{3+\log (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2})}} (18-36 x+6 x^2+(18-24 x+6 x^2) \log (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2})+(3-4 x+x^2) \log ^2(\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}))}{27-36 x+9 x^2+(18-24 x+6 x^2) \log (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2})+(3-4 x+x^2) \log ^2(\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2})} \, dx\)

Optimal. Leaf size=34 \[ e^{\frac {3}{3+\log \left (\frac {(3-x)^2 x^3}{4 \left (-x+x^2\right )^2}\right )}} x \]

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Rubi [F]  time = 2.05, 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 {e^{\frac {3}{3+\log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )}} \left (18-36 x+6 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )\right )}{27-36 x+9 x^2+\left (18-24 x+6 x^2\right ) \log \left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {9 x-6 x^2+x^3}{4-8 x+4 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3/(3 + Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]))*(18 - 36*x + 6*x^2 + (18 - 24*x + 6*x^2)*Log[(9*x
- 6*x^2 + x^3)/(4 - 8*x + 4*x^2)] + (3 - 4*x + x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]^2))/(27 - 36*x
+ 9*x^2 + (18 - 24*x + 6*x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)] + (3 - 4*x + x^2)*Log[(9*x - 6*x^2 +
x^3)/(4 - 8*x + 4*x^2)]^2),x]

[Out]

Defer[Int][E^(3/(3 + Log[((-3 + x)^2*x)/(4*(-1 + x)^2)])), x] - 3*Defer[Int][E^(3/(3 + Log[((-3 + x)^2*x)/(4*(
-1 + x)^2)]))/(3 + Log[((-3 + x)^2*x)/(4*(-1 + x)^2)])^2, x] - 18*Defer[Int][E^(3/(3 + Log[((-3 + x)^2*x)/(4*(
-1 + x)^2)]))/((-3 + x)*(3 + Log[((-3 + x)^2*x)/(4*(-1 + x)^2)])^2), x] + 6*Defer[Int][E^(3/(3 + Log[((-3 + x)
^2*x)/(4*(-1 + x)^2)]))/((-1 + x)*(3 + Log[((-3 + x)^2*x)/(4*(-1 + x)^2)])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} \left (6 \left (3-6 x+x^2\right )+6 \left (3-4 x+x^2\right ) \log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )+\left (3-4 x+x^2\right ) \log ^2\left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )}{\left (3-4 x+x^2\right ) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2} \, dx\\ &=\int \left (e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}-\frac {3 e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} \left (3+x^2\right )}{(-3+x) (-1+x) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} \left (3+x^2\right )}{(-3+x) (-1+x) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2} \, dx\right )+\int e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} \, dx\\ &=-\left (3 \int \left (\frac {e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}}{\left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2}+\frac {6 e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}}{(-3+x) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2}-\frac {2 e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}}{(-1+x) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2}\right ) \, dx\right )+\int e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} \, dx\\ &=-\left (3 \int \frac {e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}}{\left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2} \, dx\right )+6 \int \frac {e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}}{(-1+x) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2} \, dx-18 \int \frac {e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}}}{(-3+x) \left (3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )\right )^2} \, dx+\int e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 26, normalized size = 0.76 \begin {gather*} e^{\frac {3}{3+\log \left (\frac {(-3+x)^2 x}{4 (-1+x)^2}\right )}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3/(3 + Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]))*(18 - 36*x + 6*x^2 + (18 - 24*x + 6*x^2)*Log
[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)] + (3 - 4*x + x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)]^2))/(27 -
36*x + 9*x^2 + (18 - 24*x + 6*x^2)*Log[(9*x - 6*x^2 + x^3)/(4 - 8*x + 4*x^2)] + (3 - 4*x + x^2)*Log[(9*x - 6*
x^2 + x^3)/(4 - 8*x + 4*x^2)]^2),x]

[Out]

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

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fricas [A]  time = 0.65, size = 34, normalized size = 1.00 \begin {gather*} x e^{\left (\frac {3}{\log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))
+6*x^2-36*x+18)*exp(3/(log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^
2+(6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x, algorithm="fricas")

[Out]

x*e^(3/(log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1)) + 3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )^{2} + 6 \, x^{2} + 6 \, {\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - 36 \, x + 18\right )} e^{\left (\frac {3}{\log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + 3}\right )}}{{\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )^{2} + 9 \, x^{2} + 6 \, {\left (x^{2} - 4 \, x + 3\right )} \log \left (\frac {x^{3} - 6 \, x^{2} + 9 \, x}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) - 36 \, x + 27}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))
+6*x^2-36*x+18)*exp(3/(log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^
2+(6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x, algorithm="giac")

[Out]

integrate(((x^2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1))^2 + 6*x^2 + 6*(x^2 - 4*x + 3)*log(1/4*
(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1)) - 36*x + 18)*e^(3/(log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1)) + 3))/((x
^2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2 + 9*x)/(x^2 - 2*x + 1))^2 + 9*x^2 + 6*(x^2 - 4*x + 3)*log(1/4*(x^3 - 6*x^2
+ 9*x)/(x^2 - 2*x + 1)) - 36*x + 27), x)

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maple [A]  time = 0.14, size = 36, normalized size = 1.06

method	result	size
risch	\(x \,{\mathrm e}^{\frac {3}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}}\)	\(36\)
norman	\(\frac {\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right ) x \,{\mathrm e}^{\frac {3}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}}+3 x \,{\mathrm e}^{\frac {3}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}}}{\ln \left (\frac {x^{3}-6 x^{2}+9 x}{4 x^{2}-8 x +4}\right )+3}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-4*x+3)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+6*x^2-3
6*x+18)*exp(3/(ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-2
4*x+18)*ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x,method=_RETURNVERBOSE)

[Out]

x*exp(3/(ln((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, x^{3} e^{\left (-\frac {3}{2 \, \log \relax (2) + 2 \, \log \left (x - 1\right ) - 2 \, \log \left (x - 3\right ) - \log \relax (x) - 3}\right )}}{x^{2} + 3} + \frac {12 \, x^{2} e^{\left (-\frac {3}{2 \, \log \relax (2) + 2 \, \log \left (x - 1\right ) - 2 \, \log \left (x - 3\right ) - \log \relax (x) - 3}\right )}}{x^{2} + 3} - \frac {6 \, x e^{\left (-\frac {3}{2 \, \log \relax (2) + 2 \, \log \left (x - 1\right ) - 2 \, \log \left (x - 3\right ) - \log \relax (x) - 3}\right )}}{x^{2} + 3} + \int \frac {{\left (4 \, \log \relax (2)^{2} + 4 \, {\left (2 \, \log \relax (2) - 2 \, \log \left (x - 3\right ) - \log \relax (x) - 3\right )} \log \left (x - 1\right ) + 4 \, \log \left (x - 1\right )^{2} - 4 \, {\left (2 \, \log \relax (2) - \log \relax (x) - 3\right )} \log \left (x - 3\right ) + 4 \, \log \left (x - 3\right )^{2} - 2 \, {\left (2 \, \log \relax (2) - 3\right )} \log \relax (x) + \log \relax (x)^{2} - 12 \, \log \relax (2)\right )} e^{\left (-\frac {3}{2 \, \log \relax (2) + 2 \, \log \left (x - 1\right ) - 2 \, \log \left (x - 3\right ) - \log \relax (x) - 3}\right )}}{4 \, \log \relax (2)^{2} + 4 \, {\left (2 \, \log \relax (2) - 2 \, \log \left (x - 3\right ) - \log \relax (x) - 3\right )} \log \left (x - 1\right ) + 4 \, \log \left (x - 1\right )^{2} - 4 \, {\left (2 \, \log \relax (2) - \log \relax (x) - 3\right )} \log \left (x - 3\right ) + 4 \, \log \left (x - 3\right )^{2} - 2 \, {\left (2 \, \log \relax (2) - 3\right )} \log \relax (x) + \log \relax (x)^{2} - 12 \, \log \relax (2) + 9}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^2+(6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))
+6*x^2-36*x+18)*exp(3/(log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+3))/((x^2-4*x+3)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))^
2+(6*x^2-24*x+18)*log((x^3-6*x^2+9*x)/(4*x^2-8*x+4))+9*x^2-36*x+27),x, algorithm="maxima")

[Out]

-2*x^3*e^(-3/(2*log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) - 3))/(x^2 + 3) + 12*x^2*e^(-3/(2*log(2) + 2*log
(x - 1) - 2*log(x - 3) - log(x) - 3))/(x^2 + 3) - 6*x*e^(-3/(2*log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) -
3))/(x^2 + 3) + integrate((4*log(2)^2 + 4*(2*log(2) - 2*log(x - 3) - log(x) - 3)*log(x - 1) + 4*log(x - 1)^2
- 4*(2*log(2) - log(x) - 3)*log(x - 3) + 4*log(x - 3)^2 - 2*(2*log(2) - 3)*log(x) + log(x)^2 - 12*log(2))*e^(-
3/(2*log(2) + 2*log(x - 1) - 2*log(x - 3) - log(x) - 3))/(4*log(2)^2 + 4*(2*log(2) - 2*log(x - 3) - log(x) - 3
)*log(x - 1) + 4*log(x - 1)^2 - 4*(2*log(2) - log(x) - 3)*log(x - 3) + 4*log(x - 3)^2 - 2*(2*log(2) - 3)*log(x
) + log(x)^2 - 12*log(2) + 9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {3}{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )+3}}\,\left (\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )\,\left (6\,x^2-24\,x+18\right )-36\,x+{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )}^2\,\left (x^2-4\,x+3\right )+6\,x^2+18\right )}{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )\,\left (6\,x^2-24\,x+18\right )-36\,x+{\ln \left (\frac {x^3-6\,x^2+9\,x}{4\,x^2-8\,x+4}\right )}^2\,\left (x^2-4\,x+3\right )+9\,x^2+27} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3/(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4)) + 3))*(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^
2 - 24*x + 18) - 36*x + log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x^2 - 4*x + 3) + 6*x^2 + 18))/(log((9*x
- 6*x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^2 - 24*x + 18) - 36*x + log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x
^2 - 4*x + 3) + 9*x^2 + 27),x)

[Out]

int((exp(3/(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4)) + 3))*(log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^
2 - 24*x + 18) - 36*x + log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x^2 - 4*x + 3) + 6*x^2 + 18))/(log((9*x
- 6*x^2 + x^3)/(4*x^2 - 8*x + 4))*(6*x^2 - 24*x + 18) - 36*x + log((9*x - 6*x^2 + x^3)/(4*x^2 - 8*x + 4))^2*(x
^2 - 4*x + 3) + 9*x^2 + 27), x)

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sympy [A]  time = 18.87, size = 29, normalized size = 0.85 \begin {gather*} x e^{\frac {3}{\log {\left (\frac {x^{3} - 6 x^{2} + 9 x}{4 x^{2} - 8 x + 4} \right )} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-4*x+3)*ln((x**3-6*x**2+9*x)/(4*x**2-8*x+4))**2+(6*x**2-24*x+18)*ln((x**3-6*x**2+9*x)/(4*x**2-
8*x+4))+6*x**2-36*x+18)*exp(3/(ln((x**3-6*x**2+9*x)/(4*x**2-8*x+4))+3))/((x**2-4*x+3)*ln((x**3-6*x**2+9*x)/(4*
x**2-8*x+4))**2+(6*x**2-24*x+18)*ln((x**3-6*x**2+9*x)/(4*x**2-8*x+4))+9*x**2-36*x+27),x)

[Out]

x*exp(3/(log((x**3 - 6*x**2 + 9*x)/(4*x**2 - 8*x + 4)) + 3))

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