3.42.66 \(\int \frac {-36+e^{x+x^2 \log (3 x)} (-16+16 x+16 x^2+32 x^2 \log (3 x))}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {x}{-\frac {9}{4}-e^{x+x^2 \log (3 x)}+x} \]

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Rubi [F]  time = 1.69, 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 {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-36 + E^(x + x^2*Log[3*x])*(-16 + 16*x + 16*x^2 + 32*x^2*Log[3*x]))/(81 + 16*E^(2*x + 2*x^2*Log[3*x]) + E
^(x + x^2*Log[3*x])*(72 - 32*x) - 72*x + 16*x^2),x]

[Out]

-36*Defer[Int][(9 - 4*x + 4*3^x^2*E^x*x^x^2)^(-2), x] - 16*Defer[Int][E^(x + x^2*Log[3*x])/(9 - 4*x + 4*3^x^2*
E^x*x^x^2)^2, x] + 16*Defer[Int][(E^(x + x^2*Log[3*x])*x)/(9 - 4*x + 4*3^x^2*E^x*x^x^2)^2, x] + 16*Defer[Int][
(E^(x + x^2*Log[3])*x^(2 + x^2))/(9 - 4*x + 4*3^x^2*E^x*x^x^2)^2, x] + 32*Log[3*x]*Defer[Int][(E^(x + x^2*Log[
3])*x^(2 + x^2))/(9 - 4*x + 4*3^x^2*E^x*x^x^2)^2, x] - 32*Defer[Int][Defer[Int][(3^x^2*E^x*x^(2 + x^2))/(9 - 4
*x + 4*3^x^2*E^x*x^x^2)^2, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=\int \left (-\frac {36}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}-\frac {16 e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {16 e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {16\ 3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {32\ 3^{x^2} e^x x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}\right ) \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+32 \int \frac {3^{x^2} e^x x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+32 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-32 \int \frac {\int \frac {3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx}{x} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+(32 \log (3 x)) \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ &=-\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-32 \int \frac {\int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx}{x} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+(32 \log (3 x)) \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 25, normalized size = 1.09 \begin {gather*} -\frac {4 x}{9-4 x+4\ 3^{x^2} e^x x^{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 + E^(x + x^2*Log[3*x])*(-16 + 16*x + 16*x^2 + 32*x^2*Log[3*x]))/(81 + 16*E^(2*x + 2*x^2*Log[3*x
]) + E^(x + x^2*Log[3*x])*(72 - 32*x) - 72*x + 16*x^2),x]

[Out]

(-4*x)/(9 - 4*x + 4*3^x^2*E^x*x^x^2)

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fricas [A]  time = 0.60, size = 23, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3 \, x\right ) + x\right )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^2*log(3*x)+16*x^2+16*x-16)*exp(x^2*log(3*x)+x)-36)/(16*exp(x^2*log(3*x)+x)^2+(-32*x+72)*exp(x
^2*log(3*x)+x)+16*x^2-72*x+81),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.38, size = 23, normalized size = 1.00 \begin {gather*} \frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3 \, x\right ) + x\right )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^2*log(3*x)+16*x^2+16*x-16)*exp(x^2*log(3*x)+x)-36)/(16*exp(x^2*log(3*x)+x)^2+(-32*x+72)*exp(x
^2*log(3*x)+x)+16*x^2-72*x+81),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 22, normalized size = 0.96

method	result	size
risch	\(\frac {4 x}{4 x -4 \left (3 x \right )^{x^{2}} {\mathrm e}^{x}-9}\)	\(22\)
norman	\(\frac {4 x}{4 x -4 \,{\mathrm e}^{x^{2} \ln \left (3 x \right )+x}-9}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x^2*ln(3*x)+16*x^2+16*x-16)*exp(x^2*ln(3*x)+x)-36)/(16*exp(x^2*ln(3*x)+x)^2+(-32*x+72)*exp(x^2*ln(3*x
)+x)+16*x^2-72*x+81),x,method=_RETURNVERBOSE)

[Out]

4*x/(4*x-4*(3*x)^(x^2)*exp(x)-9)

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maxima [A]  time = 0.60, size = 27, normalized size = 1.17 \begin {gather*} \frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \relax (3) + x^{2} \log \relax (x) + x\right )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^2*log(3*x)+16*x^2+16*x-16)*exp(x^2*log(3*x)+x)-36)/(16*exp(x^2*log(3*x)+x)^2+(-32*x+72)*exp(x
^2*log(3*x)+x)+16*x^2-72*x+81),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.32, size = 80, normalized size = 3.48 \begin {gather*} -\frac {52\,x+72\,x^2\,\ln \left (3\,x\right )-32\,x^3\,\ln \left (3\,x\right )+20\,x^2-16\,x^3}{\left (4\,{\mathrm {e}}^x\,{\left (3\,x\right )}^{x^2}-4\,x+9\right )\,\left (5\,x+18\,x\,\ln \left (3\,x\right )-8\,x^2\,\ln \left (3\,x\right )-4\,x^2+13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + x^2*log(3*x))*(16*x + 32*x^2*log(3*x) + 16*x^2 - 16) - 36)/(16*exp(2*x + 2*x^2*log(3*x)) - 72*x -
exp(x + x^2*log(3*x))*(32*x - 72) + 16*x^2 + 81),x)

[Out]

-(52*x + 72*x^2*log(3*x) - 32*x^3*log(3*x) + 20*x^2 - 16*x^3)/((4*exp(x)*(3*x)^(x^2) - 4*x + 9)*(5*x + 18*x*lo
g(3*x) - 8*x^2*log(3*x) - 4*x^2 + 13))

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sympy [A]  time = 0.34, size = 22, normalized size = 0.96 \begin {gather*} - \frac {4 x}{- 4 x + 4 e^{x^{2} \log {\left (3 x \right )} + x} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x**2*ln(3*x)+16*x**2+16*x-16)*exp(x**2*ln(3*x)+x)-36)/(16*exp(x**2*ln(3*x)+x)**2+(-32*x+72)*exp
(x**2*ln(3*x)+x)+16*x**2-72*x+81),x)

[Out]

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

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