3.62.33 \(\int \frac {-96+6 e^4+8 x+(-48 x+3 e^4 x-4 x^2) \log ^2(\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2})}{-192 x+12 e^4 x-16 x^2+(192 x^2-12 e^4 x^2+16 x^3) \log (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2})+(-48 x^3+3 e^4 x^3-4 x^4) \log ^2(\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2})} \, dx\)

Optimal. Leaf size=29 \[ \frac {2}{-2 x+\frac {4}{\log \left (\frac {x}{\left (16-e^4+\frac {4 x}{3}\right )^2}\right )}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.57, 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 {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-96 + 6*E^4 + 8*x + (-48*x + 3*E^4*x - 4*x^2)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^
2)]^2)/(-192*x + 12*E^4*x - 16*x^2 + (192*x^2 - 12*E^4*x^2 + 16*x^3)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*
x) + 384*x + 16*x^2)] + (-48*x^3 + 3*E^4*x^3 - 4*x^4)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16
*x^2)]^2),x]

[Out]

-x^(-1) + 16*Defer[Int][1/((-48 + 3*E^4 - 4*x)*(-2 + x*Log[(9*x)/(3*E^4 - 4*(12 + x))^2])^2), x] + 4*Defer[Int
][1/(x^2*(-2 + x*Log[(9*x)/(3*E^4 - 4*(12 + x))^2])^2), x] + 2*Defer[Int][1/(x*(-2 + x*Log[(9*x)/(3*E^4 - 4*(1
2 + x))^2])^2), x] + 4*Defer[Int][1/(x^2*(-2 + x*Log[(9*x)/(3*E^4 - 4*(12 + x))^2])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{\left (-192+12 e^4\right ) x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx\\ &=\int \frac {96 \left (1-\frac {e^4}{16}\right )-8 x+x \left (48-3 e^4+4 x\right ) \log ^2\left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )}{x \left (48-3 e^4+4 x\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {2 \left (6 \left (16-e^4\right )+\left (56-3 e^4\right ) x-4 x^2\right )}{x^2 \left (48-3 e^4+4 x\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}+\frac {4}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )}\right ) \, dx\\ &=-\frac {1}{x}+2 \int \frac {6 \left (16-e^4\right )+\left (56-3 e^4\right ) x-4 x^2}{x^2 \left (48-3 e^4+4 x\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )} \, dx\\ &=-\frac {1}{x}+2 \int \left (\frac {8}{\left (-48+3 e^4-4 x\right ) \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}+\frac {2}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}+\frac {1}{x \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}\right ) \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )} \, dx\\ &=-\frac {1}{x}+2 \int \frac {1}{x \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )} \, dx+16 \int \frac {1}{\left (-48+3 e^4-4 x\right ) \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 31, normalized size = 1.07 \begin {gather*} \frac {-1-\frac {2}{-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96 + 6*E^4 + 8*x + (-48*x + 3*E^4*x - 4*x^2)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x +
16*x^2)]^2)/(-192*x + 12*E^4*x - 16*x^2 + (192*x^2 - 12*E^4*x^2 + 16*x^3)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288
- 24*x) + 384*x + 16*x^2)] + (-48*x^3 + 3*E^4*x^3 - 4*x^4)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*
x + 16*x^2)]^2),x]

[Out]

(-1 - 2/(-2 + x*Log[(9*x)/(3*E^4 - 4*(12 + x))^2]))/x

________________________________________________________________________________________

fricas [B]  time = 1.05, size = 62, normalized size = 2.14 \begin {gather*} -\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+6*exp(4)+8*x-9
6)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+1
6*x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp(4)-16*x^2-192*x),x, algorit
hm="fricas")

[Out]

-log(9*x/(16*x^2 - 24*(x + 12)*e^4 + 384*x + 9*e^8 + 2304))/(x*log(9*x/(16*x^2 - 24*(x + 12)*e^4 + 384*x + 9*e
^8 + 2304)) - 2)

________________________________________________________________________________________

giac [B]  time = 1.33, size = 66, normalized size = 2.28 \begin {gather*} -\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right ) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+6*exp(4)+8*x-9
6)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+1
6*x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp(4)-16*x^2-192*x),x, algorit
hm="giac")

[Out]

-log(9*x/(16*x^2 - 24*x*e^4 + 384*x + 9*e^8 - 288*e^4 + 2304))/(x*log(9*x/(16*x^2 - 24*x*e^4 + 384*x + 9*e^8 -
288*e^4 + 2304)) - 2)

________________________________________________________________________________________

maple [A]  time = 0.53, size = 46, normalized size = 1.59

method	result	size
risch	\(-\frac {1}{x}-\frac {2}{x \left (x \ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right )-2\right )}\)	\(46\)
norman	\(-\frac {\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right )}{x \ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right )-2}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x*exp(4)-4*x^2-48*x)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*
x^3*exp(4)-4*x^4-48*x^3)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16*x^3+19
2*x^2)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp(4)-16*x^2-192*x),x,method=_RETURNVER
BOSE)

[Out]

-1/x-2/x/(x*ln(9*x/(9*exp(8)+(-24*x-288)*exp(4)+16*x^2+384*x+2304))-2)

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 47, normalized size = 1.62 \begin {gather*} -\frac {2 \, \log \relax (3) - 2 \, \log \left (4 \, x - 3 \, e^{4} + 48\right ) + \log \relax (x)}{2 \, x \log \relax (3) - 2 \, x \log \left (4 \, x - 3 \, e^{4} + 48\right ) + x \log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+6*exp(4)+8*x-9
6)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+1
6*x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp(4)-16*x^2-192*x),x, algorit
hm="maxima")

[Out]

-(2*log(3) - 2*log(4*x - 3*e^4 + 48) + log(x))/(2*x*log(3) - 2*x*log(4*x - 3*e^4 + 48) + x*log(x) - 2)

________________________________________________________________________________________

mupad [B]  time = 6.54, size = 51, normalized size = 1.76 \begin {gather*} -\frac {2}{x\,\left (2\,x\,\ln \relax (3)-x\,\ln \left (384\,x-288\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8-24\,x\,{\mathrm {e}}^4+16\,x^2+2304\right )+x\,\ln \relax (x)-2\right )}-\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x + 6*exp(4) - log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x + 288) + 2304))^2*(48*x - 3*x*exp(4
) + 4*x^2) - 96)/(192*x - 12*x*exp(4) - log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x + 288) + 2304))*(1
92*x^2 - 12*x^2*exp(4) + 16*x^3) + 16*x^2 + log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x + 288) + 2304)
)^2*(48*x^3 - 3*x^3*exp(4) + 4*x^4)),x)

[Out]

- 2/(x*(2*x*log(3) - x*log(384*x - 288*exp(4) + 9*exp(8) - 24*x*exp(4) + 16*x^2 + 2304) + x*log(x) - 2)) - 1/x

________________________________________________________________________________________

sympy [B]  time = 0.33, size = 42, normalized size = 1.45 \begin {gather*} - \frac {2}{x^{2} \log {\left (\frac {9 x}{16 x^{2} + 384 x + \left (- 24 x - 288\right ) e^{4} + 2304 + 9 e^{8}} \right )} - 2 x} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x*exp(4)-4*x**2-48*x)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384*x+2304))**2+6*exp(4)+8*
x-96)/((3*x**3*exp(4)-4*x**4-48*x**3)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384*x+2304))**2+(-12*x**2
*exp(4)+16*x**3+192*x**2)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384*x+2304))+12*x*exp(4)-16*x**2-192*
x),x)

[Out]

-2/(x**2*log(9*x/(16*x**2 + 384*x + (-24*x - 288)*exp(4) + 2304 + 9*exp(8))) - 2*x) - 1/x

________________________________________________________________________________________