Integrand size = 119, antiderivative size = 26 \[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=2 \left (-\frac {3}{x}+x\right ) \log \left (\left (e^{25-9 e^{2 x}}+\log (x)\right )^2\right ) \] Output:
2*(x-3/x)*ln((exp(-9*exp(x)^2+25)+ln(x))^2)
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=\frac {2 \left (-3+x^2\right ) \log \left (e^{-18 e^{2 x}} \left (e^{25}+e^{9 e^{2 x}} \log (x)\right )^2\right )}{x} \] Input:
Integrate[(-12 + 4*x^2 + E^(25 - 9*E^(2*x) + 2*x)*(216*x - 72*x^3) + (E^(2 5 - 9*E^(2*x))*(6 + 2*x^2) + (6 + 2*x^2)*Log[x])*Log[E^(50 - 18*E^(2*x)) + 2*E^(25 - 9*E^(2*x))*Log[x] + Log[x]^2])/(E^(25 - 9*E^(2*x))*x^2 + x^2*Lo g[x]),x]
Output:
(2*(-3 + x^2)*Log[(E^25 + E^(9*E^(2*x))*Log[x])^2/E^(18*E^(2*x))])/x
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{2 x-9 e^{2 x}+25} \left (216 x-72 x^3\right )+4 x^2+\left (e^{25-9 e^{2 x}} \left (2 x^2+6\right )+\left (2 x^2+6\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+\log ^2(x)+2 e^{25-9 e^{2 x}} \log (x)\right )-12}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 e^{9 e^{2 x}} x^2+e^{9 e^{2 x}} x^2 \log (x) \log \left (e^{-18 e^{2 x}} \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )^2\right )+e^{25} x^2 \log \left (e^{-18 e^{2 x}} \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )^2\right )-6 e^{9 e^{2 x}}+3 e^{9 e^{2 x}} \log (x) \log \left (e^{-18 e^{2 x}} \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )^2\right )+3 e^{25} \log \left (e^{-18 e^{2 x}} \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )^2\right )\right )}{x^2 \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )}-\frac {72 e^{2 x+25} \left (x^2-3\right )}{x \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 12 \int \frac {e^{9 e^{2 x}}}{x^2 \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )}dx+12 e^{25} \int \frac {1}{x^2 \log (x) \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )}dx-4 \int \frac {e^{9 e^{2 x}}}{e^{9 e^{2 x}} \log (x)+e^{25}}dx-4 e^{25} \int \frac {1}{\log (x) \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )}dx-12 \operatorname {ExpIntegralEi}(-\log (x))+4 \operatorname {LogIntegral}(x)+2 x \log \left (e^{-18 e^{2 x}} \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )^2\right )-\frac {6 \log \left (e^{-18 e^{2 x}} \left (e^{9 e^{2 x}} \log (x)+e^{25}\right )^2\right )}{x}\) |
Input:
Int[(-12 + 4*x^2 + E^(25 - 9*E^(2*x) + 2*x)*(216*x - 72*x^3) + (E^(25 - 9* E^(2*x))*(6 + 2*x^2) + (6 + 2*x^2)*Log[x])*Log[E^(50 - 18*E^(2*x)) + 2*E^( 25 - 9*E^(2*x))*Log[x] + Log[x]^2])/(E^(25 - 9*E^(2*x))*x^2 + x^2*Log[x]), x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(24)=48\).
Time = 4.70 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85
| method | result | size |
| parallelrisch | \(\frac {4 x^{2} \ln \left ({\mathrm e}^{-18 \,{\mathrm e}^{2 x}+50}+2 \ln \left (x \right ) {\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}+\ln \left (x \right )^{2}\right )-12 \ln \left ({\mathrm e}^{-18 \,{\mathrm e}^{2 x}+50}+2 \ln \left (x \right ) {\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}+\ln \left (x \right )^{2}\right )}{2 x}\) | \(74\) |
| risch | \(\frac {4 \left (x^{2}-3\right ) \ln \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )}{x}-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )^{2}\right ) \left (x^{2} {\operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )\right )}^{2}-2 x^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )^{2}\right )+x^{2} {\operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )^{2}\right )}^{2}-3 {\operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )\right )}^{2}+6 \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )^{2}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )\right )-3 {\operatorname {csgn}\left (i \left (\ln \left (x \right )+{\mathrm e}^{-9 \,{\mathrm e}^{2 x}+25}\right )^{2}\right )}^{2}\right )}{x}\) | \(214\) |
Input:
int((((2*x^2+6)*exp(-9*exp(x)^2+25)+(2*x^2+6)*ln(x))*ln(exp(-9*exp(x)^2+25 )^2+2*ln(x)*exp(-9*exp(x)^2+25)+ln(x)^2)+(-72*x^3+216*x)*exp(x)^2*exp(-9*e xp(x)^2+25)+4*x^2-12)/(x^2*exp(-9*exp(x)^2+25)+x^2*ln(x)),x,method=_RETURN VERBOSE)
Output:
1/2*(4*ln(exp(-9*exp(x)^2+25)^2+2*ln(x)*exp(-9*exp(x)^2+25)+ln(x)^2)*x^2-1 2*ln(exp(-9*exp(x)^2+25)^2+2*ln(x)*exp(-9*exp(x)^2+25)+ln(x)^2))/x
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=\frac {2 \, {\left (x^{2} - 3\right )} \log \left ({\left (e^{\left (4 \, x\right )} \log \left (x\right )^{2} + 2 \, e^{\left (4 \, x - 9 \, e^{\left (2 \, x\right )} + 25\right )} \log \left (x\right ) + e^{\left (4 \, x - 18 \, e^{\left (2 \, x\right )} + 50\right )}\right )} e^{\left (-4 \, x\right )}\right )}{x} \] Input:
integrate((((2*x^2+6)*exp(-9*exp(x)^2+25)+(2*x^2+6)*log(x))*log(exp(-9*exp (x)^2+25)^2+2*log(x)*exp(-9*exp(x)^2+25)+log(x)^2)+(-72*x^3+216*x)*exp(x)^ 2*exp(-9*exp(x)^2+25)+4*x^2-12)/(x^2*exp(-9*exp(x)^2+25)+x^2*log(x)),x, al gorithm="fricas")
Output:
2*(x^2 - 3)*log((e^(4*x)*log(x)^2 + 2*e^(4*x - 9*e^(2*x) + 25)*log(x) + e^ (4*x - 18*e^(2*x) + 50))*e^(-4*x))/x
Time = 1.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=\frac {\left (2 x^{2} - 6\right ) \log {\left (2 e^{25 - 9 e^{2 x}} \log {\left (x \right )} + e^{50 - 18 e^{2 x}} + \log {\left (x \right )}^{2} \right )}}{x} \] Input:
integrate((((2*x**2+6)*exp(-9*exp(x)**2+25)+(2*x**2+6)*ln(x))*ln(exp(-9*ex p(x)**2+25)**2+2*ln(x)*exp(-9*exp(x)**2+25)+ln(x)**2)+(-72*x**3+216*x)*exp (x)**2*exp(-9*exp(x)**2+25)+4*x**2-12)/(x**2*exp(-9*exp(x)**2+25)+x**2*ln( x)),x)
Output:
(2*x**2 - 6)*log(2*exp(25 - 9*exp(2*x))*log(x) + exp(50 - 18*exp(2*x)) + l og(x)**2)/x
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=-\frac {4 \, {\left (9 \, {\left (x^{2} - 3\right )} e^{\left (2 \, x\right )} - {\left (x^{2} - 3\right )} \log \left (e^{\left (9 \, e^{\left (2 \, x\right )}\right )} \log \left (x\right ) + e^{25}\right )\right )}}{x} \] Input:
integrate((((2*x^2+6)*exp(-9*exp(x)^2+25)+(2*x^2+6)*log(x))*log(exp(-9*exp (x)^2+25)^2+2*log(x)*exp(-9*exp(x)^2+25)+log(x)^2)+(-72*x^3+216*x)*exp(x)^ 2*exp(-9*exp(x)^2+25)+4*x^2-12)/(x^2*exp(-9*exp(x)^2+25)+x^2*log(x)),x, al gorithm="maxima")
Output:
-4*(9*(x^2 - 3)*e^(2*x) - (x^2 - 3)*log(e^(9*e^(2*x))*log(x) + e^25))/x
\[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=\int { \frac {2 \, {\left (2 \, x^{2} - 36 \, {\left (x^{3} - 3 \, x\right )} e^{\left (2 \, x - 9 \, e^{\left (2 \, x\right )} + 25\right )} + {\left ({\left (x^{2} + 3\right )} e^{\left (-9 \, e^{\left (2 \, x\right )} + 25\right )} + {\left (x^{2} + 3\right )} \log \left (x\right )\right )} \log \left (2 \, e^{\left (-9 \, e^{\left (2 \, x\right )} + 25\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{\left (-18 \, e^{\left (2 \, x\right )} + 50\right )}\right ) - 6\right )}}{x^{2} e^{\left (-9 \, e^{\left (2 \, x\right )} + 25\right )} + x^{2} \log \left (x\right )} \,d x } \] Input:
integrate((((2*x^2+6)*exp(-9*exp(x)^2+25)+(2*x^2+6)*log(x))*log(exp(-9*exp (x)^2+25)^2+2*log(x)*exp(-9*exp(x)^2+25)+log(x)^2)+(-72*x^3+216*x)*exp(x)^ 2*exp(-9*exp(x)^2+25)+4*x^2-12)/(x^2*exp(-9*exp(x)^2+25)+x^2*log(x)),x, al gorithm="giac")
Output:
integrate(2*(2*x^2 - 36*(x^3 - 3*x)*e^(2*x - 9*e^(2*x) + 25) + ((x^2 + 3)* e^(-9*e^(2*x) + 25) + (x^2 + 3)*log(x))*log(2*e^(-9*e^(2*x) + 25)*log(x) + log(x)^2 + e^(-18*e^(2*x) + 50)) - 6)/(x^2*e^(-9*e^(2*x) + 25) + x^2*log( x)), x)
Time = 1.66 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=\ln \left ({\ln \left (x\right )}^2+2\,{\mathrm {e}}^{-9\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{25}\,\ln \left (x\right )+{\mathrm {e}}^{-18\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{50}\right )\,\left (4\,x-\frac {2\,x^3+6\,x}{x^2}\right ) \] Input:
int((log(exp(50 - 18*exp(2*x)) + log(x)^2 + 2*exp(25 - 9*exp(2*x))*log(x)) *(exp(25 - 9*exp(2*x))*(2*x^2 + 6) + log(x)*(2*x^2 + 6)) + 4*x^2 + exp(2*x )*exp(25 - 9*exp(2*x))*(216*x - 72*x^3) - 12)/(x^2*log(x) + x^2*exp(25 - 9 *exp(2*x))),x)
Output:
log(log(x)^2 + exp(-18*exp(2*x))*exp(50) + 2*exp(-9*exp(2*x))*exp(25)*log( x))*(4*x - (6*x + 2*x^3)/x^2)
\[ \int \frac {-12+4 x^2+e^{25-9 e^{2 x}+2 x} \left (216 x-72 x^3\right )+\left (e^{25-9 e^{2 x}} \left (6+2 x^2\right )+\left (6+2 x^2\right ) \log (x)\right ) \log \left (e^{50-18 e^{2 x}}+2 e^{25-9 e^{2 x}} \log (x)+\log ^2(x)\right )}{e^{25-9 e^{2 x}} x^2+x^2 \log (x)} \, dx=\int \frac {\left (\left (2 x^{2}+6\right ) {\mathrm e}^{-9 \left ({\mathrm e}^{x}\right )^{2}+25}+\left (2 x^{2}+6\right ) \mathrm {log}\left (x \right )\right ) \mathrm {log}\left (\left ({\mathrm e}^{-9 \left ({\mathrm e}^{x}\right )^{2}+25}\right )^{2}+2 \,\mathrm {log}\left (x \right ) {\mathrm e}^{-9 \left ({\mathrm e}^{x}\right )^{2}+25}+\mathrm {log}\left (x \right )^{2}\right )+\left (-72 x^{3}+216 x \right ) \left ({\mathrm e}^{x}\right )^{2} {\mathrm e}^{-9 \left ({\mathrm e}^{x}\right )^{2}+25}+4 x^{2}-12}{x^{2} {\mathrm e}^{-9 \left ({\mathrm e}^{x}\right )^{2}+25}+\mathrm {log}\left (x \right ) x^{2}}d x \] Input:
int((((2*x^2+6)*exp(-9*exp(x)^2+25)+(2*x^2+6)*log(x))*log(exp(-9*exp(x)^2+ 25)^2+2*log(x)*exp(-9*exp(x)^2+25)+log(x)^2)+(-72*x^3+216*x)*exp(x)^2*exp( -9*exp(x)^2+25)+4*x^2-12)/(x^2*exp(-9*exp(x)^2+25)+x^2*log(x)),x)
Output:
int((((2*x^2+6)*exp(-9*exp(x)^2+25)+(2*x^2+6)*log(x))*log(exp(-9*exp(x)^2+ 25)^2+2*log(x)*exp(-9*exp(x)^2+25)+log(x)^2)+(-72*x^3+216*x)*exp(x)^2*exp( -9*exp(x)^2+25)+4*x^2-12)/(x^2*exp(-9*exp(x)^2+25)+x^2*log(x)),x)